The Open Logic Text โ€” accessible offline edition

The Sequent Calculus โ€” Explore equations

The Sequent Calculus

Reading preferences

Optional display controls need JavaScript. All reading content and navigation work without it.

Equation, formal-object, proof, and reference guide

This index exposes 394 expressions, 788 native MathML variants, 144 formal objects, 896 ordered proof components, 238 source proof commands, 14 references, and all nine unsolved exercises without adding solutions.

394 expression records

Expression 1

Inline MathML variant

∧\land

Block MathML variant

∧\land

Conventional reading: conjunction

Meaning here: The logical notation read 'conjunction' names the exact connective, quantifier, identity sign, sequent arrow, or derivability relation printed by the source.

5 occurrences
  1. Occurrence 1: propositional-rules.tex, line 29, column 23
  2. Occurrence 2: proving-things.tex, line 27, column 46
  3. Occurrence 3: proving-things.tex, line 28, column 25
  4. Occurrence 4: proving-things.tex, line 28, column 53
  5. Occurrence 5: proving-things.tex, line 248, column 32

Expression 2

Inline MathML variant

¬(A→B)⇒A\lnot(!A \lif !B) \Sequent !A

Block MathML variant

¬(A→B)⇒A\lnot(!A \lif !B) \Sequent !A

Conventional reading: antecedent containing the negation of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula B, close parenthesis; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing the negation of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula B, close parenthesis; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 331, column 7

Expression 3

Inline MathML variant

Γ0∪Δ0⊆Γ∪Δ\Gamma_0 \cup \Delta_0 \subseteq \Gamma \cup \Delta

Block MathML variant

Γ0∪Δ0⊆Γ∪Δ\Gamma_0 \cup \Delta_0 \subseteq \Gamma \cup \Delta

Conventional reading: the union of Gamma sub zero, and capital Delta sub zero is a subset of the union of Gamma, and capital Delta

Meaning here: The relation read 'the union of Gamma sub zero, and capital Delta sub zero is a subset of the union of Gamma, and capital Delta' concerns the exact premise sets or formula sequences named by the source.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 124, column 7

Expression 4

Inline MathML variant

∃x(A(x))⇒A(a)\lexists[x][!A(x)] \fCenter !A(a)

Block MathML variant

∃x(A(x))⇒A(a)\lexists[x][!A(x)] \fCenter !A(a)

Conventional reading: antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument constant a

Meaning here: The first-order sequent read 'antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument constant a' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: quantifier-rules.tex, line 90, column 12

Expression 5

Inline MathML variant

∀x(A(x))⇒A(a)\lforall[x][!A(x)] \fCenter !A(a)

Block MathML variant

∀x(A(x))⇒A(a)\lforall[x][!A(x)] \fCenter !A(a)

Conventional reading: antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument constant a

Meaning here: The first-order sequent read 'antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument constant a' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proving-things-quant.tex, line 49, column 10
  2. Occurrence 2: proving-things-quant.tex, line 67, column 10

Expression 7

Inline MathML variant

¬A∨B\lnot !A \lor !B

Block MathML variant

¬A∨B\lnot !A \lor !B

Conventional reading: the disjunction of the negation of formula A and formula B

Meaning here: The first-order formula read 'the disjunction of the negation of formula A and formula B' preserves its metavariables, connectives, term arguments, and written grouping.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 76, column 12

Expression 8

Inline MathML variant

A(t)⇒∃x(A(x))!A(t) \Sequent \lexists[x][!A(x)]

Block MathML variant

A(t)⇒∃x(A(x))!A(t) \Sequent \lexists[x][!A(x)]

Conventional reading: antecedent containing formula A with argument term t; sequent arrow; succedent containing for some variable x, formula A with argument variable x

Meaning here: The first-order sequent read 'antecedent containing formula A with argument term t; sequent arrow; succedent containing for some variable x, formula A with argument variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-quantifiers.tex, line 42, column 29

Expression 9

Inline MathML variant

A(t)⇒∃x(A(x))!A(t) \fCenter \lexists[x][!A(x)]

Block MathML variant

A(t)⇒∃x(A(x))!A(t) \fCenter \lexists[x][!A(x)]

Conventional reading: antecedent containing formula A with argument term t; sequent arrow; succedent containing for some variable x, formula A with argument variable x

Meaning here: The first-order sequent read 'antecedent containing formula A with argument term t; sequent arrow; succedent containing for some variable x, formula A with argument variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-quantifiers.tex, line 47, column 14

Expression 10

Inline MathML variant

A∧B⇒A!A\land !B \fCenter !A

Block MathML variant

A∧B⇒A!A\land !B \fCenter !A

Conventional reading: antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

3 occurrences
  1. Occurrence 1: proving-things.tex, line 22, column 10
  2. Occurrence 2: proving-things.tex, line 34, column 10
  3. Occurrence 3: proving-things.tex, line 44, column 10

Expression 11

Inline MathML variant

t1=x\eq[t_1][x]

Block MathML variant

t1=x\eq[t_1][x]

Conventional reading: term t sub one is identical to variable x

Meaning here: The identity expression read 'term t sub one is identical to variable x' states equality of the named first-order terms or semantic values.

1 occurrence
  1. Occurrence 1: identity.tex, line 64, column 46

Expression 12

Inline MathML variant

¬(A∧B)⇒¬A∨¬B\lnot(!A \land !B) \Sequent \lnot !A \lor \lnot !B

Block MathML variant

¬(A∧B)⇒¬A∨¬B\lnot(!A \land !B) \Sequent \lnot !A \lor \lnot !B

Conventional reading: antecedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis; sequent arrow; succedent containing the disjunction of the negation of formula A and the negation of formula B

Meaning here: The first-order sequent read 'antecedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis; sequent arrow; succedent containing the disjunction of the negation of formula A and the negation of formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 332, column 7

Expression 13

Inline MathML variant

A⇒A∨B!A \fCenter !A \lor !B

Block MathML variant

A⇒A∨B!A \fCenter !A \lor !B

Conventional reading: antecedent containing formula A; sequent arrow; succedent containing the disjunction of formula A and formula B

Meaning here: The first-order sequent read 'antecedent containing formula A; sequent arrow; succedent containing the disjunction of formula A and formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 90, column 16

Expression 14

Inline MathML variant

A→(B→C)⇒B→(A→C)!A \lif (!B \lif !C) \Sequent !B \lif (!A \lif !C)

Block MathML variant

A→(B→C)⇒B→(A→C)!A \lif (!B \lif !C) \Sequent !B \lif (!A \lif !C)

Conventional reading: antecedent containing the conditional whose antecedent is formula A; and whose consequent is open parenthesis, the conditional whose antecedent is formula B; and whose consequent is formula C, close parenthesis; sequent arrow; succedent containing the conditional whose antecedent is formula B; and whose consequent is open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula C, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing the conditional whose antecedent is formula A; and whose consequent is open parenthesis, the conditional whose antecedent is formula B; and whose consequent is formula C, close parenthesis; sequent arrow; succedent containing the conditional whose antecedent is formula B; and whose consequent is open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula C, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 305, column 7

Expression 15

Inline MathML variant

∃x¬A(x)⇒¬∀xA(x)\lexists[x] \lnot !A(x) \fCenter \lnot \lforall[x] !A(x)

Block MathML variant

∃x¬A(x)⇒¬∀xA(x)\lexists[x] \lnot !A(x) \fCenter \lnot \lforall[x] !A(x)

Conventional reading: antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x

Meaning here: The first-order sequent read 'antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things-quant.tex, line 57, column 10

Expression 16

Inline MathML variant

M,s⊨A(t2)\Sat{M}{!A(t_2)}[s]

Block MathML variant

M,s⊨A(t2)\Sat{M}{!A(t_2)}[s]

Conventional reading: structure M, under variable assignment lowercase s, satisfies formula A with argument term t sub two

Meaning here: The satisfaction claim read 'structure M, under variable assignment lowercase s, satisfies formula A with argument term t sub two' is evaluated in the named first-order structure and, when printed, the named variable assignment.

1 occurrence
  1. Occurrence 1: soundness-identity.tex, line 40, column 1

Expression 17

Inline MathML variant

Π\Pi

Block MathML variant

Π\Pi

Conventional reading: capital Pi

Meaning here: Capital Pi denotes a second antecedent sequence in a sequent rule or derivation.

1 occurrence
  1. Occurrence 1: derivations.tex, line 72, column 19

Expression 19

Inline MathML variant

∃x(A(x))⇒∀x(A(x))\lexists[x][!A(x)] \Sequent \lforall[x][!A(x)]

Block MathML variant

∃x(A(x))⇒∀x(A(x))\lexists[x][!A(x)] \Sequent \lforall[x][!A(x)]

Conventional reading: antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing for every variable x, formula A with argument variable x

Meaning here: The first-order sequent read 'antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing for every variable x, formula A with argument variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: quantifier-rules.tex, line 101, column 10

Expression 20

Inline MathML variant

s(x)=⟦a⟧M′,ss(x) = \Value{a}{M'}[s]

Block MathML variant

s(x)=⟦a⟧M′,ss(x) = \Value{a}{M'}[s]

Conventional reading: s applied to variable x is identical to the value of constant a in structure M prime under variable assignment lowercase s

Meaning here: The semantic term read 's applied to variable x is identical to the value of constant a in structure M prime under variable assignment lowercase s' specifies an interpretation, term value, or assignment relation in a first-order structure.

1 occurrence
  1. Occurrence 1: soundness.tex, line 259, column 8

Expression 21

Inline MathML variant

Γ⇒Δ,P(a,a)\Gamma \Sequent \Delta, \Atom{\Obj P}{a,a}

Block MathML variant

Γ⇒Δ,P(a,a)\Gamma \Sequent \Delta, \Atom{\Obj P}{a,a}

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then atomic formula P applied to constant a, then constant a

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then atomic formula P applied to constant a, then constant a' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: quantifier-rules.tex, line 76, column 1

Expression 22

Inline MathML variant

A,¬A⇒!A, \lnot !A \fCenter

Block MathML variant

A,¬A⇒!A, \lnot !A \fCenter

Conventional reading: antecedent containing first formula A, then the negation of formula A; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first formula A, then the negation of formula A; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: provability-consistency.tex, line 92, column 14
  2. Occurrence 2: provability-propositional.tex, line 123, column 18

Expression 23

Inline MathML variant

(∀x(A(x))→B)⇒∃y((A(y)→B))(\lforall[x][!A(x)] \lif !B) \Sequent \lexists[y][(!A(y) \lif !B)]

Block MathML variant

(∀x(A(x))→B)⇒∃y((A(y)→B))(\lforall[x][!A(x)] \lif !B) \Sequent \lexists[y][(!A(y) \lif !B)]

Conventional reading: antecedent containing open parenthesis, the conditional whose antecedent is for every variable x, formula A with argument variable x; and whose consequent is formula B, close parenthesis; sequent arrow; succedent containing for some variable y, open parenthesis, the conditional whose antecedent is formula A with argument variable y; and whose consequent is formula B, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing open parenthesis, the conditional whose antecedent is for every variable x, formula A with argument variable x; and whose consequent is formula B, close parenthesis; sequent arrow; succedent containing for some variable y, open parenthesis, the conditional whose antecedent is formula A with argument variable y; and whose consequent is formula B, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things-quant.tex, line 104, column 7

Expression 24

Inline MathML variant

A(a),Γ⇒Δ!A(a), \Gamma \fCenter \Delta

Block MathML variant

A(a),Γ⇒Δ!A(a), \Gamma \fCenter \Delta

Conventional reading: antecedent containing first formula A with argument constant a, then Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first formula A with argument constant a, then Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: quantifier-rules.tex, line 37, column 7

Expression 25

Inline MathML variant

A⇒B!A \fCenter !B

Block MathML variant

A⇒B!A \fCenter !B

Conventional reading: antecedent containing formula A; sequent arrow; succedent containing formula B

Meaning here: The first-order sequent read 'antecedent containing formula A; sequent arrow; succedent containing formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 196, column 10

Expression 26

Inline MathML variant

⇒¬∃x(A(x))→∀x(¬A(x))\Sequent \lnot\lexists[x][!A(x)] \lif \lforall[x][\lnot !A(x)]

Block MathML variant

⇒¬∃x(A(x))→∀x(¬A(x))\Sequent \lnot\lexists[x][!A(x)] \lif \lforall[x][\lnot !A(x)]

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is the negation of for some variable x, formula A with argument variable x; and whose consequent is for every variable x, the negation of formula A with argument variable x

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is the negation of for some variable x, formula A with argument variable x; and whose consequent is for every variable x, the negation of formula A with argument variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things-quant.tex, line 94, column 7

Expression 27

Inline MathML variant

∃x(¬A(x))⇒¬∀x(A(x))\lexists[x][\lnot !A(x)] \fCenter \lnot \lforall[x][!A(x)]

Block MathML variant

∃x(¬A(x))⇒¬∀x(A(x))\lexists[x][\lnot !A(x)] \fCenter \lnot \lforall[x][!A(x)]

Conventional reading: antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x

Meaning here: The first-order sequent read 'antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proving-things-quant.tex, line 34, column 10
  2. Occurrence 2: proving-things-quant.tex, line 44, column 10

Expression 28

Inline MathML variant

M⊨A(t)\Sat{M}{!A(t)}

Block MathML variant

M⊨A(t)\Sat{M}{!A(t)}

Conventional reading: structure M satisfies formula A with argument term t

Meaning here: The satisfaction claim read 'structure M satisfies formula A with argument term t' is evaluated in the named first-order structure and, when printed, the named variable assignment.

1 occurrence
  1. Occurrence 1: soundness.tex, line 222, column 39

Expression 29

Inline MathML variant

C∈Γ!C \in \Gamma

Block MathML variant

C∈Γ!C \in \Gamma

Conventional reading: formula C is a member of Gamma

Meaning here: The relation read 'formula C is a member of Gamma' concerns the exact premise sets or formula sequences named by the source.

13 occurrences
  1. Occurrence 1: soundness.tex, line 93, column 24
  2. Occurrence 2: soundness.tex, line 97, column 54
  3. Occurrence 3: soundness.tex, line 116, column 23
  4. Occurrence 4: soundness.tex, line 122, column 17
  5. Occurrence 5: soundness.tex, line 146, column 51
  6. Occurrence 6: soundness.tex, line 150, column 66
  7. Occurrence 7: soundness.tex, line 173, column 51
  8. Occurrence 8: soundness.tex, line 176, column 17
  9. Occurrence 9: soundness.tex, line 201, column 64
  10. Occurrence 10: soundness.tex, line 219, column 50
  11. Occurrence 11: soundness.tex, line 241, column 27
  12. Occurrence 12: soundness.tex, line 245, column 57
  13. Occurrence 13: soundness.tex, line 254, column 54

Expression 30

Inline MathML variant

s=t,A(s)⇒A(t)\eq[s][t], !A(s) \fCenter !A(t)

Block MathML variant

s=t,A(s)⇒A(t)\eq[s][t], !A(s) \fCenter !A(t)

Conventional reading: antecedent containing first lowercase s is identical to term t, then formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument term t

Meaning here: The first-order sequent read 'antecedent containing first lowercase s is identical to term t, then formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument term t' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: identity.tex, line 42, column 10

Expression 31

Inline MathML variant

A[t/x]\Subst{!A}{t}{x}

Block MathML variant

A[t/x]\Subst{!A}{t}{x}

Conventional reading: the result of substituting term t for variable x in formula A

Meaning here: The substitution expression read 'the result of substituting term t for variable x in formula A' replaces the stated free variable by the stated term, subject to the source convention.

1 occurrence
  1. Occurrence 1: quantifier-rules.tex, line 59, column 41

Expression 32

Inline MathML variant

∀x(A(x))⇒A(t)\lforall[x][!A(x)] \Sequent !A(t)

Block MathML variant

∀x(A(x))⇒A(t)\lforall[x][!A(x)] \Sequent !A(t)

Conventional reading: antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument term t

Meaning here: The first-order sequent read 'antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument term t' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-quantifiers.tex, line 49, column 30

Expression 33

Inline MathML variant

Θ=A∧B,Γ\Theta = !A \land !B, \Gamma

Block MathML variant

Θ=A∧B,Γ\Theta = !A \land !B, \Gamma

Conventional reading: capital Theta equals first the conjunction of formula A and formula B, then Gamma

Meaning here: The metalevel equality read 'capital Theta equals first the conjunction of formula A and formula B, then Gamma' identifies the exact derivation count, sequence, premise family, or semantic quantity named on its two sides.

1 occurrence
  1. Occurrence 1: soundness.tex, line 141, column 7

Expression 34

Inline MathML variant

Γ⇒Δ,∃x(A)\Gamma \fCenter \Delta, \lexists[x][!A]

Block MathML variant

Γ⇒Δ,∃x(A)\Gamma \fCenter \Delta, \lexists[x][!A]

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, formula A

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: quantifier-rules.tex, line 64, column 12

Expression 35

Inline MathML variant

D⇒D!D \fCenter !D

Block MathML variant

D⇒D!D \fCenter !D

Conventional reading: antecedent containing formula D; sequent arrow; succedent containing formula D

Meaning here: The first-order sequent read 'antecedent containing formula D; sequent arrow; succedent containing formula D' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

3 occurrences
  1. Occurrence 1: derivations.tex, line 76, column 7
  2. Occurrence 2: derivations.tex, line 99, column 7
  3. Occurrence 3: derivations.tex, line 108, column 7

Expression 36

Inline MathML variant

Γ⊢A(c)\Gamma \Proves !A(c)

Block MathML variant

Γ⊢A(c)\Gamma \Proves !A(c)

Conventional reading: Gamma syntactically derives formula A with argument constant c

Meaning here: The statement read 'Gamma syntactically derives formula A with argument constant c' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: provability-quantifiers.tex, line 20, column 28

Expression 37

Inline MathML variant

∀x(¬A(x))⇒¬∃x(A(x))\lforall[x][\lnot !A(x)] \Sequent \lnot\lexists[x][!A(x)]

Block MathML variant

∀x(¬A(x))⇒¬∃x(A(x))\lforall[x][\lnot !A(x)] \Sequent \lnot\lexists[x][!A(x)]

Conventional reading: antecedent containing for every variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for some variable x, formula A with argument variable x

Meaning here: The first-order sequent read 'antecedent containing for every variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for some variable x, formula A with argument variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things-quant.tex, line 93, column 7

Expression 38

Inline MathML variant

A,¬A∨B⇒!A, \lnot !A \lor !B \Sequent \quad

Block MathML variant

A,¬A∨B⇒!A, \lnot !A \lor !B \Sequent \quad

Conventional reading: antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 177, column 57

Expression 39

Inline MathML variant

n=0n = 0

Block MathML variant

n=0n = 0

Conventional reading: n equals zero

Meaning here: The metalevel equality read 'n equals zero' identifies the exact derivation count, sequence, premise family, or semantic quantity named on its two sides.

1 occurrence
  1. Occurrence 1: rules-and-proofs.tex, line 41, column 26

Expression 40

Inline MathML variant

C,B⇒A!C, !B \fCenter !A

Block MathML variant

C,B⇒A!C, !B \fCenter !A

Conventional reading: antecedent containing first formula C, then formula B; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing first formula C, then formula B; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 60, column 12

Expression 41

Inline MathML variant

⇒¬(A∧¬A)\Sequent \lnot(!A \land \lnot !A)

Block MathML variant

⇒¬(A∧¬A)\Sequent \lnot(!A \land \lnot !A)

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and the negation of formula A, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and the negation of formula A, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 315, column 7

Expression 42

Inline MathML variant

B⇒A!B \Sequent !A

Block MathML variant

B⇒A!B \Sequent !A

Conventional reading: antecedent containing formula B; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing formula B; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proving-things.tex, line 38, column 21
  2. Occurrence 2: proving-things.tex, line 39, column 48

Expression 43

Inline MathML variant

A,Π⇒Λ!A, \Pi \fCenter \Lambda

Block MathML variant

A,Π⇒Λ!A, \Pi \fCenter \Lambda

Conventional reading: antecedent containing first formula A, then capital Pi; sequent arrow; succedent containing capital Lambda

Meaning here: The first-order sequent read 'antecedent containing first formula A, then capital Pi; sequent arrow; succedent containing capital Lambda' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: soundness.tex, line 275, column 12

Expression 44

Inline MathML variant

C⇒C!C \fCenter !C

Block MathML variant

C⇒C!C \fCenter !C

Conventional reading: antecedent containing formula C; sequent arrow; succedent containing formula C

Meaning here: The first-order sequent read 'antecedent containing formula C; sequent arrow; succedent containing formula C' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

4 occurrences
  1. Occurrence 1: derivations.tex, line 52, column 7
  2. Occurrence 2: derivations.tex, line 60, column 7
  3. Occurrence 3: derivations.tex, line 94, column 7
  4. Occurrence 4: derivations.tex, line 111, column 7

Expression 45

Inline MathML variant

M⊭A(t)\Sat/{M}{!A(t)}

Block MathML variant

M⊭A(t)\Sat/{M}{!A(t)}

Conventional reading: structure M does not satisfy formula A with argument term t

Meaning here: The satisfaction claim read 'structure M does not satisfy formula A with argument term t' is evaluated in the named first-order structure and, when printed, the named variable assignment.

2 occurrences
  1. Occurrence 1: soundness.tex, line 219, column 3
  2. Occurrence 2: soundness.tex, line 223, column 3

Expression 46

Inline MathML variant

¬B,A⇒\lnot !B, !A \fCenter

Block MathML variant

¬B,A⇒\lnot !B, !A \fCenter

Conventional reading: antecedent containing first the negation of formula B, then formula A; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first the negation of formula B, then formula A; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 198, column 10

Expression 48

Inline MathML variant

∀x(A(x)),Γ⇒Δ\lforall[x][!A(x)], \Gamma \fCenter \Delta

Block MathML variant

∀x(A(x)),Γ⇒Δ\lforall[x][!A(x)], \Gamma \fCenter \Delta

Conventional reading: antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: soundness.tex, line 214, column 14

Expression 49

Inline MathML variant

∀x(A(x))⇒A(t)\lforall[x][!A(x)] \fCenter !A(t)

Block MathML variant

∀x(A(x))⇒A(t)\lforall[x][!A(x)] \fCenter !A(t)

Conventional reading: antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument term t

Meaning here: The first-order sequent read 'antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument term t' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-quantifiers.tex, line 54, column 14

Expression 50

Inline MathML variant

Θ=A,Γ\Theta = !A, \Gamma

Block MathML variant

Θ=A,Γ\Theta = !A, \Gamma

Conventional reading: capital Theta equals first formula A, then Gamma

Meaning here: The metalevel equality read 'capital Theta equals first formula A, then Gamma' identifies the exact derivation count, sequence, premise family, or semantic quantity named on its two sides.

1 occurrence
  1. Occurrence 1: soundness.tex, line 98, column 38

Expression 51

Inline MathML variant

¬A∨¬B⇒¬(A∧B)\lnot !A \lor \lnot !B \Sequent \lnot(!A \land !B)

Block MathML variant

¬A∨¬B⇒¬(A∧B)\lnot !A \lor \lnot !B \Sequent \lnot(!A \land !B)

Conventional reading: antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 322, column 7

Expression 52

Inline MathML variant

A→B,Γ,Π⇒Δ,Λ!A \lif !B, \Gamma, \Pi \fCenter \Delta, \Lambda

Block MathML variant

A→B,Γ,Π⇒Δ,Λ!A \lif !B, \Gamma, \Pi \fCenter \Delta, \Lambda

Conventional reading: antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then Gamma, and finally capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda

Meaning here: The first-order sequent read 'antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then Gamma, and finally capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: propositional-rules.tex, line 79, column 11
  2. Occurrence 2: soundness.tex, line 316, column 15

Expression 53

Inline MathML variant

𝔐\Struct M

Block MathML variant

𝔐\Struct M

Conventional reading: structure M

Meaning here: The expression read 'structure M' denotes the named first-order structure.

11 occurrences
  1. Occurrence 1: soundness.tex, line 42, column 29
  2. Occurrence 2: soundness.tex, line 48, column 63
  3. Occurrence 3: soundness.tex, line 63, column 36
  4. Occurrence 4: soundness.tex, line 142, column 30
  5. Occurrence 5: soundness.tex, line 154, column 54
  6. Occurrence 6: soundness.tex, line 217, column 55
  7. Occurrence 7: soundness.tex, line 225, column 28
  8. Occurrence 8: soundness.tex, line 240, column 17
  9. Occurrence 9: soundness.tex, line 299, column 39
  10. Occurrence 10: soundness-identity.tex, line 19, column 25
  11. Occurrence 11: soundness-identity.tex, line 26, column 16

Expression 54

Inline MathML variant

π1\pi_1

Block MathML variant

π1\pi_1

Conventional reading: pi sub one

Meaning here: Pi sub one denotes the second cited sequent-calculus derivation.

8 occurrences
  1. Occurrence 1: proof-theoretic-notions.tex, line 112, column 45
  2. Occurrence 2: proof-theoretic-notions.tex, line 119, column 15
  3. Occurrence 3: provability-consistency.tex, line 29, column 20
  4. Occurrence 4: provability-consistency.tex, line 35, column 13
  5. Occurrence 5: provability-consistency.tex, line 57, column 17
  6. Occurrence 6: provability-consistency.tex, line 64, column 15
  7. Occurrence 7: provability-consistency.tex, line 107, column 62
  8. Occurrence 8: provability-consistency.tex, line 117, column 13

Expression 55

Inline MathML variant

A⊢B!A \Proves !B

Block MathML variant

A⊢B!A \Proves !B

Conventional reading: formula A syntactically derives formula B

Meaning here: The statement read 'formula A syntactically derives formula B' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 128, column 68

Expression 56

Inline MathML variant

A,¬A∨B⇒B!A, \lnot !A \lor !B \fCenter !B

Block MathML variant

A,¬A∨B⇒B!A, \lnot !A \lor !B \fCenter !B

Conventional reading: antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B

Meaning here: The first-order sequent read 'antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

5 occurrences
  1. Occurrence 1: proving-things.tex, line 72, column 10
  2. Occurrence 2: proving-things.tex, line 87, column 10
  3. Occurrence 3: proving-things.tex, line 105, column 10
  4. Occurrence 4: proving-things.tex, line 126, column 10
  5. Occurrence 5: proving-things.tex, line 148, column 10

Expression 57

Inline MathML variant

Γ⇒Δ,∃x(A(x))\Gamma \fCenter \Delta, \lexists[x][!A(x)]

Block MathML variant

Γ⇒Δ,∃x(A(x))\Gamma \fCenter \Delta, \lexists[x][!A(x)]

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, formula A with argument variable x

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, formula A with argument variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: quantifier-rules.tex, line 44, column 10

Expression 58

Inline MathML variant

Γ,Π⇒Δ,Λ\Gamma, \Pi \Sequent \Delta, \Lambda

Block MathML variant

Γ,Π⇒Δ,Λ\Gamma, \Pi \Sequent \Delta, \Lambda

Conventional reading: antecedent containing first Gamma, then capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda

Meaning here: The first-order sequent read 'antecedent containing first Gamma, then capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: soundness.tex, line 321, column 3
  2. Occurrence 2: soundness.tex, line 323, column 60

Expression 59

Inline MathML variant

⇒(∃x(A(x))∨∃y(B(y)))→∃z((A(z)∨B(z)))\Sequent (\lexists[x][!A(x)] \lor \lexists[y][!B(y)]) \lif \lexists[z][(!A(z) \lor !B(z))]

Block MathML variant

⇒(∃x(A(x))∨∃y(B(y)))→∃z((A(z)∨B(z)))\Sequent (\lexists[x][!A(x)] \lor \lexists[y][!B(y)]) \lif \lexists[z][(!A(z) \lor !B(z))]

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is open parenthesis, the disjunction of for some variable x, formula A with argument variable x and for some variable y, formula B with argument variable y, close parenthesis; and whose consequent is for some variable z, open parenthesis, the disjunction of formula A with argument variable z and formula B with argument variable z, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is open parenthesis, the disjunction of for some variable x, formula A with argument variable x and for some variable y, formula B with argument variable y, close parenthesis; and whose consequent is for some variable z, open parenthesis, the disjunction of formula A with argument variable z and formula B with argument variable z, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things-quant.tex, line 90, column 7

Expression 60

Inline MathML variant

¬\lnot

Block MathML variant

¬\lnot

Conventional reading: negation

Meaning here: The logical notation read 'negation' names the exact connective, quantifier, identity sign, sequent arrow, or derivability relation printed by the source.

6 occurrences
  1. Occurrence 1: propositional-rules.tex, line 15, column 23
  2. Occurrence 2: proving-things.tex, line 60, column 45
  3. Occurrence 3: proving-things.tex, line 63, column 7
  4. Occurrence 4: proving-things.tex, line 111, column 21
  5. Occurrence 5: proving-things.tex, line 165, column 27
  6. Occurrence 6: proving-things.tex, line 248, column 61

Expression 61

Inline MathML variant

ฮ“โ‡’ฮ”,AA,ฮ โ‡’ฮ›cutฮ“,ฮ โ‡’ฮ”,ฮ›\Axiom$ \Gamma \fCenter \Delta, !A$ \Axiom$ !A, \Pi \fCenter \Lambda $ \RightLabel{\Cut} \BinaryInf$ \Gamma, \Pi \fCenter \Delta, \Lambda$ \DisplayProof

Block MathML variant

ฮ“โ‡’ฮ”,AA,ฮ โ‡’ฮ›cutฮ“,ฮ โ‡’ฮ”,ฮ›\Axiom$ \Gamma \fCenter \Delta, !A$ \Axiom$ !A, \Pi \fCenter \Lambda $ \RightLabel{\Cut} \BinaryInf$ \Gamma, \Pi \fCenter \Delta, \Lambda$ \DisplayProof

Conventional reading: Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A. Premise or initial sequent: antecedent containing first formula A, then capital Pi; sequent arrow; succedent containing capital Lambda. The next inference is labeled cut rule. From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma, then capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda. The source ends this displayed proof segment here.

Meaning here: A two-premise cut inference: the first premise has A on the right, the second has A on the left, and the conclusion removes that cut formula while concatenating the remaining sides.

1 occurrence
  1. Occurrence 1: structural-rules.tex, line 71, column 1

Expression 62

Inline MathML variant

¬A(a)⇒¬∀xA(x)\lnot !A(a) \fCenter \lnot \lforall[x] !A(x)

Block MathML variant

¬A(a)⇒¬∀xA(x)\lnot !A(a) \fCenter \lnot \lforall[x] !A(x)

Conventional reading: antecedent containing the negation of formula A with argument constant a; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x

Meaning here: The first-order sequent read 'antecedent containing the negation of formula A with argument constant a; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things-quant.tex, line 55, column 10

Expression 63

Inline MathML variant

A⇒A,B!A \fCenter !A, !B

Block MathML variant

A⇒A,B!A \fCenter !A, !B

Conventional reading: antecedent containing formula A; sequent arrow; succedent containing first formula A, then formula B

Meaning here: The first-order sequent read 'antecedent containing formula A; sequent arrow; succedent containing first formula A, then formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 135, column 10

Expression 64

Inline MathML variant

LWeakening\LeftR{\Weakening}

Block MathML variant

LWeakening\LeftR{\Weakening}

Conventional reading: left weakening rule

Meaning here: The rule label read 'left weakening rule' names the exact side and logical or structural operator printed by the source.

1 occurrence
  1. Occurrence 1: derivations.tex, line 40, column 1

Expression 65

Inline MathML variant

Γ0⇒∀x(A(x))\Gamma_0 \Sequent \lforall[x][!A(x)]

Block MathML variant

Γ0⇒∀x(A(x))\Gamma_0 \Sequent \lforall[x][!A(x)]

Conventional reading: antecedent containing Gamma sub zero; sequent arrow; succedent containing for every variable x, formula A with argument variable x

Meaning here: The first-order sequent read 'antecedent containing Gamma sub zero; sequent arrow; succedent containing for every variable x, formula A with argument variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-quantifiers.tex, line 27, column 61

Expression 66

Inline MathML variant

Γ0⇒\Gamma_0 \Sequent \quad

Block MathML variant

Γ0⇒\Gamma_0 \Sequent \quad

Conventional reading: antecedent containing Gamma sub zero; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing Gamma sub zero; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

5 occurrences
  1. Occurrence 1: proof-theoretic-notions.tex, line 72, column 13
  2. Occurrence 2: proof-theoretic-notions.tex, line 74, column 32
  3. Occurrence 3: proof-theoretic-notions.tex, line 165, column 19
  4. Occurrence 4: soundness.tex, line 374, column 4
  5. Occurrence 5: soundness.tex, line 375, column 1

Expression 67

Inline MathML variant

A,Γ1⇒!A, \Gamma_1 \Sequent \quad

Block MathML variant

A,Γ1⇒!A, \Gamma_1 \Sequent \quad

Conventional reading: antecedent containing first formula A, then Gamma sub one; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first formula A, then Gamma sub one; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-consistency.tex, line 26, column 51

Expression 68

Inline MathML variant

C,D⇒D!C, !D \fCenter !D

Block MathML variant

C,D⇒D!C, !D \fCenter !D

Conventional reading: antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula D

Meaning here: The first-order sequent read 'antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula D' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

3 occurrences
  1. Occurrence 1: derivations.tex, line 78, column 10
  2. Occurrence 2: derivations.tex, line 101, column 10
  3. Occurrence 3: derivations.tex, line 110, column 10

Expression 69

Inline MathML variant

Γ,B,A,Π⇒Δ,\Gamma, !B, !A, \Pi \fCenter \Delta,

Block MathML variant

Γ,B,A,Π⇒Δ,\Gamma, !B, !A, \Pi \fCenter \Delta,

Conventional reading: antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta, followed by a printed trailing comma with no following formula

Meaning here: The first-order sequent read 'antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta, followed by a printed trailing comma with no following formula' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: derivations.tex, line 70, column 10

Expression 70

Inline MathML variant

Γ⇒A\Gamma \Sequent !A

Block MathML variant

Γ⇒A\Gamma \Sequent !A

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: provability-consistency.tex, line 52, column 28
  2. Occurrence 2: provability-consistency.tex, line 58, column 33

Expression 71

Inline MathML variant

M⊨A→B\Sat{M}{!A \lif !B}

Block MathML variant

M⊨A→B\Sat{M}{!A \lif !B}

Conventional reading: structure M satisfies the conditional whose antecedent is formula A; and whose consequent is formula B

Meaning here: The satisfaction claim read 'structure M satisfies the conditional whose antecedent is formula A; and whose consequent is formula B' is evaluated in the named first-order structure and, when printed, the named variable assignment.

1 occurrence
  1. Occurrence 1: soundness.tex, line 199, column 25

Expression 72

Inline MathML variant

⇒∀x(∀y(((x=y∧A(x))→A(y))))\Sequent \lforall[x][\lforall[y][((x = y \land !A(x)) \lif !A(y))]]

Block MathML variant

⇒∀x(∀y(((x=y∧A(x))→A(y))))\Sequent \lforall[x][\lforall[y][((x = y \land !A(x)) \lif !A(y))]]

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing for every variable x, for every variable y, open parenthesis, the conditional whose antecedent states that open parenthesis, the conjunction whose first conjunct states that variable x is identical to variable y; and whose second conjunct is formula A with argument variable x, close parenthesis; and whose consequent is formula A with argument variable y, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing for every variable x, for every variable y, open parenthesis, the conditional whose antecedent states that open parenthesis, the conjunction whose first conjunct states that variable x is identical to variable y; and whose second conjunct is formula A with argument variable x, close parenthesis; and whose consequent is formula A with argument variable y, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: identity.tex, line 70, column 7

Expression 73

Inline MathML variant

π0\pi_0

Block MathML variant

π0\pi_0

Conventional reading: pi sub zero

Meaning here: Pi sub zero denotes the first cited sequent-calculus derivation.

8 occurrences
  1. Occurrence 1: proof-theoretic-notions.tex, line 110, column 21
  2. Occurrence 2: proof-theoretic-notions.tex, line 116, column 15
  3. Occurrence 3: provability-consistency.tex, line 28, column 8
  4. Occurrence 4: provability-consistency.tex, line 32, column 13
  5. Occurrence 5: provability-consistency.tex, line 52, column 17
  6. Occurrence 6: provability-consistency.tex, line 107, column 50
  7. Occurrence 7: provability-consistency.tex, line 112, column 13
  8. Occurrence 8: provability-quantifiers.tex, line 25, column 5

Expression 75

Inline MathML variant

B→A⇒¬A→¬B!B \lif !A \Sequent \lnot !A \lif \lnot !B

Block MathML variant

B→A⇒¬A→¬B!B \lif !A \Sequent \lnot !A \lif \lnot !B

Conventional reading: antecedent containing the conditional whose antecedent is formula B; and whose consequent is formula A; sequent arrow; succedent containing the conditional whose antecedent is the negation of formula A; and whose consequent is the negation of formula B

Meaning here: The first-order sequent read 'antecedent containing the conditional whose antecedent is formula B; and whose consequent is formula A; sequent arrow; succedent containing the conditional whose antecedent is the negation of formula A; and whose consequent is the negation of formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 316, column 7

Expression 76

Inline MathML variant

L¬\LeftR{\lnot}

Block MathML variant

L¬\LeftR{\lnot}

Conventional reading: left negation rule

Meaning here: The rule label read 'left negation rule' names the exact side and logical or structural operator printed by the source.

1 occurrence
  1. Occurrence 1: provability-consistency.tex, line 53, column 1

Expression 78

Inline MathML variant

A,B⇒A∧B!A, !B \Sequent !A \land !B

Block MathML variant

A,B⇒A∧B!A, !B \Sequent !A \land !B

Conventional reading: antecedent containing first formula A, then formula B; sequent arrow; succedent containing the conjunction of formula A and formula B

Meaning here: The first-order sequent read 'antecedent containing first formula A, then formula B; sequent arrow; succedent containing the conjunction of formula A and formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 48, column 50

Expression 79

Inline MathML variant

M⊭¬A\Sat/{M}{\lnot !A}

Block MathML variant

M⊭¬A\Sat/{M}{\lnot !A}

Conventional reading: structure M does not satisfy the negation of formula A

Meaning here: The satisfaction claim read 'structure M does not satisfy the negation of formula A' is evaluated in the named first-order structure and, when printed, the named variable assignment.

1 occurrence
  1. Occurrence 1: soundness.tex, line 127, column 3

Expression 80

Inline MathML variant

M⊨A∨B\Sat{M}{!A \lor !B}

Block MathML variant

M⊨A∨B\Sat{M}{!A \lor !B}

Conventional reading: structure M satisfies the disjunction of formula A and formula B

Meaning here: The satisfaction claim read 'structure M satisfies the disjunction of formula A and formula B' is evaluated in the named first-order structure and, when printed, the named variable assignment.

1 occurrence
  1. Occurrence 1: soundness.tex, line 175, column 16

Expression 81

Inline MathML variant

A⇒A∨B!A \Sequent !A \lor !B

Block MathML variant

A⇒A∨B!A \Sequent !A \lor !B

Conventional reading: antecedent containing formula A; sequent arrow; succedent containing the disjunction of formula A and formula B

Meaning here: The first-order sequent read 'antecedent containing formula A; sequent arrow; succedent containing the disjunction of formula A and formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 85, column 25

Expression 82

Inline MathML variant

t1=t2⇒t1=t2\eq[t_1][t_2] \fCenter \eq[t_1][t_2]

Block MathML variant

t1=t2⇒t1=t2\eq[t_1][t_2] \fCenter \eq[t_1][t_2]

Conventional reading: antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub two

Meaning here: The first-order sequent read 'antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub two' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: identity.tex, line 55, column 7

Expression 83

Inline MathML variant

A(a)⇒A(a)!A(a) \fCenter !A(a)

Block MathML variant

A(a)⇒A(a)!A(a) \fCenter !A(a)

Conventional reading: antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant a

Meaning here: The first-order sequent read 'antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant a' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

3 occurrences
  1. Occurrence 1: quantifier-rules.tex, line 88, column 9
  2. Occurrence 2: quantifier-rules.tex, line 95, column 9
  3. Occurrence 3: proving-things-quant.tex, line 65, column 7

Expression 84

Inline MathML variant

M′,s⊨A(x)\Sat{M'}{!A(x)}[s]

Block MathML variant

M′,s⊨A(x)\Sat{M'}{!A(x)}[s]

Conventional reading: structure M prime, under variable assignment lowercase s, satisfies formula A with argument variable x

Meaning here: The satisfaction claim read 'structure M prime, under variable assignment lowercase s, satisfies formula A with argument variable x' is evaluated in the named first-order structure and, when printed, the named variable assignment.

1 occurrence
  1. Occurrence 1: soundness.tex, line 261, column 10

Expression 85

Inline MathML variant

Γ0′′⇒A\Gamma_0'' \Sequent !A

Block MathML variant

Γ0′′⇒A\Gamma_0'' \Sequent !A

Conventional reading: antecedent containing Gamma sub zero double prime; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing Gamma sub zero double prime; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 48, column 1

Expression 86

Inline MathML variant

¬A,Γ⇒Δ\lnot !A, \Gamma \fCenter \Delta

Block MathML variant

¬A,Γ⇒Δ\lnot !A, \Gamma \fCenter \Delta

Conventional reading: antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: propositional-rules.tex, line 20, column 10
  2. Occurrence 2: soundness.tex, line 110, column 14

Expression 88

Inline MathML variant

A(t)⊢∃x(A(x))!A(t) \Proves \lexists[x][!A(x)]

Block MathML variant

A(t)⊢∃x(A(x))!A(t) \Proves \lexists[x][!A(x)]

Conventional reading: formula A with argument term t syntactically derives for some variable x, formula A with argument variable x

Meaning here: The statement read 'formula A with argument term t syntactically derives for some variable x, formula A with argument variable x' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: provability-quantifiers.tex, line 35, column 17

Expression 89

Inline MathML variant

B⇒A∨B!B \Sequent !A \lor !B

Block MathML variant

B⇒A∨B!B \Sequent !A \lor !B

Conventional reading: antecedent containing formula B; sequent arrow; succedent containing the disjunction of formula A and formula B

Meaning here: The first-order sequent read 'antecedent containing formula B; sequent arrow; succedent containing the disjunction of formula A and formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 85, column 54

Expression 90

Inline MathML variant

==

Block MathML variant

==

Conventional reading: identity sign

Meaning here: The logical notation read 'identity sign' names the exact connective, quantifier, identity sign, sequent arrow, or derivability relation printed by the source.

1 occurrence
  1. Occurrence 1: soundness-identity.tex, line 23, column 50

Expression 91

Inline MathML variant

Γ⇒Δ,A∧B\Gamma \fCenter \Delta, !A \land !B

Block MathML variant

Γ⇒Δ,A∧B\Gamma \fCenter \Delta, !A \land !B

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

3 occurrences
  1. Occurrence 1: propositional-rules.tex, line 47, column 11
  2. Occurrence 2: derivations.tex, line 86, column 11
  3. Occurrence 3: soundness.tex, line 297, column 15

Expression 92

Inline MathML variant

s=t,A(s)⊢A(t)\eq[s][t], !A(s) \Proves !A(t)

Block MathML variant

s=t,A(s)⊢A(t)\eq[s][t], !A(s) \Proves !A(t)

Conventional reading: first lowercase s is identical to term t, then formula A with argument lowercase s syntactically derives formula A with argument term t

Meaning here: The statement read 'first lowercase s is identical to term t, then formula A with argument lowercase s syntactically derives formula A with argument term t' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: identity.tex, line 35, column 39

Expression 93

Inline MathML variant

Γ0⇒A\Gamma_0 \fCenter !A

Block MathML variant

Γ0⇒A\Gamma_0 \fCenter !A

Conventional reading: antecedent containing Gamma sub zero; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing Gamma sub zero; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

3 occurrences
  1. Occurrence 1: proof-theoretic-notions.tex, line 117, column 10
  2. Occurrence 2: provability-consistency.tex, line 33, column 8
  3. Occurrence 3: provability-consistency.tex, line 87, column 12

Expression 94

Inline MathML variant

Δ=⟨B1,…,Bn⟩\Delta = \tuple{!B_1, \dots, !B_n}

Block MathML variant

Δ=⟨B1,…,Bn⟩\Delta = \tuple{!B_1, \dots, !B_n}

Conventional reading: capital Delta equals the finite sequence first formula B sub one, continuing through the omitted intermediate entries, and finally formula B sub n

Meaning here: The metalevel equality read 'capital Delta equals the finite sequence first formula B sub one, continuing through the omitted intermediate entries, and finally formula B sub n' identifies the exact derivation count, sequence, premise family, or semantic quantity named on its two sides.

1 occurrence
  1. Occurrence 1: rules-and-proofs.tex, line 32, column 1

Expression 95

Inline MathML variant

Γ0′=⟨B,B,C⟩\Gamma_0' = \tuple{!B, !B, !C}

Block MathML variant

Γ0′=⟨B,B,C⟩\Gamma_0' = \tuple{!B, !B, !C}

Conventional reading: Gamma sub zero prime equals the finite sequence first formula B, then formula B, and finally formula C

Meaning here: The metalevel equality read 'Gamma sub zero prime equals the finite sequence first formula B, then formula B, and finally formula C' identifies the exact derivation count, sequence, premise family, or semantic quantity named on its two sides.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 50, column 11

Expression 96

Inline MathML variant

¬B,A∧B⇒\lnot !B, !A \land !B \fCenter

Block MathML variant

¬B,A∧B⇒\lnot !B, !A \land !B \fCenter

Conventional reading: antecedent containing first the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proving-things.tex, line 218, column 10
  2. Occurrence 2: proving-things.tex, line 239, column 10

Expression 97

Inline MathML variant

A,Γ0⇒!A, \Gamma_0 \Sequent \quad

Block MathML variant

A,Γ0⇒!A, \Gamma_0 \Sequent \quad

Conventional reading: antecedent containing first formula A, then Gamma sub zero; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first formula A, then Gamma sub zero; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-consistency.tex, line 108, column 4

Expression 98

Inline MathML variant

P(t,x)\Atom{\Obj P}{t,x}

Block MathML variant

P(t,x)\Atom{\Obj P}{t,x}

Conventional reading: atomic formula P applied to term t, then variable x

Meaning here: The atomic first-order formula read 'atomic formula P applied to term t, then variable x' preserves the predicate and ordered term arguments.

1 occurrence
  1. Occurrence 1: quantifier-rules.tex, line 67, column 4

Expression 99

Inline MathML variant

¬A∨B,A⇒B\lnot !A \lor !B, !A \fCenter !B

Block MathML variant

¬A∨B,A⇒B\lnot !A \lor !B, !A \fCenter !B

Conventional reading: antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B

Meaning here: The first-order sequent read 'antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

4 occurrences
  1. Occurrence 1: proving-things.tex, line 85, column 11
  2. Occurrence 2: proving-things.tex, line 103, column 11
  3. Occurrence 3: proving-things.tex, line 124, column 11
  4. Occurrence 4: proving-things.tex, line 146, column 11

Expression 100

Inline MathML variant

A(a)⇒A(a)!A(a) \Sequent !A(a)

Block MathML variant

A(a)⇒A(a)!A(a) \Sequent !A(a)

Conventional reading: antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant a

Meaning here: The first-order sequent read 'antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant a' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proving-things-quant.tex, line 24, column 16
  2. Occurrence 2: proving-things-quant.tex, line 62, column 22

Expression 101

Inline MathML variant

M⊭A∧B\Sat/{M}{!A \land !B}

Block MathML variant

M⊭A∧B\Sat/{M}{!A \land !B}

Conventional reading: structure M does not satisfy the conjunction of formula A and formula B

Meaning here: The satisfaction claim read 'structure M does not satisfy the conjunction of formula A and formula B' is evaluated in the named first-order structure and, when printed, the named variable assignment.

1 occurrence
  1. Occurrence 1: soundness.tex, line 148, column 16

Expression 102

Inline MathML variant

B,B,C⇒A!B, !B, !C \fCenter !A

Block MathML variant

B,B,C⇒A!B, !B, !C \fCenter !A

Conventional reading: antecedent containing first formula B, then formula B, and finally formula C; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing first formula B, then formula B, and finally formula C; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 56, column 10

Expression 103

Inline MathML variant

t2=t3,t1=t2⇒t1=t3\eq[t_2][t_3], \eq[t_1][t_2] \fCenter \eq[t_1][t_3]

Block MathML variant

t2=t3,t1=t2⇒t1=t3\eq[t_2][t_3], \eq[t_1][t_2] \fCenter \eq[t_1][t_3]

Conventional reading: antecedent containing first term t sub two is identical to term t sub three, then term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub three

Meaning here: The first-order sequent read 'antecedent containing first term t sub two is identical to term t sub three, then term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub three' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: identity.tex, line 59, column 10

Expression 104

Inline MathML variant

⇒A∨¬A,A∨¬A\fCenter !A \lor \lnot !A, !A \lor \lnot !A

Block MathML variant

⇒A∨¬A,A∨¬A\fCenter !A \lor \lnot !A, !A \lor \lnot !A

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then the disjunction of formula A and the negation of formula A

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then the disjunction of formula A and the negation of formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proving-things.tex, line 279, column 10
  2. Occurrence 2: proving-things.tex, line 294, column 10

Expression 105

Inline MathML variant

¬A,Γ1⇒\lnot !A, \Gamma_1 \fCenter

Block MathML variant

¬A,Γ1⇒\lnot !A, \Gamma_1 \fCenter

Conventional reading: antecedent containing first the negation of formula A, then Gamma sub one; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first the negation of formula A, then Gamma sub one; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-consistency.tex, line 118, column 8

Expression 106

Inline MathML variant

𝔐\Struct{M}

Block MathML variant

𝔐\Struct{M}

Conventional reading: structure M

Meaning here: The expression read 'structure M' denotes the named first-order structure.

25 occurrences
  1. Occurrence 1: soundness.tex, line 92, column 29
  2. Occurrence 2: soundness.tex, line 115, column 41
  3. Occurrence 3: soundness.tex, line 120, column 49
  4. Occurrence 4: soundness.tex, line 156, column 31
  5. Occurrence 5: soundness.tex, line 170, column 30
  6. Occurrence 6: soundness.tex, line 179, column 26
  7. Occurrence 7: soundness.tex, line 181, column 21
  8. Occurrence 8: soundness.tex, line 193, column 15
  9. Occurrence 9: soundness.tex, line 204, column 21
  10. Occurrence 10: soundness.tex, line 206, column 15
  11. Occurrence 11: soundness.tex, line 224, column 8
  12. Occurrence 12: soundness.tex, line 244, column 11
  13. Occurrence 13: soundness.tex, line 252, column 46
  14. Occurrence 14: soundness.tex, line 279, column 19
  15. Occurrence 15: soundness.tex, line 281, column 18
  16. Occurrence 16: soundness.tex, line 284, column 19
  17. Occurrence 17: soundness.tex, line 287, column 15
  18. Occurrence 18: soundness.tex, line 289, column 15
  19. Occurrence 19: soundness.tex, line 301, column 15
  20. Occurrence 20: soundness.tex, line 319, column 30
  21. Occurrence 21: soundness.tex, line 320, column 27
  22. Occurrence 22: soundness.tex, line 323, column 15
  23. Occurrence 23: soundness.tex, line 360, column 27
  24. Occurrence 24: soundness.tex, line 376, column 27
  25. Occurrence 25: soundness.tex, line 380, column 13

Expression 107

Inline MathML variant

A,B⇒B!A, !B \fCenter !B

Block MathML variant

A,B⇒B!A, !B \fCenter !B

Conventional reading: antecedent containing first formula A, then formula B; sequent arrow; succedent containing formula B

Meaning here: The first-order sequent read 'antecedent containing first formula A, then formula B; sequent arrow; succedent containing formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

4 occurrences
  1. Occurrence 1: proving-things.tex, line 99, column 10
  2. Occurrence 2: proving-things.tex, line 120, column 10
  3. Occurrence 3: proving-things.tex, line 142, column 10
  4. Occurrence 4: provability-propositional.tex, line 131, column 18

Expression 108

Inline MathML variant

A,Γ!A, \Gamma

Block MathML variant

A,Γ!A, \Gamma

Conventional reading: first formula A, then Gamma

Meaning here: The sequence read 'first formula A, then Gamma' preserves the formula order and any printed empty or trailing position.

1 occurrence
  1. Occurrence 1: rules-and-proofs.tex, line 47, column 64

Expression 109

Inline MathML variant

A,¬A⇒B!A, \lnot !A \fCenter !B

Block MathML variant

A,¬A⇒B!A, \lnot !A \fCenter !B

Conventional reading: antecedent containing first formula A, then the negation of formula A; sequent arrow; succedent containing formula B

Meaning here: The first-order sequent read 'antecedent containing first formula A, then the negation of formula A; sequent arrow; succedent containing formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 125, column 18

Expression 110

Inline MathML variant

A∧(B∧C)⇒(A∧B)∧C!A \land (!B \land !C) \Sequent (!A \land !B) \land !C

Block MathML variant

A∧(B∧C)⇒(A∧B)∧C!A \land (!B \land !C) \Sequent (!A \land !B) \land !C

Conventional reading: antecedent containing the conjunction of formula A and open parenthesis, the conjunction of formula B and formula C, close parenthesis; sequent arrow; succedent containing the conjunction of open parenthesis, the conjunction of formula A and formula B, close parenthesis and formula C

Meaning here: The first-order sequent read 'antecedent containing the conjunction of formula A and open parenthesis, the conjunction of formula B and formula C, close parenthesis; sequent arrow; succedent containing the conjunction of open parenthesis, the conjunction of formula A and formula B, close parenthesis and formula C' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 303, column 7

Expression 111

Inline MathML variant

A(a)⇒∀x(A(x))!A(a) \fCenter \lforall[x][!A(x)]

Block MathML variant

A(a)⇒∀x(A(x))!A(a) \fCenter \lforall[x][!A(x)]

Conventional reading: antecedent containing formula A with argument constant a; sequent arrow; succedent containing for every variable x, formula A with argument variable x

Meaning here: The first-order sequent read 'antecedent containing formula A with argument constant a; sequent arrow; succedent containing for every variable x, formula A with argument variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: quantifier-rules.tex, line 97, column 12

Expression 112

Inline MathML variant

t1=t2,Γ⇒Δ,A(t2)\eq[t_1][t_2], \Gamma \fCenter \Delta, !A(t_2)

Block MathML variant

t1=t2,Γ⇒Δ,A(t2)\eq[t_1][t_2], \Gamma \fCenter \Delta, !A(t_2)

Conventional reading: antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub two

Meaning here: The first-order sequent read 'antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub two' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: identity.tex, line 25, column 10
  2. Occurrence 2: identity.tex, line 28, column 7

Expression 113

Inline MathML variant

Γ0⇒A(c)\Gamma_0 \Sequent !A(c)

Block MathML variant

Γ0⇒A(c)\Gamma_0 \Sequent !A(c)

Conventional reading: antecedent containing Gamma sub zero; sequent arrow; succedent containing formula A with argument constant c

Meaning here: The first-order sequent read 'antecedent containing Gamma sub zero; sequent arrow; succedent containing formula A with argument constant c' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-quantifiers.tex, line 25, column 48

Expression 114

Inline MathML variant

¬A∨¬B⇒¬(A∧B)\lnot !A \lor \lnot !B \Sequent \lnot (!A \land !B)

Block MathML variant

¬A∨¬B⇒¬(A∧B)\lnot !A \lor \lnot !B \Sequent \lnot (!A \land !B)

Conventional reading: antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 155, column 50

Expression 115

Inline MathML variant

⇒A∨¬A\fCenter !A \lor \lnot !A

Block MathML variant

⇒A∨¬A\fCenter !A \lor \lnot !A

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

4 occurrences
  1. Occurrence 1: proving-things.tex, line 261, column 10
  2. Occurrence 2: proving-things.tex, line 268, column 10
  3. Occurrence 3: proving-things.tex, line 281, column 10
  4. Occurrence 4: proving-things.tex, line 296, column 10

Expression 116

Inline MathML variant

A,Γ0⇒!A, \Gamma_0 \fCenter

Block MathML variant

A,Γ0⇒!A, \Gamma_0 \fCenter

Conventional reading: antecedent containing first formula A, then Gamma sub zero; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first formula A, then Gamma sub zero; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-consistency.tex, line 113, column 8

Expression 117

Inline MathML variant

¬A⇒A→B\lnot !A \fCenter !A \lif !B

Block MathML variant

¬A⇒A→B\lnot !A \fCenter !A \lif !B

Conventional reading: antecedent containing the negation of formula A; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B

Meaning here: The first-order sequent read 'antecedent containing the negation of formula A; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 127, column 18

Expression 118

Inline MathML variant

Θ⇒Ξ\Theta \Sequent \Xi

Block MathML variant

Θ⇒Ξ\Theta \Sequent \Xi

Conventional reading: antecedent containing capital Theta; sequent arrow; succedent containing capital Xi

Meaning here: The first-order sequent read 'antecedent containing capital Theta; sequent arrow; succedent containing capital Xi' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

9 occurrences
  1. Occurrence 1: soundness.tex, line 53, column 59
  2. Occurrence 2: soundness.tex, line 54, column 21
  3. Occurrence 3: soundness.tex, line 58, column 33
  4. Occurrence 4: soundness.tex, line 75, column 48
  5. Occurrence 5: soundness.tex, line 102, column 36
  6. Occurrence 6: soundness.tex, line 119, column 62
  7. Occurrence 7: soundness.tex, line 129, column 57
  8. Occurrence 8: soundness.tex, line 180, column 47
  9. Occurrence 9: soundness.tex, line 182, column 3

Expression 119

Inline MathML variant

M⊭A\Sat/{M}{!A}

Block MathML variant

M⊭A\Sat/{M}{!A}

Conventional reading: structure M does not satisfy formula A

Meaning here: The satisfaction claim read 'structure M does not satisfy formula A' is evaluated in the named first-order structure and, when printed, the named variable assignment.

5 occurrences
  1. Occurrence 1: soundness.tex, line 45, column 1
  2. Occurrence 2: soundness.tex, line 64, column 1
  3. Occurrence 3: soundness.tex, line 145, column 3
  4. Occurrence 4: soundness.tex, line 195, column 22
  5. Occurrence 5: soundness.tex, line 282, column 33

Expression 120

Inline MathML variant

Γ⇒Δ,B\Gamma \fCenter \Delta, !B

Block MathML variant

Γ⇒Δ,B\Gamma \fCenter \Delta, !B

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

4 occurrences
  1. Occurrence 1: propositional-rules.tex, line 45, column 7
  2. Occurrence 2: propositional-rules.tex, line 66, column 7
  3. Occurrence 3: derivations.tex, line 84, column 7
  4. Occurrence 4: soundness.tex, line 295, column 12

Expression 121

Inline MathML variant

M⊭C\Sat/{M}{!C}

Block MathML variant

M⊭C\Sat/{M}{!C}

Conventional reading: structure M does not satisfy formula C

Meaning here: The satisfaction claim read 'structure M does not satisfy formula C' is evaluated in the named first-order structure and, when printed, the named variable assignment.

17 occurrences
  1. Occurrence 1: soundness.tex, line 94, column 3
  2. Occurrence 2: soundness.tex, line 97, column 6
  3. Occurrence 3: soundness.tex, line 99, column 3
  4. Occurrence 4: soundness.tex, line 117, column 3
  5. Occurrence 5: soundness.tex, line 123, column 3
  6. Occurrence 6: soundness.tex, line 129, column 3
  7. Occurrence 7: soundness.tex, line 146, column 3
  8. Occurrence 8: soundness.tex, line 150, column 3
  9. Occurrence 9: soundness.tex, line 151, column 25
  10. Occurrence 10: soundness.tex, line 173, column 3
  11. Occurrence 11: soundness.tex, line 177, column 3
  12. Occurrence 12: soundness.tex, line 197, column 3
  13. Occurrence 13: soundness.tex, line 202, column 12
  14. Occurrence 14: soundness.tex, line 219, column 26
  15. Occurrence 15: soundness.tex, line 241, column 3
  16. Occurrence 16: soundness.tex, line 247, column 3
  17. Occurrence 17: soundness.tex, line 378, column 1

Expression 122

Inline MathML variant

Γ⇒Δ,B\Gamma \Sequent \Delta, !B

Block MathML variant

Γ⇒Δ,B\Gamma \Sequent \Delta, !B

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: soundness.tex, line 304, column 58

Expression 124

Inline MathML variant

¬A,Γ0⇒\lnot !A, \Gamma_0 \Sequent \quad

Block MathML variant

¬A,Γ0⇒\lnot !A, \Gamma_0 \Sequent \quad

Conventional reading: antecedent containing first the negation of formula A, then Gamma sub zero; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first the negation of formula A, then Gamma sub zero; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-consistency.tex, line 83, column 11

Expression 125

Inline MathML variant

M⊨C\Sat{M}{!C}

Block MathML variant

M⊨C\Sat{M}{!C}

Conventional reading: structure M satisfies formula C

Meaning here: The satisfaction claim read 'structure M satisfies formula C' is evaluated in the named first-order structure and, when printed, the named variable assignment.

17 occurrences
  1. Occurrence 1: soundness.tex, line 95, column 21
  2. Occurrence 2: soundness.tex, line 100, column 17
  3. Occurrence 3: soundness.tex, line 101, column 29
  4. Occurrence 4: soundness.tex, line 118, column 12
  5. Occurrence 5: soundness.tex, line 125, column 3
  6. Occurrence 6: soundness.tex, line 147, column 7
  7. Occurrence 7: soundness.tex, line 153, column 8
  8. Occurrence 8: soundness.tex, line 174, column 7
  9. Occurrence 9: soundness.tex, line 178, column 25
  10. Occurrence 10: soundness.tex, line 198, column 7
  11. Occurrence 11: soundness.tex, line 201, column 3
  12. Occurrence 12: soundness.tex, line 203, column 25
  13. Occurrence 13: soundness.tex, line 220, column 7
  14. Occurrence 14: soundness.tex, line 241, column 51
  15. Occurrence 15: soundness.tex, line 246, column 12
  16. Occurrence 16: soundness-identity.tex, line 28, column 8
  17. Occurrence 17: soundness-identity.tex, line 32, column 14

Expression 127

Inline MathML variant

A,B⇒A∧B!A, !B \fCenter !A \land !B

Block MathML variant

A,B⇒A∧B!A, !B \fCenter !A \land !B

Conventional reading: antecedent containing first formula A, then formula B; sequent arrow; succedent containing the conjunction of formula A and formula B

Meaning here: The first-order sequent read 'antecedent containing first formula A, then formula B; sequent arrow; succedent containing the conjunction of formula A and formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 53, column 17

Expression 128

Inline MathML variant

¬A∨¬B,A∧B⇒\lnot !A \lor \lnot !B, !A \land !B \fCenter

Block MathML variant

¬A∨¬B,A∧B⇒\lnot !A \lor \lnot !B, !A \land !B \fCenter

Conventional reading: antecedent containing first the disjunction of the negation of formula A and the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first the disjunction of the negation of formula A and the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proving-things.tex, line 221, column 11
  2. Occurrence 2: proving-things.tex, line 242, column 11

Expression 129

Inline MathML variant

⇒(A→B)∨(B→C)\Sequent (!A \lif !B) \lor (!B \lif !C)

Block MathML variant

⇒(A→B)∨(B→C)\Sequent (!A \lif !B) \lor (!B \lif !C)

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing the disjunction of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula B, close parenthesis and open parenthesis, the conditional whose antecedent is formula B; and whose consequent is formula C, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing the disjunction of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula B, close parenthesis and open parenthesis, the conditional whose antecedent is formula B; and whose consequent is formula C, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 338, column 7

Expression 130

Inline MathML variant

t1=t2⇒t2=t1\eq[t_1][t_2] \fCenter \eq[t_2][t_1]

Block MathML variant

t1=t2⇒t2=t1\eq[t_1][t_2] \fCenter \eq[t_2][t_1]

Conventional reading: antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub two is identical to term t sub one

Meaning here: The first-order sequent read 'antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub two is identical to term t sub one' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: identity.tex, line 53, column 10

Expression 131

Inline MathML variant

Γ⇒A\Gamma \fCenter !A

Block MathML variant

Γ⇒A\Gamma \fCenter !A

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-consistency.tex, line 67, column 13

Expression 132

Inline MathML variant

Γ0,Δ0⇒B\Gamma_0, \Delta_0 \fCenter !B

Block MathML variant

Γ0,Δ0⇒B\Gamma_0, \Delta_0 \fCenter !B

Conventional reading: antecedent containing first Gamma sub zero, then capital Delta sub zero; sequent arrow; succedent containing formula B

Meaning here: The first-order sequent read 'antecedent containing first Gamma sub zero, then capital Delta sub zero; sequent arrow; succedent containing formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 122, column 13

Expression 133

Inline MathML variant

Γ,B,A,Π⇒Δ\Gamma, !B, !A, \Pi \fCenter \Delta

Block MathML variant

Γ,B,A,Π⇒Δ\Gamma, !B, !A, \Pi \fCenter \Delta

Conventional reading: antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: structural-rules.tex, line 56, column 10

Expression 134

Inline MathML variant

{A}∪Δ⊢B\{!A\} \cup \Delta \Proves !B

Block MathML variant

{A}∪Δ⊢B\{!A\} \cup \Delta \Proves !B

Conventional reading: the union of the set containing formula A, and capital Delta syntactically derives formula B

Meaning here: The statement read 'the union of the set containing formula A, and capital Delta syntactically derives formula B' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

2 occurrences
  1. Occurrence 1: proof-theoretic-notions.tex, line 104, column 28
  2. Occurrence 2: proof-theoretic-notions.tex, line 110, column 60

Expression 135

Inline MathML variant

M′⊨C\Sat{M'}{!C}

Block MathML variant

M′⊨C\Sat{M'}{!C}

Conventional reading: structure M prime satisfies formula C

Meaning here: The satisfaction claim read 'structure M prime satisfies formula C' is evaluated in the named first-order structure and, when printed, the named variable assignment.

1 occurrence
  1. Occurrence 1: soundness.tex, line 255, column 3

Expression 136

Inline MathML variant

A∧B,¬A∨¬B⇒!A \land !B, \lnot !A \lor \lnot !B \fCenter

Block MathML variant

A∧B,¬A∨¬B⇒!A \land !B, \lnot !A \lor \lnot !B \fCenter

Conventional reading: antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

5 occurrences
  1. Occurrence 1: proving-things.tex, line 171, column 10
  2. Occurrence 2: proving-things.tex, line 185, column 10
  3. Occurrence 3: proving-things.tex, line 204, column 10
  4. Occurrence 4: proving-things.tex, line 223, column 10
  5. Occurrence 5: proving-things.tex, line 244, column 10

Expression 137

Inline MathML variant

aM′=s(x)\Assign{a}{M'} = s(x)

Block MathML variant

aM′=s(x)\Assign{a}{M'} = s(x)

Conventional reading: the interpretation of constant a in structure M prime equals s applied to variable x

Meaning here: The semantic term read 'the interpretation of constant a in structure M prime equals s applied to variable x' specifies an interpretation, term value, or assignment relation in a first-order structure.

1 occurrence
  1. Occurrence 1: soundness.tex, line 253, column 3

Expression 138

Inline MathML variant

Γ\Gamma

Block MathML variant

Γ\Gamma

Conventional reading: Gamma

Meaning here: Gamma is context-sensitive: it denotes a sequent antecedent sequence or a premise set, as stated by every exact occurrence record.

34 occurrences
  1. Occurrence 1: rules-and-proofs.tex, line 23, column 7
  2. Occurrence 2: rules-and-proofs.tex, line 24, column 42
  3. Occurrence 3: rules-and-proofs.tex, line 38, column 43
  4. Occurrence 4: rules-and-proofs.tex, line 39, column 25
  5. Occurrence 5: rules-and-proofs.tex, line 46, column 4
  6. Occurrence 6: rules-and-proofs.tex, line 47, column 50
  7. Occurrence 7: rules-and-proofs.tex, line 49, column 4
  8. Occurrence 8: derivations.tex, line 49, column 1
  9. Occurrence 9: derivations.tex, line 72, column 6
  10. Occurrence 10: derivations.tex, line 89, column 57
  11. Occurrence 11: proof-theoretic-notions.tex, line 38, column 15
  12. Occurrence 12: proof-theoretic-notions.tex, line 41, column 61
  13. Occurrence 13: proof-theoretic-notions.tex, line 70, column 20
  14. Occurrence 14: proof-theoretic-notions.tex, line 72, column 43
  15. Occurrence 15: proof-theoretic-notions.tex, line 135, column 1
  16. Occurrence 16: proof-theoretic-notions.tex, line 152, column 35
  17. Occurrence 17: proof-theoretic-notions.tex, line 153, column 22
  18. Occurrence 18: proof-theoretic-notions.tex, line 163, column 14
  19. Occurrence 19: proof-theoretic-notions.tex, line 166, column 24
  20. Occurrence 20: provability-consistency.tex, line 21, column 22
  21. Occurrence 21: provability-consistency.tex, line 42, column 50
  22. Occurrence 22: provability-consistency.tex, line 76, column 58
  23. Occurrence 23: provability-consistency.tex, line 97, column 14
  24. Occurrence 24: provability-consistency.tex, line 102, column 22
  25. Occurrence 25: provability-consistency.tex, line 123, column 50
  26. Occurrence 26: provability-quantifiers.tex, line 20, column 4
  27. Occurrence 27: provability-quantifiers.tex, line 28, column 50
  28. Occurrence 28: soundness.tex, line 237, column 21
  29. Occurrence 29: soundness.tex, line 255, column 46
  30. Occurrence 30: soundness.tex, line 368, column 4
  31. Occurrence 31: soundness.tex, line 372, column 44
  32. Occurrence 32: soundness.tex, line 379, column 41
  33. Occurrence 33: soundness.tex, line 380, column 52
  34. Occurrence 34: soundness.tex, line 381, column 1

Expression 140

Inline MathML variant

A,Γ⇒Δ!A, \Gamma \fCenter \Delta

Block MathML variant

A,Γ⇒Δ!A, \Gamma \fCenter \Delta

Conventional reading: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

8 occurrences
  1. Occurrence 1: propositional-rules.tex, line 23, column 7
  2. Occurrence 2: propositional-rules.tex, line 33, column 7
  3. Occurrence 3: propositional-rules.tex, line 54, column 7
  4. Occurrence 4: structural-rules.tex, line 28, column 10
  5. Occurrence 5: structural-rules.tex, line 42, column 10
  6. Occurrence 6: derivations.tex, line 44, column 10
  7. Occurrence 7: soundness.tex, line 83, column 14
  8. Occurrence 8: soundness.tex, line 137, column 12

Expression 141

Inline MathML variant

s(x)=⟦t1⟧Ms(x) = \Value{t_1}{M}

Block MathML variant

s(x)=⟦t1⟧Ms(x) = \Value{t_1}{M}

Conventional reading: s applied to variable x is identical to the value of term t sub one in structure M

Meaning here: The semantic term read 's applied to variable x is identical to the value of term t sub one in structure M' specifies an interpretation, term value, or assignment relation in a first-order structure.

1 occurrence
  1. Occurrence 1: soundness-identity.tex, line 34, column 1

Expression 142

Inline MathML variant

∀x((A(x)→B))⇒∃y(A(y))→B\lforall[x][(!A(x) \lif !B)] \Sequent \lexists[y][!A(y)] \lif !B

Block MathML variant

∀x((A(x)→B))⇒∃y(A(y))→B\lforall[x][(!A(x) \lif !B)] \Sequent \lexists[y][!A(y)] \lif !B

Conventional reading: antecedent containing for every variable x, open parenthesis, the conditional whose antecedent is formula A with argument variable x; and whose consequent is formula B, close parenthesis; sequent arrow; succedent containing the conditional whose antecedent is for some variable y, formula A with argument variable y; and whose consequent is formula B

Meaning here: The first-order sequent read 'antecedent containing for every variable x, open parenthesis, the conditional whose antecedent is formula A with argument variable x; and whose consequent is formula B, close parenthesis; sequent arrow; succedent containing the conditional whose antecedent is for some variable y, formula A with argument variable y; and whose consequent is formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things-quant.tex, line 92, column 7

Expression 143

Inline MathML variant

⇒Δ\quad \Sequent \Delta

Block MathML variant

⇒Δ\quad \Sequent \Delta

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: rules-and-proofs.tex, line 39, column 59

Expression 144

Inline MathML variant

⇒A\quad \Sequent !A

Block MathML variant

⇒A\quad \Sequent !A

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 32, column 30

Expression 146

Inline MathML variant

Γ⇒Δ,A(a)\Gamma \fCenter \Delta, !A(a)

Block MathML variant

Γ⇒Δ,A(a)\Gamma \fCenter \Delta, !A(a)

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument constant a

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument constant a' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: quantifier-rules.tex, line 21, column 7
  2. Occurrence 2: soundness.tex, line 232, column 12

Expression 147

Inline MathML variant

A,Γ⇒Δ,B!A, \Gamma \fCenter \Delta, !B

Block MathML variant

A,Γ⇒Δ,B!A, \Gamma \fCenter \Delta, !B

Conventional reading: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing first capital Delta, then formula B

Meaning here: The first-order sequent read 'antecedent containing first formula A, then Gamma; sequent arrow; succedent containing first capital Delta, then formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: propositional-rules.tex, line 82, column 7
  2. Occurrence 2: soundness.tex, line 187, column 12

Expression 148

Inline MathML variant

t2t_2

Block MathML variant

t2t_2

Conventional reading: term t sub two

Meaning here: The symbol read 'term t sub two' receives its exact term, variable, constant, or assignment role from each bound source occurrence.

1 occurrence
  1. Occurrence 1: identity.tex, line 20, column 36

Expression 149

Inline MathML variant

empty antecedent⇒t=t{} \Sequent \eq[t][t]

Block MathML variant

empty antecedent⇒t=t{} \Sequent \eq[t][t]

Conventional reading: antecedent containing the empty sequence; sequent arrow; succedent containing term t is identical to term t

Meaning here: The first-order sequent read 'antecedent containing the empty sequence; sequent arrow; succedent containing term t is identical to term t' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: identity.tex, line 17, column 31
  2. Occurrence 2: soundness-identity.tex, line 18, column 30

Expression 150

Inline MathML variant

A⇒B,A!A \fCenter !B, !A

Block MathML variant

A⇒B,A!A \fCenter !B, !A

Conventional reading: antecedent containing formula A; sequent arrow; succedent containing first formula B, then formula A

Meaning here: The first-order sequent read 'antecedent containing formula A; sequent arrow; succedent containing first formula B, then formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proving-things.tex, line 115, column 10
  2. Occurrence 2: proving-things.tex, line 137, column 10

Expression 152

Inline MathML variant

M′,s⊨A(a)\Sat{M'}{!A(a)}[s]

Block MathML variant

M′,s⊨A(a)\Sat{M'}{!A(a)}[s]

Conventional reading: structure M prime, under variable assignment lowercase s, satisfies formula A with argument constant a

Meaning here: The satisfaction claim read 'structure M prime, under variable assignment lowercase s, satisfies formula A with argument constant a' is evaluated in the named first-order structure and, when printed, the named variable assignment.

1 occurrence
  1. Occurrence 1: soundness.tex, line 258, column 3

Expression 154

Inline MathML variant

C,D⇒C∧D!C, !D \fCenter !C \land !D

Block MathML variant

C,D⇒C∧D!C, !D \fCenter !C \land !D

Conventional reading: antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula C and formula D

Meaning here: The first-order sequent read 'antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula C and formula D' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: derivations.tex, line 103, column 11

Expression 155

Inline MathML variant

⇒∃x((A(x)→∀y(A(y))))\Sequent \lexists[x][(!A(x) \lif \lforall[y][!A(y)])]

Block MathML variant

⇒∃x((A(x)→∀y(A(y))))\Sequent \lexists[x][(!A(x) \lif \lforall[y][!A(y)])]

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing for some variable x, open parenthesis, the conditional whose antecedent is formula A with argument variable x; and whose consequent is for every variable y, formula A with argument variable y, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing for some variable x, open parenthesis, the conditional whose antecedent is formula A with argument variable x; and whose consequent is for every variable y, formula A with argument variable y, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things-quant.tex, line 105, column 7

Expression 156

Inline MathML variant

A,Δ0⇒B!A, \Delta_0 \fCenter !B

Block MathML variant

A,Δ0⇒B!A, \Delta_0 \fCenter !B

Conventional reading: antecedent containing first formula A, then capital Delta sub zero; sequent arrow; succedent containing formula B

Meaning here: The first-order sequent read 'antecedent containing first formula A, then capital Delta sub zero; sequent arrow; succedent containing formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 120, column 10

Expression 157

Inline MathML variant

Γ∪Δ⊢B\Gamma \cup \Delta \Proves !B

Block MathML variant

Γ∪Δ⊢B\Gamma \cup \Delta \Proves !B

Conventional reading: the union of Gamma, and capital Delta syntactically derives formula B

Meaning here: The statement read 'the union of Gamma, and capital Delta syntactically derives formula B' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

2 occurrences
  1. Occurrence 1: proof-theoretic-notions.tex, line 105, column 11
  2. Occurrence 2: proof-theoretic-notions.tex, line 125, column 7

Expression 158

Inline MathML variant

C∈Δ,A→B!C \in \Delta, !A \lif !B

Block MathML variant

C∈Δ,A→B!C \in \Delta, !A \lif !B

Conventional reading: formula C is a member of first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B

Meaning here: The relation read 'formula C is a member of first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B' concerns the exact premise sets or formula sequences named by the source.

1 occurrence
  1. Occurrence 1: soundness.tex, line 200, column 21

Expression 159

Inline MathML variant

¬A,A∧B⇒\lnot !A, !A \land !B \fCenter

Block MathML variant

¬A,A∧B⇒\lnot !A, !A \land !B \fCenter

Conventional reading: antecedent containing first the negation of formula A, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first the negation of formula A, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proving-things.tex, line 215, column 10
  2. Occurrence 2: proving-things.tex, line 233, column 10

Expression 160

Inline MathML variant

LExchange\LeftR{\Exchange}

Block MathML variant

LExchange\LeftR{\Exchange}

Conventional reading: left exchange rule

Meaning here: The rule label read 'left exchange rule' names the exact side and logical or structural operator printed by the source.

1 occurrence
  1. Occurrence 1: derivations.tex, line 56, column 36

Expression 161

Inline MathML variant

LK\Log{LK}

Block MathML variant

LK\Log{LK}

Conventional reading: the sequent calculus L K

Meaning here: L K is the classical sequent calculus defined and used in this chapter.

23 occurrences
  1. Occurrence 1: rules-and-proofs.tex, line 66, column 50
  2. Occurrence 2: derivations.tex, line 23, column 14
  3. Occurrence 3: derivations.tex, line 24, column 10
  4. Occurrence 4: derivations.tex, line 34, column 36
  5. Occurrence 5: derivations.tex, line 34, column 52
  6. Occurrence 6: proving-things.tex, line 16, column 9
  7. Occurrence 7: proving-things.tex, line 46, column 23
  8. Occurrence 8: proving-things.tex, line 51, column 9
  9. Occurrence 9: proving-things.tex, line 155, column 9
  10. Occurrence 10: proving-things-quant.tex, line 14, column 9
  11. Occurrence 11: proof-theoretic-notions.tex, line 32, column 4
  12. Occurrence 12: proof-theoretic-notions.tex, line 40, column 46
  13. Occurrence 13: proof-theoretic-notions.tex, line 71, column 53
  14. Occurrence 14: proof-theoretic-notions.tex, line 74, column 1
  15. Occurrence 15: proof-theoretic-notions.tex, line 164, column 52
  16. Occurrence 16: provability-consistency.tex, line 26, column 1
  17. Occurrence 17: provability-consistency.tex, line 27, column 27
  18. Occurrence 18: provability-consistency.tex, line 28, column 24
  19. Occurrence 19: provability-consistency.tex, line 107, column 23
  20. Occurrence 20: provability-quantifiers.tex, line 25, column 19
  21. Occurrence 21: soundness.tex, line 53, column 36
  22. Occurrence 22: identity.tex, line 47, column 1
  23. Occurrence 23: soundness-identity.tex, line 14, column 1

Expression 162

Inline MathML variant

⇒A\fCenter !A

Block MathML variant

⇒A\fCenter !A

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 259, column 10

Expression 163

Inline MathML variant

x=t1\eq[x][t_1]

Block MathML variant

x=t1\eq[x][t_1]

Conventional reading: variable x is identical to term t sub one

Meaning here: The identity expression read 'variable x is identical to term t sub one' states equality of the named first-order terms or semantic values.

1 occurrence
  1. Occurrence 1: identity.tex, line 63, column 52

Expression 164

Inline MathML variant

A(t)⇒A(t)!A(t) \fCenter !A(t)

Block MathML variant

A(t)⇒A(t)!A(t) \fCenter !A(t)

Conventional reading: antecedent containing formula A with argument term t; sequent arrow; succedent containing formula A with argument term t

Meaning here: The first-order sequent read 'antecedent containing formula A with argument term t; sequent arrow; succedent containing formula A with argument term t' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: provability-quantifiers.tex, line 45, column 11
  2. Occurrence 2: provability-quantifiers.tex, line 52, column 11

Expression 165

Inline MathML variant

⇒¬∃x(∀y(((A(x,y)→¬A(y,y))∧(¬A(y,y)→A(x,y)))))\Sequent \lnot\lexists[x][\lforall[y][((!A(x,y) \lif \lnot !A(y,y)) \land (\lnot !A(y,y) \lif !A(x,y)))]]

Block MathML variant

⇒¬∃x(∀y(((A(x,y)→¬A(y,y))∧(¬A(y,y)→A(x,y)))))\Sequent \lnot\lexists[x][\lforall[y][((!A(x,y) \lif \lnot !A(y,y)) \land (\lnot !A(y,y) \lif !A(x,y)))]]

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing the negation of for some variable x, for every variable y, open parenthesis, the conjunction of open parenthesis, the conditional whose antecedent is formula A with arguments variable x and variable y; and whose consequent is the negation of formula A with arguments variable y and variable y, close parenthesis and open parenthesis, the conditional whose antecedent is the negation of formula A with arguments variable y and variable y; and whose consequent is formula A with arguments variable x and variable y, close parenthesis, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing the negation of for some variable x, for every variable y, open parenthesis, the conjunction of open parenthesis, the conditional whose antecedent is formula A with arguments variable x and variable y; and whose consequent is the negation of formula A with arguments variable y and variable y, close parenthesis and open parenthesis, the conditional whose antecedent is the negation of formula A with arguments variable y and variable y; and whose consequent is formula A with arguments variable x and variable y, close parenthesis, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things-quant.tex, line 95, column 7

Expression 166

Inline MathML variant

A,Γ⇒Δ,B!A, \Gamma \Sequent \Delta, !B

Block MathML variant

A,Γ⇒Δ,B!A, \Gamma \Sequent \Delta, !B

Conventional reading: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing first capital Delta, then formula B

Meaning here: The first-order sequent read 'antecedent containing first formula A, then Gamma; sequent arrow; succedent containing first capital Delta, then formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: soundness.tex, line 193, column 64

Expression 167

Inline MathML variant

A→B⇒¬A∨B!A \lif !B \Sequent \lnot !A \lor !B

Block MathML variant

A→B⇒¬A∨B!A \lif !B \Sequent \lnot !A \lor !B

Conventional reading: antecedent containing the conditional whose antecedent is formula A; and whose consequent is formula B; sequent arrow; succedent containing the disjunction of the negation of formula A and formula B

Meaning here: The first-order sequent read 'antecedent containing the conditional whose antecedent is formula A; and whose consequent is formula B; sequent arrow; succedent containing the disjunction of the negation of formula A and formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 333, column 7

Expression 168

Inline MathML variant

Γ⊢¬A\Gamma \Proves \lnot !A

Block MathML variant

Γ⊢¬A\Gamma \Proves \lnot !A

Conventional reading: Gamma syntactically derives the negation of formula A

Meaning here: The statement read 'Gamma syntactically derives the negation of formula A' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: provability-consistency.tex, line 72, column 12

Expression 169

Inline MathML variant

M⊭A→B\Sat/{M}{!A \lif !B}

Block MathML variant

M⊭A→B\Sat/{M}{!A \lif !B}

Conventional reading: structure M does not satisfy the conditional whose antecedent is formula A; and whose consequent is formula B

Meaning here: The satisfaction claim read 'structure M does not satisfy the conditional whose antecedent is formula A; and whose consequent is formula B' is evaluated in the named first-order structure and, when printed, the named variable assignment.

2 occurrences
  1. Occurrence 1: soundness.tex, line 322, column 3
  2. Occurrence 2: soundness.tex, line 329, column 3

Expression 170

Inline MathML variant

C∈Γ0!C \in \Gamma_0

Block MathML variant

C∈Γ0!C \in \Gamma_0

Conventional reading: formula C is a member of Gamma sub zero

Meaning here: The relation read 'formula C is a member of Gamma sub zero' concerns the exact premise sets or formula sequences named by the source.

1 occurrence
  1. Occurrence 1: soundness.tex, line 377, column 10

Expression 171

Inline MathML variant

Γ⇒Δ,A∨B\Gamma \fCenter \Delta, !A \lor !B

Block MathML variant

Γ⇒Δ,A∨B\Gamma \fCenter \Delta, !A \lor !B

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

3 occurrences
  1. Occurrence 1: propositional-rules.tex, line 63, column 10
  2. Occurrence 2: propositional-rules.tex, line 68, column 10
  3. Occurrence 3: soundness.tex, line 167, column 14

Expression 172

Inline MathML variant

D,C⇒C!D, !C \fCenter !C

Block MathML variant

D,C⇒C!D, !C \fCenter !C

Conventional reading: antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C

Meaning here: The first-order sequent read 'antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

4 occurrences
  1. Occurrence 1: derivations.tex, line 54, column 10
  2. Occurrence 2: derivations.tex, line 62, column 10
  3. Occurrence 3: derivations.tex, line 96, column 10
  4. Occurrence 4: derivations.tex, line 113, column 10

Expression 173

Inline MathML variant

B,¬A∨B⇒!B, \lnot !A \lor !B \Sequent \quad

Block MathML variant

B,¬A∨B⇒!B, \lnot !A \lor !B \Sequent \quad

Conventional reading: antecedent containing first formula B, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first formula B, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 178, column 49

Expression 174

Inline MathML variant

¬A,Γ⇒\lnot !A, \Gamma \Sequent \quad

Block MathML variant

¬A,Γ⇒\lnot !A, \Gamma \Sequent \quad

Conventional reading: antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: provability-consistency.tex, line 53, column 52
  2. Occurrence 2: provability-consistency.tex, line 57, column 28

Expression 175

Inline MathML variant

A,Δ0⇒B!A, \Delta_0 \Sequent !B

Block MathML variant

A,Δ0⇒B!A, \Delta_0 \Sequent !B

Conventional reading: antecedent containing first formula A, then capital Delta sub zero; sequent arrow; succedent containing formula B

Meaning here: The first-order sequent read 'antecedent containing first formula A, then capital Delta sub zero; sequent arrow; succedent containing formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 112, column 56

Expression 176

Inline MathML variant

=\eq

Block MathML variant

=\eq

Conventional reading: identity sign

Meaning here: The logical notation read 'identity sign' names the exact connective, quantifier, identity sign, sequent arrow, or derivability relation printed by the source.

8 occurrences
  1. Occurrence 1: identity.tex, line 16, column 35
  2. Occurrence 2: identity.tex, line 20, column 15
  3. Occurrence 3: identity.tex, line 24, column 13
  4. Occurrence 4: identity.tex, line 29, column 13
  5. Occurrence 5: identity.tex, line 41, column 13
  6. Occurrence 6: identity.tex, line 47, column 24
  7. Occurrence 7: identity.tex, line 52, column 13
  8. Occurrence 8: identity.tex, line 58, column 13

Expression 177

Inline MathML variant

A∧B⇒B!A \land !B \fCenter !B

Block MathML variant

A∧B⇒B!A \land !B \fCenter !B

Conventional reading: antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula B

Meaning here: The first-order sequent read 'antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proving-things.tex, line 237, column 10
  2. Occurrence 2: provability-propositional.tex, line 46, column 16

Expression 178

Inline MathML variant

A1,…,An⊢B!A_1, \dots, !A_n \Proves !B

Block MathML variant

A1,…,An⊢B!A_1, \dots, !A_n \Proves !B

Conventional reading: first formula A sub one, continuing through the omitted intermediate formulas, and finally formula A sub n syntactically derives formula B

Meaning here: The statement read 'first formula A sub one, continuing through the omitted intermediate formulas, and finally formula A sub n syntactically derives formula B' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 129, column 64

Expression 179

Inline MathML variant

Γ⇒Δ,∀x(P(a,x))\Gamma \Sequent \Delta, \lforall[x][\Atom{\Obj P}{a,x}]

Block MathML variant

Γ⇒Δ,∀x(P(a,x))\Gamma \Sequent \Delta, \lforall[x][\Atom{\Obj P}{a,x}]

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for every variable x, atomic formula P applied to constant a, then variable x

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for every variable x, atomic formula P applied to constant a, then variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: quantifier-rules.tex, line 75, column 7

Expression 180

Inline MathML variant

M⊨∀x(A(x))\Sat{M}{\lforall[x][!A(x)]}

Block MathML variant

M⊨∀x(A(x))\Sat{M}{\lforall[x][!A(x)]}

Conventional reading: structure M satisfies for every variable x, formula A with argument variable x

Meaning here: The satisfaction claim read 'structure M satisfies for every variable x, formula A with argument variable x' is evaluated in the named first-order structure and, when printed, the named variable assignment.

3 occurrences
  1. Occurrence 1: soundness.tex, line 222, column 3
  2. Occurrence 2: soundness.tex, line 240, column 34
  3. Occurrence 3: soundness.tex, line 248, column 3

Expression 181

Inline MathML variant

⇒A∨¬A,A\fCenter !A \lor \lnot !A, !A

Block MathML variant

⇒A∨¬A,A\fCenter !A \lor \lnot !A, !A

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then formula A

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proving-things.tex, line 277, column 10
  2. Occurrence 2: proving-things.tex, line 292, column 10

Expression 182

Inline MathML variant

Γ0={B,C}\Gamma_0 = \{!B, !C\}

Block MathML variant

Γ0={B,C}\Gamma_0 = \{!B, !C\}

Conventional reading: Gamma sub zero equals the set containing first formula B, then formula C

Meaning here: The metalevel equality read 'Gamma sub zero equals the set containing first formula B, then formula C' identifies the exact derivation count, sequence, premise family, or semantic quantity named on its two sides.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 49, column 49

Expression 183

Inline MathML variant

Γ⇒Δ,A→B\Gamma \Sequent \Delta, !A \lif !B

Block MathML variant

Γ⇒Δ,A→B\Gamma \Sequent \Delta, !A \lif !B

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: soundness.tex, line 204, column 60
  2. Occurrence 2: soundness.tex, line 206, column 59

Expression 184

Inline MathML variant

C,C,B⇒A!C, !C, !B \fCenter !A

Block MathML variant

C,C,B⇒A!C, !C, !B \fCenter !A

Conventional reading: antecedent containing first formula C, then formula C, and finally formula B; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing first formula C, then formula C, and finally formula B; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 62, column 12

Expression 185

Inline MathML variant

B⊢A→B!B \Proves !A \lif !B

Block MathML variant

B⊢A→B!B \Proves !A \lif !B

Conventional reading: formula B syntactically derives the conditional whose antecedent is formula A; and whose consequent is formula B

Meaning here: The statement read 'formula B syntactically derives the conditional whose antecedent is formula A; and whose consequent is formula B' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 103, column 44

Expression 186

Inline MathML variant

A∨B,¬A,¬B!A \lor !B, \lnot !A, \lnot !B

Block MathML variant

A∨B,¬A,¬B!A \lor !B, \lnot !A, \lnot !B

Conventional reading: first the disjunction of formula A and formula B, then the negation of formula A, and finally the negation of formula B

Meaning here: The sequence read 'first the disjunction of formula A and formula B, then the negation of formula A, and finally the negation of formula B' preserves the formula order and any printed empty or trailing position.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 60, column 9

Expression 187

Inline MathML variant

(A→B)→A⇒A(!A \lif !B) \lif !A \Sequent !A

Block MathML variant

(A→B)→A⇒A(!A \lif !B) \lif !A \Sequent !A

Conventional reading: antecedent containing the conditional whose antecedent is open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula B, close parenthesis; and whose consequent is formula A; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing the conditional whose antecedent is open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula B, close parenthesis; and whose consequent is formula A; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 337, column 7

Expression 188

Inline MathML variant

A(x)!A(x)

Block MathML variant

A(x)!A(x)

Conventional reading: formula A with argument variable x

Meaning here: The first-order formula read 'formula A with argument variable x' preserves its metavariables, connectives, term arguments, and written grouping.

9 occurrences
  1. Occurrence 1: quantifier-rules.tex, line 58, column 52
  2. Occurrence 2: provability-quantifiers.tex, line 20, column 16
  3. Occurrence 3: provability-quantifiers.tex, line 28, column 62
  4. Occurrence 4: soundness.tex, line 209, column 16
  5. Occurrence 5: soundness.tex, line 229, column 16
  6. Occurrence 6: soundness.tex, line 237, column 12
  7. Occurrence 7: soundness.tex, line 261, column 60
  8. Occurrence 8: identity.tex, line 64, column 1
  9. Occurrence 9: identity.tex, line 64, column 32

Expression 189

Inline MathML variant

Γ0′′=⟨C,C,B⟩\Gamma_0'' = \tuple{!C, !C, !B}

Block MathML variant

Γ0′′=⟨C,C,B⟩\Gamma_0'' = \tuple{!C, !C, !B}

Conventional reading: Gamma sub zero double prime equals the finite sequence first formula C, then formula C, and finally formula B

Meaning here: The metalevel equality read 'Gamma sub zero double prime equals the finite sequence first formula C, then formula C, and finally formula B' identifies the exact derivation count, sequence, premise family, or semantic quantity named on its two sides.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 50, column 48

Expression 190

Inline MathML variant

A!A

Block MathML variant

A!A

Conventional reading: formula A

Meaning here: The first-order formula read 'formula A' preserves its metavariables, connectives, term arguments, and written grouping.

26 occurrences
  1. Occurrence 1: rules-and-proofs.tex, line 47, column 25
  2. Occurrence 2: rules-and-proofs.tex, line 48, column 37
  3. Occurrence 3: rules-and-proofs.tex, line 60, column 53
  4. Occurrence 4: rules-and-proofs.tex, line 69, column 44
  5. Occurrence 5: quantifier-rules.tex, line 57, column 33
  6. Occurrence 6: quantifier-rules.tex, line 66, column 36
  7. Occurrence 7: quantifier-rules.tex, line 66, column 48
  8. Occurrence 8: quantifier-rules.tex, line 74, column 36
  9. Occurrence 9: derivations.tex, line 46, column 63
  10. Occurrence 10: derivations.tex, line 73, column 1
  11. Occurrence 11: derivations.tex, line 90, column 30
  12. Occurrence 12: derivations.tex, line 105, column 51
  13. Occurrence 13: proving-things.tex, line 180, column 12
  14. Occurrence 14: proving-things-quant.tex, line 63, column 21
  15. Occurrence 15: proof-theoretic-notions.tex, line 31, column 16
  16. Occurrence 16: proof-theoretic-notions.tex, line 33, column 8
  17. Occurrence 17: proof-theoretic-notions.tex, line 37, column 16
  18. Occurrence 18: proof-theoretic-notions.tex, line 41, column 30
  19. Occurrence 19: proof-theoretic-notions.tex, line 136, column 12
  20. Occurrence 20: soundness.tex, line 22, column 27
  21. Occurrence 21: soundness.tex, line 133, column 46
  22. Occurrence 22: soundness.tex, line 158, column 65
  23. Occurrence 23: soundness.tex, line 161, column 46
  24. Occurrence 24: soundness.tex, line 183, column 52
  25. Occurrence 25: soundness.tex, line 348, column 22
  26. Occurrence 26: soundness.tex, line 361, column 52

Expression 191

Inline MathML variant

⊥⇒\lfalse \Sequent \quad

Block MathML variant

⊥⇒\lfalse \Sequent \quad

Conventional reading: antecedent containing falsum; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing falsum; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: rules-and-proofs.tex, line 58, column 24

Expression 192

Inline MathML variant

B,Π⇒Λ!B, \Pi \fCenter \Lambda

Block MathML variant

B,Π⇒Λ!B, \Pi \fCenter \Lambda

Conventional reading: antecedent containing first formula B, then capital Pi; sequent arrow; succedent containing capital Lambda

Meaning here: The first-order sequent read 'antecedent containing first formula B, then capital Pi; sequent arrow; succedent containing capital Lambda' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: propositional-rules.tex, line 77, column 7
  2. Occurrence 2: soundness.tex, line 314, column 12

Expression 193

Inline MathML variant

A∧B⇒A!A \land !B \Sequent !A

Block MathML variant

A∧B⇒A!A \land !B \Sequent !A

Conventional reading: antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

3 occurrences
  1. Occurrence 1: proving-things.tex, line 16, column 51
  2. Occurrence 2: proving-things.tex, line 46, column 64
  3. Occurrence 3: provability-propositional.tex, line 37, column 23

Expression 194

Inline MathML variant

B⊢A∨B!B \Proves !A \lor !B

Block MathML variant

B⊢A∨B!B \Proves !A \lor !B

Conventional reading: formula B syntactically derives the disjunction of formula A and formula B

Meaning here: The statement read 'formula B syntactically derives the disjunction of formula A and formula B' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 61, column 42

Expression 195

Inline MathML variant

C,D⇒C∧D!C, !D \Sequent !C \land !D

Block MathML variant

C,D⇒C∧D!C, !D \Sequent !C \land !D

Conventional reading: antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula C and formula D

Meaning here: The first-order sequent read 'antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula C and formula D' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: derivations.tex, line 91, column 52

Expression 196

Inline MathML variant

M⊨A(t2)\Sat{M}{!A(t_2)}

Block MathML variant

M⊨A(t2)\Sat{M}{!A(t_2)}

Conventional reading: structure M satisfies formula A with argument term t sub two

Meaning here: The satisfaction claim read 'structure M satisfies formula A with argument term t sub two' is evaluated in the named first-order structure and, when printed, the named variable assignment.

2 occurrences
  1. Occurrence 1: soundness-identity.tex, line 28, column 50
  2. Occurrence 2: soundness-identity.tex, line 41, column 1

Expression 197

Inline MathML variant

s=uivxs\varAssign{s}{s}{x}

Block MathML variant

s=uivxs\varAssign{s}{s}{x}

Conventional reading: variable assignment lowercase s agrees with variable assignment lowercase s except possibly at variable x

Meaning here: The semantic term read 'variable assignment lowercase s agrees with variable assignment lowercase s except possibly at variable x' specifies an interpretation, term value, or assignment relation in a first-order structure.

2 occurrences
  1. Occurrence 1: soundness.tex, line 258, column 58
  2. Occurrence 2: soundness-identity.tex, line 35, column 30

Expression 198

Inline MathML variant

R∧\RightR{\land}

Block MathML variant

R∧\RightR{\land}

Conventional reading: right conjunction rule

Meaning here: The rule label read 'right conjunction rule' names the exact side and logical or structural operator printed by the source.

1 occurrence
  1. Occurrence 1: derivations.tex, line 81, column 12

Expression 199

Inline MathML variant

t1=t2⇒t1=t1\eq[t_1][t_2] \fCenter \eq[t_1][t_1]

Block MathML variant

t1=t2⇒t1=t1\eq[t_1][t_2] \fCenter \eq[t_1][t_1]

Conventional reading: antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub one

Meaning here: The first-order sequent read 'antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub one' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: identity.tex, line 51, column 10

Expression 200

Inline MathML variant

A(t),Γ⇒Δ!A(t), \Gamma \fCenter \Delta

Block MathML variant

A(t),Γ⇒Δ!A(t), \Gamma \fCenter \Delta

Conventional reading: antecedent containing first formula A with argument term t, then Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first formula A with argument term t, then Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: quantifier-rules.tex, line 16, column 7
  2. Occurrence 2: soundness.tex, line 212, column 12

Expression 201

Inline MathML variant

A⇒A!A \fCenter!A

Block MathML variant

A⇒A!A \fCenter!A

Conventional reading: antecedent containing formula A; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing formula A; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 229, column 7

Expression 202

Inline MathML variant

⇒A,¬A\fCenter !A, \lnot !A

Block MathML variant

⇒A,¬A\fCenter !A, \lnot !A

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing first formula A, then the negation of formula A

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing first formula A, then the negation of formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proving-things.tex, line 288, column 10
  2. Occurrence 2: provability-consistency.tex, line 62, column 12

Expression 203

Inline MathML variant

¬(A1∧…∧Am)\lnot(!A_1 \land \dots \land !A_m)

Block MathML variant

¬(A1∧…∧Am)\lnot(!A_1 \land \dots \land !A_m)

Conventional reading: the negation of open parenthesis, the conjunction beginning with formula A sub one, continuing through the omitted intermediate conjunctions, and ending with formula A sub m, close parenthesis

Meaning here: The first-order formula read 'the negation of open parenthesis, the conjunction beginning with formula A sub one, continuing through the omitted intermediate conjunctions, and ending with formula A sub m, close parenthesis' preserves its metavariables, connectives, term arguments, and written grouping.

1 occurrence
  1. Occurrence 1: rules-and-proofs.tex, line 42, column 1

Expression 204

Inline MathML variant

⇒¬∀x(A(x))→∃x(¬A(x))\Sequent \lnot\lforall[x][!A(x)] \lif \lexists[x][\lnot!A(x)]

Block MathML variant

⇒¬∀x(A(x))→∃x(¬A(x))\Sequent \lnot\lforall[x][!A(x)] \lif \lexists[x][\lnot!A(x)]

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is the negation of for every variable x, formula A with argument variable x; and whose consequent is for some variable x, the negation of formula A with argument variable x

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is the negation of for every variable x, formula A with argument variable x; and whose consequent is for some variable x, the negation of formula A with argument variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things-quant.tex, line 103, column 7

Expression 205

Inline MathML variant

¬A⇒A→B\lnot !A \Sequent !A \lif !B

Block MathML variant

¬A⇒A→B\lnot !A \Sequent !A \lif !B

Conventional reading: antecedent containing the negation of formula A; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B

Meaning here: The first-order sequent read 'antecedent containing the negation of formula A; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 116, column 25

Expression 206

Inline MathML variant

m=0m = 0

Block MathML variant

m=0m = 0

Conventional reading: m equals zero

Meaning here: The metalevel equality read 'm equals zero' identifies the exact derivation count, sequence, premise family, or semantic quantity named on its two sides.

1 occurrence
  1. Occurrence 1: rules-and-proofs.tex, line 39, column 50

Expression 207

Inline MathML variant

t2=t3,t1=t2⇒t1=t2\eq[t_2][t_3], \eq[t_1][t_2] \fCenter \eq[t_1][t_2]

Block MathML variant

t2=t3,t1=t2⇒t1=t2\eq[t_2][t_3], \eq[t_1][t_2] \fCenter \eq[t_1][t_2]

Conventional reading: antecedent containing first term t sub two is identical to term t sub three, then term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub two

Meaning here: The first-order sequent read 'antecedent containing first term t sub two is identical to term t sub three, then term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub two' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: identity.tex, line 57, column 10

Expression 208

Inline MathML variant

Γ,Δ,Π,Λ\Gamma, \Delta, \Pi, \Lambda

Block MathML variant

Γ,Δ,Π,Λ\Gamma, \Delta, \Pi, \Lambda

Conventional reading: first Gamma, then capital Delta, then capital Pi, and finally capital Lambda

Meaning here: The sequence read 'first Gamma, then capital Delta, then capital Pi, and finally capital Lambda' preserves the formula order and any printed empty or trailing position.

1 occurrence
  1. Occurrence 1: rules-and-proofs.tex, line 15, column 24

Expression 209

Inline MathML variant

Γ,Δ\Gamma, \Delta

Block MathML variant

Γ,Δ\Gamma, \Delta

Conventional reading: first Gamma, then capital Delta

Meaning here: The sequence read 'first Gamma, then capital Delta' preserves the formula order and any printed empty or trailing position.

1 occurrence
  1. Occurrence 1: rules-and-proofs.tex, line 49, column 69

Expression 211

Inline MathML variant

⇒(A→¬A)→¬A\Sequent (!A \lif \lnot !A) \lif \lnot !A

Block MathML variant

⇒(A→¬A)→¬A\Sequent (!A \lif \lnot !A) \lif \lnot !A

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is open parenthesis, the conditional whose antecedent is formula A; and whose consequent is the negation of formula A, close parenthesis; and whose consequent is the negation of formula A

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is open parenthesis, the conditional whose antecedent is formula A; and whose consequent is the negation of formula A, close parenthesis; and whose consequent is the negation of formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 317, column 7

Expression 212

Inline MathML variant

Γ⊢Ai\Gamma \Proves !A_i

Block MathML variant

Γ⊢Ai\Gamma \Proves !A_i

Conventional reading: Gamma syntactically derives formula A sub i

Meaning here: The statement read 'Gamma syntactically derives formula A sub i' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 130, column 29

Expression 213

Inline MathML variant

⇒¬(A→B)→¬B\Sequent \lnot(!A \lif !B) \lif \lnot !B

Block MathML variant

⇒¬(A→B)→¬B\Sequent \lnot(!A \lif !B) \lif \lnot !B

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is the negation of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula B, close parenthesis; and whose consequent is the negation of formula B

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is the negation of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula B, close parenthesis; and whose consequent is the negation of formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 318, column 7

Expression 214

Inline MathML variant

A(t),Γ⇒Δ!A(t), \Gamma \Sequent \Delta

Block MathML variant

A(t),Γ⇒Δ!A(t), \Gamma \Sequent \Delta

Conventional reading: antecedent containing first formula A with argument term t, then Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first formula A with argument term t, then Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: soundness.tex, line 218, column 21

Expression 215

Inline MathML variant

A∧¬C⇒¬(A→C)!A \land \lnot !C \Sequent \lnot (!A \lif !C)

Block MathML variant

A∧¬C⇒¬(A→C)!A \land \lnot !C \Sequent \lnot (!A \lif !C)

Conventional reading: antecedent containing the conjunction of formula A and the negation of formula C; sequent arrow; succedent containing the negation of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula C, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing the conjunction of formula A and the negation of formula C; sequent arrow; succedent containing the negation of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula C, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 320, column 7

Expression 216

Inline MathML variant

Γ,A,B,Π⇒Δ\Gamma, !A, !B, \Pi \fCenter \Delta

Block MathML variant

Γ,A,B,Π⇒Δ\Gamma, !A, !B, \Pi \fCenter \Delta

Conventional reading: antecedent containing first Gamma, then formula A, then formula B, and finally capital Pi; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first Gamma, then formula A, then formula B, and finally capital Pi; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: structural-rules.tex, line 54, column 7
  2. Occurrence 2: derivations.tex, line 68, column 7

Expression 217

Inline MathML variant

Γ⇒Δ,A[t/x]\Gamma \fCenter \Delta, \Subst{!A}{t}{x}

Block MathML variant

Γ⇒Δ,A[t/x]\Gamma \fCenter \Delta, \Subst{!A}{t}{x}

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the result of substituting term t for variable x in formula A

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the result of substituting term t for variable x in formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: quantifier-rules.tex, line 62, column 9

Expression 218

Inline MathML variant

Γ0,Γ1⇒\Gamma_0,\Gamma_1 \fCenter

Block MathML variant

Γ0,Γ1⇒\Gamma_0,\Gamma_1 \fCenter

Conventional reading: antecedent containing first Gamma sub zero, then Gamma sub one; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first Gamma sub zero, then Gamma sub one; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-consistency.tex, line 38, column 11

Expression 220

Inline MathML variant

A,A→B⊢B!A, !A \lif !B \Proves !B

Block MathML variant

A,A→B⊢B!A, !A \lif !B \Proves !B

Conventional reading: first formula A, then the conditional whose antecedent is formula A; and whose consequent is formula B syntactically derives formula B

Meaning here: The statement read 'first formula A, then the conditional whose antecedent is formula A; and whose consequent is formula B syntactically derives formula B' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

2 occurrences
  1. Occurrence 1: provability-propositional.tex, line 22, column 19
  2. Occurrence 2: provability-propositional.tex, line 101, column 46

Expression 221

Inline MathML variant

t1=t2,Γ⇒Δ,A(t2)\eq[t_1][t_2], \Gamma \Sequent \Delta, !A(t_2)

Block MathML variant

t1=t2,Γ⇒Δ,A(t2)\eq[t_1][t_2], \Gamma \Sequent \Delta, !A(t_2)

Conventional reading: antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub two

Meaning here: The first-order sequent read 'antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub two' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: soundness-identity.tex, line 25, column 4

Expression 222

Inline MathML variant

Γ⇒Δ,∀x(A(x))\Gamma \fCenter \Delta, \lforall[x][!A(x)]

Block MathML variant

Γ⇒Δ,∀x(A(x))\Gamma \fCenter \Delta, \lforall[x][!A(x)]

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for every variable x, formula A with argument variable x

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for every variable x, formula A with argument variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: quantifier-rules.tex, line 23, column 10
  2. Occurrence 2: soundness.tex, line 234, column 14

Expression 223

Inline MathML variant

∀x(A(x)),¬A(a)⇒\lforall[x][!A(x)], \lnot !A(a) \fCenter

Block MathML variant

∀x(A(x)),¬A(a)⇒\lforall[x][!A(x)], \lnot !A(a) \fCenter

Conventional reading: antecedent containing first for every variable x, formula A with argument variable x, then the negation of formula A with argument constant a; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first for every variable x, formula A with argument variable x, then the negation of formula A with argument constant a; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proving-things-quant.tex, line 53, column 10
  2. Occurrence 2: proving-things-quant.tex, line 71, column 10

Expression 224

Inline MathML variant

⇒A∨¬A\quad \Sequent !A \lor \lnot !A

Block MathML variant

⇒A∨¬A\quad \Sequent !A \lor \lnot !A

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 255, column 7

Expression 225

Inline MathML variant

C,D⇒D∧C!C, !D \fCenter !D \land !C

Block MathML variant

C,D⇒D∧C!C, !D \fCenter !D \land !C

Conventional reading: antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula D and formula C

Meaning here: The first-order sequent read 'antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula D and formula C' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: derivations.tex, line 117, column 11

Expression 226

Inline MathML variant

Γ0′⇒A\Gamma_0' \Sequent !A

Block MathML variant

Γ0′⇒A\Gamma_0' \Sequent !A

Conventional reading: antecedent containing Gamma sub zero prime; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing Gamma sub zero prime; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proof-theoretic-notions.tex, line 41, column 1
  2. Occurrence 2: proof-theoretic-notions.tex, line 47, column 9

Expression 227

Inline MathML variant

C!C

Block MathML variant

C!C

Conventional reading: formula C

Meaning here: The first-order formula read 'formula C' preserves its metavariables, connectives, term arguments, and written grouping.

6 occurrences
  1. Occurrence 1: derivations.tex, line 49, column 32
  2. Occurrence 2: derivations.tex, line 72, column 49
  3. Occurrence 3: derivations.tex, line 73, column 38
  4. Occurrence 4: derivations.tex, line 90, column 38
  5. Occurrence 5: derivations.tex, line 106, column 33
  6. Occurrence 6: soundness.tex, line 379, column 25

Expression 228

Inline MathML variant

¬A∈Γ\lnot !A \in \Gamma

Block MathML variant

¬A∈Γ\lnot !A \in \Gamma

Conventional reading: the negation of formula A is a member of Gamma

Meaning here: The relation read 'the negation of formula A is a member of Gamma' concerns the exact premise sets or formula sequences named by the source.

3 occurrences
  1. Occurrence 1: provability-consistency.tex, line 76, column 30
  2. Occurrence 2: provability-consistency.tex, line 81, column 35
  3. Occurrence 3: provability-consistency.tex, line 96, column 9

Expression 229

Inline MathML variant

M⊨A(t1)\Sat{M}{!A(t_1)}

Block MathML variant

M⊨A(t1)\Sat{M}{!A(t_1)}

Conventional reading: structure M satisfies formula A with argument term t sub one

Meaning here: The satisfaction claim read 'structure M satisfies formula A with argument term t sub one' is evaluated in the named first-order structure and, when printed, the named variable assignment.

1 occurrence
  1. Occurrence 1: soundness-identity.tex, line 32, column 35

Expression 230

Inline MathML variant

Γ⇒Δ,A\Gamma \fCenter \Delta, !A

Block MathML variant

Γ⇒Δ,A\Gamma \fCenter \Delta, !A

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

14 occurrences
  1. Occurrence 1: propositional-rules.tex, line 18, column 7
  2. Occurrence 2: propositional-rules.tex, line 44, column 7
  3. Occurrence 3: propositional-rules.tex, line 61, column 7
  4. Occurrence 4: propositional-rules.tex, line 76, column 7
  5. Occurrence 5: structural-rules.tex, line 33, column 10
  6. Occurrence 6: structural-rules.tex, line 47, column 10
  7. Occurrence 7: derivations.tex, line 83, column 7
  8. Occurrence 8: soundness.tex, line 88, column 14
  9. Occurrence 9: soundness.tex, line 108, column 12
  10. Occurrence 10: soundness.tex, line 165, column 12
  11. Occurrence 11: soundness.tex, line 273, column 12
  12. Occurrence 12: soundness.tex, line 293, column 12
  13. Occurrence 13: soundness.tex, line 302, column 54
  14. Occurrence 14: soundness.tex, line 312, column 12

Expression 232

Inline MathML variant

Ξ=Δ,A∨B\Xi = \Delta, !A \lor !B

Block MathML variant

Ξ=Δ,A∨B\Xi = \Delta, !A \lor !B

Conventional reading: capital Xi equals first capital Delta, then the disjunction of formula A and formula B

Meaning here: The metalevel equality read 'capital Xi equals first capital Delta, then the disjunction of formula A and formula B' identifies the exact derivation count, sequence, premise family, or semantic quantity named on its two sides.

1 occurrence
  1. Occurrence 1: soundness.tex, line 169, column 29

Expression 233

Inline MathML variant

Γ∪{¬A}\Gamma \cup \{\lnot !A\}

Block MathML variant

Γ∪{¬A}\Gamma \cup \{\lnot !A\}

Conventional reading: the union of Gamma, and the set containing the negation of formula A

Meaning here: The relation read 'the union of Gamma, and the set containing the negation of formula A' concerns the exact premise sets or formula sequences named by the source.

4 occurrences
  1. Occurrence 1: provability-consistency.tex, line 47, column 25
  2. Occurrence 2: provability-consistency.tex, line 54, column 24
  3. Occurrence 3: provability-consistency.tex, line 56, column 4
  4. Occurrence 4: provability-consistency.tex, line 101, column 31

Expression 234

Inline MathML variant

Γ⊢A\Gamma \Proves {!A}

Block MathML variant

Γ⊢A\Gamma \Proves {!A}

Conventional reading: Gamma syntactically derives formula A

Meaning here: The statement read 'Gamma syntactically derives formula A' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 135, column 30

Expression 235

Inline MathML variant

A∨B,Γ⇒Δ!A \lor !B, \Gamma \fCenter \Delta

Block MathML variant

A∨B,Γ⇒Δ!A \lor !B, \Gamma \fCenter \Delta

Conventional reading: antecedent containing first the disjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first the disjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: propositional-rules.tex, line 57, column 11

Expression 236

Inline MathML variant

A∈Δ!A \in \Delta

Block MathML variant

A∈Δ!A \in \Delta

Conventional reading: formula A is a member of capital Delta

Meaning here: The relation read 'formula A is a member of capital Delta' concerns the exact premise sets or formula sequences named by the source.

1 occurrence
  1. Occurrence 1: soundness.tex, line 46, column 47

Expression 237

Inline MathML variant

Γ0,¬A⇒\Gamma_0, \lnot !A \fCenter

Block MathML variant

Γ0,¬A⇒\Gamma_0, \lnot !A \fCenter

Conventional reading: antecedent containing first Gamma sub zero, then the negation of formula A; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first Gamma sub zero, then the negation of formula A; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-consistency.tex, line 94, column 15

Expression 238

Inline MathML variant

Γ1,A⇒\Gamma_1, !A \Sequent \quad

Block MathML variant

Γ1,A⇒\Gamma_1, !A \Sequent \quad

Conventional reading: antecedent containing first Gamma sub one, then formula A; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first Gamma sub one, then formula A; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-consistency.tex, line 28, column 53

Expression 239

Inline MathML variant

Γ0⊆Γ\Gamma_0 \subseteq \Gamma

Block MathML variant

Γ0⊆Γ\Gamma_0 \subseteq \Gamma

Conventional reading: Gamma sub zero is a subset of Gamma

Meaning here: The relation read 'Gamma sub zero is a subset of Gamma' concerns the exact premise sets or formula sequences named by the source.

16 occurrences
  1. Occurrence 1: proof-theoretic-notions.tex, line 39, column 15
  2. Occurrence 2: proof-theoretic-notions.tex, line 71, column 15
  3. Occurrence 3: proof-theoretic-notions.tex, line 73, column 41
  4. Occurrence 4: proof-theoretic-notions.tex, line 95, column 54
  5. Occurrence 5: proof-theoretic-notions.tex, line 109, column 43
  6. Occurrence 6: proof-theoretic-notions.tex, line 150, column 62
  7. Occurrence 7: proof-theoretic-notions.tex, line 160, column 7
  8. Occurrence 8: proof-theoretic-notions.tex, line 164, column 14
  9. Occurrence 9: provability-consistency.tex, line 41, column 7
  10. Occurrence 10: provability-consistency.tex, line 96, column 35
  11. Occurrence 11: provability-consistency.tex, line 106, column 23
  12. Occurrence 12: provability-consistency.tex, line 122, column 7
  13. Occurrence 13: provability-quantifiers.tex, line 26, column 17
  14. Occurrence 14: soundness.tex, line 357, column 52
  15. Occurrence 15: soundness.tex, line 373, column 24
  16. Occurrence 16: soundness.tex, line 378, column 51

Expression 240

Inline MathML variant

¬A∨B⇒A→B\lnot !A \lor !B \Sequent !A \lif !B

Block MathML variant

¬A∨B⇒A→B\lnot !A \lor !B \Sequent !A \lif !B

Conventional reading: antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B

Meaning here: The first-order sequent read 'antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 51, column 51

Expression 241

Inline MathML variant

∀x(A(x))⊢A(t)\lforall[x][!A(x)] \Proves !A(t)

Block MathML variant

∀x(A(x))⊢A(t)\lforall[x][!A(x)] \Proves !A(t)

Conventional reading: for every variable x, formula A with argument variable x syntactically derives formula A with argument term t

Meaning here: The statement read 'for every variable x, formula A with argument variable x syntactically derives formula A with argument term t' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: provability-quantifiers.tex, line 36, column 18

Expression 242

Inline MathML variant

∃x(A(x)),Γ⇒Δ\lexists[x][!A(x)], \Gamma \fCenter \Delta

Block MathML variant

∃x(A(x)),Γ⇒Δ\lexists[x][!A(x)], \Gamma \fCenter \Delta

Conventional reading: antecedent containing first for some variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first for some variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: quantifier-rules.tex, line 39, column 10

Expression 243

Inline MathML variant

B,¬A,¬B⇒!B, \lnot !A, \lnot !B \fCenter

Block MathML variant

B,¬A,¬B⇒!B, \lnot !A, \lnot !B \fCenter

Conventional reading: antecedent containing first formula B, then the negation of formula A, and finally the negation of formula B; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first formula B, then the negation of formula A, and finally the negation of formula B; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 79, column 16

Expression 244

Inline MathML variant

⇒¬(A∨B)→(¬A∧¬B)\Sequent \lnot(!A \lor !B) \lif (\lnot !A \land \lnot !B)

Block MathML variant

⇒¬(A∨B)→(¬A∧¬B)\Sequent \lnot(!A \lor !B) \lif (\lnot !A \land \lnot !B)

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is the negation of open parenthesis, the disjunction of formula A and formula B, close parenthesis; and whose consequent is open parenthesis, the conjunction of the negation of formula A and the negation of formula B, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is the negation of open parenthesis, the disjunction of formula A and formula B, close parenthesis; and whose consequent is open parenthesis, the conjunction of the negation of formula A and the negation of formula B, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 324, column 7

Expression 245

Inline MathML variant

B⇒A→B!B \Sequent !A \lif !B

Block MathML variant

B⇒A→B!B \Sequent !A \lif !B

Conventional reading: antecedent containing formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B

Meaning here: The first-order sequent read 'antecedent containing formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 116, column 60

Expression 246

Inline MathML variant

Γ⇒Δ,P(t,t)\Gamma \Sequent \Delta, \Atom{\Obj P}{t,t}

Block MathML variant

Γ⇒Δ,P(t,t)\Gamma \Sequent \Delta, \Atom{\Obj P}{t,t}

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then atomic formula P applied to term t, then term t

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then atomic formula P applied to term t, then term t' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: quantifier-rules.tex, line 68, column 39

Expression 248

Inline MathML variant

Γ0,Γ1⇒\Gamma_0, \Gamma_1 \fCenter

Block MathML variant

Γ0,Γ1⇒\Gamma_0, \Gamma_1 \fCenter

Conventional reading: antecedent containing first Gamma sub zero, then Gamma sub one; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first Gamma sub zero, then Gamma sub one; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-consistency.tex, line 120, column 11

Expression 249

Inline MathML variant

t1=t2,Γ⇒Δ,A(t1)\eq[t_1][t_2], \Gamma \Sequent \Delta, !A(t_1)

Block MathML variant

t1=t2,Γ⇒Δ,A(t1)\eq[t_1][t_2], \Gamma \Sequent \Delta, !A(t_1)

Conventional reading: antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub one

Meaning here: The first-order sequent read 'antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub one' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: soundness-identity.tex, line 24, column 4

Expression 250

Inline MathML variant

Γ⊨A\Gamma \Entails !A

Block MathML variant

Γ⊨A\Gamma \Entails !A

Conventional reading: Gamma semantically entails formula A

Meaning here: The statement read 'Gamma semantically entails formula A' is first-order semantic consequence over the named structures and assignments.

1 occurrence
  1. Occurrence 1: soundness.tex, line 353, column 29

Expression 251

Inline MathML variant

¬A,A⇒\lnot !A, !A \fCenter

Block MathML variant

¬A,A⇒\lnot !A, !A \fCenter

Conventional reading: antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

4 occurrences
  1. Occurrence 1: proving-things.tex, line 193, column 10
  2. Occurrence 2: provability-consistency.tex, line 90, column 14
  3. Occurrence 3: provability-propositional.tex, line 72, column 16
  4. Occurrence 4: provability-propositional.tex, line 121, column 18

Expression 252

Inline MathML variant

¬A∈Θ\lnot !A \in \Theta

Block MathML variant

¬A∈Θ\lnot !A \in \Theta

Conventional reading: the negation of formula A is a member of capital Theta

Meaning here: The relation read 'the negation of formula A is a member of capital Theta' concerns the exact premise sets or formula sequences named by the source.

1 occurrence
  1. Occurrence 1: soundness.tex, line 127, column 55

Expression 253

Inline MathML variant

Γ=⟨A1,…,Am⟩\Gamma = \tuple{!A_1, \dots, !A_m}

Block MathML variant

Γ=⟨A1,…,Am⟩\Gamma = \tuple{!A_1, \dots, !A_m}

Conventional reading: Gamma equals the finite sequence first formula A sub one, continuing through the omitted intermediate entries, and finally formula A sub m

Meaning here: The metalevel equality read 'Gamma equals the finite sequence first formula A sub one, continuing through the omitted intermediate entries, and finally formula A sub m' identifies the exact derivation count, sequence, premise family, or semantic quantity named on its two sides.

1 occurrence
  1. Occurrence 1: rules-and-proofs.tex, line 31, column 30

Expression 254

Inline MathML variant

(A∨B)→C⇒A→C(!A \lor !B) \lif !C \Sequent !A \lif !C

Block MathML variant

(A∨B)→C⇒A→C(!A \lor !B) \lif !C \Sequent !A \lif !C

Conventional reading: antecedent containing the conditional whose antecedent is open parenthesis, the disjunction of formula A and formula B, close parenthesis; and whose consequent is formula C; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula C

Meaning here: The first-order sequent read 'antecedent containing the conditional whose antecedent is open parenthesis, the disjunction of formula A and formula B, close parenthesis; and whose consequent is formula C; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula C' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 313, column 7

Expression 255

Inline MathML variant

A,¬A,¬B⇒!A, \lnot !A, \lnot !B \fCenter

Block MathML variant

A,¬A,¬B⇒!A, \lnot !A, \lnot !B \fCenter

Conventional reading: antecedent containing first formula A, then the negation of formula A, and finally the negation of formula B; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first formula A, then the negation of formula A, and finally the negation of formula B; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 74, column 16

Expression 256

Inline MathML variant

∀x(A(x)),Γ⇒Δ\lforall[x][!A(x)],\Gamma \fCenter \Delta

Block MathML variant

∀x(A(x)),Γ⇒Δ\lforall[x][!A(x)],\Gamma \fCenter \Delta

Conventional reading: antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: quantifier-rules.tex, line 18, column 10

Expression 257

Inline MathML variant

M,s⊨A(t1)\Sat{M}{!A(t_1)}[s]

Block MathML variant

M,s⊨A(t1)\Sat{M}{!A(t_1)}[s]

Conventional reading: structure M, under variable assignment lowercase s, satisfies formula A with argument term t sub one

Meaning here: The satisfaction claim read 'structure M, under variable assignment lowercase s, satisfies formula A with argument term t sub one' is evaluated in the named first-order structure and, when printed, the named variable assignment.

1 occurrence
  1. Occurrence 1: soundness-identity.tex, line 35, column 1

Expression 259

Inline MathML variant

⇒t1=t1\fCenter \eq[t_1][t_1]

Block MathML variant

⇒t1=t1\fCenter \eq[t_1][t_1]

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing term t sub one is identical to term t sub one

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing term t sub one is identical to term t sub one' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: identity.tex, line 49, column 7

Expression 260

Inline MathML variant

Γ,A\Gamma, !A

Block MathML variant

Γ,A\Gamma, !A

Conventional reading: first Gamma, then formula A

Meaning here: The sequence read 'first Gamma, then formula A' preserves the formula order and any printed empty or trailing position.

1 occurrence
  1. Occurrence 1: rules-and-proofs.tex, line 46, column 54

Expression 261

Inline MathML variant

A,B⊢A∧B!A, !B \Proves !A \land !B

Block MathML variant

A,B⊢A∧B!A, !B \Proves !A \land !B

Conventional reading: first formula A, then formula B syntactically derives the conjunction of formula A and formula B

Meaning here: The statement read 'first formula A, then formula B syntactically derives the conjunction of formula A and formula B' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 30, column 47

Expression 263

Inline MathML variant

M′⊨A(a)\Sat{M'}{!A(a)}

Block MathML variant

M′⊨A(a)\Sat{M'}{!A(a)}

Conventional reading: structure M prime satisfies formula A with argument constant a

Meaning here: The satisfaction claim read 'structure M prime satisfies formula A with argument constant a' is evaluated in the named first-order structure and, when printed, the named variable assignment.

1 occurrence
  1. Occurrence 1: soundness.tex, line 257, column 3

Expression 264

Inline MathML variant

t1=t2,Γ⇒Δ,A(t1)\eq[t_1][t_2], \Gamma \fCenter \Delta, !A(t_1)

Block MathML variant

t1=t2,Γ⇒Δ,A(t1)\eq[t_1][t_2], \Gamma \fCenter \Delta, !A(t_1)

Conventional reading: antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub one

Meaning here: The first-order sequent read 'antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub one' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: identity.tex, line 23, column 7
  2. Occurrence 2: identity.tex, line 30, column 10

Expression 265

Inline MathML variant

∃x(¬A(x))⇒¬∀x(A(x))\lexists[x][ \lnot !A(x)] \fCenter \lnot \lforall[x][!A(x)]

Block MathML variant

∃x(¬A(x))⇒¬∀x(A(x))\lexists[x][ \lnot !A(x)] \fCenter \lnot \lforall[x][!A(x)]

Conventional reading: antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x

Meaning here: The first-order sequent read 'antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things-quant.tex, line 75, column 10

Expression 266

Inline MathML variant

Γ⇒\Gamma \Sequent \quad

Block MathML variant

Γ⇒\Gamma \Sequent \quad

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: rules-and-proofs.tex, line 41, column 35

Expression 267

Inline MathML variant

C∈Δ!C \in \Delta

Block MathML variant

C∈Δ!C \in \Delta

Conventional reading: formula C is a member of capital Delta

Meaning here: The relation read 'formula C is a member of capital Delta' concerns the exact premise sets or formula sequences named by the source.

16 occurrences
  1. Occurrence 1: soundness.tex, line 94, column 59
  2. Occurrence 2: soundness.tex, line 100, column 63
  3. Occurrence 3: soundness.tex, line 117, column 59
  4. Occurrence 4: soundness.tex, line 124, column 32
  5. Occurrence 5: soundness.tex, line 147, column 53
  6. Occurrence 6: soundness.tex, line 152, column 50
  7. Occurrence 7: soundness.tex, line 174, column 53
  8. Occurrence 8: soundness.tex, line 177, column 66
  9. Occurrence 9: soundness.tex, line 198, column 53
  10. Occurrence 10: soundness.tex, line 203, column 8
  11. Occurrence 11: soundness.tex, line 220, column 30
  12. Occurrence 12: soundness.tex, line 242, column 8
  13. Occurrence 13: soundness.tex, line 246, column 39
  14. Occurrence 14: soundness.tex, line 256, column 3
  15. Occurrence 15: soundness-identity.tex, line 28, column 31
  16. Occurrence 16: soundness-identity.tex, line 31, column 68

Expression 268

Inline MathML variant

Γ⇒Δ,A,A\Gamma \fCenter \Delta, !A, !A

Block MathML variant

Γ⇒Δ,A,A\Gamma \fCenter \Delta, !A, !A

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A, and finally formula A

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A, and finally formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: structural-rules.tex, line 45, column 7

Expression 269

Inline MathML variant

¬A,Γ⇒Δ\lnot !A, \Gamma \Sequent \Delta

Block MathML variant

¬A,Γ⇒Δ\lnot !A, \Gamma \Sequent \Delta

Conventional reading: antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: soundness.tex, line 105, column 3

Expression 270

Inline MathML variant

∃x(A(x))⇒∀x(A(x))\lexists[x][!A(x)] \fCenter \lforall[x][!A(x)]

Block MathML variant

∃x(A(x))⇒∀x(A(x))\lexists[x][!A(x)] \fCenter \lforall[x][!A(x)]

Conventional reading: antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing for every variable x, formula A with argument variable x

Meaning here: The first-order sequent read 'antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing for every variable x, formula A with argument variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: quantifier-rules.tex, line 92, column 12
  2. Occurrence 2: quantifier-rules.tex, line 99, column 12

Expression 271

Inline MathML variant

A⇒A!A \Sequent !A

Block MathML variant

A⇒A!A \Sequent !A

Conventional reading: antecedent containing formula A; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing formula A; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

5 occurrences
  1. Occurrence 1: rules-and-proofs.tex, line 56, column 11
  2. Occurrence 2: proving-things.tex, line 37, column 64
  3. Occurrence 3: proving-things.tex, line 38, column 48
  4. Occurrence 4: proof-theoretic-notions.tex, line 84, column 21
  5. Occurrence 5: soundness.tex, line 62, column 40

Expression 272

Inline MathML variant

Π⇒Λ\Pi \Sequent \Lambda

Block MathML variant

Π⇒Λ\Pi \Sequent \Lambda

Conventional reading: antecedent containing capital Pi; sequent arrow; succedent containing capital Lambda

Meaning here: The first-order sequent read 'antecedent containing capital Pi; sequent arrow; succedent containing capital Lambda' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: soundness.tex, line 325, column 15

Expression 273

Inline MathML variant

A⇒¬¬A!A \Sequent \lnot\lnot !A

Block MathML variant

A⇒¬¬A!A \Sequent \lnot\lnot !A

Conventional reading: antecedent containing formula A; sequent arrow; succedent containing the negation of the negation of formula A

Meaning here: The first-order sequent read 'antecedent containing formula A; sequent arrow; succedent containing the negation of the negation of formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 306, column 7

Expression 275

Inline MathML variant

B∈Γ0!B \in \Gamma_0

Block MathML variant

B∈Γ0!B \in \Gamma_0

Conventional reading: formula B is a member of Gamma sub zero

Meaning here: The relation read 'formula B is a member of Gamma sub zero' concerns the exact premise sets or formula sequences named by the source.

1 occurrence
  1. Occurrence 1: soundness.tex, line 361, column 19

Expression 276

Inline MathML variant

Γ0∪Γ1⊆Γ\Gamma_0 \cup \Gamma_1 \subseteq \Gamma

Block MathML variant

Γ0∪Γ1⊆Γ\Gamma_0 \cup \Gamma_1 \subseteq \Gamma

Conventional reading: the union of Gamma sub zero, and Gamma sub one is a subset of Gamma

Meaning here: The relation read 'the union of Gamma sub zero, and Gamma sub one is a subset of Gamma' concerns the exact premise sets or formula sequences named by the source.

2 occurrences
  1. Occurrence 1: provability-consistency.tex, line 42, column 1
  2. Occurrence 2: provability-consistency.tex, line 123, column 1

Expression 277

Inline MathML variant

M⊭B\Sat/{M}{!B}

Block MathML variant

M⊭B\Sat/{M}{!B}

Conventional reading: structure M does not satisfy formula B

Meaning here: The satisfaction claim read 'structure M does not satisfy formula B' is evaluated in the named first-order structure and, when printed, the named variable assignment.

1 occurrence
  1. Occurrence 1: soundness.tex, line 328, column 3

Expression 278

Inline MathML variant

(A1∧⋯∧Am)→(B1∨⋯∨Bn)(!A_1 \land \cdots \land !A_m) \lif (!B_1 \lor \cdots \lor !B_n)

Block MathML variant

(A1∧⋯∧Am)→(B1∨⋯∨Bn)(!A_1 \land \cdots \land !A_m) \lif (!B_1 \lor \cdots \lor !B_n)

Conventional reading: the conditional whose antecedent is open parenthesis, the conjunction beginning with formula A sub one, continuing through the omitted intermediate conjunctions, and ending with formula A sub m, close parenthesis; and whose consequent is open parenthesis, the disjunction beginning with formula B sub one, continuing through the omitted intermediate disjunctions, and ending with formula B sub n, close parenthesis

Meaning here: The first-order formula read 'the conditional whose antecedent is open parenthesis, the conjunction beginning with formula A sub one, continuing through the omitted intermediate conjunctions, and ending with formula A sub m, close parenthesis; and whose consequent is open parenthesis, the disjunction beginning with formula B sub one, continuing through the omitted intermediate disjunctions, and ending with formula B sub n, close parenthesis' preserves its metavariables, connectives, term arguments, and written grouping.

1 occurrence
  1. Occurrence 1: rules-and-proofs.tex, line 34, column 1

Expression 279

Inline MathML variant

∃x(A(x))∧∀y(∀z(((A(y)∧A(z))→y=z)))⇒∃x((A(x)∧∀y((A(y)→y=x))))\lexists[x][!A(x)] \land \lforall[y][\lforall[z][((!A(y) \land !A(z)) \lif y = z)]] \Sequent \lexists[x][(!A(x) \land \lforall[y][(!A(y) \lif y = x)])]

Block MathML variant

∃x(A(x))∧∀y(∀z(((A(y)∧A(z))→y=z)))⇒∃x((A(x)∧∀y((A(y)→y=x))))\lexists[x][!A(x)] \land \lforall[y][\lforall[z][((!A(y) \land !A(z)) \lif y = z)]] \Sequent \lexists[x][(!A(x) \land \lforall[y][(!A(y) \lif y = x)])]

Conventional reading: antecedent containing the conjunction whose first conjunct is for some variable x, formula A with argument variable x; and whose second conjunct states that for every variable y, for every variable z, open parenthesis, the conditional whose antecedent is open parenthesis, the conjunction of formula A with argument variable y and formula A with argument variable z, close parenthesis; and whose consequent states that variable y is identical to variable z, close parenthesis; sequent arrow; succedent containing for some variable x, open parenthesis, the conjunction whose first conjunct is formula A with argument variable x; and whose second conjunct states that for every variable y, open parenthesis, the conditional whose antecedent is formula A with argument variable y; and whose consequent states that variable y is identical to variable x, close parenthesis, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing the conjunction whose first conjunct is for some variable x, formula A with argument variable x; and whose second conjunct states that for every variable y, for every variable z, open parenthesis, the conditional whose antecedent is open parenthesis, the conjunction of formula A with argument variable y and formula A with argument variable z, close parenthesis; and whose consequent states that variable y is identical to variable z, close parenthesis; sequent arrow; succedent containing for some variable x, open parenthesis, the conjunction whose first conjunct is formula A with argument variable x; and whose second conjunct states that for every variable y, open parenthesis, the conditional whose antecedent is formula A with argument variable y; and whose consequent states that variable y is identical to variable x, close parenthesis, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: identity.tex, line 71, column 7

Expression 280

Inline MathML variant

t1=t2,t2=t3⇒t1=t3\eq[t_1][t_2], \eq[t_2][t_3] \fCenter \eq[t_1][t_3]

Block MathML variant

t1=t2,t2=t3⇒t1=t3\eq[t_1][t_2], \eq[t_2][t_3] \fCenter \eq[t_1][t_3]

Conventional reading: antecedent containing first term t sub one is identical to term t sub two, then term t sub two is identical to term t sub three; sequent arrow; succedent containing term t sub one is identical to term t sub three

Meaning here: The first-order sequent read 'antecedent containing first term t sub one is identical to term t sub two, then term t sub two is identical to term t sub three; sequent arrow; succedent containing term t sub one is identical to term t sub three' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: identity.tex, line 61, column 10

Expression 281

Inline MathML variant

t1t_1

Block MathML variant

t1t_1

Conventional reading: term t sub one

Meaning here: The symbol read 'term t sub one' receives its exact term, variable, constant, or assignment role from each bound source occurrence.

1 occurrence
  1. Occurrence 1: identity.tex, line 20, column 26

Expression 283

Inline MathML variant

Γ0⇒¬A\Gamma_0 \fCenter \lnot !A

Block MathML variant

Γ0⇒¬A\Gamma_0 \fCenter \lnot !A

Conventional reading: antecedent containing Gamma sub zero; sequent arrow; succedent containing the negation of formula A

Meaning here: The first-order sequent read 'antecedent containing Gamma sub zero; sequent arrow; succedent containing the negation of formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-consistency.tex, line 115, column 10

Expression 284

Inline MathML variant

⇒(∀x(A(x))∧∀y(B(y)))→∀z((A(z)∧B(z)))\Sequent (\lforall[x][!A(x)] \land \lforall[y][!B(y)]) \lif \lforall[z][(!A(z) \land !B(z))]

Block MathML variant

⇒(∀x(A(x))∧∀y(B(y)))→∀z((A(z)∧B(z)))\Sequent (\lforall[x][!A(x)] \land \lforall[y][!B(y)]) \lif \lforall[z][(!A(z) \land !B(z))]

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is open parenthesis, the conjunction of for every variable x, formula A with argument variable x and for every variable y, formula B with argument variable y, close parenthesis; and whose consequent is for every variable z, open parenthesis, the conjunction of formula A with argument variable z and formula B with argument variable z, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is open parenthesis, the conjunction of for every variable x, formula A with argument variable x and for every variable y, formula B with argument variable y, close parenthesis; and whose consequent is for every variable z, open parenthesis, the conjunction of formula A with argument variable z and formula B with argument variable z, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things-quant.tex, line 88, column 7

Expression 285

Inline MathML variant

B⇒A∨B!B \fCenter !A \lor !B

Block MathML variant

B⇒A∨B!B \fCenter !A \lor !B

Conventional reading: antecedent containing formula B; sequent arrow; succedent containing the disjunction of formula A and formula B

Meaning here: The first-order sequent read 'antecedent containing formula B; sequent arrow; succedent containing the disjunction of formula A and formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 94, column 16

Expression 286

Inline MathML variant

M⊨t1=t2\Sat{M}{\eq[t_1][t_2]}

Block MathML variant

M⊨t1=t2\Sat{M}{\eq[t_1][t_2]}

Conventional reading: structure M satisfies term t sub one is identical to term t sub two

Meaning here: The satisfaction claim read 'structure M satisfies term t sub one is identical to term t sub two' is evaluated in the named first-order structure and, when printed, the named variable assignment.

3 occurrences
  1. Occurrence 1: soundness-identity.tex, line 27, column 17
  2. Occurrence 2: soundness-identity.tex, line 31, column 1
  3. Occurrence 3: soundness-identity.tex, line 37, column 1

Expression 287

Inline MathML variant

M,s⊨A(x)\Sat{M}{!A(x)}[s]

Block MathML variant

M,s⊨A(x)\Sat{M}{!A(x)}[s]

Conventional reading: structure M, under variable assignment lowercase s, satisfies formula A with argument variable x

Meaning here: The satisfaction claim read 'structure M, under variable assignment lowercase s, satisfies formula A with argument variable x' is evaluated in the named first-order structure and, when printed, the named variable assignment.

4 occurrences
  1. Occurrence 1: soundness.tex, line 250, column 3
  2. Occurrence 2: soundness.tex, line 262, column 42
  3. Occurrence 3: soundness.tex, line 264, column 3
  4. Occurrence 4: soundness-identity.tex, line 36, column 38

Expression 288

Inline MathML variant

∃x(¬A(x))⇒¬∀x(A(x))\lexists[x][\lnot !A(x)] \Sequent \lnot \lforall[x][!A(x)]

Block MathML variant

∃x(¬A(x))⇒¬∀x(A(x))\lexists[x][\lnot !A(x)] \Sequent \lnot \lforall[x][!A(x)]

Conventional reading: antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x

Meaning here: The first-order sequent read 'antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things-quant.tex, line 14, column 50

Expression 289

Inline MathML variant

A(a)⇒A(b)!A(a) \Sequent !A(b)

Block MathML variant

A(a)⇒A(b)!A(a) \Sequent !A(b)

Conventional reading: antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant b

Meaning here: The first-order sequent read 'antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant b' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things-quant.tex, line 28, column 28

Expression 290

Inline MathML variant

{A}⊆Γ\{!A\} \subseteq \Gamma

Block MathML variant

{A}⊆Γ\{!A\} \subseteq \Gamma

Conventional reading: the set containing formula A is a subset of Gamma

Meaning here: The relation read 'the set containing formula A is a subset of Gamma' concerns the exact premise sets or formula sequences named by the source.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 84, column 60

Expression 291

Inline MathML variant

⊬A\Proves/ !A

Block MathML variant

⊬A\Proves/ !A

Conventional reading: no premises does not syntactically derive formula A

Meaning here: The statement read 'no premises does not syntactically derive formula A' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 33, column 30

Expression 292

Inline MathML variant

¬A\lnot !A

Block MathML variant

¬A\lnot !A

Conventional reading: the negation of formula A

Meaning here: The first-order formula read 'the negation of formula A' preserves its metavariables, connectives, term arguments, and written grouping.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 283, column 62

Expression 293

Inline MathML variant

Γ⇒Δ,A\Gamma \Sequent \Delta, !A

Block MathML variant

Γ⇒Δ,A\Gamma \Sequent \Delta, !A

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

5 occurrences
  1. Occurrence 1: soundness.tex, line 77, column 26
  2. Occurrence 2: soundness.tex, line 104, column 21
  3. Occurrence 3: soundness.tex, line 114, column 42
  4. Occurrence 4: soundness.tex, line 171, column 9
  5. Occurrence 5: soundness.tex, line 325, column 46

Expression 295

Inline MathML variant

B,A⇒B!B, !A \fCenter !B

Block MathML variant

B,A⇒B!B, !A \fCenter !B

Conventional reading: antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B

Meaning here: The first-order sequent read 'antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

4 occurrences
  1. Occurrence 1: proving-things.tex, line 83, column 10
  2. Occurrence 2: proving-things.tex, line 101, column 10
  3. Occurrence 3: proving-things.tex, line 122, column 10
  4. Occurrence 4: proving-things.tex, line 144, column 10

Expression 296

Inline MathML variant

¬A,A⇒B\lnot !A, !A \fCenter !B

Block MathML variant

¬A,A⇒B\lnot !A, !A \fCenter !B

Conventional reading: antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B

Meaning here: The first-order sequent read 'antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

4 occurrences
  1. Occurrence 1: proving-things.tex, line 81, column 10
  2. Occurrence 2: proving-things.tex, line 96, column 10
  3. Occurrence 3: proving-things.tex, line 117, column 10
  4. Occurrence 4: proving-things.tex, line 139, column 10

Expression 297

Inline MathML variant

Θ=Γ\Theta = \Gamma

Block MathML variant

Θ=Γ\Theta = \Gamma

Conventional reading: capital Theta equals Gamma

Meaning here: The metalevel equality read 'capital Theta equals Gamma' identifies the exact derivation count, sequence, premise family, or semantic quantity named on its two sides.

1 occurrence
  1. Occurrence 1: soundness.tex, line 169, column 7

Expression 298

Inline MathML variant

Γ,Π⇒Δ,Λ\Gamma, \Pi \fCenter \Delta, \Lambda

Block MathML variant

Γ,Π⇒Δ,Λ\Gamma, \Pi \fCenter \Delta, \Lambda

Conventional reading: antecedent containing first Gamma, then capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda

Meaning here: The first-order sequent read 'antecedent containing first Gamma, then capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: soundness.tex, line 277, column 15

Expression 299

Inline MathML variant

¬A⊢A→B\lnot !A \Proves !A \lif !B

Block MathML variant

¬A⊢A→B\lnot !A \Proves !A \lif !B

Conventional reading: the negation of formula A syntactically derives the conditional whose antecedent is formula A; and whose consequent is formula B

Meaning here: The statement read 'the negation of formula A syntactically derives the conditional whose antecedent is formula A; and whose consequent is formula B' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 103, column 10

Expression 300

Inline MathML variant

aa

Block MathML variant

aa

Conventional reading: constant a

Meaning here: The symbol read 'constant a' receives its exact term, variable, constant, or assignment role from each bound source occurrence.

14 occurrences
  1. Occurrence 1: quantifier-rules.tex, line 28, column 35
  2. Occurrence 2: quantifier-rules.tex, line 30, column 6
  3. Occurrence 3: quantifier-rules.tex, line 31, column 67
  4. Occurrence 4: quantifier-rules.tex, line 48, column 34
  5. Occurrence 5: quantifier-rules.tex, line 49, column 66
  6. Occurrence 6: quantifier-rules.tex, line 74, column 14
  7. Occurrence 7: quantifier-rules.tex, line 83, column 14
  8. Occurrence 8: proving-things-quant.tex, line 26, column 26
  9. Occurrence 9: proving-things-quant.tex, line 62, column 67
  10. Occurrence 10: proving-things-quant.tex, line 80, column 68
  11. Occurrence 11: soundness.tex, line 229, column 42
  12. Occurrence 12: soundness.tex, line 236, column 57
  13. Occurrence 13: soundness.tex, line 255, column 24
  14. Occurrence 14: soundness.tex, line 261, column 38

Expression 301

Inline MathML variant

Γ⊢A\Gamma \Proves !A

Block MathML variant

Γ⊢A\Gamma \Proves !A

Conventional reading: Gamma syntactically derives formula A

Meaning here: The statement read 'Gamma syntactically derives formula A' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

16 occurrences
  1. Occurrence 1: proof-theoretic-notions.tex, line 38, column 25
  2. Occurrence 2: proof-theoretic-notions.tex, line 80, column 26
  3. Occurrence 3: proof-theoretic-notions.tex, line 90, column 34
  4. Occurrence 4: proof-theoretic-notions.tex, line 95, column 9
  5. Occurrence 5: proof-theoretic-notions.tex, line 104, column 4
  6. Occurrence 6: proof-theoretic-notions.tex, line 109, column 4
  7. Occurrence 7: proof-theoretic-notions.tex, line 128, column 44
  8. Occurrence 8: proof-theoretic-notions.tex, line 150, column 12
  9. Occurrence 9: proof-theoretic-notions.tex, line 159, column 14
  10. Occurrence 10: provability-consistency.tex, line 20, column 6
  11. Occurrence 11: provability-consistency.tex, line 47, column 1
  12. Occurrence 12: provability-consistency.tex, line 51, column 15
  13. Occurrence 13: provability-consistency.tex, line 76, column 6
  14. Occurrence 14: provability-consistency.tex, line 81, column 11
  15. Occurrence 15: soundness.tex, line 353, column 4
  16. Occurrence 16: soundness.tex, line 357, column 4

Expression 302

Inline MathML variant

A∨B,¬B⇒A!A \lor !B, \lnot !B \Sequent !A

Block MathML variant

A∨B,¬B⇒A!A \lor !B, \lnot !B \Sequent !A

Conventional reading: antecedent containing first the disjunction of formula A and formula B, then the negation of formula B; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing first the disjunction of formula A and formula B, then the negation of formula B; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 321, column 7

Expression 303

Inline MathML variant

Γ0⊢A\Gamma_0 \Proves !A

Block MathML variant

Γ0⊢A\Gamma_0 \Proves !A

Conventional reading: Gamma sub zero syntactically derives formula A

Meaning here: The statement read 'Gamma sub zero syntactically derives formula A' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

2 occurrences
  1. Occurrence 1: proof-theoretic-notions.tex, line 151, column 33
  2. Occurrence 2: proof-theoretic-notions.tex, line 161, column 55

Expression 304

Inline MathML variant

A∨¬A!A \lor \lnot !A

Block MathML variant

A∨¬A!A \lor \lnot !A

Conventional reading: the disjunction of formula A and the negation of formula A

Meaning here: The first-order formula read 'the disjunction of formula A and the negation of formula A' preserves its metavariables, connectives, term arguments, and written grouping.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 273, column 55

Expression 305

Inline MathML variant

¬A,Γ1⇒\lnot !A, \Gamma_1 \Sequent \quad

Block MathML variant

¬A,Γ1⇒\lnot !A, \Gamma_1 \Sequent \quad

Conventional reading: antecedent containing first the negation of formula A, then Gamma sub one; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first the negation of formula A, then Gamma sub one; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-consistency.tex, line 108, column 38

Expression 307

Inline MathML variant

M⊨t=t\Sat{M}{\eq[t][t]}

Block MathML variant

M⊨t=t\Sat{M}{\eq[t][t]}

Conventional reading: structure M satisfies term t is identical to term t

Meaning here: The satisfaction claim read 'structure M satisfies term t is identical to term t' is evaluated in the named first-order structure and, when printed, the named variable assignment.

1 occurrence
  1. Occurrence 1: soundness-identity.tex, line 19, column 38

Expression 308

Inline MathML variant

Γ⇒Δ,∃x(P(t,x))\Gamma \Sequent \Delta, \lexists[x][\Atom{\Obj P}{t,x}]

Block MathML variant

Γ⇒Δ,∃x(P(t,x))\Gamma \Sequent \Delta, \lexists[x][\Atom{\Obj P}{t,x}]

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, atomic formula P applied to term t, then variable x

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, atomic formula P applied to term t, then variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: quantifier-rules.tex, line 67, column 42

Expression 309

Inline MathML variant

A→C⇒¬(A∧¬C)!A \lif !C \Sequent \lnot (!A \land \lnot !C)

Block MathML variant

A→C⇒¬(A∧¬C)!A \lif !C \Sequent \lnot (!A \land \lnot !C)

Conventional reading: antecedent containing the conditional whose antecedent is formula A; and whose consequent is formula C; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and the negation of formula C, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing the conditional whose antecedent is formula A; and whose consequent is formula C; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and the negation of formula C, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 319, column 7

Expression 310

Inline MathML variant

A∧B⊢A!A \land !B \Proves !A

Block MathML variant

A∧B⊢A!A \land !B \Proves !A

Conventional reading: the conjunction of formula A and formula B syntactically derives formula A

Meaning here: The statement read 'the conjunction of formula A and formula B syntactically derives formula A' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

2 occurrences
  1. Occurrence 1: provability-propositional.tex, line 21, column 57
  2. Occurrence 2: provability-propositional.tex, line 28, column 51

Expression 312

Inline MathML variant

B⇒A→B!B \fCenter !A \lif !B

Block MathML variant

B⇒A→B!B \fCenter !A \lif !B

Conventional reading: antecedent containing formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B

Meaning here: The first-order sequent read 'antecedent containing formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 133, column 18

Expression 313

Inline MathML variant

(A→C)∧(B→C)⇒(A∨B)→C(!A \lif !C) \land (!B \lif !C) \Sequent (!A \lor !B) \lif !C

Block MathML variant

(A→C)∧(B→C)⇒(A∨B)→C(!A \lif !C) \land (!B \lif !C) \Sequent (!A \lor !B) \lif !C

Conventional reading: antecedent containing the conjunction of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula C, close parenthesis and open parenthesis, the conditional whose antecedent is formula B; and whose consequent is formula C, close parenthesis; sequent arrow; succedent containing the conditional whose antecedent is open parenthesis, the disjunction of formula A and formula B, close parenthesis; and whose consequent is formula C

Meaning here: The first-order sequent read 'antecedent containing the conjunction of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula C, close parenthesis and open parenthesis, the conditional whose antecedent is formula B; and whose consequent is formula C, close parenthesis; sequent arrow; succedent containing the conditional whose antecedent is open parenthesis, the disjunction of formula A and formula B, close parenthesis; and whose consequent is formula C' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 314, column 7

Expression 314

Inline MathML variant

A∧B,Γ⇒Δ!A \land !B, \Gamma \fCenter \Delta

Block MathML variant

A∧B,Γ⇒Δ!A \land !B, \Gamma \fCenter \Delta

Conventional reading: antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

3 occurrences
  1. Occurrence 1: propositional-rules.tex, line 35, column 10
  2. Occurrence 2: propositional-rules.tex, line 40, column 10
  3. Occurrence 3: soundness.tex, line 139, column 14

Expression 315

Inline MathML variant

Γ0\Gamma_0

Block MathML variant

Γ0\Gamma_0

Conventional reading: Gamma sub zero

Meaning here: Gamma sub zero denotes a finite premise set; when the source writes it as a sequent antecedent, the source tacitly chooses a sequence containing its members.

7 occurrences
  1. Occurrence 1: proof-theoretic-notions.tex, line 40, column 25
  2. Occurrence 2: proof-theoretic-notions.tex, line 52, column 4
  3. Occurrence 3: proof-theoretic-notions.tex, line 64, column 31
  4. Occurrence 4: proof-theoretic-notions.tex, line 66, column 46
  5. Occurrence 5: proof-theoretic-notions.tex, line 97, column 54
  6. Occurrence 6: proof-theoretic-notions.tex, line 165, column 55
  7. Occurrence 7: provability-consistency.tex, line 25, column 18

Expression 317

Inline MathML variant

B!B

Block MathML variant

B!B

Conventional reading: formula B

Meaning here: The first-order formula read 'formula B' preserves its metavariables, connectives, term arguments, and written grouping.

11 occurrences
  1. Occurrence 1: rules-and-proofs.tex, line 69, column 56
  2. Occurrence 2: derivations.tex, line 73, column 10
  3. Occurrence 3: derivations.tex, line 90, column 47
  4. Occurrence 4: derivations.tex, line 106, column 19
  5. Occurrence 5: proving-things.tex, line 180, column 21
  6. Occurrence 6: soundness.tex, line 133, column 59
  7. Occurrence 7: soundness.tex, line 158, column 30
  8. Occurrence 8: soundness.tex, line 159, column 6
  9. Occurrence 9: soundness.tex, line 161, column 59
  10. Occurrence 10: soundness.tex, line 183, column 17
  11. Occurrence 11: soundness.tex, line 183, column 60

Expression 318

Inline MathML variant

⇒¬A\fCenter \lnot !A

Block MathML variant

⇒¬A\fCenter \lnot !A

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing the negation of formula A

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing the negation of formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 266, column 10

Expression 319

Inline MathML variant

⟦t1⟧M=⟦t2⟧M\Value{t_1}{M} = \Value{t_2}{M}

Block MathML variant

⟦t1⟧M=⟦t2⟧M\Value{t_1}{M} = \Value{t_2}{M}

Conventional reading: the value of term t sub one in structure M is identical to the value of term t sub two in structure M

Meaning here: The semantic term read 'the value of term t sub one in structure M is identical to the value of term t sub two in structure M' specifies an interpretation, term value, or assignment relation in a first-order structure.

1 occurrence
  1. Occurrence 1: soundness-identity.tex, line 37, column 35

Expression 320

Inline MathML variant

⇒A,A∨¬A\fCenter !A, !A \lor \lnot !A

Block MathML variant

⇒A,A∨¬A\fCenter !A, !A \lor \lnot !A

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing first formula A, then the disjunction of formula A and the negation of formula A

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing first formula A, then the disjunction of formula A and the negation of formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 290, column 10

Expression 321

Inline MathML variant

C∈Γ~!C \in \Gamma

Block MathML variant

C∈Γ~!C \in \Gamma

Conventional reading: formula C is a member of Gamma

Meaning here: The relation read 'formula C is a member of Gamma' concerns the exact premise sets or formula sequences named by the source.

1 occurrence
  1. Occurrence 1: soundness.tex, line 197, column 51

Expression 322

Inline MathML variant

A,Γ⇒Δ!A, \Gamma \Sequent \Delta

Block MathML variant

A,Γ⇒Δ!A, \Gamma \Sequent \Delta

Conventional reading: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: soundness.tex, line 76, column 13
  2. Occurrence 2: soundness.tex, line 144, column 3

Expression 325

Inline MathML variant

¬B,B⇒\lnot !B, !B \fCenter

Block MathML variant

¬B,B⇒\lnot !B, !B \fCenter

Conventional reading: antecedent containing first the negation of formula B, then formula B; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first the negation of formula B, then formula B; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 77, column 16

Expression 326

Inline MathML variant

A⊢A∨B!A \Proves !A \lor !B

Block MathML variant

A⊢A∨B!A \Proves !A \lor !B

Conventional reading: formula A syntactically derives the disjunction of formula A and formula B

Meaning here: The statement read 'formula A syntactically derives the disjunction of formula A and formula B' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 61, column 14

Expression 327

Inline MathML variant

⇒(¬A∧¬B)→¬(A∨B)\Sequent (\lnot !A \land \lnot !B) \lif\lnot(!A \lor !B)

Block MathML variant

⇒(¬A∧¬B)→¬(A∨B)\Sequent (\lnot !A \land \lnot !B) \lif\lnot(!A \lor !B)

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is open parenthesis, the conjunction of the negation of formula A and the negation of formula B, close parenthesis; and whose consequent is the negation of open parenthesis, the disjunction of formula A and formula B, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is open parenthesis, the conjunction of the negation of formula A and the negation of formula B, close parenthesis; and whose consequent is the negation of open parenthesis, the disjunction of formula A and formula B, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 323, column 7

Expression 328

Inline MathML variant

Γ⇒Δ,A∨B\Gamma \Sequent \Delta, !A \lor !B

Block MathML variant

Γ⇒Δ,A∨B\Gamma \Sequent \Delta, !A \lor !B

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: soundness.tex, line 180, column 3

Expression 329

Inline MathML variant

A→B,A⇒B!A \lif !B, !A \fCenter !B

Block MathML variant

A→B,A⇒B!A \lif !B, !A \fCenter !B

Conventional reading: antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then formula A; sequent arrow; succedent containing formula B

Meaning here: The first-order sequent read 'antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then formula A; sequent arrow; succedent containing formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 114, column 19

Expression 330

Inline MathML variant

A∧B⇒B!A \land !B \Sequent !B

Block MathML variant

A∧B⇒B!A \land !B \Sequent !B

Conventional reading: antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula B

Meaning here: The first-order sequent read 'antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 37, column 53

Expression 331

Inline MathML variant

B1∨…∨Bn!B_1 \lor \dots \lor !B_n

Block MathML variant

B1∨…∨Bn!B_1 \lor \dots \lor !B_n

Conventional reading: the disjunction beginning with formula B sub one, continuing through the omitted intermediate disjunctions, and ending with formula B sub n

Meaning here: The first-order formula read 'the disjunction beginning with formula B sub one, continuing through the omitted intermediate disjunctions, and ending with formula B sub n' preserves its metavariables, connectives, term arguments, and written grouping.

1 occurrence
  1. Occurrence 1: rules-and-proofs.tex, line 40, column 28

Expression 332

Inline MathML variant

Γ0′′\Gamma_0''

Block MathML variant

Γ0′′\Gamma_0''

Conventional reading: Gamma sub zero double prime

Meaning here: Gamma sub zero double prime denotes the next reordered antecedent sequence in that construction.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 48, column 34

Expression 333

Inline MathML variant

A⇒!A \fCenter

Block MathML variant

A⇒!A \fCenter

Conventional reading: antecedent containing formula A; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing formula A; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 264, column 10

Expression 334

Inline MathML variant

M⊭∀x(A(x))\Sat/{M}{\lforall[x][!A(x)]}

Block MathML variant

M⊭∀x(A(x))\Sat/{M}{\lforall[x][!A(x)]}

Conventional reading: structure M does not satisfy for every variable x, formula A with argument variable x

Meaning here: The satisfaction claim read 'structure M does not satisfy for every variable x, formula A with argument variable x' is evaluated in the named first-order structure and, when printed, the named variable assignment.

1 occurrence
  1. Occurrence 1: soundness.tex, line 223, column 22

Expression 335

Inline MathML variant

M⊨A\Sat{M}{!A}

Block MathML variant

M⊨A\Sat{M}{!A}

Conventional reading: structure M satisfies formula A

Meaning here: The satisfaction claim read 'structure M satisfies formula A' is evaluated in the named first-order structure and, when printed, the named variable assignment.

9 occurrences
  1. Occurrence 1: soundness.tex, line 46, column 1
  2. Occurrence 2: soundness.tex, line 65, column 1
  3. Occurrence 3: soundness.tex, line 119, column 3
  4. Occurrence 4: soundness.tex, line 126, column 21
  5. Occurrence 5: soundness.tex, line 172, column 3
  6. Occurrence 6: soundness.tex, line 283, column 11
  7. Occurrence 7: soundness.tex, line 304, column 3
  8. Occurrence 8: soundness.tex, line 326, column 25
  9. Occurrence 9: soundness.tex, line 363, column 1

Expression 337

Inline MathML variant

Γ⇒Δ\Gamma \Sequent \Delta

Block MathML variant

Γ⇒Δ\Gamma \Sequent \Delta

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

8 occurrences
  1. Occurrence 1: rules-and-proofs.tex, line 20, column 1
  2. Occurrence 2: rules-and-proofs.tex, line 32, column 44
  3. Occurrence 3: soundness.tex, line 44, column 1
  4. Occurrence 4: soundness.tex, line 90, column 28
  5. Occurrence 5: soundness.tex, line 157, column 14
  6. Occurrence 6: soundness.tex, line 285, column 28
  7. Occurrence 7: soundness.tex, line 301, column 54
  8. Occurrence 8: soundness.tex, line 324, column 55

Expression 338

Inline MathML variant

¬A∨¬B,A⇒\lnot !A \lor \lnot !B, !A \fCenter

Block MathML variant

¬A∨¬B,A⇒\lnot !A \lor \lnot !B, !A \fCenter

Conventional reading: antecedent containing first the disjunction of the negation of formula A and the negation of formula B, then formula A; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first the disjunction of the negation of formula A and the negation of formula B, then formula A; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 200, column 11

Expression 339

Inline MathML variant

(A∧B)→C⇒(A→C)∨(B→C)(!A \land !B) \lif !C \Sequent (!A \lif !C) \lor (!B \lif !C)

Block MathML variant

(A∧B)→C⇒(A→C)∨(B→C)(!A \land !B) \lif !C \Sequent (!A \lif !C) \lor (!B \lif !C)

Conventional reading: antecedent containing the conditional whose antecedent is open parenthesis, the conjunction of formula A and formula B, close parenthesis; and whose consequent is formula C; sequent arrow; succedent containing the disjunction of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula C, close parenthesis and open parenthesis, the conditional whose antecedent is formula B; and whose consequent is formula C, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing the conditional whose antecedent is open parenthesis, the conjunction of formula A and formula B, close parenthesis; and whose consequent is formula C; sequent arrow; succedent containing the disjunction of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula C, close parenthesis and open parenthesis, the conditional whose antecedent is formula B; and whose consequent is formula C, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 336, column 7

Expression 340

Inline MathML variant

B⇒B!B \fCenter !B

Block MathML variant

B⇒B!B \fCenter !B

Conventional reading: antecedent containing formula B; sequent arrow; succedent containing formula B

Meaning here: The first-order sequent read 'antecedent containing formula B; sequent arrow; succedent containing formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

10 occurrences
  1. Occurrence 1: proving-things.tex, line 97, column 7
  2. Occurrence 2: proving-things.tex, line 118, column 7
  3. Occurrence 3: proving-things.tex, line 140, column 7
  4. Occurrence 4: proving-things.tex, line 235, column 7
  5. Occurrence 5: provability-propositional.tex, line 44, column 13
  6. Occurrence 6: provability-propositional.tex, line 51, column 13
  7. Occurrence 7: provability-propositional.tex, line 75, column 13
  8. Occurrence 8: provability-propositional.tex, line 92, column 13
  9. Occurrence 9: provability-propositional.tex, line 112, column 15
  10. Occurrence 10: provability-propositional.tex, line 129, column 15

Expression 341

Inline MathML variant

A,Γ1⇒!A, \Gamma_1 \fCenter

Block MathML variant

A,Γ1⇒!A, \Gamma_1 \fCenter

Conventional reading: antecedent containing first formula A, then Gamma sub one; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first formula A, then Gamma sub one; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-consistency.tex, line 36, column 8

Expression 342

Inline MathML variant

⇒¬¬A→A\Sequent \lnot \lnot !A \lif !A

Block MathML variant

⇒¬¬A→A\Sequent \lnot \lnot !A \lif !A

Conventional reading: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is the negation of the negation of formula A; and whose consequent is formula A

Meaning here: The first-order sequent read 'antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is the negation of the negation of formula A; and whose consequent is formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 334, column 7

Expression 343

Inline MathML variant

A(s)⇒A(s)!A(s) \fCenter !A(s)

Block MathML variant

A(s)⇒A(s)!A(s) \fCenter !A(s)

Conventional reading: antecedent containing formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument lowercase s

Meaning here: The first-order sequent read 'antecedent containing formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument lowercase s' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: identity.tex, line 38, column 7

Expression 344

Inline MathML variant

Γ1⊆Γ\Gamma_1 \subseteq \Gamma

Block MathML variant

Γ1⊆Γ\Gamma_1 \subseteq \Gamma

Conventional reading: Gamma sub one is a subset of Gamma

Meaning here: The relation read 'Gamma sub one is a subset of Gamma' concerns the exact premise sets or formula sequences named by the source.

4 occurrences
  1. Occurrence 1: provability-consistency.tex, line 25, column 33
  2. Occurrence 2: provability-consistency.tex, line 41, column 39
  3. Occurrence 3: provability-consistency.tex, line 106, column 55
  4. Occurrence 4: provability-consistency.tex, line 122, column 39

Expression 345

Inline MathML variant

A,A,Γ⇒Δ!A, !A, \Gamma \fCenter \Delta

Block MathML variant

A,A,Γ⇒Δ!A, !A, \Gamma \fCenter \Delta

Conventional reading: antecedent containing first formula A, then formula A, and finally Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first formula A, then formula A, and finally Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: structural-rules.tex, line 40, column 7

Expression 346

Inline MathML variant

C,D!C, !D

Block MathML variant

C,D!C, !D

Conventional reading: first formula C, then formula D

Meaning here: The sequence read 'first formula C, then formula D' preserves the formula order and any printed empty or trailing position.

1 occurrence
  1. Occurrence 1: derivations.tex, line 90, column 1

Expression 347

Inline MathML variant

Δ0⊆Δ\Delta_0 \subseteq \Delta

Block MathML variant

Δ0⊆Δ\Delta_0 \subseteq \Delta

Conventional reading: capital Delta sub zero is a subset of capital Delta

Meaning here: The relation read 'capital Delta sub zero is a subset of capital Delta' concerns the exact premise sets or formula sequences named by the source.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 111, column 54

Expression 348

Inline MathML variant

M′⊭C\Sat/{M'}{!C}

Block MathML variant

M′⊭C\Sat/{M'}{!C}

Conventional reading: structure M prime does not satisfy formula C

Meaning here: The satisfaction claim read 'structure M prime does not satisfy formula C' is evaluated in the named first-order structure and, when printed, the named variable assignment.

1 occurrence
  1. Occurrence 1: soundness.tex, line 256, column 20

Expression 349

Inline MathML variant

A∨B,¬A,¬B⇒!A \lor !B, \lnot !A, \lnot !B \fCenter

Block MathML variant

A∨B,¬A,¬B⇒!A \lor !B, \lnot !A, \lnot !B \fCenter

Conventional reading: antecedent containing first the disjunction of formula A and formula B, then the negation of formula A, and finally the negation of formula B; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first the disjunction of formula A and formula B, then the negation of formula A, and finally the negation of formula B; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 81, column 17

Expression 350

Inline MathML variant

A∧B⇒A!A \land !B \fCenter !A

Block MathML variant

A∧B⇒A!A \land !B \fCenter !A

Conventional reading: antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proving-things.tex, line 231, column 10
  2. Occurrence 2: provability-propositional.tex, line 42, column 16

Expression 351

Inline MathML variant

s(x)=⟦t2⟧Ms(x) = \Value{t_2}{M}

Block MathML variant

s(x)=⟦t2⟧Ms(x) = \Value{t_2}{M}

Conventional reading: s applied to variable x is identical to the value of term t sub two in structure M

Meaning here: The semantic term read 's applied to variable x is identical to the value of term t sub two in structure M' specifies an interpretation, term value, or assignment relation in a first-order structure.

1 occurrence
  1. Occurrence 1: soundness-identity.tex, line 38, column 11

Expression 352

Inline MathML variant

A∨B,¬A,¬B⇒!A \lor !B, \lnot !A, \lnot !B \Sequent

Block MathML variant

A∨B,¬A,¬B⇒!A \lor !B, \lnot !A, \lnot !B \Sequent

Conventional reading: antecedent containing first the disjunction of formula A and formula B, then the negation of formula A, and finally the negation of formula B; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first the disjunction of formula A and formula B, then the negation of formula A, and finally the negation of formula B; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 67, column 48

Expression 353

Inline MathML variant

tt

Block MathML variant

tt

Conventional reading: term t

Meaning here: The symbol read 'term t' receives its exact term, variable, constant, or assignment role from each bound source occurrence.

9 occurrences
  1. Occurrence 1: quantifier-rules.tex, line 27, column 22
  2. Occurrence 2: quantifier-rules.tex, line 48, column 8
  3. Occurrence 3: quantifier-rules.tex, line 66, column 11
  4. Occurrence 4: quantifier-rules.tex, line 71, column 33
  5. Occurrence 5: quantifier-rules.tex, line 81, column 10
  6. Occurrence 6: soundness.tex, line 209, column 42
  7. Occurrence 7: identity.tex, line 17, column 4
  8. Occurrence 8: identity.tex, line 35, column 12
  9. Occurrence 9: soundness-identity.tex, line 20, column 20

Expression 354

Inline MathML variant

¬A,Γ⇒\lnot !A, \Gamma \fCenter

Block MathML variant

¬A,Γ⇒\lnot !A, \Gamma \fCenter

Conventional reading: antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-consistency.tex, line 65, column 10

Expression 355

Inline MathML variant

Γ⇒Δ,B,A,Λ\Gamma \fCenter \Delta, !B, !A, \Lambda

Block MathML variant

Γ⇒Δ,B,A,Λ\Gamma \fCenter \Delta, !B, !A, \Lambda

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B, then formula A, and finally capital Lambda

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B, then formula A, and finally capital Lambda' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: structural-rules.tex, line 61, column 10

Expression 356

Inline MathML variant

A⇒A!A \fCenter !A

Block MathML variant

A⇒A!A \fCenter !A

Conventional reading: antecedent containing formula A; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing formula A; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

12 occurrences
  1. Occurrence 1: proving-things.tex, line 42, column 7
  2. Occurrence 2: proving-things.tex, line 133, column 7
  3. Occurrence 3: proving-things.tex, line 191, column 7
  4. Occurrence 4: proving-things.tex, line 286, column 7
  5. Occurrence 5: provability-consistency.tex, line 60, column 9
  6. Occurrence 6: provability-consistency.tex, line 88, column 11
  7. Occurrence 7: provability-propositional.tex, line 40, column 13
  8. Occurrence 8: provability-propositional.tex, line 50, column 13
  9. Occurrence 9: provability-propositional.tex, line 70, column 13
  10. Occurrence 10: provability-propositional.tex, line 88, column 13
  11. Occurrence 11: provability-propositional.tex, line 111, column 15
  12. Occurrence 12: provability-propositional.tex, line 119, column 15

Expression 357

Inline MathML variant

Γ⊢∀x(A(x))\Gamma \Proves \lforall[x][!A(x)]

Block MathML variant

Γ⊢∀x(A(x))\Gamma \Proves \lforall[x][!A(x)]

Conventional reading: Gamma syntactically derives for every variable x, formula A with argument variable x

Meaning here: The statement read 'Gamma syntactically derives for every variable x, formula A with argument variable x' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: provability-quantifiers.tex, line 20, column 57

Expression 358

Inline MathML variant

A(t)!A(t)

Block MathML variant

A(t)!A(t)

Conventional reading: formula A with argument term t

Meaning here: The first-order formula read 'formula A with argument term t' preserves its metavariables, connectives, term arguments, and written grouping.

1 occurrence
  1. Occurrence 1: quantifier-rules.tex, line 59, column 15

Expression 359

Inline MathML variant

C∈Θ!C \in \Theta

Block MathML variant

C∈Θ!C \in \Theta

Conventional reading: formula C is a member of capital Theta

Meaning here: The relation read 'formula C is a member of capital Theta' concerns the exact premise sets or formula sequences named by the source.

6 occurrences
  1. Occurrence 1: soundness.tex, line 98, column 8
  2. Occurrence 2: soundness.tex, line 99, column 51
  3. Occurrence 3: soundness.tex, line 123, column 47
  4. Occurrence 4: soundness.tex, line 128, column 21
  5. Occurrence 5: soundness.tex, line 149, column 12
  6. Occurrence 6: soundness.tex, line 152, column 3

Expression 360

Inline MathML variant

Γ∪{A}\Gamma \cup \{!A\}

Block MathML variant

Γ∪{A}\Gamma \cup \{!A\}

Conventional reading: the union of Gamma, and the set containing formula A

Meaning here: The relation read 'the union of Gamma, and the set containing formula A' concerns the exact premise sets or formula sequences named by the source.

3 occurrences
  1. Occurrence 1: provability-consistency.tex, line 20, column 30
  2. Occurrence 2: provability-consistency.tex, line 72, column 42
  3. Occurrence 3: provability-consistency.tex, line 101, column 6

Expression 361

Inline MathML variant

B,Π⇒Λ!B, \Pi \Sequent \Lambda

Block MathML variant

B,Π⇒Λ!B, \Pi \Sequent \Lambda

Conventional reading: antecedent containing first formula B, then capital Pi; sequent arrow; succedent containing capital Lambda

Meaning here: The first-order sequent read 'antecedent containing first formula B, then capital Pi; sequent arrow; succedent containing capital Lambda' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: soundness.tex, line 327, column 3

Expression 362

Inline MathML variant

M⊨A∧B\Sat{M}{!A \land !B}

Block MathML variant

M⊨A∧B\Sat{M}{!A \land !B}

Conventional reading: structure M satisfies the conjunction of formula A and formula B

Meaning here: The satisfaction claim read 'structure M satisfies the conjunction of formula A and formula B' is evaluated in the named first-order structure and, when printed, the named variable assignment.

1 occurrence
  1. Occurrence 1: soundness.tex, line 307, column 3

Expression 363

Inline MathML variant

Θ=¬A,Γ\Theta = \lnot !A, \Gamma

Block MathML variant

Θ=¬A,Γ\Theta = \lnot !A, \Gamma

Conventional reading: capital Theta equals first the negation of formula A, then Gamma

Meaning here: The metalevel equality read 'capital Theta equals first the negation of formula A, then Gamma' identifies the exact derivation count, sequence, premise family, or semantic quantity named on its two sides.

1 occurrence
  1. Occurrence 1: soundness.tex, line 112, column 7

Expression 364

Inline MathML variant

Π∖Λ\Pi \setminus \Lambda

Block MathML variant

Π∖Λ\Pi \setminus \Lambda

Conventional reading: capital Pi set difference capital Lambda

Meaning here: The relation read 'capital Pi set difference capital Lambda' concerns the exact premise sets or formula sequences named by the source.

1 occurrence
  1. Occurrence 1: soundness.tex, line 288, column 28

Expression 365

Inline MathML variant

A→B,A⇒B!A \lif !B, !A \Sequent !B

Block MathML variant

A→B,A⇒B!A \lif !B, !A \Sequent !B

Conventional reading: antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then formula A; sequent arrow; succedent containing formula B

Meaning here: The first-order sequent read 'antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then formula A; sequent arrow; succedent containing formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 109, column 23

Expression 366

Inline MathML variant

¬A∨B⇒A→B\lnot !A \lor !B \fCenter !A \lif !B

Block MathML variant

¬A∨B⇒A→B\lnot !A \lor !B \fCenter !A \lif !B

Conventional reading: antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B

Meaning here: The first-order sequent read 'antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

6 occurrences
  1. Occurrence 1: proving-things.tex, line 57, column 10
  2. Occurrence 2: proving-things.tex, line 73, column 37
  3. Occurrence 3: proving-things.tex, line 89, column 10
  4. Occurrence 4: proving-things.tex, line 107, column 10
  5. Occurrence 5: proving-things.tex, line 128, column 10
  6. Occurrence 6: proving-things.tex, line 150, column 10

Expression 367

Inline MathML variant

A∨(B∨C)⇒(A∨B)∨C!A \lor (!B \lor !C) \Sequent (!A \lor !B) \lor !C

Block MathML variant

A∨(B∨C)⇒(A∨B)∨C!A \lor (!B \lor !C) \Sequent (!A \lor !B) \lor !C

Conventional reading: antecedent containing the disjunction of formula A and open parenthesis, the disjunction of formula B and formula C, close parenthesis; sequent arrow; succedent containing the disjunction of open parenthesis, the disjunction of formula A and formula B, close parenthesis and formula C

Meaning here: The first-order sequent read 'antecedent containing the disjunction of formula A and open parenthesis, the disjunction of formula B and formula C, close parenthesis; sequent arrow; succedent containing the disjunction of open parenthesis, the disjunction of formula A and formula B, close parenthesis and formula C' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 304, column 7

Expression 369

Inline MathML variant

A→B,¬A→B⇒B!A \lif !B, \lnot !A \lif !B \Sequent !B

Block MathML variant

A→B,¬A→B⇒B!A \lif !B, \lnot !A \lif !B \Sequent !B

Conventional reading: antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then the conditional whose antecedent is the negation of formula A; and whose consequent is formula B; sequent arrow; succedent containing formula B

Meaning here: The first-order sequent read 'antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then the conditional whose antecedent is the negation of formula A; and whose consequent is formula B; sequent arrow; succedent containing formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proving-things.tex, line 335, column 7

Expression 370

Inline MathML variant

∀x(A(x)),Γ⇒Δ\lforall[x][!A(x)], \Gamma \Sequent \Delta

Block MathML variant

∀x(A(x)),Γ⇒Δ\lforall[x][!A(x)], \Gamma \Sequent \Delta

Conventional reading: antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

3 occurrences
  1. Occurrence 1: soundness.tex, line 216, column 39
  2. Occurrence 2: soundness.tex, line 224, column 36
  3. Occurrence 3: soundness.tex, line 226, column 3

Expression 371

Inline MathML variant

C⇒C!C \Sequent !C

Block MathML variant

C⇒C!C \Sequent !C

Conventional reading: antecedent containing formula C; sequent arrow; succedent containing formula C

Meaning here: The first-order sequent read 'antecedent containing formula C; sequent arrow; succedent containing formula C' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: derivations.tex, line 38, column 30
  2. Occurrence 2: derivations.tex, line 48, column 29

Expression 372

Inline MathML variant

Δ\Delta

Block MathML variant

Δ\Delta

Conventional reading: capital Delta

Meaning here: Capital Delta is context-sensitive: it denotes a sequent succedent sequence or a premise set, as stated by every exact occurrence record.

10 occurrences
  1. Occurrence 1: rules-and-proofs.tex, line 23, column 20
  2. Occurrence 2: rules-and-proofs.tex, line 25, column 26
  3. Occurrence 3: rules-and-proofs.tex, line 39, column 1
  4. Occurrence 4: rules-and-proofs.tex, line 41, column 1
  5. Occurrence 5: rules-and-proofs.tex, line 49, column 18
  6. Occurrence 6: derivations.tex, line 49, column 14
  7. Occurrence 7: derivations.tex, line 72, column 37
  8. Occurrence 8: derivations.tex, line 90, column 11
  9. Occurrence 9: proof-theoretic-notions.tex, line 98, column 25
  10. Occurrence 10: soundness.tex, line 237, column 34

Expression 373

Inline MathML variant

π\pi

Block MathML variant

π\pi

Conventional reading: pi

Meaning here: Pi denotes the sequent-calculus derivation currently under discussion.

13 occurrences
  1. Occurrence 1: provability-consistency.tex, line 82, column 22
  2. Occurrence 2: provability-consistency.tex, line 86, column 17
  3. Occurrence 3: soundness.tex, line 58, column 5
  4. Occurrence 4: soundness.tex, line 59, column 46
  5. Occurrence 5: soundness.tex, line 61, column 42
  6. Occurrence 6: soundness.tex, line 134, column 53
  7. Occurrence 7: soundness.tex, line 162, column 50
  8. Occurrence 8: soundness.tex, line 184, column 49
  9. Occurrence 9: soundness.tex, line 209, column 56
  10. Occurrence 10: soundness.tex, line 229, column 56
  11. Occurrence 11: soundness.tex, line 270, column 41
  12. Occurrence 12: soundness.tex, line 290, column 51
  13. Occurrence 13: soundness.tex, line 309, column 49

Expression 374

Inline MathML variant

Γ⇒Δ,¬A\Gamma \fCenter \Delta, \lnot !A

Block MathML variant

Γ⇒Δ,¬A\Gamma \fCenter \Delta, \lnot !A

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the negation of formula A

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the negation of formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: propositional-rules.tex, line 25, column 10

Expression 376

Inline MathML variant

A∧B,Γ⇒Δ!A \land !B, \Gamma \Sequent \Delta

Block MathML variant

A∧B,Γ⇒Δ!A \land !B, \Gamma \Sequent \Delta

Conventional reading: antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: soundness.tex, line 155, column 34

Expression 377

Inline MathML variant

C,D⇒C!C, !D \fCenter !C

Block MathML variant

C,D⇒C!C, !D \fCenter !C

Conventional reading: antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula C

Meaning here: The first-order sequent read 'antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula C' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

3 occurrences
  1. Occurrence 1: derivations.tex, line 64, column 10
  2. Occurrence 2: derivations.tex, line 98, column 10
  3. Occurrence 3: derivations.tex, line 115, column 10

Expression 378

Inline MathML variant

D,C⇒C!D, !C \Sequent !C

Block MathML variant

D,C⇒C!D, !C \Sequent !C

Conventional reading: antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C

Meaning here: The first-order sequent read 'antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: derivations.tex, line 50, column 1

Expression 379

Inline MathML variant

Γ⇒Δ,A,B,Λ\Gamma \fCenter \Delta, !A, !B, \Lambda

Block MathML variant

Γ⇒Δ,A,B,Λ\Gamma \fCenter \Delta, !A, !B, \Lambda

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A, then formula B, and finally capital Lambda

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A, then formula B, and finally capital Lambda' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: structural-rules.tex, line 59, column 7

Expression 380

Inline MathML variant

¬A∨¬B⇒¬(A∧B)\lnot !A \lor \lnot !B \fCenter \lnot (!A \land !B)

Block MathML variant

¬A∨¬B⇒¬(A∧B)\lnot !A \lor \lnot !B \fCenter \lnot (!A \land !B)

Conventional reading: antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis

Meaning here: The first-order sequent read 'antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

6 occurrences
  1. Occurrence 1: proving-things.tex, line 162, column 10
  2. Occurrence 2: proving-things.tex, line 173, column 10
  3. Occurrence 3: proving-things.tex, line 187, column 10
  4. Occurrence 4: proving-things.tex, line 206, column 10
  5. Occurrence 5: proving-things.tex, line 225, column 10
  6. Occurrence 6: proving-things.tex, line 246, column 10

Expression 382

Inline MathML variant

A∧B⊢B!A \land !B \Proves !B

Block MathML variant

A∧B⊢B!A \land !B \Proves !B

Conventional reading: the conjunction of formula A and formula B syntactically derives formula B

Meaning here: The statement read 'the conjunction of formula A and formula B syntactically derives formula B' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: provability-propositional.tex, line 29, column 13

Expression 383

Inline MathML variant

Γ⊬A\Gamma \Proves/ !A

Block MathML variant

Γ⊬A\Gamma \Proves/ !A

Conventional reading: Gamma does not syntactically derive formula A

Meaning here: The statement read 'Gamma does not syntactically derive formula A' is first-order syntactic derivability in the L K sequent calculus, including the stated premises and formula scope.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 42, column 10

Expression 384

Inline MathML variant

B,C⇒A!B, !C \fCenter !A

Block MathML variant

B,C⇒A!B, !C \fCenter !A

Conventional reading: antecedent containing first formula B, then formula C; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing first formula B, then formula C; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: proof-theoretic-notions.tex, line 58, column 12

Expression 385

Inline MathML variant

¬A(a),∀x(A(x))⇒\lnot !A(a), \lforall[x][!A(x)] \fCenter

Block MathML variant

¬A(a),∀x(A(x))⇒\lnot !A(a), \lforall[x][!A(x)] \fCenter

Conventional reading: antecedent containing first the negation of formula A with argument constant a, then for every variable x, formula A with argument variable x; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first the negation of formula A with argument constant a, then for every variable x, formula A with argument variable x; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proving-things-quant.tex, line 51, column 10
  2. Occurrence 2: proving-things-quant.tex, line 69, column 10

Expression 387

Inline MathML variant

Γ0⇒A\Gamma_0 \Sequent !A

Block MathML variant

Γ0⇒A\Gamma_0 \Sequent !A

Conventional reading: antecedent containing Gamma sub zero; sequent arrow; succedent containing formula A

Meaning here: The first-order sequent read 'antecedent containing Gamma sub zero; sequent arrow; succedent containing formula A' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

9 occurrences
  1. Occurrence 1: proof-theoretic-notions.tex, line 65, column 20
  2. Occurrence 2: proof-theoretic-notions.tex, line 96, column 29
  3. Occurrence 3: proof-theoretic-notions.tex, line 98, column 57
  4. Occurrence 4: proof-theoretic-notions.tex, line 110, column 32
  5. Occurrence 5: proof-theoretic-notions.tex, line 160, column 57
  6. Occurrence 6: provability-consistency.tex, line 26, column 24
  7. Occurrence 7: provability-consistency.tex, line 27, column 56
  8. Occurrence 8: provability-consistency.tex, line 82, column 41
  9. Occurrence 9: soundness.tex, line 358, column 38

Expression 388

Inline MathML variant

Γ⇒Δ,A→B\Gamma \fCenter \Delta, !A \lif !B

Block MathML variant

Γ⇒Δ,A→B\Gamma \fCenter \Delta, !A \lif !B

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: propositional-rules.tex, line 84, column 10
  2. Occurrence 2: soundness.tex, line 189, column 14

Expression 389

Inline MathML variant

B,Γ⇒Δ!B, \Gamma \fCenter \Delta

Block MathML variant

B,Γ⇒Δ!B, \Gamma \fCenter \Delta

Conventional reading: antecedent containing first formula B, then Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing first formula B, then Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: propositional-rules.tex, line 38, column 7
  2. Occurrence 2: propositional-rules.tex, line 55, column 7

Expression 390

Inline MathML variant

Γ⇒Δ\Gamma \fCenter \Delta

Block MathML variant

Γ⇒Δ\Gamma \fCenter \Delta

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing capital Delta

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing capital Delta' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

5 occurrences
  1. Occurrence 1: structural-rules.tex, line 26, column 7
  2. Occurrence 2: structural-rules.tex, line 31, column 7
  3. Occurrence 3: derivations.tex, line 42, column 7
  4. Occurrence 4: soundness.tex, line 81, column 12
  5. Occurrence 5: soundness.tex, line 86, column 12

Expression 391

Inline MathML variant

A,¬A∨¬B⇒!A, \lnot !A \lor \lnot !B \fCenter

Block MathML variant

A,¬A∨¬B⇒!A, \lnot !A \lor \lnot !B \fCenter

Conventional reading: antecedent containing first formula A, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas

Meaning here: The first-order sequent read 'antecedent containing first formula A, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proving-things.tex, line 183, column 10
  2. Occurrence 2: proving-things.tex, line 202, column 10

Expression 392

Inline MathML variant

¬A(a)⇒¬∀x(A(x))\lnot !A(a) \fCenter \lnot \lforall[x][!A(x)]

Block MathML variant

¬A(a)⇒¬∀x(A(x))\lnot !A(a) \fCenter \lnot \lforall[x][!A(x)]

Conventional reading: antecedent containing the negation of formula A with argument constant a; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x

Meaning here: The first-order sequent read 'antecedent containing the negation of formula A with argument constant a; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

2 occurrences
  1. Occurrence 1: proving-things-quant.tex, line 42, column 10
  2. Occurrence 2: proving-things-quant.tex, line 73, column 10

Expression 393

Inline MathML variant

s=t,A(s)⇒A(s)\eq[s][t], !A(s) \fCenter !A(s)

Block MathML variant

s=t,A(s)⇒A(s)\eq[s][t], !A(s) \fCenter !A(s)

Conventional reading: antecedent containing first lowercase s is identical to term t, then formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument lowercase s

Meaning here: The first-order sequent read 'antecedent containing first lowercase s is identical to term t, then formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument lowercase s' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: identity.tex, line 40, column 10

Expression 394

Inline MathML variant

Γ⇒Δ,A(t)\Gamma \fCenter \Delta, !A(t)

Block MathML variant

Γ⇒Δ,A(t)\Gamma \fCenter \Delta, !A(t)

Conventional reading: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t

Meaning here: The first-order sequent read 'antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t' preserves the ordered antecedent and succedent exactly. It is a syntactic sequent; validity is asserted only where the surrounding source says so.

1 occurrence
  1. Occurrence 1: quantifier-rules.tex, line 42, column 7

144 formal objects

Definition: Sequent โ€” line 18

Definition: Sequent โ€” line 18. A complete source-order spoken rendering of this definition; it preserves the source claim and does not add a solution or proof.

  1. A sequent is an expression of the form antecedent containing Gamma; sequent arrow; succedent containing capital Delta where Gamma and capital Delta are finite (possibly empty) sequences of sentences of the language L.
  2. Gamma is called the antecedent, while capital Delta is the succedent.

Source: content/first-order-logic/sequent-calculus/rules-and-proofs.tex, line 18.

Definition: Initial Sequent โ€” line 52

Definition: Initial Sequent โ€” line 52. A complete source-order spoken rendering of this definition; it preserves the source claim and does not add a solution or proof.

  1. An initial sequent is a sequent of one of the following forms: Next item: antecedent containing formula A; sequent arrow; succedent containing formula A.
  2. antecedent containing falsum; sequent arrow; succedent containing no formulas for any sentence formula A in the language.

Source: content/first-order-logic/sequent-calculus/rules-and-proofs.tex, line 52.

Definition of negation sequent rules โ€” line 17

Definition of negation sequent rules โ€” line 17. This rule definition gives every premise, inference label, and conclusion in source order for negation.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  2. The next inference is labeled left negation rule.
  3. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  4. The source ends this displayed proof segment here.
  5. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  6. The next inference is labeled right negation rule.
  7. From the immediately preceding branch using the right negation rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the negation of formula A.
  8. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/propositional-rules.tex, line 17.

Proof tree concluding antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 21

Proof tree concluding antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 21. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are left negation rule. Its last stated proof component is: From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing capital Delta.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  2. The next inference is labeled left negation rule.
  3. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/propositional-rules.tex, line 21.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the negation of formula A โ€” line 26

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the negation of formula A โ€” line 26. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are right negation rule. Its last stated proof component is: From the immediately preceding branch using the right negation rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the negation of formula A.

  1. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled right negation rule.
  3. From the immediately preceding branch using the right negation rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the negation of formula A.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/propositional-rules.tex, line 26.

Definition of conjunction sequent rules โ€” line 31

Definition of conjunction sequent rules โ€” line 31. This rule definition gives every premise, inference label, and conclusion in source order for conjunction.

  1. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled left conjunction rule.
  3. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  4. The source ends this displayed proof segment here.
  5. Premise or initial sequent: antecedent containing first formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  6. The next inference is labeled left conjunction rule.
  7. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  8. The source ends this displayed proof segment here.
  9. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  10. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  11. The next inference is labeled right conjunction rule.
  12. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B.
  13. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/propositional-rules.tex, line 31.

Rule table for conjunction โ€” line 32

Rule table for conjunction โ€” line 32. This table presents its sequent-calculus rule diagrams in source row order for conjunction. Every premise, rule label, conclusion, and display break is linearized.

  1. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled left conjunction rule.
  3. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  4. The source ends this displayed proof segment here.
  5. Premise or initial sequent: antecedent containing first formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  6. The next inference is labeled left conjunction rule.
  7. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  8. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/propositional-rules.tex, line 32.

Proof tree concluding antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 36

Proof tree concluding antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 36. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are left conjunction rule. Its last stated proof component is: From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.

  1. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled left conjunction rule.
  3. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/propositional-rules.tex, line 36.

Proof tree concluding antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 41

Proof tree concluding antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 41. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are left conjunction rule. Its last stated proof component is: From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.

  1. Premise or initial sequent: antecedent containing first formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled left conjunction rule.
  3. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/propositional-rules.tex, line 41.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B โ€” line 48

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B โ€” line 48. This source proof tree has 5 explicitly ordered components. Its inference labels, in order, are right conjunction rule. Its last stated proof component is: From the two immediately preceding branches using the right conjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  2. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  3. The next inference is labeled right conjunction rule.
  4. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B.
  5. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/propositional-rules.tex, line 48.

Definition of disjunction sequent rules โ€” line 53

Definition of disjunction sequent rules โ€” line 53. This rule definition gives every premise, inference label, and conclusion in source order for disjunction.

  1. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  2. Premise or initial sequent: antecedent containing first formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  3. The next inference is labeled left disjunction rule.
  4. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  5. The source ends this displayed proof segment here.
  6. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  7. The next inference is labeled right disjunction rule.
  8. From the immediately preceding branch using the right disjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B.
  9. The source ends this displayed proof segment here.
  10. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  11. The next inference is labeled right disjunction rule.
  12. From the immediately preceding branch using the right disjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B.
  13. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/propositional-rules.tex, line 53.

Proof tree concluding antecedent containing first the disjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 58

Proof tree concluding antecedent containing first the disjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 58. This source proof tree has 5 explicitly ordered components. Its inference labels, in order, are left disjunction rule. Its last stated proof component is: From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.

  1. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  2. Premise or initial sequent: antecedent containing first formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  3. The next inference is labeled left disjunction rule.
  4. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  5. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/propositional-rules.tex, line 58.

Rule table for disjunction โ€” line 60

Rule table for disjunction โ€” line 60. This table presents its sequent-calculus rule diagrams in source row order for disjunction. Every premise, rule label, conclusion, and display break is linearized.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  2. The next inference is labeled right disjunction rule.
  3. From the immediately preceding branch using the right disjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B.
  4. The source ends this displayed proof segment here.
  5. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  6. The next inference is labeled right disjunction rule.
  7. From the immediately preceding branch using the right disjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B.
  8. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/propositional-rules.tex, line 60.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B โ€” line 64

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B โ€” line 64. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are right disjunction rule. Its last stated proof component is: From the immediately preceding branch using the right disjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  2. The next inference is labeled right disjunction rule.
  3. From the immediately preceding branch using the right disjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/propositional-rules.tex, line 64.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B โ€” line 69

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B โ€” line 69. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are right disjunction rule. Its last stated proof component is: From the immediately preceding branch using the right disjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  2. The next inference is labeled right disjunction rule.
  3. From the immediately preceding branch using the right disjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/propositional-rules.tex, line 69.

Definition of conditional sequent rules โ€” line 75

Definition of conditional sequent rules โ€” line 75. This rule definition gives every premise, inference label, and conclusion in source order for conditional.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  2. Premise or initial sequent: antecedent containing first formula B, then capital Pi; sequent arrow; succedent containing capital Lambda.
  3. The next inference is labeled left conditional rule.
  4. From the two immediately preceding branches using the left conditional rule, infer antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then Gamma, and finally capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda.
  5. The source ends this displayed proof segment here.
  6. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  7. The next inference is labeled right conditional rule.
  8. From the immediately preceding branch using the right conditional rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B.
  9. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/propositional-rules.tex, line 75.

Proof tree concluding antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then Gamma, and finally capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda โ€” line 80

Proof tree concluding antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then Gamma, and finally capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda โ€” line 80. This source proof tree has 5 explicitly ordered components. Its inference labels, in order, are left conditional rule. Its last stated proof component is: From the two immediately preceding branches using the left conditional rule, infer antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then Gamma, and finally capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  2. Premise or initial sequent: antecedent containing first formula B, then capital Pi; sequent arrow; succedent containing capital Lambda.
  3. The next inference is labeled left conditional rule.
  4. From the two immediately preceding branches using the left conditional rule, infer antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then Gamma, and finally capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda.
  5. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/propositional-rules.tex, line 80.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 85

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 85. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are right conditional rule. Its last stated proof component is: From the immediately preceding branch using the right conditional rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B.

  1. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  2. The next inference is labeled right conditional rule.
  3. From the immediately preceding branch using the right conditional rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/propositional-rules.tex, line 85.

Definition of universal quantifier sequent rules โ€” line 15

Definition of universal quantifier sequent rules โ€” line 15. This rule definition gives every premise, inference label, and conclusion in source order for universal quantifier.

  1. Premise or initial sequent: antecedent containing first formula A with argument term t, then Gamma; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled left universal quantifier rule.
  3. From the immediately preceding branch using the left universal quantifier rule, infer antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta.
  4. The source ends this displayed proof segment here.
  5. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument constant a.
  6. The next inference is labeled right universal quantifier rule.
  7. From the immediately preceding branch using the right universal quantifier rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for every variable x, formula A with argument variable x.
  8. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/quantifier-rules.tex, line 15.

Proof tree concluding antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 19

Proof tree concluding antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 19. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are left universal quantifier rule. Its last stated proof component is: From the immediately preceding branch using the left universal quantifier rule, infer antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta.

  1. Premise or initial sequent: antecedent containing first formula A with argument term t, then Gamma; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled left universal quantifier rule.
  3. From the immediately preceding branch using the left universal quantifier rule, infer antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/quantifier-rules.tex, line 19.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for every variable x, formula A with argument variable x โ€” line 24

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for every variable x, formula A with argument variable x โ€” line 24. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are right universal quantifier rule. Its last stated proof component is: From the immediately preceding branch using the right universal quantifier rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for every variable x, formula A with argument variable x.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument constant a.
  2. The next inference is labeled right universal quantifier rule.
  3. From the immediately preceding branch using the right universal quantifier rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for every variable x, formula A with argument variable x.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/quantifier-rules.tex, line 24.

Definition of existential quantifier sequent rules โ€” line 36

Definition of existential quantifier sequent rules โ€” line 36. This rule definition gives every premise, inference label, and conclusion in source order for existential quantifier.

  1. Premise or initial sequent: antecedent containing first formula A with argument constant a, then Gamma; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled left existential quantifier rule.
  3. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing first for some variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta.
  4. The source ends this displayed proof segment here.
  5. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t.
  6. The next inference is labeled right existential quantifier rule.
  7. From the immediately preceding branch using the right existential quantifier rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, formula A with argument variable x.
  8. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/quantifier-rules.tex, line 36.

Proof tree concluding antecedent containing first for some variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 40

Proof tree concluding antecedent containing first for some variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 40. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are left existential quantifier rule. Its last stated proof component is: From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing first for some variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta.

  1. Premise or initial sequent: antecedent containing first formula A with argument constant a, then Gamma; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled left existential quantifier rule.
  3. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing first for some variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/quantifier-rules.tex, line 40.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, formula A with argument variable x โ€” line 45

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, formula A with argument variable x โ€” line 45. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are right existential quantifier rule. Its last stated proof component is: From the immediately preceding branch using the right existential quantifier rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, formula A with argument variable x.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t.
  2. The next inference is labeled right existential quantifier rule.
  3. From the immediately preceding branch using the right existential quantifier rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, formula A with argument variable x.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/quantifier-rules.tex, line 45.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, formula A โ€” line 61

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, formula A โ€” line 61. This source proof tree has 3 explicitly ordered components. Its inference labels, in order, are right existential quantifier rule. Its last stated proof component is: From the immediately preceding branch using the right existential quantifier rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, formula A.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the result of substituting term t for variable x in formula A.
  2. The next inference is labeled right existential quantifier rule.
  3. From the immediately preceding branch using the right existential quantifier rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, formula A.

Source: content/first-order-logic/sequent-calculus/quantifier-rules.tex, line 61.

Proof tree concluding antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing for every variable x, formula A with argument variable x โ€” line 87

Proof tree concluding antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing for every variable x, formula A with argument variable x โ€” line 87. This source proof tree has 11 explicitly ordered components. Its inference labels, in order, are left existential quantifier rule, marked with an asterisk to identify this intentionally invalid inference, then right universal quantifier rule, then right universal quantifier rule, marked with an asterisk to identify this intentionally invalid inference, then left existential quantifier rule. Its last stated proof component is: From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing for every variable x, formula A with argument variable x.

  1. Premise or initial sequent: antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant a.
  2. The next inference is labeled left existential quantifier rule, marked with an asterisk to identify this intentionally invalid inference.
  3. From the immediately preceding branch using the left existential quantifier rule, marked with an asterisk to identify this intentionally invalid inference, infer antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument constant a.
  4. The next inference is labeled right universal quantifier rule.
  5. From the immediately preceding branch using the right universal quantifier rule, infer antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing for every variable x, formula A with argument variable x.
  6. The source ends this displayed proof segment here.
  7. Premise or initial sequent: antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant a.
  8. The next inference is labeled right universal quantifier rule, marked with an asterisk to identify this intentionally invalid inference.
  9. From the immediately preceding branch using the right universal quantifier rule, marked with an asterisk to identify this intentionally invalid inference, infer antecedent containing formula A with argument constant a; sequent arrow; succedent containing for every variable x, formula A with argument variable x.
  10. The next inference is labeled left existential quantifier rule.
  11. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing for every variable x, formula A with argument variable x.

Source: content/first-order-logic/sequent-calculus/quantifier-rules.tex, line 87.

Proof tree concluding antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing for every variable x, formula A with argument variable x โ€” line 93

Proof tree concluding antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing for every variable x, formula A with argument variable x โ€” line 93. This source proof tree has 6 explicitly ordered components. Its inference labels, in order, are left existential quantifier rule, marked with an asterisk to identify this intentionally invalid inference, then right universal quantifier rule. Its last stated proof component is: From the immediately preceding branch using the right universal quantifier rule, infer antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing for every variable x, formula A with argument variable x.

  1. Premise or initial sequent: antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant a.
  2. The next inference is labeled left existential quantifier rule, marked with an asterisk to identify this intentionally invalid inference.
  3. From the immediately preceding branch using the left existential quantifier rule, marked with an asterisk to identify this intentionally invalid inference, infer antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument constant a.
  4. The next inference is labeled right universal quantifier rule.
  5. From the immediately preceding branch using the right universal quantifier rule, infer antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing for every variable x, formula A with argument variable x.
  6. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/quantifier-rules.tex, line 93.

Definition of Weakening sequent rules โ€” line 25

Definition of Weakening sequent rules โ€” line 25. This rule definition gives every premise, inference label, and conclusion in source order for Weakening.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled left weakening rule.
  3. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  4. The source ends this displayed proof segment here.
  5. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  6. The next inference is labeled right weakening rule.
  7. From the immediately preceding branch using the right weakening rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  8. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/structural-rules.tex, line 25.

Proof tree concluding antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 29

Proof tree concluding antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 29. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are left weakening rule. Its last stated proof component is: From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled left weakening rule.
  3. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/structural-rules.tex, line 29.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A โ€” line 34

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A โ€” line 34. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are right weakening rule. Its last stated proof component is: From the immediately preceding branch using the right weakening rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled right weakening rule.
  3. From the immediately preceding branch using the right weakening rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/structural-rules.tex, line 34.

Definition of Contraction sequent rules โ€” line 39

Definition of Contraction sequent rules โ€” line 39. This rule definition gives every premise, inference label, and conclusion in source order for Contraction.

  1. Premise or initial sequent: antecedent containing first formula A, then formula A, and finally Gamma; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled left contraction rule.
  3. From the immediately preceding branch using the left contraction rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  4. The source ends this displayed proof segment here.
  5. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A, and finally formula A.
  6. The next inference is labeled right contraction rule.
  7. From the immediately preceding branch using the right contraction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  8. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/structural-rules.tex, line 39.

Proof tree concluding antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 43

Proof tree concluding antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 43. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are left contraction rule. Its last stated proof component is: From the immediately preceding branch using the left contraction rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.

  1. Premise or initial sequent: antecedent containing first formula A, then formula A, and finally Gamma; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled left contraction rule.
  3. From the immediately preceding branch using the left contraction rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/structural-rules.tex, line 43.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A โ€” line 48

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A โ€” line 48. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are right contraction rule. Its last stated proof component is: From the immediately preceding branch using the right contraction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A, and finally formula A.
  2. The next inference is labeled right contraction rule.
  3. From the immediately preceding branch using the right contraction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/structural-rules.tex, line 48.

Definition of Exchange sequent rules โ€” line 53

Definition of Exchange sequent rules โ€” line 53. This rule definition gives every premise, inference label, and conclusion in source order for Exchange.

  1. Premise or initial sequent: antecedent containing first Gamma, then formula A, then formula B, and finally capital Pi; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled left exchange rule.
  3. From the immediately preceding branch using the left exchange rule, infer antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta.
  4. The source ends this displayed proof segment here.
  5. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A, then formula B, and finally capital Lambda.
  6. The next inference is labeled right exchange rule.
  7. From the immediately preceding branch using the right exchange rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B, then formula A, and finally capital Lambda.
  8. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/structural-rules.tex, line 53.

Proof tree concluding antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta โ€” line 57

Proof tree concluding antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta โ€” line 57. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are left exchange rule. Its last stated proof component is: From the immediately preceding branch using the left exchange rule, infer antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta.

  1. Premise or initial sequent: antecedent containing first Gamma, then formula A, then formula B, and finally capital Pi; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled left exchange rule.
  3. From the immediately preceding branch using the left exchange rule, infer antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/structural-rules.tex, line 57.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B, then formula A, and finally capital Lambda โ€” line 62

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B, then formula A, and finally capital Lambda โ€” line 62. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are right exchange rule. Its last stated proof component is: From the immediately preceding branch using the right exchange rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B, then formula A, and finally capital Lambda.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A, then formula B, and finally capital Lambda.
  2. The next inference is labeled right exchange rule.
  3. From the immediately preceding branch using the right exchange rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B, then formula A, and finally capital Lambda.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/structural-rules.tex, line 62.

Definition of Exchange sequent rules โ€” line 70

Definition of Exchange sequent rules โ€” line 70. This rule definition gives every premise, inference label, and conclusion in source order for Exchange.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  2. Premise or initial sequent: antecedent containing first formula A, then capital Pi; sequent arrow; succedent containing capital Lambda.
  3. The next inference is labeled cut rule.
  4. From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma, then capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda.
  5. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/structural-rules.tex, line 70.

Proof tree concluding antecedent containing first Gamma, then capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda โ€” line 76

Proof tree concluding antecedent containing first Gamma, then capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda โ€” line 76. This source proof tree has 5 explicitly ordered components. Its inference labels, in order, are cut rule. Its last stated proof component is: From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma, then capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  2. Premise or initial sequent: antecedent containing first formula A, then capital Pi; sequent arrow; succedent containing capital Lambda.
  3. The next inference is labeled cut rule.
  4. From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma, then capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda.
  5. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/structural-rules.tex, line 76.

Definition: L K derivation โ€” line 23

Definition: L K derivation โ€” line 23. A complete source-order spoken rendering of this definition; it preserves the source claim and does not add a solution or proof.

  1. An L K derivation of a sequent S is a finite tree of sequents satisfying the following conditions: Next item: The topmost sequents of the tree are initial sequents.
  2. Next item: The bottommost sequent of the tree is S.
  3. Next item: Every sequent in the tree except S is a premise of a correct application of an inference rule whose conclusion stands directly below that sequent in the tree.
  4. We then say that S is the end-sequent of the derivation and that S is derivable in L K (or L K derivable).

Source: content/first-order-logic/sequent-calculus/derivations.tex, line 23.

Example: Every initial sequent, e.g., antecedent containing formula C; sequent arrow; succedentโ€ฆ โ€” line 37

Example: Every initial sequent, e.g., antecedent containing formula C; sequent arrow; succedentโ€ฆ โ€” line 37. A complete source-order spoken rendering of this example; it preserves the source claim and does not add a solution or proof.

  1. Every initial sequent, e.g., antecedent containing formula C; sequent arrow; succedent containing formula C is a derivation.
  2. We can obtain a new derivation from this by applying, say, the left weakening rule, Proof diagram.
  3. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  4. The next inference is labeled left weakening rule.
  5. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  6. End proof diagram.
  7. The rule, however, is meant to be general: we can replace the formula A in the rule with any sentence, e.g., also with formula D.
  8. If the premise matches our initial sequent antecedent containing formula C; sequent arrow; succedent containing formula C, that means that both Gamma and capital Delta are just formula C, and the conclusion would then be antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.
  9. So, the following is a derivation: Proof diagram.
  10. Premise or initial sequent: antecedent containing formula C; sequent arrow; succedent containing formula C.
  11. The next inference is labeled left weakening rule.
  12. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.
  13. End proof diagram.
  14. We can now apply another rule, say left exchange rule, which allows us to switch two sentences on the left.
  15. So, the following is also a correct derivation: Proof diagram.
  16. Premise or initial sequent: antecedent containing formula C; sequent arrow; succedent containing formula C.
  17. The next inference is labeled left weakening rule.
  18. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.
  19. The next inference is labeled left exchange rule.
  20. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula C.
  21. End proof diagram.
  22. In this application of the rule, which was given as Proof diagram.
  23. Premise or initial sequent: antecedent containing first Gamma, then formula A, then formula B, and finally capital Pi; sequent arrow; succedent containing capital Delta.
  24. The next inference is labeled left exchange rule.
  25. From the immediately preceding branch using the left exchange rule, infer antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta, followed by a printed trailing comma with no following formula. Source disclosure. Preserve and speak the printed trailing comma as having no following formula, then disclose that it appears to be stray punctuation.
  26. End proof diagram.
  27. both Gamma and capital Pi were empty, capital Delta is formula C, and the roles of formula A and formula B are played by formula D and formula C, respectively.
  28. In much the same way, we also see that Proof diagram.
  29. Premise or initial sequent: antecedent containing formula D; sequent arrow; succedent containing formula D.
  30. The next inference is labeled left weakening rule.
  31. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula D.
  32. End proof diagram.
  33. is a derivation.
  34. Now we can take these two derivations, and combine them using right conjunction rule.
  35. That rule was Proof diagram.
  36. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  37. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  38. The next inference is labeled right conjunction rule.
  39. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B.
  40. End proof diagram.
  41. In our case, the premises must match the last sequents of the derivations ending in the premises.
  42. That means that Gamma is first formula C, then formula D, capital Delta is empty, formula A is formula C and formula B is formula D.
  43. So the conclusion, if the inference should be correct, is antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula C and formula D.
  44. Proof diagram.
  45. Premise or initial sequent: antecedent containing formula C; sequent arrow; succedent containing formula C.
  46. The next inference is labeled left weakening rule.
  47. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.
  48. The next inference is labeled left exchange rule.
  49. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula C.
  50. Premise or initial sequent: antecedent containing formula D; sequent arrow; succedent containing formula D.
  51. The next inference is labeled left weakening rule.
  52. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula D.
  53. The next inference is labeled right conjunction rule.
  54. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula C and formula D.
  55. End proof diagram.
  56. Of course, we can also reverse the premises, then formula A would be formula D and formula B would be formula C.
  57. Proof diagram.
  58. Premise or initial sequent: antecedent containing formula D; sequent arrow; succedent containing formula D.
  59. The next inference is labeled left weakening rule.
  60. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula D.
  61. Premise or initial sequent: antecedent containing formula C; sequent arrow; succedent containing formula C.
  62. The next inference is labeled left weakening rule.
  63. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.
  64. The next inference is labeled left exchange rule.
  65. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula C.
  66. The next inference is labeled right conjunction rule.
  67. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula D and formula C.
  68. End proof diagram.

Source: content/first-order-logic/sequent-calculus/derivations.tex, line 37.

Proof tree concluding antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 41

Proof tree concluding antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 41. This source proof tree has 3 explicitly ordered components. Its inference labels, in order, are left weakening rule. Its last stated proof component is: From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled left weakening rule.
  3. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.

Source: content/first-order-logic/sequent-calculus/derivations.tex, line 41.

Proof tree concluding antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C โ€” line 51

Proof tree concluding antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C โ€” line 51. This source proof tree has 3 explicitly ordered components. Its inference labels, in order, are left weakening rule. Its last stated proof component is: From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.

  1. Premise or initial sequent: antecedent containing formula C; sequent arrow; succedent containing formula C.
  2. The next inference is labeled left weakening rule.
  3. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.

Source: content/first-order-logic/sequent-calculus/derivations.tex, line 51.

Proof tree concluding antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula C โ€” line 59

Proof tree concluding antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula C โ€” line 59. This source proof tree has 5 explicitly ordered components. Its inference labels, in order, are left weakening rule, then left exchange rule. Its last stated proof component is: From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula C.

  1. Premise or initial sequent: antecedent containing formula C; sequent arrow; succedent containing formula C.
  2. The next inference is labeled left weakening rule.
  3. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.
  4. The next inference is labeled left exchange rule.
  5. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula C.

Source: content/first-order-logic/sequent-calculus/derivations.tex, line 59.

Proof tree concluding antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta, followed by a printed trailing comma with no following formula โ€” line 67

Proof tree concluding antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta, followed by a printed trailing comma with no following formula โ€” line 67. This source proof tree has 3 explicitly ordered components. Its inference labels, in order, are left exchange rule. Its last stated proof component is: From the immediately preceding branch using the left exchange rule, infer antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta, followed by a printed trailing comma with no following formula.

  1. Premise or initial sequent: antecedent containing first Gamma, then formula A, then formula B, and finally capital Pi; sequent arrow; succedent containing capital Delta.
  2. The next inference is labeled left exchange rule.
  3. From the immediately preceding branch using the left exchange rule, infer antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta, followed by a printed trailing comma with no following formula. Source disclosure. Preserve and speak the printed trailing comma as having no following formula, then disclose that it appears to be stray punctuation.

Source: content/first-order-logic/sequent-calculus/derivations.tex, line 67.

Proof tree concluding antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula D โ€” line 75

Proof tree concluding antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula D โ€” line 75. This source proof tree has 3 explicitly ordered components. Its inference labels, in order, are left weakening rule. Its last stated proof component is: From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula D.

  1. Premise or initial sequent: antecedent containing formula D; sequent arrow; succedent containing formula D.
  2. The next inference is labeled left weakening rule.
  3. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula D.

Source: content/first-order-logic/sequent-calculus/derivations.tex, line 75.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B โ€” line 82

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B โ€” line 82. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are right conjunction rule. Its last stated proof component is: From the two immediately preceding branches using the right conjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B.

  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  2. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  3. The next inference is labeled right conjunction rule.
  4. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B.

Source: content/first-order-logic/sequent-calculus/derivations.tex, line 82.

Proof tree concluding antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula C and formula D โ€” line 93

Proof tree concluding antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula C and formula D โ€” line 93. This source proof tree has 10 explicitly ordered components. Its inference labels, in order, are left weakening rule, then left exchange rule, then left weakening rule, then right conjunction rule. Its last stated proof component is: From the two immediately preceding branches using the right conjunction rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula C and formula D.

  1. Premise or initial sequent: antecedent containing formula C; sequent arrow; succedent containing formula C.
  2. The next inference is labeled left weakening rule.
  3. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.
  4. The next inference is labeled left exchange rule.
  5. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula C.
  6. Premise or initial sequent: antecedent containing formula D; sequent arrow; succedent containing formula D.
  7. The next inference is labeled left weakening rule.
  8. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula D.
  9. The next inference is labeled right conjunction rule.
  10. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula C and formula D.

Source: content/first-order-logic/sequent-calculus/derivations.tex, line 93.

Proof tree concluding antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula D and formula C โ€” line 107

Proof tree concluding antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula D and formula C โ€” line 107. This source proof tree has 10 explicitly ordered components. Its inference labels, in order, are left weakening rule, then left weakening rule, then left exchange rule, then right conjunction rule. Its last stated proof component is: From the two immediately preceding branches using the right conjunction rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula D and formula C.

  1. Premise or initial sequent: antecedent containing formula D; sequent arrow; succedent containing formula D.
  2. The next inference is labeled left weakening rule.
  3. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula D.
  4. Premise or initial sequent: antecedent containing formula C; sequent arrow; succedent containing formula C.
  5. The next inference is labeled left weakening rule.
  6. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.
  7. The next inference is labeled left exchange rule.
  8. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula C.
  9. The next inference is labeled right conjunction rule.
  10. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula D and formula C.

Source: content/first-order-logic/sequent-calculus/derivations.tex, line 107.

Example: Give an L K derivation for the sequent antecedent containing the conjunction of formula A andโ€ฆ โ€” line 15

Example: Give an L K derivation for the sequent antecedent containing the conjunction of formula A andโ€ฆ โ€” line 15. A complete source-order spoken rendering of this example; it preserves the source claim and does not add a solution or proof.

  1. Give an L K derivation for the sequent antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A..
  2. We begin by writing the desired end-sequent at the bottom of the derivation.
  3. Proof diagram.
  4. An upper premise slot is intentionally blank in this staged diagram.
  5. From the immediately preceding branch, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.
  6. End proof diagram.
  7. Next, we need to figure out what kind of inference could have a lower sequent of this form.
  8. This could be a structural rule, but it is a good idea to start by looking for a logical rule.
  9. The only logical connective occurring in the lower sequent is conjunction, so we're looking for a conjunction rule, and since the conjunction symbol occurs in the antecedent, we're looking at the left conjunction rule.
  10. Proof diagram.
  11. An upper premise slot is intentionally blank in this staged diagram.
  12. The next inference is labeled left conjunction rule.
  13. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.
  14. End proof diagram.
  15. There are two options for what could have been the upper sequent of the left conjunction rule inference: we could have an upper sequent of antecedent containing formula A; sequent arrow; succedent containing formula A, or of antecedent containing formula B; sequent arrow; succedent containing formula A.
  16. Clearly, antecedent containing formula A; sequent arrow; succedent containing formula A is an initial sequent (which is a good thing), while antecedent containing formula B; sequent arrow; succedent containing formula A is not derivable in general.
  17. We fill in the upper sequent: Proof diagram.
  18. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  19. The next inference is labeled left conjunction rule.
  20. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.
  21. End proof diagram.
  22. We now have a correct L K derivation of the sequent antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 15.

Proof tree concluding antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A โ€” line 20

Proof tree concluding antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A โ€” line 20. This source proof tree has 2 explicitly ordered components. Its inference labels, in order, are none because this is a staged placeholder. Its last stated proof component is: From the immediately preceding branch, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. From the immediately preceding branch, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 20.

Proof tree concluding antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A โ€” line 31

Proof tree concluding antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A โ€” line 31. This source proof tree has 3 explicitly ordered components. Its inference labels, in order, are left conjunction rule. Its last stated proof component is: From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. The next inference is labeled left conjunction rule.
  3. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 31.

Proof tree concluding antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A โ€” line 41

Proof tree concluding antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A โ€” line 41. This source proof tree has 3 explicitly ordered components. Its inference labels, in order, are left conjunction rule. Its last stated proof component is: From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.

  1. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  2. The next inference is labeled left conjunction rule.
  3. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 41.

Example: Give an L K derivation for the sequent antecedent containing the disjunction of the negation ofโ€ฆ โ€” line 50

Example: Give an L K derivation for the sequent antecedent containing the disjunction of the negation ofโ€ฆ โ€” line 50. A complete source-order spoken rendering of this example; it preserves the source claim and does not add a solution or proof.

  1. Give an L K derivation for the sequent antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B..
  2. Begin by writing the desired end-sequent at the bottom of the derivation.
  3. Proof diagram.
  4. An upper premise slot is intentionally blank in this staged diagram.
  5. From the immediately preceding branch, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  6. End proof diagram.
  7. To find a logical rule that could give us this end-sequent, we look at the logical connectives in the end-sequent: negation, disjunction, and conditional.
  8. We only care at the moment about disjunction and conditional because they are main operators of sentences in the end-sequent, while negation is inside the scope of another connective, so we will take care of it later.
  9. Our options for logical rules for the final inference are therefore the left disjunction rule and the right conditional rule.
  10. We could pick either rule, really, but let's pick the right conditional rule (if for no reason other than it allows us to put off splitting into two branches).
  11. According to the form of right conditional rule inferences which can yield the lower sequent, this must look like: Proof diagram.
  12. An upper premise slot is intentionally blank in this staged diagram.
  13. From the immediately preceding branch, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  14. The next inference is labeled right conditional rule.
  15. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  16. End proof diagram..
  17. If we move the disjunction of the negation of formula A and formula B to the outside of the antecedent, we can apply the left disjunction rule.
  18. According to the schema, this must split into two upper sequents as follows: Proof diagram.
  19. An upper premise slot is intentionally blank in this staged diagram.
  20. From the immediately preceding branch, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B.
  21. An upper premise slot is intentionally blank in this staged diagram.
  22. From the immediately preceding branch, infer antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B.
  23. The next inference is labeled left disjunction rule.
  24. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B.
  25. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended.
  26. From the immediately preceding branch using the right exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  27. The next inference is labeled right conditional rule.
  28. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  29. End proof diagram.
  30. Remember that we are trying to wind our way up to initial sequents; we seem to be pretty close!
  31. The right branch is just one weakening and one exchange away from an initial sequent and then it is done: Proof diagram.
  32. An upper premise slot is intentionally blank in this staged diagram.
  33. From the immediately preceding branch, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B.
  34. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  35. The next inference is labeled left weakening rule.
  36. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then formula B; sequent arrow; succedent containing formula B.
  37. The next inference is labeled left exchange rule.
  38. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B.
  39. The next inference is labeled left disjunction rule.
  40. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B.
  41. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended.
  42. From the immediately preceding branch using the right exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  43. The next inference is labeled right conditional rule.
  44. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  45. End proof diagram..
  46. Now looking at the left branch, the only logical connective in any sentence is the negation symbol in the antecedent sentences, so we're looking at an instance of the left negation rule.
  47. Proof diagram.
  48. An upper premise slot is intentionally blank in this staged diagram.
  49. From the immediately preceding branch, infer antecedent containing formula A; sequent arrow; succedent containing first formula B, then formula A.
  50. The next inference is labeled left negation rule.
  51. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B.
  52. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  53. The next inference is labeled left weakening rule.
  54. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then formula B; sequent arrow; succedent containing formula B.
  55. The next inference is labeled left exchange rule.
  56. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B.
  57. The next inference is labeled left disjunction rule.
  58. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B.
  59. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended.
  60. From the immediately preceding branch using the right exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  61. The next inference is labeled right conditional rule.
  62. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  63. End proof diagram.
  64. Similarly to how we finished off the right branch, we are just one weakening and one exchange away from finishing off this left branch as well.
  65. Proof diagram.
  66. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  67. The next inference is labeled right weakening rule.
  68. From the immediately preceding branch using the right weakening rule, infer antecedent containing formula A; sequent arrow; succedent containing first formula A, then formula B.
  69. The next inference is labeled right exchange rule.
  70. From the immediately preceding branch using the right exchange rule, infer antecedent containing formula A; sequent arrow; succedent containing first formula B, then formula A.
  71. The next inference is labeled left negation rule.
  72. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B.
  73. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  74. The next inference is labeled left weakening rule.
  75. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then formula B; sequent arrow; succedent containing formula B.
  76. The next inference is labeled left exchange rule.
  77. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B.
  78. The next inference is labeled left disjunction rule.
  79. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B.
  80. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended.
  81. From the immediately preceding branch using the right exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  82. The next inference is labeled right conditional rule.
  83. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  84. End proof diagram.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 50.

Proof tree concluding antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 55

Proof tree concluding antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 55. This source proof tree has 2 explicitly ordered components. Its inference labels, in order, are none because this is a staged placeholder. Its last stated proof component is: From the immediately preceding branch, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. From the immediately preceding branch, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 55.

Proof tree concluding antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 70

Proof tree concluding antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 70. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are right conditional rule. Its last stated proof component is: From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. From the immediately preceding branch, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  3. The next inference is labeled right conditional rule.
  4. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 70.

Proof tree concluding antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 79

Proof tree concluding antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 79. This source proof tree has 10 explicitly ordered components. Its inference labels, in order, are left disjunction rule, then right exchange rule, then right conditional rule. Its last stated proof component is: From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. From the immediately preceding branch, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B.
  3. An upper premise slot is intentionally blank in this staged diagram.
  4. From the immediately preceding branch, infer antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B.
  5. The next inference is labeled left disjunction rule.
  6. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B.
  7. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended.
  8. From the immediately preceding branch using the right exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  9. The next inference is labeled right conditional rule.
  10. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 79.

Proof tree concluding antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 94

Proof tree concluding antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 94. This source proof tree has 13 explicitly ordered components. Its inference labels, in order, are left weakening rule, then left exchange rule, then left disjunction rule, then right exchange rule, then right conditional rule. Its last stated proof component is: From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. From the immediately preceding branch, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B.
  3. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  4. The next inference is labeled left weakening rule.
  5. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then formula B; sequent arrow; succedent containing formula B.
  6. The next inference is labeled left exchange rule.
  7. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B.
  8. The next inference is labeled left disjunction rule.
  9. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B.
  10. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended.
  11. From the immediately preceding branch using the right exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  12. The next inference is labeled right conditional rule.
  13. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 94.

Proof tree concluding antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 113

Proof tree concluding antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 113. This source proof tree has 15 explicitly ordered components. Its inference labels, in order, are left negation rule, then left weakening rule, then left exchange rule, then left disjunction rule, then right exchange rule, then right conditional rule. Its last stated proof component is: From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. From the immediately preceding branch, infer antecedent containing formula A; sequent arrow; succedent containing first formula B, then formula A.
  3. The next inference is labeled left negation rule.
  4. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B.
  5. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  6. The next inference is labeled left weakening rule.
  7. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then formula B; sequent arrow; succedent containing formula B.
  8. The next inference is labeled left exchange rule.
  9. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B.
  10. The next inference is labeled left disjunction rule.
  11. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B.
  12. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended.
  13. From the immediately preceding branch using the right exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  14. The next inference is labeled right conditional rule.
  15. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 113.

Proof tree concluding antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 132

Proof tree concluding antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 132. This source proof tree has 18 explicitly ordered components. Its inference labels, in order, are right weakening rule, then right exchange rule, then left negation rule, then left weakening rule, then left exchange rule, then left disjunction rule, then right exchange rule, then right conditional rule. Its last stated proof component is: From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.

  1. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  2. The next inference is labeled right weakening rule.
  3. From the immediately preceding branch using the right weakening rule, infer antecedent containing formula A; sequent arrow; succedent containing first formula A, then formula B.
  4. The next inference is labeled right exchange rule.
  5. From the immediately preceding branch using the right exchange rule, infer antecedent containing formula A; sequent arrow; succedent containing first formula B, then formula A.
  6. The next inference is labeled left negation rule.
  7. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B.
  8. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  9. The next inference is labeled left weakening rule.
  10. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then formula B; sequent arrow; succedent containing formula B.
  11. The next inference is labeled left exchange rule.
  12. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B.
  13. The next inference is labeled left disjunction rule.
  14. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B.
  15. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended.
  16. From the immediately preceding branch using the right exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  17. The next inference is labeled right conditional rule.
  18. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 132.

Example: Give an L K derivation of the sequent antecedent containing the disjunction of the negation ofโ€ฆ โ€” line 154

Example: Give an L K derivation of the sequent antecedent containing the disjunction of the negation ofโ€ฆ โ€” line 154. A complete source-order spoken rendering of this example; it preserves the source claim and does not add a solution or proof.

  1. Give an L K derivation of the sequent antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  2. Using the techniques from above, we start by writing the desired end-sequent at the bottom.
  3. Proof diagram.
  4. An upper premise slot is intentionally blank in this staged diagram.
  5. From the immediately preceding branch, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  6. End proof diagram.
  7. The available main connectives of sentences in the end-sequent are the disjunction symbol and the negation symbol.
  8. It would work to apply either the left disjunction rule or the right negation rule here, but we start with the right negation rule because it avoids splitting up into two branches for a moment: Proof diagram.
  9. An upper premise slot is intentionally blank in this staged diagram.
  10. From the immediately preceding branch, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  11. The next inference is labeled right negation rule.
  12. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  13. End proof diagram.
  14. Now we have a choice of whether to look at the left conjunction rule or the left disjunction rule.
  15. Let's see what happens when we apply the left conjunction rule: we have a choice to start with either the sequent antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing no formulas or the sequent antecedent containing first formula B, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing no formulas.
  16. Since the derivation is symmetric with regards to formula A and formula B, let's go with the former: Proof diagram.
  17. An upper premise slot is intentionally blank in this staged diagram.
  18. From the immediately preceding branch, infer antecedent containing first formula A, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  19. The next inference is labeled left conjunction rule.
  20. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  21. The next inference is labeled right negation rule.
  22. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  23. End proof diagram.
  24. Continuing to fill in the derivation, we see that we run into a problem: Proof diagram.
  25. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  26. The next inference is labeled left negation rule.
  27. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing no formulas.
  28. An upper premise slot is intentionally blank in this staged diagram.
  29. The next inference is labeled question mark placeholder for an inference rule not yet chosen.
  30. From the immediately preceding branch using the question mark placeholder for an inference rule not yet chosen, infer antecedent containing formula A; sequent arrow; succedent containing formula B.
  31. The next inference is labeled left negation rule.
  32. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula B, then formula A; sequent arrow; succedent containing no formulas.
  33. The next inference is labeled left disjunction rule.
  34. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and the negation of formula B, then formula A; sequent arrow; succedent containing no formulas.
  35. The next inference is labeled left exchange rule.
  36. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  37. The next inference is labeled left conjunction rule.
  38. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  39. The next inference is labeled right negation rule.
  40. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  41. End proof diagram.
  42. The top of the right branch cannot be reduced any further, and it cannot be brought by way of structural inferences to an initial sequent, so this is not the right path to take.
  43. So clearly, it was a mistake to apply the left conjunction rule above.
  44. Going back to what we had before and carrying out the left disjunction rule instead, we get Proof diagram.
  45. An upper premise slot is intentionally blank in this staged diagram.
  46. From the immediately preceding branch, infer antecedent containing first the negation of formula A, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  47. An upper premise slot is intentionally blank in this staged diagram.
  48. From the immediately preceding branch, infer antecedent containing first the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  49. The next inference is labeled left disjunction rule.
  50. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  51. The next inference is labeled left exchange rule.
  52. From the immediately preceding branch using the left exchange rule, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  53. The next inference is labeled right negation rule.
  54. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  55. End proof diagram.
  56. Completing each branch as we've done before, we get Proof diagram.
  57. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  58. The next inference is labeled left conjunction rule.
  59. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.
  60. The next inference is labeled left negation rule.
  61. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  62. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  63. The next inference is labeled left conjunction rule.
  64. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula B.
  65. The next inference is labeled left negation rule.
  66. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  67. The next inference is labeled left disjunction rule.
  68. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  69. The next inference is labeled left exchange rule.
  70. From the immediately preceding branch using the left exchange rule, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  71. The next inference is labeled right negation rule.
  72. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  73. End proof diagram.
  74. (We could have carried out the conjunction rules lower than the negation rules in these steps and still obtained a correct derivation).

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 154.

Proof tree concluding antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis โ€” line 160

Proof tree concluding antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis โ€” line 160. This source proof tree has 2 explicitly ordered components. Its inference labels, in order, are none because this is a staged placeholder. Its last stated proof component is: From the immediately preceding branch, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. From the immediately preceding branch, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 160.

Proof tree concluding antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis โ€” line 169

Proof tree concluding antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis โ€” line 169. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are right negation rule. Its last stated proof component is: From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. From the immediately preceding branch, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  3. The next inference is labeled right negation rule.
  4. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 169.

Proof tree concluding antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis โ€” line 181

Proof tree concluding antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis โ€” line 181. This source proof tree has 6 explicitly ordered components. Its inference labels, in order, are left conjunction rule, then right negation rule. Its last stated proof component is: From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. From the immediately preceding branch, infer antecedent containing first formula A, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  3. The next inference is labeled left conjunction rule.
  4. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  5. The next inference is labeled right negation rule.
  6. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 181.

Proof tree concluding antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis โ€” line 190

Proof tree concluding antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis โ€” line 190. This source proof tree has 16 explicitly ordered components. Its inference labels, in order, are left negation rule, then question mark placeholder for an inference rule not yet chosen, then left negation rule, then left disjunction rule, then left exchange rule, then left conjunction rule, then right negation rule. Its last stated proof component is: From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.

  1. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  2. The next inference is labeled left negation rule.
  3. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing no formulas.
  4. An upper premise slot is intentionally blank in this staged diagram.
  5. The next inference is labeled question mark placeholder for an inference rule not yet chosen.
  6. From the immediately preceding branch using the question mark placeholder for an inference rule not yet chosen, infer antecedent containing formula A; sequent arrow; succedent containing formula B.
  7. The next inference is labeled left negation rule.
  8. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula B, then formula A; sequent arrow; succedent containing no formulas.
  9. The next inference is labeled left disjunction rule.
  10. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and the negation of formula B, then formula A; sequent arrow; succedent containing no formulas.
  11. The next inference is labeled left exchange rule.
  12. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  13. The next inference is labeled left conjunction rule.
  14. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  15. The next inference is labeled right negation rule.
  16. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 190.

Proof tree concluding antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis โ€” line 213

Proof tree concluding antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis โ€” line 213. This source proof tree has 10 explicitly ordered components. Its inference labels, in order, are left disjunction rule, then left exchange rule, then right negation rule. Its last stated proof component is: From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. From the immediately preceding branch, infer antecedent containing first the negation of formula A, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  3. An upper premise slot is intentionally blank in this staged diagram.
  4. From the immediately preceding branch, infer antecedent containing first the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  5. The next inference is labeled left disjunction rule.
  6. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  7. The next inference is labeled left exchange rule.
  8. From the immediately preceding branch using the left exchange rule, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  9. The next inference is labeled right negation rule.
  10. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 213.

Proof tree concluding antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis โ€” line 228

Proof tree concluding antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis โ€” line 228. This source proof tree has 16 explicitly ordered components. Its inference labels, in order, are left conjunction rule, then left negation rule, then left conjunction rule, then left negation rule, then left disjunction rule, then left exchange rule, then right negation rule. Its last stated proof component is: From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.

  1. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  2. The next inference is labeled left conjunction rule.
  3. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.
  4. The next inference is labeled left negation rule.
  5. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  6. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  7. The next inference is labeled left conjunction rule.
  8. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula B.
  9. The next inference is labeled left negation rule.
  10. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  11. The next inference is labeled left disjunction rule.
  12. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  13. The next inference is labeled left exchange rule.
  14. From the immediately preceding branch using the left exchange rule, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  15. The next inference is labeled right negation rule.
  16. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 228.

Example: So far we haven't used the contraction rule, but it is sometimes required โ€” line 252

Example: So far we haven't used the contraction rule, but it is sometimes required โ€” line 252. A complete source-order spoken rendering of this example; it preserves the source claim and does not add a solution or proof.

  1. So far we haven't used the contraction rule, but it is sometimes required.
  2. Here's an example where that happens.
  3. Suppose we want to prove antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.
  4. Applying right disjunction rule backwards would give us one of these two derivations: Proof diagram.
  5. An upper premise slot is intentionally blank in this staged diagram.
  6. From the immediately preceding branch, infer antecedent containing no formulas; sequent arrow; succedent containing formula A.
  7. The next inference is labeled right disjunction rule.
  8. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.
  9. The source ends this displayed proof segment here.
  10. An upper premise slot is intentionally blank in this staged diagram.
  11. From the immediately preceding branch, infer antecedent containing formula A; sequent arrow; succedent containing no formulas.
  12. The next inference is labeled right negation rule.
  13. From the immediately preceding branch using the right negation rule, infer antecedent containing no formulas; sequent arrow; succedent containing the negation of formula A.
  14. The next inference is labeled right disjunction rule.
  15. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.
  16. End proof diagram.
  17. Neither of these of course ends in an initial sequent.
  18. The trick is to realize that the contraction rule allows us to combine two copies of a sentence into oneโ€”and when we're searching for a proof, i.e., going from bottom to top, we can keep a copy of the disjunction of formula A and the negation of formula A in the premise, e.g., Proof diagram.
  19. An upper premise slot is intentionally blank in this staged diagram.
  20. From the immediately preceding branch, infer antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then formula A.
  21. The next inference is labeled right disjunction rule.
  22. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then the disjunction of formula A and the negation of formula A.
  23. The next inference is labeled right contraction rule.
  24. From the immediately preceding branch using the right contraction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.
  25. End proof diagram.
  26. Now we can apply right disjunction rule a second time, and also get the negation of formula A, which leads to a complete derivation.
  27. Proof diagram.
  28. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  29. The next inference is labeled right negation rule.
  30. From the immediately preceding branch using the right negation rule, infer antecedent containing no formulas; sequent arrow; succedent containing first formula A, then the negation of formula A.
  31. The next inference is labeled right disjunction rule.
  32. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing first formula A, then the disjunction of formula A and the negation of formula A.
  33. The next inference is labeled right exchange rule.
  34. From the immediately preceding branch using the right exchange rule, infer antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then formula A.
  35. The next inference is labeled right disjunction rule.
  36. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then the disjunction of formula A and the negation of formula A.
  37. The next inference is labeled right contraction rule.
  38. From the immediately preceding branch using the right contraction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.
  39. End proof diagram.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 252.

Proof tree concluding antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A โ€” line 257

Proof tree concluding antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A โ€” line 257. This source proof tree has 11 explicitly ordered components. Its inference labels, in order, are right disjunction rule, then right negation rule, then right disjunction rule. Its last stated proof component is: From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. From the immediately preceding branch, infer antecedent containing no formulas; sequent arrow; succedent containing formula A.
  3. The next inference is labeled right disjunction rule.
  4. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.
  5. The source ends this displayed proof segment here.
  6. An upper premise slot is intentionally blank in this staged diagram.
  7. From the immediately preceding branch, infer antecedent containing formula A; sequent arrow; succedent containing no formulas.
  8. The next inference is labeled right negation rule.
  9. From the immediately preceding branch using the right negation rule, infer antecedent containing no formulas; sequent arrow; succedent containing the negation of formula A.
  10. The next inference is labeled right disjunction rule.
  11. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 257.

Proof tree concluding antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A โ€” line 262

Proof tree concluding antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A โ€” line 262. This source proof tree has 5 explicitly ordered components. Its inference labels, in order, are right disjunction rule. Its last stated proof component is: From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. From the immediately preceding branch, infer antecedent containing no formulas; sequent arrow; succedent containing formula A.
  3. The next inference is labeled right disjunction rule.
  4. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.
  5. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 262.

Proof tree concluding antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A โ€” line 275

Proof tree concluding antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A โ€” line 275. This source proof tree has 6 explicitly ordered components. Its inference labels, in order, are right disjunction rule, then right contraction rule. Its last stated proof component is: From the immediately preceding branch using the right contraction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. From the immediately preceding branch, infer antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then formula A.
  3. The next inference is labeled right disjunction rule.
  4. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then the disjunction of formula A and the negation of formula A.
  5. The next inference is labeled right contraction rule.
  6. From the immediately preceding branch using the right contraction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 275.

Proof tree concluding antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A โ€” line 285

Proof tree concluding antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A โ€” line 285. This source proof tree has 11 explicitly ordered components. Its inference labels, in order, are right negation rule, then right disjunction rule, then right exchange rule, then right disjunction rule, then right contraction rule. Its last stated proof component is: From the immediately preceding branch using the right contraction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.

  1. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  2. The next inference is labeled right negation rule.
  3. From the immediately preceding branch using the right negation rule, infer antecedent containing no formulas; sequent arrow; succedent containing first formula A, then the negation of formula A.
  4. The next inference is labeled right disjunction rule.
  5. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing first formula A, then the disjunction of formula A and the negation of formula A.
  6. The next inference is labeled right exchange rule.
  7. From the immediately preceding branch using the right exchange rule, infer antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then formula A.
  8. The next inference is labeled right disjunction rule.
  9. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then the disjunction of formula A and the negation of formula A.
  10. The next inference is labeled right contraction rule.
  11. From the immediately preceding branch using the right contraction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 285.

Exercise: Give derivations of the following sequents: Next item: antecedent containing the conjunction ofโ€ฆ โ€” line 300

Exercise: Give derivations of the following sequents: Next item: antecedent containing the conjunction ofโ€ฆ โ€” line 300. A complete source-order spoken rendering of this exercise; it preserves the source claim and does not add a solution or proof.

Unsolved source exercise; no solution added.

  1. Give derivations of the following sequents: Next item: antecedent containing the conjunction of formula A and open parenthesis, the conjunction of formula B and formula C, close parenthesis; sequent arrow; succedent containing the conjunction of open parenthesis, the conjunction of formula A and formula B, close parenthesis and formula C.
  2. Next item: antecedent containing the disjunction of formula A and open parenthesis, the disjunction of formula B and formula C, close parenthesis; sequent arrow; succedent containing the disjunction of open parenthesis, the disjunction of formula A and formula B, close parenthesis and formula C.
  3. Next item: antecedent containing the conditional whose antecedent is formula A; and whose consequent is open parenthesis, the conditional whose antecedent is formula B; and whose consequent is formula C, close parenthesis; sequent arrow; succedent containing the conditional whose antecedent is formula B; and whose consequent is open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula C, close parenthesis.
  4. Next item: antecedent containing formula A; sequent arrow; succedent containing the negation of the negation of formula A.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 300.

Exercise: Give derivations of the following sequents: Next item: antecedent containing the conditionalโ€ฆ โ€” line 310

Exercise: Give derivations of the following sequents: Next item: antecedent containing the conditionalโ€ฆ โ€” line 310. A complete source-order spoken rendering of this exercise; it preserves the source claim and does not add a solution or proof.

Unsolved source exercise; no solution added.

  1. Give derivations of the following sequents: Next item: antecedent containing the conditional whose antecedent is open parenthesis, the disjunction of formula A and formula B, close parenthesis; and whose consequent is formula C; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula C.
  2. Next item: antecedent containing the conjunction of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula C, close parenthesis and open parenthesis, the conditional whose antecedent is formula B; and whose consequent is formula C, close parenthesis; sequent arrow; succedent containing the conditional whose antecedent is open parenthesis, the disjunction of formula A and formula B, close parenthesis; and whose consequent is formula C.
  3. Next item: antecedent containing no formulas; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and the negation of formula A, close parenthesis.
  4. Next item: antecedent containing the conditional whose antecedent is formula B; and whose consequent is formula A; sequent arrow; succedent containing the conditional whose antecedent is the negation of formula A; and whose consequent is the negation of formula B.
  5. Next item: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is open parenthesis, the conditional whose antecedent is formula A; and whose consequent is the negation of formula A, close parenthesis; and whose consequent is the negation of formula A.
  6. Next item: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is the negation of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula B, close parenthesis; and whose consequent is the negation of formula B.
  7. Next item: antecedent containing the conditional whose antecedent is formula A; and whose consequent is formula C; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and the negation of formula C, close parenthesis.
  8. Next item: antecedent containing the conjunction of formula A and the negation of formula C; sequent arrow; succedent containing the negation of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula C, close parenthesis.
  9. Next item: antecedent containing first the disjunction of formula A and formula B, then the negation of formula B; sequent arrow; succedent containing formula A.
  10. Next item: antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  11. Next item: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is open parenthesis, the conjunction of the negation of formula A and the negation of formula B, close parenthesis; and whose consequent is the negation of open parenthesis, the disjunction of formula A and formula B, close parenthesis.
  12. Next item: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is the negation of open parenthesis, the disjunction of formula A and formula B, close parenthesis; and whose consequent is open parenthesis, the conjunction of the negation of formula A and the negation of formula B, close parenthesis.

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 310.

Exercise: Give derivations of the following sequents: Next item: antecedent containing the negation ofโ€ฆ โ€” line 328

Exercise: Give derivations of the following sequents: Next item: antecedent containing the negation ofโ€ฆ โ€” line 328. A complete source-order spoken rendering of this exercise; it preserves the source claim and does not add a solution or proof.

Unsolved source exercise; no solution added.

  1. Give derivations of the following sequents: Next item: antecedent containing the negation of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula B, close parenthesis; sequent arrow; succedent containing formula A.
  2. Next item: antecedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis; sequent arrow; succedent containing the disjunction of the negation of formula A and the negation of formula B.
  3. Next item: antecedent containing the conditional whose antecedent is formula A; and whose consequent is formula B; sequent arrow; succedent containing the disjunction of the negation of formula A and formula B.
  4. Next item: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is the negation of the negation of formula A; and whose consequent is formula A.
  5. Next item: antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then the conditional whose antecedent is the negation of formula A; and whose consequent is formula B; sequent arrow; succedent containing formula B.
  6. Next item: antecedent containing the conditional whose antecedent is open parenthesis, the conjunction of formula A and formula B, close parenthesis; and whose consequent is formula C; sequent arrow; succedent containing the disjunction of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula C, close parenthesis and open parenthesis, the conditional whose antecedent is formula B; and whose consequent is formula C, close parenthesis.
  7. Next item: antecedent containing the conditional whose antecedent is open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula B, close parenthesis; and whose consequent is formula A; sequent arrow; succedent containing formula A.
  8. Next item: antecedent containing no formulas; sequent arrow; succedent containing the disjunction of open parenthesis, the conditional whose antecedent is formula A; and whose consequent is formula B, close parenthesis and open parenthesis, the conditional whose antecedent is formula B; and whose consequent is formula C, close parenthesis.
  9. (These all require the right contraction rule.)

Source: content/first-order-logic/sequent-calculus/proving-things.tex, line 328.

Example: Give an L K derivation of the sequent antecedent containing for some variable x, the negationโ€ฆ โ€” line 13

Example: Give an L K derivation of the sequent antecedent containing for some variable x, the negationโ€ฆ โ€” line 13. A complete source-order spoken rendering of this example; it preserves the source claim and does not add a solution or proof.

  1. Give an L K derivation of the sequent antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x..
  2. When dealing with quantifiers, we have to make sure not to violate the eigenvariable condition, and sometimes this requires us to play around with the order of carrying out certain inferences.
  3. In general, it helps to try and take care of rules subject to the eigenvariable condition first (they will be lower down in the finished proof).
  4. Also, it is a good idea to try and look ahead and try to guess what the initial sequent might look like.
  5. In our case, it will have to be something like antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant a.
  6. That means that when we are ``reversing'' the quantifier rules, we will have to pick the same termโ€”what we will call constant a โ€”for both the universal quantifier and the existential quantifier rule.
  7. If we picked different terms for each rule, we would end up with something like antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant b, which, of course, is not derivable..
  8. Starting as usual, we write Proof diagram.
  9. An upper premise slot is intentionally blank in this staged diagram.
  10. From the immediately preceding branch, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  11. End proof diagram.
  12. We could either carry out the left existential quantifier rule or the right negation rule.
  13. Since the left existential quantifier rule is subject to the eigenvariable condition, it's a good idea to take care of it sooner rather than later, so we'll do that one first.
  14. Proof diagram.
  15. An upper premise slot is intentionally blank in this staged diagram.
  16. From the immediately preceding branch, infer antecedent containing the negation of formula A with argument constant a; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  17. The next inference is labeled left existential quantifier rule.
  18. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  19. End proof diagram.
  20. Applying the left negation rule and right negation rules backwards, we get Proof diagram.
  21. An upper premise slot is intentionally blank in this staged diagram.
  22. From the immediately preceding branch, infer antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument constant a.
  23. The next inference is labeled left negation rule.
  24. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A with argument constant a, then for every variable x, formula A with argument variable x; sequent arrow; succedent containing no formulas.
  25. The next inference is labeled left exchange rule.
  26. From the immediately preceding branch using the left exchange rule, infer antecedent containing first for every variable x, formula A with argument variable x, then the negation of formula A with argument constant a; sequent arrow; succedent containing no formulas.
  27. The next inference is labeled right negation rule.
  28. From the immediately preceding branch using the right negation rule, infer antecedent containing the negation of formula A with argument constant a; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  29. The next inference is labeled left existential quantifier rule.
  30. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  31. End proof diagram.
  32. At this point, our only option is to carry out the left universal quantifier rule.
  33. Since this rule is not subject to the eigenvariable restriction, we're in the clear.
  34. Remember, we want to try and obtain an initial sequent (of the form antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant a ), so we should choose constant a as our argument for formula A when we apply the rule.
  35. Proof diagram.
  36. Premise or initial sequent: antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant a.
  37. The next inference is labeled left universal quantifier rule.
  38. From the immediately preceding branch using the left universal quantifier rule, infer antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument constant a.
  39. The next inference is labeled left negation rule.
  40. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A with argument constant a, then for every variable x, formula A with argument variable x; sequent arrow; succedent containing no formulas.
  41. The next inference is labeled left exchange rule.
  42. From the immediately preceding branch using the left exchange rule, infer antecedent containing first for every variable x, formula A with argument variable x, then the negation of formula A with argument constant a; sequent arrow; succedent containing no formulas.
  43. The next inference is labeled right negation rule.
  44. From the immediately preceding branch using the right negation rule, infer antecedent containing the negation of formula A with argument constant a; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  45. The next inference is labeled left existential quantifier rule.
  46. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  47. End proof diagram.
  48. It is important, especially when dealing with quantifiers, to double check at this point that the eigenvariable condition has not been violated.
  49. Since the only rule we applied that is subject to the eigenvariable condition was left existential quantifier rule, and the eigenvariable constant a does not occur in its lower sequent (the end-sequent), this is a correct derivation.

Source: content/first-order-logic/sequent-calculus/proving-things-quant.tex, line 13.

Proof tree concluding antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x โ€” line 32

Proof tree concluding antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x โ€” line 32. This source proof tree has 2 explicitly ordered components. Its inference labels, in order, are none because this is a staged placeholder. Its last stated proof component is: From the immediately preceding branch, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. From the immediately preceding branch, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.

Source: content/first-order-logic/sequent-calculus/proving-things-quant.tex, line 32.

Proof tree concluding antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x โ€” line 40

Proof tree concluding antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x โ€” line 40. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are left existential quantifier rule. Its last stated proof component is: From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. From the immediately preceding branch, infer antecedent containing the negation of formula A with argument constant a; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  3. The next inference is labeled left existential quantifier rule.
  4. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.

Source: content/first-order-logic/sequent-calculus/proving-things-quant.tex, line 40.

Proof tree concluding antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x โ€” line 47

Proof tree concluding antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x โ€” line 47. This source proof tree has 10 explicitly ordered components. Its inference labels, in order, are left negation rule, then left exchange rule, then right negation rule, then left existential quantifier rule. Its last stated proof component is: From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. From the immediately preceding branch, infer antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument constant a.
  3. The next inference is labeled left negation rule.
  4. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A with argument constant a, then for every variable x, formula A with argument variable x; sequent arrow; succedent containing no formulas.
  5. The next inference is labeled left exchange rule.
  6. From the immediately preceding branch using the left exchange rule, infer antecedent containing first for every variable x, formula A with argument variable x, then the negation of formula A with argument constant a; sequent arrow; succedent containing no formulas.
  7. The next inference is labeled right negation rule.
  8. From the immediately preceding branch using the right negation rule, infer antecedent containing the negation of formula A with argument constant a; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  9. The next inference is labeled left existential quantifier rule.
  10. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.

Source: content/first-order-logic/sequent-calculus/proving-things-quant.tex, line 47.

Proof tree concluding antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x โ€” line 64

Proof tree concluding antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x โ€” line 64. This source proof tree has 11 explicitly ordered components. Its inference labels, in order, are left universal quantifier rule, then left negation rule, then left exchange rule, then right negation rule, then left existential quantifier rule. Its last stated proof component is: From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.

  1. Premise or initial sequent: antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant a.
  2. The next inference is labeled left universal quantifier rule.
  3. From the immediately preceding branch using the left universal quantifier rule, infer antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument constant a.
  4. The next inference is labeled left negation rule.
  5. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A with argument constant a, then for every variable x, formula A with argument variable x; sequent arrow; succedent containing no formulas.
  6. The next inference is labeled left exchange rule.
  7. From the immediately preceding branch using the left exchange rule, infer antecedent containing first for every variable x, formula A with argument variable x, then the negation of formula A with argument constant a; sequent arrow; succedent containing no formulas.
  8. The next inference is labeled right negation rule.
  9. From the immediately preceding branch using the right negation rule, infer antecedent containing the negation of formula A with argument constant a; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  10. The next inference is labeled left existential quantifier rule.
  11. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.

Source: content/first-order-logic/sequent-calculus/proving-things-quant.tex, line 64.

Exercise: Give derivations of the following sequents: Next item: antecedent containing no formulas;โ€ฆ โ€” line 85

Exercise: Give derivations of the following sequents: Next item: antecedent containing no formulas;โ€ฆ โ€” line 85. A complete source-order spoken rendering of this exercise; it preserves the source claim and does not add a solution or proof.

Unsolved source exercise; no solution added.

  1. Give derivations of the following sequents: Next item: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is open parenthesis, the conjunction of for every variable x, formula A with argument variable x and for every variable y, formula B with argument variable y, close parenthesis; and whose consequent is for every variable z, open parenthesis, the conjunction of formula A with argument variable z and formula B with argument variable z, close parenthesis.
  2. Next item: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is open parenthesis, the disjunction of for some variable x, formula A with argument variable x and for some variable y, formula B with argument variable y, close parenthesis; and whose consequent is for some variable z, open parenthesis, the disjunction of formula A with argument variable z and formula B with argument variable z, close parenthesis.
  3. Next item: antecedent containing for every variable x, open parenthesis, the conditional whose antecedent is formula A with argument variable x; and whose consequent is formula B, close parenthesis; sequent arrow; succedent containing the conditional whose antecedent is for some variable y, formula A with argument variable y; and whose consequent is formula B.
  4. Next item: antecedent containing for every variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for some variable x, formula A with argument variable x.
  5. Next item: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is the negation of for some variable x, formula A with argument variable x; and whose consequent is for every variable x, the negation of formula A with argument variable x.
  6. Next item: antecedent containing no formulas; sequent arrow; succedent containing the negation of for some variable x, for every variable y, open parenthesis, the conjunction of open parenthesis, the conditional whose antecedent is formula A with arguments variable x and variable y; and whose consequent is the negation of formula A with arguments variable y and variable y, close parenthesis and open parenthesis, the conditional whose antecedent is the negation of formula A with arguments variable y and variable y; and whose consequent is formula A with arguments variable x and variable y, close parenthesis, close parenthesis.

Source: content/first-order-logic/sequent-calculus/proving-things-quant.tex, line 85.

Exercise: Give derivations of the following sequents: Next item: antecedent containing no formulas;โ€ฆ โ€” line 100

Exercise: Give derivations of the following sequents: Next item: antecedent containing no formulas;โ€ฆ โ€” line 100. A complete source-order spoken rendering of this exercise; it preserves the source claim and does not add a solution or proof.

Unsolved source exercise; no solution added.

  1. Give derivations of the following sequents: Next item: antecedent containing no formulas; sequent arrow; succedent containing the conditional whose antecedent is the negation of for every variable x, formula A with argument variable x; and whose consequent is for some variable x, the negation of formula A with argument variable x.
  2. Next item: antecedent containing open parenthesis, the conditional whose antecedent is for every variable x, formula A with argument variable x; and whose consequent is formula B, close parenthesis; sequent arrow; succedent containing for some variable y, open parenthesis, the conditional whose antecedent is formula A with argument variable y; and whose consequent is formula B, close parenthesis.
  3. Next item: antecedent containing no formulas; sequent arrow; succedent containing for some variable x, open parenthesis, the conditional whose antecedent is formula A with argument variable x; and whose consequent is for every variable y, formula A with argument variable y, close parenthesis.
  4. (These all require the right contraction rule.)

Source: content/first-order-logic/sequent-calculus/proving-things-quant.tex, line 100.

Definition: Theorems โ€” line 30

Definition: Theorems โ€” line 30. A complete source-order spoken rendering of this definition; it preserves the source claim and does not add a solution or proof.

  1. A sentence, formula A, is a theorem if there is a derivation in L K of the sequent antecedent containing no formulas; sequent arrow; succedent containing formula A.
  2. We write no premises syntactically derives formula A if formula A is a theorem and no premises does not syntactically derive formula A if it is not.

Source: content/first-order-logic/sequent-calculus/proof-theoretic-notions.tex, line 30.

Definition: derivability โ€” line 36

Definition: derivability โ€” line 36. A complete source-order spoken rendering of this definition; it preserves the source claim and does not add a solution or proof.

  1. A sentence, formula A, is derivable from a set of sentences Gamma, Gamma syntactically derives formula A, if and only if there is a finite subset Gamma sub zero of Gamma and a sequence Gamma sub zero prime of the sentences in Gamma sub zero such that L K derives antecedent containing Gamma sub zero prime; sequent arrow; succedent containing formula A.
  2. If formula A is not derivable from Gamma we write Gamma does not syntactically derive formula A.

Source: content/first-order-logic/sequent-calculus/proof-theoretic-notions.tex, line 36.

Proof tree concluding antecedent containing first formula C, then formula C, and finally formula B; sequent arrow; succedent containing formula A โ€” line 54

Proof tree concluding antecedent containing first formula C, then formula C, and finally formula B; sequent arrow; succedent containing formula A โ€” line 54. This source proof tree has 8 explicitly ordered components. Its inference labels, in order, are left contraction rule, then left exchange rule, then left weakening rule. Its last stated proof component is: From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula C, then formula C, and finally formula B; sequent arrow; succedent containing formula A.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. Previously derived premise with its intervening derivation omitted: antecedent containing first formula B, then formula B, and finally formula C; sequent arrow; succedent containing formula A.
  3. The next inference is labeled left contraction rule.
  4. From the immediately preceding branch using the left contraction rule, infer antecedent containing first formula B, then formula C; sequent arrow; succedent containing formula A.
  5. The next inference is labeled left exchange rule.
  6. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula C, then formula B; sequent arrow; succedent containing formula A.
  7. The next inference is labeled left weakening rule.
  8. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula C, then formula C, and finally formula B; sequent arrow; succedent containing formula A.

Source: content/first-order-logic/sequent-calculus/proof-theoretic-notions.tex, line 54.

Definition: Consistency โ€” line 69

Definition: Consistency โ€” line 69. A complete source-order spoken rendering of this definition; it preserves the source claim and does not add a solution or proof.

  1. A set of sentences Gamma is inconsistent if and only if there is a finite subset Gamma sub zero of Gamma such that L K derives antecedent containing Gamma sub zero; sequent arrow; succedent containing no formulas.
  2. If Gamma is not inconsistent, i.e., if for every finite Gamma sub zero is a subset of Gamma, L K does not derive antecedent containing Gamma sub zero; sequent arrow; succedent containing no formulas, we say it is consistent.

Source: content/first-order-logic/sequent-calculus/proof-theoretic-notions.tex, line 69.

Proposition: Transitivity โ€” line 102

Proposition: Transitivity โ€” line 102. A complete source-order spoken rendering of this proposition; it preserves the source claim and does not add a solution or proof.

  1. If Gamma syntactically derives formula A and the union of the set containing formula A, and capital Delta syntactically derives formula B, then the union of Gamma, and capital Delta syntactically derives formula B.

Source: content/first-order-logic/sequent-calculus/proof-theoretic-notions.tex, line 102.

Proof tree concluding antecedent containing first Gamma sub zero, then capital Delta sub zero; sequent arrow; succedent containing formula B โ€” line 114

Proof tree concluding antecedent containing first Gamma sub zero, then capital Delta sub zero; sequent arrow; succedent containing formula B โ€” line 114. This source proof tree has 8 explicitly ordered components. Its inference labels, in order, are derivation pi sub zero, then derivation pi sub one, then cut rule. Its last stated proof component is: From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma sub zero, then capital Delta sub zero; sequent arrow; succedent containing formula B.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. The next inference is labeled derivation pi sub zero.
  3. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma sub zero; sequent arrow; succedent containing formula A.
  4. An upper premise slot is intentionally blank in this staged diagram.
  5. The next inference is labeled derivation pi sub one.
  6. Previously derived premise with its intervening derivation omitted: antecedent containing first formula A, then capital Delta sub zero; sequent arrow; succedent containing formula B.
  7. The next inference is labeled cut rule.
  8. From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma sub zero, then capital Delta sub zero; sequent arrow; succedent containing formula B.

Source: content/first-order-logic/sequent-calculus/proof-theoretic-notions.tex, line 114.

Proposition: Gamma is inconsistent if and only if Gamma syntactically derives formula A for every sentenceโ€ฆ โ€” line 133

Proposition: Gamma is inconsistent if and only if Gamma syntactically derives formula A for every sentenceโ€ฆ โ€” line 133. A complete source-order spoken rendering of this proposition; it preserves the source claim and does not add a solution or proof.

  1. Gamma is inconsistent if and only if Gamma syntactically derives formula A for every sentence formula A.

Source: content/first-order-logic/sequent-calculus/proof-theoretic-notions.tex, line 133.

Exercise: Prove the proposition that Gamma is inconsistent if and only if Gamma syntactically derivesโ€ฆ โ€” line 143

Exercise: Prove the proposition that Gamma is inconsistent if and only if Gamma syntactically derivesโ€ฆ โ€” line 143. A complete source-order spoken rendering of this exercise; it preserves the source claim and does not add a solution or proof.

Unsolved source exercise; no solution added.

  1. Prove the proposition that Gamma is inconsistent if and only if Gamma syntactically derives every sentence

Source: content/first-order-logic/sequent-calculus/proof-theoretic-notions.tex, line 143.

Proposition: Compactness โ€” line 147

Proposition: Compactness โ€” line 147. A complete source-order spoken rendering of this proposition; it preserves the source claim and does not add a solution or proof.

  1. Next item: If Gamma syntactically derives formula A then there is a finite subset Gamma sub zero of Gamma such that Gamma sub zero syntactically derives formula A.
  2. Next item: If every finite subset of Gamma is consistent, then Gamma is consistent.

Source: content/first-order-logic/sequent-calculus/proof-theoretic-notions.tex, line 147.

Proposition: If Gamma syntactically derives formula A and the union of Gamma, and the set containing formulaโ€ฆ โ€” line 19

Proposition: If Gamma syntactically derives formula A and the union of Gamma, and the set containing formulaโ€ฆ โ€” line 19. A complete source-order spoken rendering of this proposition; it preserves the source claim and does not add a solution or proof.

  1. If Gamma syntactically derives formula A and the union of Gamma, and the set containing formula A is inconsistent, then Gamma is inconsistent.

Source: content/first-order-logic/sequent-calculus/provability-consistency.tex, line 19.

Proof tree concluding antecedent containing first Gamma sub zero, then Gamma sub one; sequent arrow; succedent containing no formulas โ€” line 30

Proof tree concluding antecedent containing first Gamma sub zero, then Gamma sub one; sequent arrow; succedent containing no formulas โ€” line 30. This source proof tree has 8 explicitly ordered components. Its inference labels, in order, are derivation pi sub zero, then derivation pi sub one, then cut rule. Its last stated proof component is: From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma sub zero, then Gamma sub one; sequent arrow; succedent containing no formulas.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. The next inference is labeled derivation pi sub zero.
  3. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma sub zero; sequent arrow; succedent containing formula A.
  4. An upper premise slot is intentionally blank in this staged diagram.
  5. The next inference is labeled derivation pi sub one.
  6. Previously derived premise with its intervening derivation omitted: antecedent containing first formula A, then Gamma sub one; sequent arrow; succedent containing no formulas.
  7. The next inference is labeled cut rule.
  8. From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma sub zero, then Gamma sub one; sequent arrow; succedent containing no formulas.

Source: content/first-order-logic/sequent-calculus/provability-consistency.tex, line 30.

Proposition: Gamma syntactically derives formula A if and only if the union of Gamma, and the set containingโ€ฆ โ€” line 45

Proposition: Gamma syntactically derives formula A if and only if the union of Gamma, and the set containingโ€ฆ โ€” line 45. A complete source-order spoken rendering of this proposition; it preserves the source claim and does not add a solution or proof.

  1. Gamma syntactically derives formula A if and only if the union of Gamma, and the set containing the negation of formula A is inconsistent.

Source: content/first-order-logic/sequent-calculus/provability-consistency.tex, line 45.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing formula A โ€” line 59

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing formula A โ€” line 59. This source proof tree has 8 explicitly ordered components. Its inference labels, in order, are right negation rule, then derivation pi sub one, then cut rule. Its last stated proof component is: From the two immediately preceding branches using the cut rule, infer antecedent containing Gamma; sequent arrow; succedent containing formula A.

  1. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  2. The next inference is labeled right negation rule.
  3. From the immediately preceding branch using the right negation rule, infer antecedent containing no formulas; sequent arrow; succedent containing first formula A, then the negation of formula A.
  4. An upper premise slot is intentionally blank in this staged diagram.
  5. The next inference is labeled derivation pi sub one.
  6. Previously derived premise with its intervening derivation omitted: antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing no formulas.
  7. The next inference is labeled cut rule.
  8. From the two immediately preceding branches using the cut rule, infer antecedent containing Gamma; sequent arrow; succedent containing formula A.

Source: content/first-order-logic/sequent-calculus/provability-consistency.tex, line 59.

Exercise: Prove that Gamma syntactically derives the negation of formula A if and only if the union ofโ€ฆ โ€” line 71

Exercise: Prove that Gamma syntactically derives the negation of formula A if and only if the union ofโ€ฆ โ€” line 71. A complete source-order spoken rendering of this exercise; it preserves the source claim and does not add a solution or proof.

Unsolved source exercise; no solution added.

  1. Prove that Gamma syntactically derives the negation of formula A if and only if the union of Gamma, and the set containing formula A is inconsistent.

Source: content/first-order-logic/sequent-calculus/provability-consistency.tex, line 71.

Proposition: If Gamma syntactically derives formula A and the negation of formula A is a member of Gamma,โ€ฆ โ€” line 75

Proposition: If Gamma syntactically derives formula A and the negation of formula A is a member of Gamma,โ€ฆ โ€” line 75. A complete source-order spoken rendering of this proposition; it preserves the source claim and does not add a solution or proof.

  1. If Gamma syntactically derives formula A and the negation of formula A is a member of Gamma, then Gamma is inconsistent.

Source: content/first-order-logic/sequent-calculus/provability-consistency.tex, line 75.

Proof tree concluding antecedent containing first Gamma sub zero, then the negation of formula A; sequent arrow; succedent containing no formulas โ€” line 84

Proof tree concluding antecedent containing first Gamma sub zero, then the negation of formula A; sequent arrow; succedent containing no formulas โ€” line 84. This source proof tree has 10 explicitly ordered components. Its inference labels, in order, are derivation pi, then left negation rule, then left exchange rule, then cut rule. Its last stated proof component is: From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma sub zero, then the negation of formula A; sequent arrow; succedent containing no formulas.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. The next inference is labeled derivation pi.
  3. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma sub zero; sequent arrow; succedent containing formula A.
  4. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  5. The next inference is labeled left negation rule.
  6. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing no formulas.
  7. The next inference is labeled left exchange rule.
  8. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula A, then the negation of formula A; sequent arrow; succedent containing no formulas.
  9. The next inference is labeled cut rule.
  10. From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma sub zero, then the negation of formula A; sequent arrow; succedent containing no formulas.

Source: content/first-order-logic/sequent-calculus/provability-consistency.tex, line 84.

Proposition: If the union of Gamma, and the set containing formula A and the union of Gamma, and the setโ€ฆ โ€” line 100

Proposition: If the union of Gamma, and the set containing formula A and the union of Gamma, and the setโ€ฆ โ€” line 100. A complete source-order spoken rendering of this proposition; it preserves the source claim and does not add a solution or proof.

  1. If the union of Gamma, and the set containing formula A and the union of Gamma, and the set containing the negation of formula A are both inconsistent, then Gamma is inconsistent.

Source: content/first-order-logic/sequent-calculus/provability-consistency.tex, line 100.

Proof tree concluding antecedent containing first Gamma sub zero, then Gamma sub one; sequent arrow; succedent containing no formulas โ€” line 110

Proof tree concluding antecedent containing first Gamma sub zero, then Gamma sub one; sequent arrow; succedent containing no formulas โ€” line 110. This source proof tree has 10 explicitly ordered components. Its inference labels, in order, are derivation pi sub zero, then right negation rule, then derivation pi sub one, then cut rule. Its last stated proof component is: From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma sub zero, then Gamma sub one; sequent arrow; succedent containing no formulas.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. The next inference is labeled derivation pi sub zero.
  3. Previously derived premise with its intervening derivation omitted: antecedent containing first formula A, then Gamma sub zero; sequent arrow; succedent containing no formulas.
  4. The next inference is labeled right negation rule.
  5. From the immediately preceding branch using the right negation rule, infer antecedent containing Gamma sub zero; sequent arrow; succedent containing the negation of formula A.
  6. An upper premise slot is intentionally blank in this staged diagram.
  7. The next inference is labeled derivation pi sub one.
  8. Previously derived premise with its intervening derivation omitted: antecedent containing first the negation of formula A, then Gamma sub one; sequent arrow; succedent containing no formulas.
  9. The next inference is labeled cut rule.
  10. From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma sub zero, then Gamma sub one; sequent arrow; succedent containing no formulas.

Source: content/first-order-logic/sequent-calculus/provability-consistency.tex, line 110.

Proposition: Next item: Both the conjunction of formula A and formula B syntactically derives formula A andโ€ฆ โ€” line 26

Proposition: Next item: Both the conjunction of formula A and formula B syntactically derives formula A andโ€ฆ โ€” line 26. A complete source-order spoken rendering of this proposition; it preserves the source claim and does not add a solution or proof.

  1. Next item: Both the conjunction of formula A and formula B syntactically derives formula A and the conjunction of formula A and formula B syntactically derives formula B.
  2. Next item: first formula A, then formula B syntactically derives the conjunction of formula A and formula B.

Source: content/first-order-logic/sequent-calculus/provability-propositional.tex, line 26.

Proof tree concluding antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula B โ€” line 39

Proof tree concluding antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula B โ€” line 39. This source proof tree has 7 explicitly ordered components. Its inference labels, in order, are left conjunction rule, then left conjunction rule. Its last stated proof component is: From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula B.

  1. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  2. The next inference is labeled left conjunction rule.
  3. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.
  4. The source ends this displayed proof segment here.
  5. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  6. The next inference is labeled left conjunction rule.
  7. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula B.

Source: content/first-order-logic/sequent-calculus/provability-propositional.tex, line 39.

Proof tree concluding antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A โ€” line 43

Proof tree concluding antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A โ€” line 43. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are left conjunction rule. Its last stated proof component is: From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.

  1. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  2. The next inference is labeled left conjunction rule.
  3. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/provability-propositional.tex, line 43.

Proof tree concluding antecedent containing first formula A, then formula B; sequent arrow; succedent containing the conjunction of formula A and formula B โ€” line 49

Proof tree concluding antecedent containing first formula A, then formula B; sequent arrow; succedent containing the conjunction of formula A and formula B โ€” line 49. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are right conjunction rule. Its last stated proof component is: From the two immediately preceding branches using the right conjunction rule, infer antecedent containing first formula A, then formula B; sequent arrow; succedent containing the conjunction of formula A and formula B.

  1. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  2. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  3. The next inference is labeled right conjunction rule.
  4. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing first formula A, then formula B; sequent arrow; succedent containing the conjunction of formula A and formula B.

Source: content/first-order-logic/sequent-calculus/provability-propositional.tex, line 49.

Proposition: Next item: first the disjunction of formula A and formula B, then the negation of formula A,โ€ฆ โ€” line 58

Proposition: Next item: first the disjunction of formula A and formula B, then the negation of formula A,โ€ฆ โ€” line 58. A complete source-order spoken rendering of this proposition; it preserves the source claim and does not add a solution or proof.

  1. Next item: first the disjunction of formula A and formula B, then the negation of formula A, and finally the negation of formula B is inconsistent.
  2. Next item: Both formula A syntactically derives the disjunction of formula A and formula B and formula B syntactically derives the disjunction of formula A and formula B.

Source: content/first-order-logic/sequent-calculus/provability-propositional.tex, line 58.

Proof tree concluding antecedent containing first the disjunction of formula A and formula B, then the negation of formula A, and finally the negation of formula B; sequent arrow; succedent containing no formulas โ€” line 69

Proof tree concluding antecedent containing first the disjunction of formula A and formula B, then the negation of formula A, and finally the negation of formula B; sequent arrow; succedent containing no formulas โ€” line 69. This source proof tree has 12 explicitly ordered components. Its inference labels, in order, are left negation rule, then left negation rule, then left disjunction rule. Its last stated proof component is: From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of formula A and formula B, then the negation of formula A, and finally the negation of formula B; sequent arrow; succedent containing no formulas.

  1. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  2. The next inference is labeled left negation rule.
  3. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing no formulas.
  4. A double inference line abbreviates one or more weakening, contraction, or exchange steps.
  5. From the immediately preceding branch, infer antecedent containing first formula A, then the negation of formula A, and finally the negation of formula B; sequent arrow; succedent containing no formulas.
  6. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  7. The next inference is labeled left negation rule.
  8. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula B, then formula B; sequent arrow; succedent containing no formulas.
  9. A double inference line abbreviates one or more weakening, contraction, or exchange steps.
  10. From the immediately preceding branch, infer antecedent containing first formula B, then the negation of formula A, and finally the negation of formula B; sequent arrow; succedent containing no formulas.
  11. The next inference is labeled left disjunction rule.
  12. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of formula A and formula B, then the negation of formula A, and finally the negation of formula B; sequent arrow; succedent containing no formulas.

Source: content/first-order-logic/sequent-calculus/provability-propositional.tex, line 69.

Proof tree concluding antecedent containing formula B; sequent arrow; succedent containing the disjunction of formula A and formula B โ€” line 87

Proof tree concluding antecedent containing formula B; sequent arrow; succedent containing the disjunction of formula A and formula B โ€” line 87. This source proof tree has 7 explicitly ordered components. Its inference labels, in order, are right disjunction rule, then right disjunction rule. Its last stated proof component is: From the immediately preceding branch using the right disjunction rule, infer antecedent containing formula B; sequent arrow; succedent containing the disjunction of formula A and formula B.

  1. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  2. The next inference is labeled right disjunction rule.
  3. From the immediately preceding branch using the right disjunction rule, infer antecedent containing formula A; sequent arrow; succedent containing the disjunction of formula A and formula B.
  4. The source ends this displayed proof segment here.
  5. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  6. The next inference is labeled right disjunction rule.
  7. From the immediately preceding branch using the right disjunction rule, infer antecedent containing formula B; sequent arrow; succedent containing the disjunction of formula A and formula B.

Source: content/first-order-logic/sequent-calculus/provability-propositional.tex, line 87.

Proof tree concluding antecedent containing formula A; sequent arrow; succedent containing the disjunction of formula A and formula B โ€” line 91

Proof tree concluding antecedent containing formula A; sequent arrow; succedent containing the disjunction of formula A and formula B โ€” line 91. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are right disjunction rule. Its last stated proof component is: From the immediately preceding branch using the right disjunction rule, infer antecedent containing formula A; sequent arrow; succedent containing the disjunction of formula A and formula B.

  1. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  2. The next inference is labeled right disjunction rule.
  3. From the immediately preceding branch using the right disjunction rule, infer antecedent containing formula A; sequent arrow; succedent containing the disjunction of formula A and formula B.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/provability-propositional.tex, line 91.

Proposition: Next item: first formula A, then the conditional whose antecedent is formula A; and whoseโ€ฆ โ€” line 99

Proposition: Next item: first formula A, then the conditional whose antecedent is formula A; and whoseโ€ฆ โ€” line 99. A complete source-order spoken rendering of this proposition; it preserves the source claim and does not add a solution or proof.

  1. Next item: first formula A, then the conditional whose antecedent is formula A; and whose consequent is formula B syntactically derives formula B.
  2. Next item: Both the negation of formula A syntactically derives the conditional whose antecedent is formula A; and whose consequent is formula B and formula B syntactically derives the conditional whose antecedent is formula A; and whose consequent is formula B.

Source: content/first-order-logic/sequent-calculus/provability-propositional.tex, line 99.

Proof tree concluding antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then formula A; sequent arrow; succedent containing formula B โ€” line 110

Proof tree concluding antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then formula A; sequent arrow; succedent containing formula B โ€” line 110. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are left conditional rule. Its last stated proof component is: From the two immediately preceding branches using the left conditional rule, infer antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then formula A; sequent arrow; succedent containing formula B.

  1. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  2. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  3. The next inference is labeled left conditional rule.
  4. From the two immediately preceding branches using the left conditional rule, infer antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then formula A; sequent arrow; succedent containing formula B.

Source: content/first-order-logic/sequent-calculus/provability-propositional.tex, line 110.

Proof tree concluding antecedent containing formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 118

Proof tree concluding antecedent containing formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 118. This source proof tree has 15 explicitly ordered components. Its inference labels, in order, are left negation rule, then left exchange rule, then right weakening rule, then right conditional rule, then left weakening rule, then right conditional rule. Its last stated proof component is: From the immediately preceding branch using the right conditional rule, infer antecedent containing formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.

  1. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  2. The next inference is labeled left negation rule.
  3. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing no formulas.
  4. The next inference is labeled left exchange rule.
  5. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula A, then the negation of formula A; sequent arrow; succedent containing no formulas.
  6. The next inference is labeled right weakening rule.
  7. From the immediately preceding branch using the right weakening rule, infer antecedent containing first formula A, then the negation of formula A; sequent arrow; succedent containing formula B.
  8. The next inference is labeled right conditional rule.
  9. From the immediately preceding branch using the right conditional rule, infer antecedent containing the negation of formula A; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  10. The source ends this displayed proof segment here.
  11. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  12. The next inference is labeled left weakening rule.
  13. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then formula B; sequent arrow; succedent containing formula B.
  14. The next inference is labeled right conditional rule.
  15. From the immediately preceding branch using the right conditional rule, infer antecedent containing formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.

Source: content/first-order-logic/sequent-calculus/provability-propositional.tex, line 118.

Proof tree concluding antecedent containing the negation of formula A; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 128

Proof tree concluding antecedent containing the negation of formula A; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 128. This source proof tree has 10 explicitly ordered components. Its inference labels, in order, are left negation rule, then left exchange rule, then right weakening rule, then right conditional rule. Its last stated proof component is: From the immediately preceding branch using the right conditional rule, infer antecedent containing the negation of formula A; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.

  1. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  2. The next inference is labeled left negation rule.
  3. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing no formulas.
  4. The next inference is labeled left exchange rule.
  5. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula A, then the negation of formula A; sequent arrow; succedent containing no formulas.
  6. The next inference is labeled right weakening rule.
  7. From the immediately preceding branch using the right weakening rule, infer antecedent containing first formula A, then the negation of formula A; sequent arrow; succedent containing formula B.
  8. The next inference is labeled right conditional rule.
  9. From the immediately preceding branch using the right conditional rule, infer antecedent containing the negation of formula A; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  10. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/provability-propositional.tex, line 128.

Theorem: If constant c is a constant not occurring in Gamma or formula A with argument variable x andโ€ฆ โ€” line 18

Theorem: If constant c is a constant not occurring in Gamma or formula A with argument variable x andโ€ฆ โ€” line 18. A complete source-order spoken rendering of this theorem; it preserves the source claim and does not add a solution or proof.

  1. If constant c is a constant not occurring in Gamma or formula A with argument variable x and Gamma syntactically derives formula A with argument constant c, then Gamma syntactically derives for every variable x, formula A with argument variable x.

Source: content/first-order-logic/sequent-calculus/provability-quantifiers.tex, line 18.

Proposition: prvEx,prvAll formula A with argument term t syntactically derives for some variable x, formulaโ€ฆ โ€” line 32

Proposition: prvEx,prvAll formula A with argument term t syntactically derives for some variable x, formulaโ€ฆ โ€” line 32. A complete source-order spoken rendering of this proposition; it preserves the source claim and does not add a solution or proof.

  1. prvEx,prvAll formula A with argument term t syntactically derives for some variable x, formula A with argument variable x.
  2. for every variable x, formula A with argument variable x syntactically derives formula A with argument term t.

Source: content/first-order-logic/sequent-calculus/provability-quantifiers.tex, line 32.

Proof tree concluding antecedent containing formula A with argument term t; sequent arrow; succedent containing for some variable x, formula A with argument variable x โ€” line 44

Proof tree concluding antecedent containing formula A with argument term t; sequent arrow; succedent containing for some variable x, formula A with argument variable x โ€” line 44. This source proof tree has 3 explicitly ordered components. Its inference labels, in order, are right existential quantifier rule. Its last stated proof component is: From the immediately preceding branch using the right existential quantifier rule, infer antecedent containing formula A with argument term t; sequent arrow; succedent containing for some variable x, formula A with argument variable x.

  1. Premise or initial sequent: antecedent containing formula A with argument term t; sequent arrow; succedent containing formula A with argument term t.
  2. The next inference is labeled right existential quantifier rule.
  3. From the immediately preceding branch using the right existential quantifier rule, infer antecedent containing formula A with argument term t; sequent arrow; succedent containing for some variable x, formula A with argument variable x.

Source: content/first-order-logic/sequent-calculus/provability-quantifiers.tex, line 44.

Proof tree concluding antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument term t โ€” line 51

Proof tree concluding antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument term t โ€” line 51. This source proof tree has 3 explicitly ordered components. Its inference labels, in order, are left universal quantifier rule. Its last stated proof component is: From the immediately preceding branch using the left universal quantifier rule, infer antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument term t.

  1. Premise or initial sequent: antecedent containing formula A with argument term t; sequent arrow; succedent containing formula A with argument term t.
  2. The next inference is labeled left universal quantifier rule.
  3. From the immediately preceding branch using the left universal quantifier rule, infer antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument term t.

Source: content/first-order-logic/sequent-calculus/provability-quantifiers.tex, line 51.

Definition: A structure M satisfies a sequent antecedent containing Gamma; sequent arrow;โ€ฆ โ€” line 41

Definition: A structure M satisfies a sequent antecedent containing Gamma; sequent arrow;โ€ฆ โ€” line 41. A complete source-order spoken rendering of this definition; it preserves the source claim and does not add a solution or proof.

  1. A structure M satisfies a sequent antecedent containing Gamma; sequent arrow; succedent containing capital Delta if and only if either structure M does not satisfy formula A for some formula A in Gamma or structure M satisfies formula A for some formula A in capital Delta..
  2. A sequent is valid if and only if every structure M satisfies it.

Source: content/first-order-logic/sequent-calculus/soundness.tex, line 41.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A โ€” line 79

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A โ€” line 79. This source proof tree has 9 explicitly ordered components. Its inference labels, in order, are left weakening rule, then right weakening rule. Its last stated proof component is: From the immediately preceding branch using the right weakening rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  3. The next inference is labeled left weakening rule.
  4. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  5. The source ends this displayed proof segment here.
  6. An upper premise slot is intentionally blank in this staged diagram.
  7. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  8. The next inference is labeled right weakening rule.
  9. From the immediately preceding branch using the right weakening rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.

Source: content/first-order-logic/sequent-calculus/soundness.tex, line 79.

Proof tree concluding antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 84

Proof tree concluding antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 84. This source proof tree has 5 explicitly ordered components. Its inference labels, in order, are left weakening rule. Its last stated proof component is: From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  3. The next inference is labeled left weakening rule.
  4. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  5. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/soundness.tex, line 84.

Proof tree concluding antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 106

Proof tree concluding antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 106. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are left negation rule. Its last stated proof component is: From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing capital Delta.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  3. The next inference is labeled left negation rule.
  4. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing capital Delta.

Source: content/first-order-logic/sequent-calculus/soundness.tex, line 106.

Proof tree concluding antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 135

Proof tree concluding antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 135. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are left conjunction rule. Its last stated proof component is: From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. Previously derived premise with its intervening derivation omitted: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  3. The next inference is labeled left conjunction rule.
  4. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.

Source: content/first-order-logic/sequent-calculus/soundness.tex, line 135.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B โ€” line 163

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B โ€” line 163. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are right disjunction rule. Its last stated proof component is: From the immediately preceding branch using the right disjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  3. The next inference is labeled right disjunction rule.
  4. From the immediately preceding branch using the right disjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B.

Source: content/first-order-logic/sequent-calculus/soundness.tex, line 163.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 185

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B โ€” line 185. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are right conditional rule. Its last stated proof component is: From the immediately preceding branch using the right conditional rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. Previously derived premise with its intervening derivation omitted: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  3. The next inference is labeled right conditional rule.
  4. From the immediately preceding branch using the right conditional rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B.

Source: content/first-order-logic/sequent-calculus/soundness.tex, line 185.

Proof tree concluding antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 210

Proof tree concluding antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta โ€” line 210. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are left universal quantifier rule. Its last stated proof component is: From the immediately preceding branch using the left universal quantifier rule, infer antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. Previously derived premise with its intervening derivation omitted: antecedent containing first formula A with argument term t, then Gamma; sequent arrow; succedent containing capital Delta.
  3. The next inference is labeled left universal quantifier rule.
  4. From the immediately preceding branch using the left universal quantifier rule, infer antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta.

Source: content/first-order-logic/sequent-calculus/soundness.tex, line 210.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for every variable x, formula A with argument variable x โ€” line 230

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for every variable x, formula A with argument variable x โ€” line 230. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are right universal quantifier rule. Its last stated proof component is: From the immediately preceding branch using the right universal quantifier rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for every variable x, formula A with argument variable x.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument constant a.
  3. The next inference is labeled right universal quantifier rule.
  4. From the immediately preceding branch using the right universal quantifier rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for every variable x, formula A with argument variable x.

Source: content/first-order-logic/sequent-calculus/soundness.tex, line 230.

Proof tree concluding antecedent containing first Gamma, then capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda โ€” line 271

Proof tree concluding antecedent containing first Gamma, then capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda โ€” line 271. This source proof tree has 6 explicitly ordered components. Its inference labels, in order, are cut rule. Its last stated proof component is: From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma, then capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  3. An upper premise slot is intentionally blank in this staged diagram.
  4. Previously derived premise with its intervening derivation omitted: antecedent containing first formula A, then capital Pi; sequent arrow; succedent containing capital Lambda.
  5. The next inference is labeled cut rule.
  6. From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma, then capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda.

Source: content/first-order-logic/sequent-calculus/soundness.tex, line 271.

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B โ€” line 291

Proof tree concluding antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B โ€” line 291. This source proof tree has 6 explicitly ordered components. Its inference labels, in order, are right conjunction rule. Its last stated proof component is: From the two immediately preceding branches using the right conjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  3. An upper premise slot is intentionally blank in this staged diagram.
  4. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  5. The next inference is labeled right conjunction rule.
  6. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B.

Source: content/first-order-logic/sequent-calculus/soundness.tex, line 291.

Proof tree concluding antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then Gamma, and finally capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda โ€” line 310

Proof tree concluding antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then Gamma, and finally capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda โ€” line 310. This source proof tree has 6 explicitly ordered components. Its inference labels, in order, are left conditional rule. Its last stated proof component is: From the two immediately preceding branches using the left conditional rule, infer antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then Gamma, and finally capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda.

  1. An upper premise slot is intentionally blank in this staged diagram.
  2. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  3. An upper premise slot is intentionally blank in this staged diagram.
  4. Previously derived premise with its intervening derivation omitted: antecedent containing first formula B, then capital Pi; sequent arrow; succedent containing capital Lambda.
  5. The next inference is labeled left conditional rule.
  6. From the two immediately preceding branches using the left conditional rule, infer antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then Gamma, and finally capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda.

Source: content/first-order-logic/sequent-calculus/soundness.tex, line 310.

Corollary: If Gamma syntactically derives formula A then Gamma semantically entails formula A โ€” line 351

Corollary: If Gamma syntactically derives formula A then Gamma semantically entails formula A โ€” line 351. A complete source-order spoken rendering of this corollary; it preserves the source claim and does not add a solution or proof.

  1. If Gamma syntactically derives formula A then Gamma semantically entails formula A.

Source: content/first-order-logic/sequent-calculus/soundness.tex, line 351.

Definition of derivation with identity sequent rules โ€” line 22

Definition of derivation with identity sequent rules โ€” line 22. This rule definition gives every premise, inference label, and conclusion in source order for derivation with identity.

  1. Premise or initial sequent: antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub one.
  2. The next inference is labeled identity rule.
  3. From the immediately preceding branch using the identity rule, infer antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub two.
  4. The source ends this displayed proof segment here.
  5. Premise or initial sequent: antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub two.
  6. The next inference is labeled identity rule.
  7. From the immediately preceding branch using the identity rule, infer antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub one.
  8. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/identity.tex, line 22.

Proof tree concluding antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub two โ€” line 26

Proof tree concluding antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub two โ€” line 26. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are identity rule. Its last stated proof component is: From the immediately preceding branch using the identity rule, infer antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub two.

  1. Premise or initial sequent: antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub one.
  2. The next inference is labeled identity rule.
  3. From the immediately preceding branch using the identity rule, infer antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub two.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/identity.tex, line 26.

Proof tree concluding antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub one โ€” line 31

Proof tree concluding antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub one โ€” line 31. This source proof tree has 4 explicitly ordered components. Its inference labels, in order, are identity rule. Its last stated proof component is: From the immediately preceding branch using the identity rule, infer antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub one.

  1. Premise or initial sequent: antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub two.
  2. The next inference is labeled identity rule.
  3. From the immediately preceding branch using the identity rule, infer antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub one.
  4. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/identity.tex, line 31.

Example: If lowercase s and term t are closed terms, then first lowercase s is identical to term t, thenโ€ฆ โ€” line 34

Example: If lowercase s and term t are closed terms, then first lowercase s is identical to term t, thenโ€ฆ โ€” line 34. A complete source-order spoken rendering of this example; it preserves the source claim and does not add a solution or proof.

  1. If lowercase s and term t are closed terms, then first lowercase s is identical to term t, then formula A with argument lowercase s syntactically derives formula A with argument term t: Proof diagram.
  2. Premise or initial sequent: antecedent containing formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument lowercase s.
  3. The next inference is labeled left weakening rule.
  4. From the immediately preceding branch using the left weakening rule, infer antecedent containing first lowercase s is identical to term t, then formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument lowercase s.
  5. The next inference is labeled identity rule.
  6. From the immediately preceding branch using the identity rule, infer antecedent containing first lowercase s is identical to term t, then formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument term t.
  7. End proof diagram.
  8. This may be familiar as the principle of substitutability of identicals, or Leibniz' Law..
  9. L K proves that identity sign is symmetric and transitive: Proof diagram.
  10. Premise or initial sequent: antecedent containing no formulas; sequent arrow; succedent containing term t sub one is identical to term t sub one.
  11. The next inference is labeled left weakening rule.
  12. From the immediately preceding branch using the left weakening rule, infer antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub one.
  13. The next inference is labeled identity rule.
  14. From the immediately preceding branch using the identity rule, infer antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub two is identical to term t sub one.
  15. The source ends this displayed proof segment here.
  16. Premise or initial sequent: antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub two.
  17. The next inference is labeled left weakening rule.
  18. From the immediately preceding branch using the left weakening rule, infer antecedent containing first term t sub two is identical to term t sub three, then term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub two.
  19. The next inference is labeled identity rule.
  20. From the immediately preceding branch using the identity rule, infer antecedent containing first term t sub two is identical to term t sub three, then term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub three.
  21. The next inference is labeled left exchange rule.
  22. From the immediately preceding branch using the left exchange rule, infer antecedent containing first term t sub one is identical to term t sub two, then term t sub two is identical to term t sub three; sequent arrow; succedent containing term t sub one is identical to term t sub three.
  23. End proof diagram.
  24. In the derivation on the left, the formula variable x is identical to term t sub one is our formula A with argument variable x.
  25. On the right, we take formula A with argument variable x to be term t sub one is identical to variable x.

Source: content/first-order-logic/sequent-calculus/identity.tex, line 34.

Proof tree concluding antecedent containing first lowercase s is identical to term t, then formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument term t โ€” line 37

Proof tree concluding antecedent containing first lowercase s is identical to term t, then formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument term t โ€” line 37. This source proof tree has 5 explicitly ordered components. Its inference labels, in order, are left weakening rule, then identity rule. Its last stated proof component is: From the immediately preceding branch using the identity rule, infer antecedent containing first lowercase s is identical to term t, then formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument term t.

  1. Premise or initial sequent: antecedent containing formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument lowercase s.
  2. The next inference is labeled left weakening rule.
  3. From the immediately preceding branch using the left weakening rule, infer antecedent containing first lowercase s is identical to term t, then formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument lowercase s.
  4. The next inference is labeled identity rule.
  5. From the immediately preceding branch using the identity rule, infer antecedent containing first lowercase s is identical to term t, then formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument term t.

Source: content/first-order-logic/sequent-calculus/identity.tex, line 37.

Proof tree concluding antecedent containing first term t sub one is identical to term t sub two, then term t sub two is identical to term t sub three; sequent arrow; succedent containing term t sub one is identical to term t sub three โ€” line 48

Proof tree concluding antecedent containing first term t sub one is identical to term t sub two, then term t sub two is identical to term t sub three; sequent arrow; succedent containing term t sub one is identical to term t sub three โ€” line 48. This source proof tree has 13 explicitly ordered components. Its inference labels, in order, are left weakening rule, then identity rule, then left weakening rule, then identity rule, then left exchange rule. Its last stated proof component is: From the immediately preceding branch using the left exchange rule, infer antecedent containing first term t sub one is identical to term t sub two, then term t sub two is identical to term t sub three; sequent arrow; succedent containing term t sub one is identical to term t sub three.

  1. Premise or initial sequent: antecedent containing no formulas; sequent arrow; succedent containing term t sub one is identical to term t sub one.
  2. The next inference is labeled left weakening rule.
  3. From the immediately preceding branch using the left weakening rule, infer antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub one.
  4. The next inference is labeled identity rule.
  5. From the immediately preceding branch using the identity rule, infer antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub two is identical to term t sub one.
  6. The source ends this displayed proof segment here.
  7. Premise or initial sequent: antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub two.
  8. The next inference is labeled left weakening rule.
  9. From the immediately preceding branch using the left weakening rule, infer antecedent containing first term t sub two is identical to term t sub three, then term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub two.
  10. The next inference is labeled identity rule.
  11. From the immediately preceding branch using the identity rule, infer antecedent containing first term t sub two is identical to term t sub three, then term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub three.
  12. The next inference is labeled left exchange rule.
  13. From the immediately preceding branch using the left exchange rule, infer antecedent containing first term t sub one is identical to term t sub two, then term t sub two is identical to term t sub three; sequent arrow; succedent containing term t sub one is identical to term t sub three.

Source: content/first-order-logic/sequent-calculus/identity.tex, line 48.

Proof tree concluding antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub two is identical to term t sub one โ€” line 54

Proof tree concluding antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub two is identical to term t sub one โ€” line 54. This source proof tree has 6 explicitly ordered components. Its inference labels, in order, are left weakening rule, then identity rule. Its last stated proof component is: From the immediately preceding branch using the identity rule, infer antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub two is identical to term t sub one.

  1. Premise or initial sequent: antecedent containing no formulas; sequent arrow; succedent containing term t sub one is identical to term t sub one.
  2. The next inference is labeled left weakening rule.
  3. From the immediately preceding branch using the left weakening rule, infer antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub one.
  4. The next inference is labeled identity rule.
  5. From the immediately preceding branch using the identity rule, infer antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub two is identical to term t sub one.
  6. The source ends this displayed proof segment here.

Source: content/first-order-logic/sequent-calculus/identity.tex, line 54.

Exercise: Give derivations of the following sequents: Next item: antecedent containing no formulas;โ€ฆ โ€” line 67

Exercise: Give derivations of the following sequents: Next item: antecedent containing no formulas;โ€ฆ โ€” line 67. A complete source-order spoken rendering of this exercise; it preserves the source claim and does not add a solution or proof.

Unsolved source exercise; no solution added.

  1. Give derivations of the following sequents: Next item: antecedent containing no formulas; sequent arrow; succedent containing for every variable x, for every variable y, open parenthesis, the conditional whose antecedent states that open parenthesis, the conjunction whose first conjunct states that variable x is identical to variable y; and whose second conjunct is formula A with argument variable x, close parenthesis; and whose consequent is formula A with argument variable y, close parenthesis Next item: antecedent containing the conjunction whose first conjunct is for some variable x, formula A with argument variable x; and whose second conjunct states that for every variable y, for every variable z, open parenthesis, the conditional whose antecedent is open parenthesis, the conjunction of formula A with argument variable y and formula A with argument variable z, close parenthesis; and whose consequent states that variable y is identical to variable z, close parenthesis; sequent arrow; succedent containing for some variable x, open parenthesis, the conjunction whose first conjunct is formula A with argument variable x; and whose second conjunct states that for every variable y, open parenthesis, the conditional whose antecedent is formula A with argument variable y; and whose consequent states that variable y is identical to variable x, close parenthesis, close parenthesis

Source: content/first-order-logic/sequent-calculus/identity.tex, line 67.

14 source references

  1. the proposition characterizing inconsistency by derivability of every sentence โ€” source line 144.
  2. the proposition relating substitution to the semantic value of terms โ€” source line 221.
  3. the proposition giving the satisfaction clauses for quantifiers โ€” source line 249.
  4. the corollary that sentence truth is independent of variable assignment โ€” source line 254.
  5. the proposition linking satisfaction of a sentence with truth in a structure โ€” source line 257.
  6. the proposition on assignment extensionality for formulas โ€” source line 260.
  7. the proposition on extensionality of first-order evaluation โ€” source line 262.
  8. the sequent-calculus soundness theorem โ€” source line 336.
  9. the sequent-calculus soundness theorem โ€” source line 359.
  10. the sequent-calculus soundness theorem โ€” source line 374.
  11. the proposition linking satisfaction of a sentence with truth in a structure โ€” source line 34.
  12. the proposition on assignment extensionality for formulas โ€” source line 36.
  13. the proposition on assignment extensionality for formulas โ€” source line 39.
  14. the proposition linking satisfaction of a sentence with truth in a structure โ€” source line 40.

896 ordered proof components

Open the complete component ledger
  1. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  2. The next inference is labeled left negation rule.
  3. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  4. The source ends this displayed proof segment here.
  5. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  6. The next inference is labeled right negation rule.
  7. From the immediately preceding branch using the right negation rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the negation of formula A.
  8. The source ends this displayed proof segment here.
  9. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  10. The next inference is labeled left negation rule.
  11. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  12. The source ends this displayed proof segment here.
  13. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  14. The next inference is labeled right negation rule.
  15. From the immediately preceding branch using the right negation rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the negation of formula A.
  16. The source ends this displayed proof segment here.
  17. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  18. The next inference is labeled left conjunction rule.
  19. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  20. The source ends this displayed proof segment here.
  21. Premise or initial sequent: antecedent containing first formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  22. The next inference is labeled left conjunction rule.
  23. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  24. The source ends this displayed proof segment here.
  25. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  26. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  27. The next inference is labeled right conjunction rule.
  28. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B.
  29. The source ends this displayed proof segment here.
  30. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  31. The next inference is labeled left conjunction rule.
  32. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  33. The source ends this displayed proof segment here.
  34. Premise or initial sequent: antecedent containing first formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  35. The next inference is labeled left conjunction rule.
  36. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  37. The source ends this displayed proof segment here.
  38. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  39. The next inference is labeled left conjunction rule.
  40. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  41. The source ends this displayed proof segment here.
  42. Premise or initial sequent: antecedent containing first formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  43. The next inference is labeled left conjunction rule.
  44. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  45. The source ends this displayed proof segment here.
  46. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  47. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  48. The next inference is labeled right conjunction rule.
  49. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B.
  50. The source ends this displayed proof segment here.
  51. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  52. Premise or initial sequent: antecedent containing first formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  53. The next inference is labeled left disjunction rule.
  54. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  55. The source ends this displayed proof segment here.
  56. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  57. The next inference is labeled right disjunction rule.
  58. From the immediately preceding branch using the right disjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B.
  59. The source ends this displayed proof segment here.
  60. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  61. The next inference is labeled right disjunction rule.
  62. From the immediately preceding branch using the right disjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B.
  63. The source ends this displayed proof segment here.
  64. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  65. Premise or initial sequent: antecedent containing first formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  66. The next inference is labeled left disjunction rule.
  67. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  68. The source ends this displayed proof segment here.
  69. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  70. The next inference is labeled right disjunction rule.
  71. From the immediately preceding branch using the right disjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B.
  72. The source ends this displayed proof segment here.
  73. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  74. The next inference is labeled right disjunction rule.
  75. From the immediately preceding branch using the right disjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B.
  76. The source ends this displayed proof segment here.
  77. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  78. The next inference is labeled right disjunction rule.
  79. From the immediately preceding branch using the right disjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B.
  80. The source ends this displayed proof segment here.
  81. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  82. The next inference is labeled right disjunction rule.
  83. From the immediately preceding branch using the right disjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B.
  84. The source ends this displayed proof segment here.
  85. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  86. Premise or initial sequent: antecedent containing first formula B, then capital Pi; sequent arrow; succedent containing capital Lambda.
  87. The next inference is labeled left conditional rule.
  88. From the two immediately preceding branches using the left conditional rule, infer antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then Gamma, and finally capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda.
  89. The source ends this displayed proof segment here.
  90. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  91. The next inference is labeled right conditional rule.
  92. From the immediately preceding branch using the right conditional rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B.
  93. The source ends this displayed proof segment here.
  94. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  95. Premise or initial sequent: antecedent containing first formula B, then capital Pi; sequent arrow; succedent containing capital Lambda.
  96. The next inference is labeled left conditional rule.
  97. From the two immediately preceding branches using the left conditional rule, infer antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then Gamma, and finally capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda.
  98. The source ends this displayed proof segment here.
  99. Premise or initial sequent: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  100. The next inference is labeled right conditional rule.
  101. From the immediately preceding branch using the right conditional rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B.
  102. The source ends this displayed proof segment here.
  103. Premise or initial sequent: antecedent containing first formula A with argument term t, then Gamma; sequent arrow; succedent containing capital Delta.
  104. The next inference is labeled left universal quantifier rule.
  105. From the immediately preceding branch using the left universal quantifier rule, infer antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta.
  106. The source ends this displayed proof segment here.
  107. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument constant a.
  108. The next inference is labeled right universal quantifier rule.
  109. From the immediately preceding branch using the right universal quantifier rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for every variable x, formula A with argument variable x.
  110. The source ends this displayed proof segment here.
  111. Premise or initial sequent: antecedent containing first formula A with argument term t, then Gamma; sequent arrow; succedent containing capital Delta.
  112. The next inference is labeled left universal quantifier rule.
  113. From the immediately preceding branch using the left universal quantifier rule, infer antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta.
  114. The source ends this displayed proof segment here.
  115. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument constant a.
  116. The next inference is labeled right universal quantifier rule.
  117. From the immediately preceding branch using the right universal quantifier rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for every variable x, formula A with argument variable x.
  118. The source ends this displayed proof segment here.
  119. Premise or initial sequent: antecedent containing first formula A with argument constant a, then Gamma; sequent arrow; succedent containing capital Delta.
  120. The next inference is labeled left existential quantifier rule.
  121. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing first for some variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta.
  122. The source ends this displayed proof segment here.
  123. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t.
  124. The next inference is labeled right existential quantifier rule.
  125. From the immediately preceding branch using the right existential quantifier rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, formula A with argument variable x.
  126. The source ends this displayed proof segment here.
  127. Premise or initial sequent: antecedent containing first formula A with argument constant a, then Gamma; sequent arrow; succedent containing capital Delta.
  128. The next inference is labeled left existential quantifier rule.
  129. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing first for some variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta.
  130. The source ends this displayed proof segment here.
  131. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t.
  132. The next inference is labeled right existential quantifier rule.
  133. From the immediately preceding branch using the right existential quantifier rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, formula A with argument variable x.
  134. The source ends this displayed proof segment here.
  135. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the result of substituting term t for variable x in formula A.
  136. The next inference is labeled right existential quantifier rule.
  137. From the immediately preceding branch using the right existential quantifier rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for some variable x, formula A.
  138. Premise or initial sequent: antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant a.
  139. The next inference is labeled left existential quantifier rule, marked with an asterisk to identify this intentionally invalid inference.
  140. From the immediately preceding branch using the left existential quantifier rule, marked with an asterisk to identify this intentionally invalid inference, infer antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument constant a.
  141. The next inference is labeled right universal quantifier rule.
  142. From the immediately preceding branch using the right universal quantifier rule, infer antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing for every variable x, formula A with argument variable x.
  143. The source ends this displayed proof segment here.
  144. Premise or initial sequent: antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant a.
  145. The next inference is labeled right universal quantifier rule, marked with an asterisk to identify this intentionally invalid inference.
  146. From the immediately preceding branch using the right universal quantifier rule, marked with an asterisk to identify this intentionally invalid inference, infer antecedent containing formula A with argument constant a; sequent arrow; succedent containing for every variable x, formula A with argument variable x.
  147. The next inference is labeled left existential quantifier rule.
  148. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing for every variable x, formula A with argument variable x.
  149. Premise or initial sequent: antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant a.
  150. The next inference is labeled left existential quantifier rule, marked with an asterisk to identify this intentionally invalid inference.
  151. From the immediately preceding branch using the left existential quantifier rule, marked with an asterisk to identify this intentionally invalid inference, infer antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument constant a.
  152. The next inference is labeled right universal quantifier rule.
  153. From the immediately preceding branch using the right universal quantifier rule, infer antecedent containing for some variable x, formula A with argument variable x; sequent arrow; succedent containing for every variable x, formula A with argument variable x.
  154. The source ends this displayed proof segment here.
  155. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  156. The next inference is labeled left weakening rule.
  157. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  158. The source ends this displayed proof segment here.
  159. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  160. The next inference is labeled right weakening rule.
  161. From the immediately preceding branch using the right weakening rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  162. The source ends this displayed proof segment here.
  163. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  164. The next inference is labeled left weakening rule.
  165. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  166. The source ends this displayed proof segment here.
  167. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  168. The next inference is labeled right weakening rule.
  169. From the immediately preceding branch using the right weakening rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  170. The source ends this displayed proof segment here.
  171. Premise or initial sequent: antecedent containing first formula A, then formula A, and finally Gamma; sequent arrow; succedent containing capital Delta.
  172. The next inference is labeled left contraction rule.
  173. From the immediately preceding branch using the left contraction rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  174. The source ends this displayed proof segment here.
  175. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A, and finally formula A.
  176. The next inference is labeled right contraction rule.
  177. From the immediately preceding branch using the right contraction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  178. The source ends this displayed proof segment here.
  179. Premise or initial sequent: antecedent containing first formula A, then formula A, and finally Gamma; sequent arrow; succedent containing capital Delta.
  180. The next inference is labeled left contraction rule.
  181. From the immediately preceding branch using the left contraction rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  182. The source ends this displayed proof segment here.
  183. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A, and finally formula A.
  184. The next inference is labeled right contraction rule.
  185. From the immediately preceding branch using the right contraction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  186. The source ends this displayed proof segment here.
  187. Premise or initial sequent: antecedent containing first Gamma, then formula A, then formula B, and finally capital Pi; sequent arrow; succedent containing capital Delta.
  188. The next inference is labeled left exchange rule.
  189. From the immediately preceding branch using the left exchange rule, infer antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta.
  190. The source ends this displayed proof segment here.
  191. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A, then formula B, and finally capital Lambda.
  192. The next inference is labeled right exchange rule.
  193. From the immediately preceding branch using the right exchange rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B, then formula A, and finally capital Lambda.
  194. The source ends this displayed proof segment here.
  195. Premise or initial sequent: antecedent containing first Gamma, then formula A, then formula B, and finally capital Pi; sequent arrow; succedent containing capital Delta.
  196. The next inference is labeled left exchange rule.
  197. From the immediately preceding branch using the left exchange rule, infer antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta.
  198. The source ends this displayed proof segment here.
  199. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A, then formula B, and finally capital Lambda.
  200. The next inference is labeled right exchange rule.
  201. From the immediately preceding branch using the right exchange rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B, then formula A, and finally capital Lambda.
  202. The source ends this displayed proof segment here.
  203. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  204. Premise or initial sequent: antecedent containing first formula A, then capital Pi; sequent arrow; succedent containing capital Lambda.
  205. The next inference is labeled cut rule.
  206. From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma, then capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda.
  207. The source ends this displayed proof segment here.
  208. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  209. Premise or initial sequent: antecedent containing first formula A, then capital Pi; sequent arrow; succedent containing capital Lambda.
  210. The next inference is labeled cut rule.
  211. From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma, then capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda.
  212. The source ends this displayed proof segment here.
  213. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  214. The next inference is labeled left weakening rule.
  215. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  216. Premise or initial sequent: antecedent containing formula C; sequent arrow; succedent containing formula C.
  217. The next inference is labeled left weakening rule.
  218. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.
  219. Premise or initial sequent: antecedent containing formula C; sequent arrow; succedent containing formula C.
  220. The next inference is labeled left weakening rule.
  221. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.
  222. The next inference is labeled left exchange rule.
  223. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula C.
  224. Premise or initial sequent: antecedent containing first Gamma, then formula A, then formula B, and finally capital Pi; sequent arrow; succedent containing capital Delta.
  225. The next inference is labeled left exchange rule.
  226. From the immediately preceding branch using the left exchange rule, infer antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta, followed by a printed trailing comma with no following formula. Source disclosure. Preserve and speak the printed trailing comma as having no following formula, then disclose that it appears to be stray punctuation.
  227. Premise or initial sequent: antecedent containing formula D; sequent arrow; succedent containing formula D.
  228. The next inference is labeled left weakening rule.
  229. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula D.
  230. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  231. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  232. The next inference is labeled right conjunction rule.
  233. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B.
  234. Premise or initial sequent: antecedent containing formula C; sequent arrow; succedent containing formula C.
  235. The next inference is labeled left weakening rule.
  236. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.
  237. The next inference is labeled left exchange rule.
  238. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula C.
  239. Premise or initial sequent: antecedent containing formula D; sequent arrow; succedent containing formula D.
  240. The next inference is labeled left weakening rule.
  241. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula D.
  242. The next inference is labeled right conjunction rule.
  243. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula C and formula D.
  244. Premise or initial sequent: antecedent containing formula D; sequent arrow; succedent containing formula D.
  245. The next inference is labeled left weakening rule.
  246. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula D.
  247. Premise or initial sequent: antecedent containing formula C; sequent arrow; succedent containing formula C.
  248. The next inference is labeled left weakening rule.
  249. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.
  250. The next inference is labeled left exchange rule.
  251. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula C.
  252. The next inference is labeled right conjunction rule.
  253. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula D and formula C.
  254. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  255. The next inference is labeled left weakening rule.
  256. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  257. Premise or initial sequent: antecedent containing formula C; sequent arrow; succedent containing formula C.
  258. The next inference is labeled left weakening rule.
  259. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.
  260. Premise or initial sequent: antecedent containing formula C; sequent arrow; succedent containing formula C.
  261. The next inference is labeled left weakening rule.
  262. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.
  263. The next inference is labeled left exchange rule.
  264. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula C.
  265. Premise or initial sequent: antecedent containing first Gamma, then formula A, then formula B, and finally capital Pi; sequent arrow; succedent containing capital Delta.
  266. The next inference is labeled left exchange rule.
  267. From the immediately preceding branch using the left exchange rule, infer antecedent containing first Gamma, then formula B, then formula A, and finally capital Pi; sequent arrow; succedent containing capital Delta, followed by a printed trailing comma with no following formula. Source disclosure. Preserve and speak the printed trailing comma as having no following formula, then disclose that it appears to be stray punctuation.
  268. Premise or initial sequent: antecedent containing formula D; sequent arrow; succedent containing formula D.
  269. The next inference is labeled left weakening rule.
  270. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula D.
  271. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  272. Premise or initial sequent: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  273. The next inference is labeled right conjunction rule.
  274. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B.
  275. Premise or initial sequent: antecedent containing formula C; sequent arrow; succedent containing formula C.
  276. The next inference is labeled left weakening rule.
  277. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.
  278. The next inference is labeled left exchange rule.
  279. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula C.
  280. Premise or initial sequent: antecedent containing formula D; sequent arrow; succedent containing formula D.
  281. The next inference is labeled left weakening rule.
  282. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula D.
  283. The next inference is labeled right conjunction rule.
  284. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula C and formula D.
  285. Premise or initial sequent: antecedent containing formula D; sequent arrow; succedent containing formula D.
  286. The next inference is labeled left weakening rule.
  287. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula D.
  288. Premise or initial sequent: antecedent containing formula C; sequent arrow; succedent containing formula C.
  289. The next inference is labeled left weakening rule.
  290. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula D, then formula C; sequent arrow; succedent containing formula C.
  291. The next inference is labeled left exchange rule.
  292. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing formula C.
  293. The next inference is labeled right conjunction rule.
  294. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing first formula C, then formula D; sequent arrow; succedent containing the conjunction of formula D and formula C.
  295. An upper premise slot is intentionally blank in this staged diagram.
  296. From the immediately preceding branch, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.
  297. An upper premise slot is intentionally blank in this staged diagram.
  298. The next inference is labeled left conjunction rule.
  299. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.
  300. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  301. The next inference is labeled left conjunction rule.
  302. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.
  303. An upper premise slot is intentionally blank in this staged diagram.
  304. From the immediately preceding branch, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.
  305. An upper premise slot is intentionally blank in this staged diagram.
  306. The next inference is labeled left conjunction rule.
  307. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.
  308. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  309. The next inference is labeled left conjunction rule.
  310. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.
  311. An upper premise slot is intentionally blank in this staged diagram.
  312. From the immediately preceding branch, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  313. An upper premise slot is intentionally blank in this staged diagram.
  314. From the immediately preceding branch, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  315. The next inference is labeled right conditional rule.
  316. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  317. An upper premise slot is intentionally blank in this staged diagram.
  318. From the immediately preceding branch, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B.
  319. An upper premise slot is intentionally blank in this staged diagram.
  320. From the immediately preceding branch, infer antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B.
  321. The next inference is labeled left disjunction rule.
  322. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B.
  323. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended.
  324. From the immediately preceding branch using the right exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  325. The next inference is labeled right conditional rule.
  326. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  327. An upper premise slot is intentionally blank in this staged diagram.
  328. From the immediately preceding branch, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B.
  329. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  330. The next inference is labeled left weakening rule.
  331. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then formula B; sequent arrow; succedent containing formula B.
  332. The next inference is labeled left exchange rule.
  333. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B.
  334. The next inference is labeled left disjunction rule.
  335. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B.
  336. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended.
  337. From the immediately preceding branch using the right exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  338. The next inference is labeled right conditional rule.
  339. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  340. An upper premise slot is intentionally blank in this staged diagram.
  341. From the immediately preceding branch, infer antecedent containing formula A; sequent arrow; succedent containing first formula B, then formula A.
  342. The next inference is labeled left negation rule.
  343. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B.
  344. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  345. The next inference is labeled left weakening rule.
  346. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then formula B; sequent arrow; succedent containing formula B.
  347. The next inference is labeled left exchange rule.
  348. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B.
  349. The next inference is labeled left disjunction rule.
  350. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B.
  351. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended.
  352. From the immediately preceding branch using the right exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  353. The next inference is labeled right conditional rule.
  354. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  355. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  356. The next inference is labeled right weakening rule.
  357. From the immediately preceding branch using the right weakening rule, infer antecedent containing formula A; sequent arrow; succedent containing first formula A, then formula B.
  358. The next inference is labeled right exchange rule.
  359. From the immediately preceding branch using the right exchange rule, infer antecedent containing formula A; sequent arrow; succedent containing first formula B, then formula A.
  360. The next inference is labeled left negation rule.
  361. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B.
  362. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  363. The next inference is labeled left weakening rule.
  364. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then formula B; sequent arrow; succedent containing formula B.
  365. The next inference is labeled left exchange rule.
  366. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B.
  367. The next inference is labeled left disjunction rule.
  368. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B.
  369. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended.
  370. From the immediately preceding branch using the right exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  371. The next inference is labeled right conditional rule.
  372. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  373. An upper premise slot is intentionally blank in this staged diagram.
  374. From the immediately preceding branch, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  375. An upper premise slot is intentionally blank in this staged diagram.
  376. From the immediately preceding branch, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  377. The next inference is labeled right conditional rule.
  378. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  379. An upper premise slot is intentionally blank in this staged diagram.
  380. From the immediately preceding branch, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B.
  381. An upper premise slot is intentionally blank in this staged diagram.
  382. From the immediately preceding branch, infer antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B.
  383. The next inference is labeled left disjunction rule.
  384. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B.
  385. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended.
  386. From the immediately preceding branch using the right exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  387. The next inference is labeled right conditional rule.
  388. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  389. An upper premise slot is intentionally blank in this staged diagram.
  390. From the immediately preceding branch, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B.
  391. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  392. The next inference is labeled left weakening rule.
  393. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then formula B; sequent arrow; succedent containing formula B.
  394. The next inference is labeled left exchange rule.
  395. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B.
  396. The next inference is labeled left disjunction rule.
  397. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B.
  398. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended.
  399. From the immediately preceding branch using the right exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  400. The next inference is labeled right conditional rule.
  401. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  402. An upper premise slot is intentionally blank in this staged diagram.
  403. From the immediately preceding branch, infer antecedent containing formula A; sequent arrow; succedent containing first formula B, then formula A.
  404. The next inference is labeled left negation rule.
  405. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B.
  406. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  407. The next inference is labeled left weakening rule.
  408. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then formula B; sequent arrow; succedent containing formula B.
  409. The next inference is labeled left exchange rule.
  410. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B.
  411. The next inference is labeled left disjunction rule.
  412. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B.
  413. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended.
  414. From the immediately preceding branch using the right exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  415. The next inference is labeled right conditional rule.
  416. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  417. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  418. The next inference is labeled right weakening rule.
  419. From the immediately preceding branch using the right weakening rule, infer antecedent containing formula A; sequent arrow; succedent containing first formula A, then formula B.
  420. The next inference is labeled right exchange rule.
  421. From the immediately preceding branch using the right exchange rule, infer antecedent containing formula A; sequent arrow; succedent containing first formula B, then formula A.
  422. The next inference is labeled left negation rule.
  423. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing formula B.
  424. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  425. The next inference is labeled left weakening rule.
  426. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then formula B; sequent arrow; succedent containing formula B.
  427. The next inference is labeled left exchange rule.
  428. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula B, then formula A; sequent arrow; succedent containing formula B.
  429. The next inference is labeled left disjunction rule.
  430. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and formula B, then formula A; sequent arrow; succedent containing formula B.
  431. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended.
  432. From the immediately preceding branch using the right exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing formula B.
  433. The next inference is labeled right conditional rule.
  434. From the immediately preceding branch using the right conditional rule, infer antecedent containing the disjunction of the negation of formula A and formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  435. An upper premise slot is intentionally blank in this staged diagram.
  436. From the immediately preceding branch, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  437. An upper premise slot is intentionally blank in this staged diagram.
  438. From the immediately preceding branch, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  439. The next inference is labeled right negation rule.
  440. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  441. An upper premise slot is intentionally blank in this staged diagram.
  442. From the immediately preceding branch, infer antecedent containing first formula A, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  443. The next inference is labeled left conjunction rule.
  444. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  445. The next inference is labeled right negation rule.
  446. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  447. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  448. The next inference is labeled left negation rule.
  449. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing no formulas.
  450. An upper premise slot is intentionally blank in this staged diagram.
  451. The next inference is labeled question mark placeholder for an inference rule not yet chosen.
  452. From the immediately preceding branch using the question mark placeholder for an inference rule not yet chosen, infer antecedent containing formula A; sequent arrow; succedent containing formula B.
  453. The next inference is labeled left negation rule.
  454. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula B, then formula A; sequent arrow; succedent containing no formulas.
  455. The next inference is labeled left disjunction rule.
  456. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and the negation of formula B, then formula A; sequent arrow; succedent containing no formulas.
  457. The next inference is labeled left exchange rule.
  458. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  459. The next inference is labeled left conjunction rule.
  460. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  461. The next inference is labeled right negation rule.
  462. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  463. An upper premise slot is intentionally blank in this staged diagram.
  464. From the immediately preceding branch, infer antecedent containing first the negation of formula A, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  465. An upper premise slot is intentionally blank in this staged diagram.
  466. From the immediately preceding branch, infer antecedent containing first the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  467. The next inference is labeled left disjunction rule.
  468. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  469. The next inference is labeled left exchange rule.
  470. From the immediately preceding branch using the left exchange rule, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  471. The next inference is labeled right negation rule.
  472. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  473. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  474. The next inference is labeled left conjunction rule.
  475. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.
  476. The next inference is labeled left negation rule.
  477. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  478. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  479. The next inference is labeled left conjunction rule.
  480. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula B.
  481. The next inference is labeled left negation rule.
  482. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  483. The next inference is labeled left disjunction rule.
  484. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  485. The next inference is labeled left exchange rule.
  486. From the immediately preceding branch using the left exchange rule, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  487. The next inference is labeled right negation rule.
  488. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  489. An upper premise slot is intentionally blank in this staged diagram.
  490. From the immediately preceding branch, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  491. An upper premise slot is intentionally blank in this staged diagram.
  492. From the immediately preceding branch, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  493. The next inference is labeled right negation rule.
  494. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  495. An upper premise slot is intentionally blank in this staged diagram.
  496. From the immediately preceding branch, infer antecedent containing first formula A, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  497. The next inference is labeled left conjunction rule.
  498. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  499. The next inference is labeled right negation rule.
  500. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  501. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  502. The next inference is labeled left negation rule.
  503. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing no formulas.
  504. An upper premise slot is intentionally blank in this staged diagram.
  505. The next inference is labeled question mark placeholder for an inference rule not yet chosen.
  506. From the immediately preceding branch using the question mark placeholder for an inference rule not yet chosen, infer antecedent containing formula A; sequent arrow; succedent containing formula B.
  507. The next inference is labeled left negation rule.
  508. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula B, then formula A; sequent arrow; succedent containing no formulas.
  509. The next inference is labeled left disjunction rule.
  510. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and the negation of formula B, then formula A; sequent arrow; succedent containing no formulas.
  511. The next inference is labeled left exchange rule.
  512. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula A, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  513. The next inference is labeled left conjunction rule.
  514. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  515. The next inference is labeled right negation rule.
  516. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  517. An upper premise slot is intentionally blank in this staged diagram.
  518. From the immediately preceding branch, infer antecedent containing first the negation of formula A, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  519. An upper premise slot is intentionally blank in this staged diagram.
  520. From the immediately preceding branch, infer antecedent containing first the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  521. The next inference is labeled left disjunction rule.
  522. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  523. The next inference is labeled left exchange rule.
  524. From the immediately preceding branch using the left exchange rule, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  525. The next inference is labeled right negation rule.
  526. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  527. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  528. The next inference is labeled left conjunction rule.
  529. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.
  530. The next inference is labeled left negation rule.
  531. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  532. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  533. The next inference is labeled left conjunction rule.
  534. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula B.
  535. The next inference is labeled left negation rule.
  536. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  537. The next inference is labeled left disjunction rule.
  538. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of the negation of formula A and the negation of formula B, then the conjunction of formula A and formula B; sequent arrow; succedent containing no formulas.
  539. The next inference is labeled left exchange rule.
  540. From the immediately preceding branch using the left exchange rule, infer antecedent containing first the conjunction of formula A and formula B, then the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing no formulas.
  541. The next inference is labeled right negation rule.
  542. From the immediately preceding branch using the right negation rule, infer antecedent containing the disjunction of the negation of formula A and the negation of formula B; sequent arrow; succedent containing the negation of open parenthesis, the conjunction of formula A and formula B, close parenthesis.
  543. An upper premise slot is intentionally blank in this staged diagram.
  544. From the immediately preceding branch, infer antecedent containing no formulas; sequent arrow; succedent containing formula A.
  545. The next inference is labeled right disjunction rule.
  546. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.
  547. The source ends this displayed proof segment here.
  548. An upper premise slot is intentionally blank in this staged diagram.
  549. From the immediately preceding branch, infer antecedent containing formula A; sequent arrow; succedent containing no formulas.
  550. The next inference is labeled right negation rule.
  551. From the immediately preceding branch using the right negation rule, infer antecedent containing no formulas; sequent arrow; succedent containing the negation of formula A.
  552. The next inference is labeled right disjunction rule.
  553. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.
  554. An upper premise slot is intentionally blank in this staged diagram.
  555. From the immediately preceding branch, infer antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then formula A.
  556. The next inference is labeled right disjunction rule.
  557. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then the disjunction of formula A and the negation of formula A.
  558. The next inference is labeled right contraction rule.
  559. From the immediately preceding branch using the right contraction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.
  560. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  561. The next inference is labeled right negation rule.
  562. From the immediately preceding branch using the right negation rule, infer antecedent containing no formulas; sequent arrow; succedent containing first formula A, then the negation of formula A.
  563. The next inference is labeled right disjunction rule.
  564. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing first formula A, then the disjunction of formula A and the negation of formula A.
  565. The next inference is labeled right exchange rule.
  566. From the immediately preceding branch using the right exchange rule, infer antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then formula A.
  567. The next inference is labeled right disjunction rule.
  568. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then the disjunction of formula A and the negation of formula A.
  569. The next inference is labeled right contraction rule.
  570. From the immediately preceding branch using the right contraction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.
  571. An upper premise slot is intentionally blank in this staged diagram.
  572. From the immediately preceding branch, infer antecedent containing no formulas; sequent arrow; succedent containing formula A.
  573. The next inference is labeled right disjunction rule.
  574. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.
  575. The source ends this displayed proof segment here.
  576. An upper premise slot is intentionally blank in this staged diagram.
  577. From the immediately preceding branch, infer antecedent containing formula A; sequent arrow; succedent containing no formulas.
  578. The next inference is labeled right negation rule.
  579. From the immediately preceding branch using the right negation rule, infer antecedent containing no formulas; sequent arrow; succedent containing the negation of formula A.
  580. The next inference is labeled right disjunction rule.
  581. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.
  582. An upper premise slot is intentionally blank in this staged diagram.
  583. From the immediately preceding branch, infer antecedent containing no formulas; sequent arrow; succedent containing formula A.
  584. The next inference is labeled right disjunction rule.
  585. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.
  586. The source ends this displayed proof segment here.
  587. An upper premise slot is intentionally blank in this staged diagram.
  588. From the immediately preceding branch, infer antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then formula A.
  589. The next inference is labeled right disjunction rule.
  590. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then the disjunction of formula A and the negation of formula A.
  591. The next inference is labeled right contraction rule.
  592. From the immediately preceding branch using the right contraction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.
  593. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  594. The next inference is labeled right negation rule.
  595. From the immediately preceding branch using the right negation rule, infer antecedent containing no formulas; sequent arrow; succedent containing first formula A, then the negation of formula A.
  596. The next inference is labeled right disjunction rule.
  597. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing first formula A, then the disjunction of formula A and the negation of formula A.
  598. The next inference is labeled right exchange rule.
  599. From the immediately preceding branch using the right exchange rule, infer antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then formula A.
  600. The next inference is labeled right disjunction rule.
  601. From the immediately preceding branch using the right disjunction rule, infer antecedent containing no formulas; sequent arrow; succedent containing first the disjunction of formula A and the negation of formula A, then the disjunction of formula A and the negation of formula A.
  602. The next inference is labeled right contraction rule.
  603. From the immediately preceding branch using the right contraction rule, infer antecedent containing no formulas; sequent arrow; succedent containing the disjunction of formula A and the negation of formula A.
  604. An upper premise slot is intentionally blank in this staged diagram.
  605. From the immediately preceding branch, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  606. An upper premise slot is intentionally blank in this staged diagram.
  607. From the immediately preceding branch, infer antecedent containing the negation of formula A with argument constant a; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  608. The next inference is labeled left existential quantifier rule.
  609. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  610. An upper premise slot is intentionally blank in this staged diagram.
  611. From the immediately preceding branch, infer antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument constant a.
  612. The next inference is labeled left negation rule.
  613. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A with argument constant a, then for every variable x, formula A with argument variable x; sequent arrow; succedent containing no formulas.
  614. The next inference is labeled left exchange rule.
  615. From the immediately preceding branch using the left exchange rule, infer antecedent containing first for every variable x, formula A with argument variable x, then the negation of formula A with argument constant a; sequent arrow; succedent containing no formulas.
  616. The next inference is labeled right negation rule.
  617. From the immediately preceding branch using the right negation rule, infer antecedent containing the negation of formula A with argument constant a; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  618. The next inference is labeled left existential quantifier rule.
  619. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  620. Premise or initial sequent: antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant a.
  621. The next inference is labeled left universal quantifier rule.
  622. From the immediately preceding branch using the left universal quantifier rule, infer antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument constant a.
  623. The next inference is labeled left negation rule.
  624. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A with argument constant a, then for every variable x, formula A with argument variable x; sequent arrow; succedent containing no formulas.
  625. The next inference is labeled left exchange rule.
  626. From the immediately preceding branch using the left exchange rule, infer antecedent containing first for every variable x, formula A with argument variable x, then the negation of formula A with argument constant a; sequent arrow; succedent containing no formulas.
  627. The next inference is labeled right negation rule.
  628. From the immediately preceding branch using the right negation rule, infer antecedent containing the negation of formula A with argument constant a; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  629. The next inference is labeled left existential quantifier rule.
  630. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  631. An upper premise slot is intentionally blank in this staged diagram.
  632. From the immediately preceding branch, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  633. An upper premise slot is intentionally blank in this staged diagram.
  634. From the immediately preceding branch, infer antecedent containing the negation of formula A with argument constant a; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  635. The next inference is labeled left existential quantifier rule.
  636. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  637. An upper premise slot is intentionally blank in this staged diagram.
  638. From the immediately preceding branch, infer antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument constant a.
  639. The next inference is labeled left negation rule.
  640. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A with argument constant a, then for every variable x, formula A with argument variable x; sequent arrow; succedent containing no formulas.
  641. The next inference is labeled left exchange rule.
  642. From the immediately preceding branch using the left exchange rule, infer antecedent containing first for every variable x, formula A with argument variable x, then the negation of formula A with argument constant a; sequent arrow; succedent containing no formulas.
  643. The next inference is labeled right negation rule.
  644. From the immediately preceding branch using the right negation rule, infer antecedent containing the negation of formula A with argument constant a; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  645. The next inference is labeled left existential quantifier rule.
  646. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  647. Premise or initial sequent: antecedent containing formula A with argument constant a; sequent arrow; succedent containing formula A with argument constant a.
  648. The next inference is labeled left universal quantifier rule.
  649. From the immediately preceding branch using the left universal quantifier rule, infer antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument constant a.
  650. The next inference is labeled left negation rule.
  651. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A with argument constant a, then for every variable x, formula A with argument variable x; sequent arrow; succedent containing no formulas.
  652. The next inference is labeled left exchange rule.
  653. From the immediately preceding branch using the left exchange rule, infer antecedent containing first for every variable x, formula A with argument variable x, then the negation of formula A with argument constant a; sequent arrow; succedent containing no formulas.
  654. The next inference is labeled right negation rule.
  655. From the immediately preceding branch using the right negation rule, infer antecedent containing the negation of formula A with argument constant a; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  656. The next inference is labeled left existential quantifier rule.
  657. From the immediately preceding branch using the left existential quantifier rule, infer antecedent containing for some variable x, the negation of formula A with argument variable x; sequent arrow; succedent containing the negation of for every variable x, formula A with argument variable x.
  658. An upper premise slot is intentionally blank in this staged diagram.
  659. Previously derived premise with its intervening derivation omitted: antecedent containing first formula B, then formula B, and finally formula C; sequent arrow; succedent containing formula A.
  660. The next inference is labeled left contraction rule.
  661. From the immediately preceding branch using the left contraction rule, infer antecedent containing first formula B, then formula C; sequent arrow; succedent containing formula A.
  662. The next inference is labeled left exchange rule.
  663. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula C, then formula B; sequent arrow; succedent containing formula A.
  664. The next inference is labeled left weakening rule.
  665. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula C, then formula C, and finally formula B; sequent arrow; succedent containing formula A.
  666. An upper premise slot is intentionally blank in this staged diagram.
  667. The next inference is labeled derivation pi sub zero.
  668. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma sub zero; sequent arrow; succedent containing formula A.
  669. An upper premise slot is intentionally blank in this staged diagram.
  670. The next inference is labeled derivation pi sub one.
  671. Previously derived premise with its intervening derivation omitted: antecedent containing first formula A, then capital Delta sub zero; sequent arrow; succedent containing formula B.
  672. The next inference is labeled cut rule.
  673. From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma sub zero, then capital Delta sub zero; sequent arrow; succedent containing formula B.
  674. An upper premise slot is intentionally blank in this staged diagram.
  675. The next inference is labeled derivation pi sub zero.
  676. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma sub zero; sequent arrow; succedent containing formula A.
  677. An upper premise slot is intentionally blank in this staged diagram.
  678. The next inference is labeled derivation pi sub one.
  679. Previously derived premise with its intervening derivation omitted: antecedent containing first formula A, then Gamma sub one; sequent arrow; succedent containing no formulas.
  680. The next inference is labeled cut rule.
  681. From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma sub zero, then Gamma sub one; sequent arrow; succedent containing no formulas.
  682. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  683. The next inference is labeled right negation rule.
  684. From the immediately preceding branch using the right negation rule, infer antecedent containing no formulas; sequent arrow; succedent containing first formula A, then the negation of formula A.
  685. An upper premise slot is intentionally blank in this staged diagram.
  686. The next inference is labeled derivation pi sub one.
  687. Previously derived premise with its intervening derivation omitted: antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing no formulas.
  688. The next inference is labeled cut rule.
  689. From the two immediately preceding branches using the cut rule, infer antecedent containing Gamma; sequent arrow; succedent containing formula A.
  690. An upper premise slot is intentionally blank in this staged diagram.
  691. The next inference is labeled derivation pi.
  692. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma sub zero; sequent arrow; succedent containing formula A.
  693. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  694. The next inference is labeled left negation rule.
  695. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing no formulas.
  696. The next inference is labeled left exchange rule.
  697. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula A, then the negation of formula A; sequent arrow; succedent containing no formulas.
  698. The next inference is labeled cut rule.
  699. From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma sub zero, then the negation of formula A; sequent arrow; succedent containing no formulas.
  700. An upper premise slot is intentionally blank in this staged diagram.
  701. The next inference is labeled derivation pi sub zero.
  702. Previously derived premise with its intervening derivation omitted: antecedent containing first formula A, then Gamma sub zero; sequent arrow; succedent containing no formulas.
  703. The next inference is labeled right negation rule.
  704. From the immediately preceding branch using the right negation rule, infer antecedent containing Gamma sub zero; sequent arrow; succedent containing the negation of formula A.
  705. An upper premise slot is intentionally blank in this staged diagram.
  706. The next inference is labeled derivation pi sub one.
  707. Previously derived premise with its intervening derivation omitted: antecedent containing first the negation of formula A, then Gamma sub one; sequent arrow; succedent containing no formulas.
  708. The next inference is labeled cut rule.
  709. From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma sub zero, then Gamma sub one; sequent arrow; succedent containing no formulas.
  710. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  711. The next inference is labeled left conjunction rule.
  712. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.
  713. The source ends this displayed proof segment here.
  714. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  715. The next inference is labeled left conjunction rule.
  716. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula B.
  717. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  718. The next inference is labeled left conjunction rule.
  719. From the immediately preceding branch using the left conjunction rule, infer antecedent containing the conjunction of formula A and formula B; sequent arrow; succedent containing formula A.
  720. The source ends this displayed proof segment here.
  721. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  722. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  723. The next inference is labeled right conjunction rule.
  724. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing first formula A, then formula B; sequent arrow; succedent containing the conjunction of formula A and formula B.
  725. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  726. The next inference is labeled left negation rule.
  727. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing no formulas.
  728. A double inference line abbreviates one or more weakening, contraction, or exchange steps.
  729. From the immediately preceding branch, infer antecedent containing first formula A, then the negation of formula A, and finally the negation of formula B; sequent arrow; succedent containing no formulas.
  730. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  731. The next inference is labeled left negation rule.
  732. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula B, then formula B; sequent arrow; succedent containing no formulas.
  733. A double inference line abbreviates one or more weakening, contraction, or exchange steps.
  734. From the immediately preceding branch, infer antecedent containing first formula B, then the negation of formula A, and finally the negation of formula B; sequent arrow; succedent containing no formulas.
  735. The next inference is labeled left disjunction rule.
  736. From the two immediately preceding branches using the left disjunction rule, infer antecedent containing first the disjunction of formula A and formula B, then the negation of formula A, and finally the negation of formula B; sequent arrow; succedent containing no formulas.
  737. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  738. The next inference is labeled right disjunction rule.
  739. From the immediately preceding branch using the right disjunction rule, infer antecedent containing formula A; sequent arrow; succedent containing the disjunction of formula A and formula B.
  740. The source ends this displayed proof segment here.
  741. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  742. The next inference is labeled right disjunction rule.
  743. From the immediately preceding branch using the right disjunction rule, infer antecedent containing formula B; sequent arrow; succedent containing the disjunction of formula A and formula B.
  744. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  745. The next inference is labeled right disjunction rule.
  746. From the immediately preceding branch using the right disjunction rule, infer antecedent containing formula A; sequent arrow; succedent containing the disjunction of formula A and formula B.
  747. The source ends this displayed proof segment here.
  748. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  749. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  750. The next inference is labeled left conditional rule.
  751. From the two immediately preceding branches using the left conditional rule, infer antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then formula A; sequent arrow; succedent containing formula B.
  752. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  753. The next inference is labeled left negation rule.
  754. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing no formulas.
  755. The next inference is labeled left exchange rule.
  756. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula A, then the negation of formula A; sequent arrow; succedent containing no formulas.
  757. The next inference is labeled right weakening rule.
  758. From the immediately preceding branch using the right weakening rule, infer antecedent containing first formula A, then the negation of formula A; sequent arrow; succedent containing formula B.
  759. The next inference is labeled right conditional rule.
  760. From the immediately preceding branch using the right conditional rule, infer antecedent containing the negation of formula A; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  761. The source ends this displayed proof segment here.
  762. Premise or initial sequent: antecedent containing formula B; sequent arrow; succedent containing formula B.
  763. The next inference is labeled left weakening rule.
  764. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then formula B; sequent arrow; succedent containing formula B.
  765. The next inference is labeled right conditional rule.
  766. From the immediately preceding branch using the right conditional rule, infer antecedent containing formula B; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  767. Premise or initial sequent: antecedent containing formula A; sequent arrow; succedent containing formula A.
  768. The next inference is labeled left negation rule.
  769. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then formula A; sequent arrow; succedent containing no formulas.
  770. The next inference is labeled left exchange rule.
  771. From the immediately preceding branch using the left exchange rule, infer antecedent containing first formula A, then the negation of formula A; sequent arrow; succedent containing no formulas.
  772. The next inference is labeled right weakening rule.
  773. From the immediately preceding branch using the right weakening rule, infer antecedent containing first formula A, then the negation of formula A; sequent arrow; succedent containing formula B.
  774. The next inference is labeled right conditional rule.
  775. From the immediately preceding branch using the right conditional rule, infer antecedent containing the negation of formula A; sequent arrow; succedent containing the conditional whose antecedent is formula A; and whose consequent is formula B.
  776. The source ends this displayed proof segment here.
  777. Premise or initial sequent: antecedent containing formula A with argument term t; sequent arrow; succedent containing formula A with argument term t.
  778. The next inference is labeled right existential quantifier rule.
  779. From the immediately preceding branch using the right existential quantifier rule, infer antecedent containing formula A with argument term t; sequent arrow; succedent containing for some variable x, formula A with argument variable x.
  780. Premise or initial sequent: antecedent containing formula A with argument term t; sequent arrow; succedent containing formula A with argument term t.
  781. The next inference is labeled left universal quantifier rule.
  782. From the immediately preceding branch using the left universal quantifier rule, infer antecedent containing for every variable x, formula A with argument variable x; sequent arrow; succedent containing formula A with argument term t.
  783. An upper premise slot is intentionally blank in this staged diagram.
  784. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  785. The next inference is labeled left weakening rule.
  786. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  787. The source ends this displayed proof segment here.
  788. An upper premise slot is intentionally blank in this staged diagram.
  789. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  790. The next inference is labeled right weakening rule.
  791. From the immediately preceding branch using the right weakening rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  792. An upper premise slot is intentionally blank in this staged diagram.
  793. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing capital Delta.
  794. The next inference is labeled left weakening rule.
  795. From the immediately preceding branch using the left weakening rule, infer antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  796. The source ends this displayed proof segment here.
  797. An upper premise slot is intentionally blank in this staged diagram.
  798. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  799. The next inference is labeled left negation rule.
  800. From the immediately preceding branch using the left negation rule, infer antecedent containing first the negation of formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  801. An upper premise slot is intentionally blank in this staged diagram.
  802. Previously derived premise with its intervening derivation omitted: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing capital Delta.
  803. The next inference is labeled left conjunction rule.
  804. From the immediately preceding branch using the left conjunction rule, infer antecedent containing first the conjunction of formula A and formula B, then Gamma; sequent arrow; succedent containing capital Delta.
  805. An upper premise slot is intentionally blank in this staged diagram.
  806. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  807. The next inference is labeled right disjunction rule.
  808. From the immediately preceding branch using the right disjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the disjunction of formula A and formula B.
  809. An upper premise slot is intentionally blank in this staged diagram.
  810. Previously derived premise with its intervening derivation omitted: antecedent containing first formula A, then Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  811. The next inference is labeled right conditional rule.
  812. From the immediately preceding branch using the right conditional rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conditional whose antecedent is formula A; and whose consequent is formula B.
  813. An upper premise slot is intentionally blank in this staged diagram.
  814. Previously derived premise with its intervening derivation omitted: antecedent containing first formula A with argument term t, then Gamma; sequent arrow; succedent containing capital Delta.
  815. The next inference is labeled left universal quantifier rule.
  816. From the immediately preceding branch using the left universal quantifier rule, infer antecedent containing first for every variable x, formula A with argument variable x, then Gamma; sequent arrow; succedent containing capital Delta.
  817. An upper premise slot is intentionally blank in this staged diagram.
  818. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument constant a.
  819. The next inference is labeled right universal quantifier rule.
  820. From the immediately preceding branch using the right universal quantifier rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then for every variable x, formula A with argument variable x.
  821. An upper premise slot is intentionally blank in this staged diagram.
  822. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  823. An upper premise slot is intentionally blank in this staged diagram.
  824. Previously derived premise with its intervening derivation omitted: antecedent containing first formula A, then capital Pi; sequent arrow; succedent containing capital Lambda.
  825. The next inference is labeled cut rule.
  826. From the two immediately preceding branches using the cut rule, infer antecedent containing first Gamma, then capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda.
  827. An upper premise slot is intentionally blank in this staged diagram.
  828. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  829. An upper premise slot is intentionally blank in this staged diagram.
  830. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula B.
  831. The next inference is labeled right conjunction rule.
  832. From the two immediately preceding branches using the right conjunction rule, infer antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then the conjunction of formula A and formula B.
  833. An upper premise slot is intentionally blank in this staged diagram.
  834. Previously derived premise with its intervening derivation omitted: antecedent containing Gamma; sequent arrow; succedent containing first capital Delta, then formula A.
  835. An upper premise slot is intentionally blank in this staged diagram.
  836. Previously derived premise with its intervening derivation omitted: antecedent containing first formula B, then capital Pi; sequent arrow; succedent containing capital Lambda.
  837. The next inference is labeled left conditional rule.
  838. From the two immediately preceding branches using the left conditional rule, infer antecedent containing first the conditional whose antecedent is formula A; and whose consequent is formula B, then Gamma, and finally capital Pi; sequent arrow; succedent containing first capital Delta, then capital Lambda.
  839. Premise or initial sequent: antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub one.
  840. The next inference is labeled identity rule.
  841. From the immediately preceding branch using the identity rule, infer antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub two.
  842. The source ends this displayed proof segment here.
  843. Premise or initial sequent: antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub two.
  844. The next inference is labeled identity rule.
  845. From the immediately preceding branch using the identity rule, infer antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub one.
  846. The source ends this displayed proof segment here.
  847. Premise or initial sequent: antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub one.
  848. The next inference is labeled identity rule.
  849. From the immediately preceding branch using the identity rule, infer antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub two.
  850. The source ends this displayed proof segment here.
  851. Premise or initial sequent: antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub two.
  852. The next inference is labeled identity rule.
  853. From the immediately preceding branch using the identity rule, infer antecedent containing first term t sub one is identical to term t sub two, then Gamma; sequent arrow; succedent containing first capital Delta, then formula A with argument term t sub one.
  854. The source ends this displayed proof segment here.
  855. Premise or initial sequent: antecedent containing formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument lowercase s.
  856. The next inference is labeled left weakening rule.
  857. From the immediately preceding branch using the left weakening rule, infer antecedent containing first lowercase s is identical to term t, then formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument lowercase s.
  858. The next inference is labeled identity rule.
  859. From the immediately preceding branch using the identity rule, infer antecedent containing first lowercase s is identical to term t, then formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument term t.
  860. Premise or initial sequent: antecedent containing no formulas; sequent arrow; succedent containing term t sub one is identical to term t sub one.
  861. The next inference is labeled left weakening rule.
  862. From the immediately preceding branch using the left weakening rule, infer antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub one.
  863. The next inference is labeled identity rule.
  864. From the immediately preceding branch using the identity rule, infer antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub two is identical to term t sub one.
  865. The source ends this displayed proof segment here.
  866. Premise or initial sequent: antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub two.
  867. The next inference is labeled left weakening rule.
  868. From the immediately preceding branch using the left weakening rule, infer antecedent containing first term t sub two is identical to term t sub three, then term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub two.
  869. The next inference is labeled identity rule.
  870. From the immediately preceding branch using the identity rule, infer antecedent containing first term t sub two is identical to term t sub three, then term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub three.
  871. The next inference is labeled left exchange rule.
  872. From the immediately preceding branch using the left exchange rule, infer antecedent containing first term t sub one is identical to term t sub two, then term t sub two is identical to term t sub three; sequent arrow; succedent containing term t sub one is identical to term t sub three.
  873. Premise or initial sequent: antecedent containing formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument lowercase s.
  874. The next inference is labeled left weakening rule.
  875. From the immediately preceding branch using the left weakening rule, infer antecedent containing first lowercase s is identical to term t, then formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument lowercase s.
  876. The next inference is labeled identity rule.
  877. From the immediately preceding branch using the identity rule, infer antecedent containing first lowercase s is identical to term t, then formula A with argument lowercase s; sequent arrow; succedent containing formula A with argument term t.
  878. Premise or initial sequent: antecedent containing no formulas; sequent arrow; succedent containing term t sub one is identical to term t sub one.
  879. The next inference is labeled left weakening rule.
  880. From the immediately preceding branch using the left weakening rule, infer antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub one.
  881. The next inference is labeled identity rule.
  882. From the immediately preceding branch using the identity rule, infer antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub two is identical to term t sub one.
  883. The source ends this displayed proof segment here.
  884. Premise or initial sequent: antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub two.
  885. The next inference is labeled left weakening rule.
  886. From the immediately preceding branch using the left weakening rule, infer antecedent containing first term t sub two is identical to term t sub three, then term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub two.
  887. The next inference is labeled identity rule.
  888. From the immediately preceding branch using the identity rule, infer antecedent containing first term t sub two is identical to term t sub three, then term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub three.
  889. The next inference is labeled left exchange rule.
  890. From the immediately preceding branch using the left exchange rule, infer antecedent containing first term t sub one is identical to term t sub two, then term t sub two is identical to term t sub three; sequent arrow; succedent containing term t sub one is identical to term t sub three.
  891. Premise or initial sequent: antecedent containing no formulas; sequent arrow; succedent containing term t sub one is identical to term t sub one.
  892. The next inference is labeled left weakening rule.
  893. From the immediately preceding branch using the left weakening rule, infer antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub one is identical to term t sub one.
  894. The next inference is labeled identity rule.
  895. From the immediately preceding branch using the identity rule, infer antecedent containing term t sub one is identical to term t sub two; sequent arrow; succedent containing term t sub two is identical to term t sub one.
  896. The source ends this displayed proof segment here.

238 proof commands

Open the complete command binding index
  1. The next inference is labeled left negation rule. source line 19.
  2. The source ends this displayed proof segment here. source line 21.
  3. The next inference is labeled right negation rule. source line 24.
  4. The source ends this displayed proof segment here. source line 26.
  5. The next inference is labeled left conjunction rule. source line 34.
  6. The source ends this displayed proof segment here. source line 36.
  7. The next inference is labeled left conjunction rule. source line 39.
  8. The source ends this displayed proof segment here. source line 41.
  9. The next inference is labeled right conjunction rule. source line 46.
  10. The source ends this displayed proof segment here. source line 48.
  11. The next inference is labeled left disjunction rule. source line 56.
  12. The source ends this displayed proof segment here. source line 58.
  13. The next inference is labeled right disjunction rule. source line 62.
  14. The source ends this displayed proof segment here. source line 64.
  15. The next inference is labeled right disjunction rule. source line 67.
  16. The source ends this displayed proof segment here. source line 69.
  17. The next inference is labeled left conditional rule. source line 78.
  18. The source ends this displayed proof segment here. source line 80.
  19. The next inference is labeled right conditional rule. source line 83.
  20. The source ends this displayed proof segment here. source line 85.
  21. The next inference is labeled left universal quantifier rule. source line 17.
  22. The source ends this displayed proof segment here. source line 19.
  23. The next inference is labeled right universal quantifier rule. source line 22.
  24. The source ends this displayed proof segment here. source line 24.
  25. The next inference is labeled left existential quantifier rule. source line 38.
  26. The source ends this displayed proof segment here. source line 40.
  27. The next inference is labeled right existential quantifier rule. source line 43.
  28. The source ends this displayed proof segment here. source line 45.
  29. The next inference is labeled right existential quantifier rule. source line 63.
  30. The next inference is labeled left existential quantifier rule, marked with an asterisk to identify this intentionally invalid inference. source line 89.
  31. The next inference is labeled right universal quantifier rule. source line 91.
  32. The source ends this displayed proof segment here. source line 93.
  33. The next inference is labeled right universal quantifier rule, marked with an asterisk to identify this intentionally invalid inference. source line 96.
  34. The next inference is labeled left existential quantifier rule. source line 98.
  35. The next inference is labeled left weakening rule. source line 27.
  36. The source ends this displayed proof segment here. source line 29.
  37. The next inference is labeled right weakening rule. source line 32.
  38. The source ends this displayed proof segment here. source line 34.
  39. The next inference is labeled left contraction rule. source line 41.
  40. The source ends this displayed proof segment here. source line 43.
  41. The next inference is labeled right contraction rule. source line 46.
  42. The source ends this displayed proof segment here. source line 48.
  43. The next inference is labeled left exchange rule. source line 55.
  44. The source ends this displayed proof segment here. source line 57.
  45. The next inference is labeled right exchange rule. source line 60.
  46. The source ends this displayed proof segment here. source line 62.
  47. The next inference is labeled cut rule. source line 74.
  48. The source ends this displayed proof segment here. source line 76.
  49. The next inference is labeled left weakening rule. source line 43.
  50. The next inference is labeled left weakening rule. source line 53.
  51. The next inference is labeled left weakening rule. source line 61.
  52. The next inference is labeled left exchange rule. source line 63.
  53. The next inference is labeled left exchange rule. source line 69.
  54. The next inference is labeled left weakening rule. source line 77.
  55. The next inference is labeled right conjunction rule. source line 85.
  56. The next inference is labeled left weakening rule. source line 95.
  57. The next inference is labeled left exchange rule. source line 97.
  58. The next inference is labeled left weakening rule. source line 100.
  59. The next inference is labeled right conjunction rule. source line 102.
  60. The next inference is labeled left weakening rule. source line 109.
  61. The next inference is labeled left weakening rule. source line 112.
  62. The next inference is labeled left exchange rule. source line 114.
  63. The next inference is labeled right conjunction rule. source line 116.
  64. An upper premise slot is intentionally blank in this staged diagram. source line 21.
  65. An upper premise slot is intentionally blank in this staged diagram. source line 32.
  66. The next inference is labeled left conjunction rule. source line 33.
  67. The next inference is labeled left conjunction rule. source line 43.
  68. An upper premise slot is intentionally blank in this staged diagram. source line 56.
  69. An upper premise slot is intentionally blank in this staged diagram. source line 71.
  70. The next inference is labeled right conditional rule. source line 73.
  71. An upper premise slot is intentionally blank in this staged diagram. source line 80.
  72. An upper premise slot is intentionally blank in this staged diagram. source line 82.
  73. The next inference is labeled left disjunction rule. source line 84.
  74. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended. source line 86.
  75. The next inference is labeled right conditional rule. source line 88.
  76. An upper premise slot is intentionally blank in this staged diagram. source line 95.
  77. The next inference is labeled left weakening rule. source line 98.
  78. The next inference is labeled left exchange rule. source line 100.
  79. The next inference is labeled left disjunction rule. source line 102.
  80. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended. source line 104.
  81. The next inference is labeled right conditional rule. source line 106.
  82. An upper premise slot is intentionally blank in this staged diagram. source line 114.
  83. The next inference is labeled left negation rule. source line 116.
  84. The next inference is labeled left weakening rule. source line 119.
  85. The next inference is labeled left exchange rule. source line 121.
  86. The next inference is labeled left disjunction rule. source line 123.
  87. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended. source line 125.
  88. The next inference is labeled right conditional rule. source line 127.
  89. The next inference is labeled right weakening rule. source line 134.
  90. The next inference is labeled right exchange rule. source line 136.
  91. The next inference is labeled left negation rule. source line 138.
  92. The next inference is labeled left weakening rule. source line 141.
  93. The next inference is labeled left exchange rule. source line 143.
  94. The next inference is labeled left disjunction rule. source line 145.
  95. The next inference is labeled right exchange rule. Source disclosure. Preserve and speak the printed right-exchange label, then disclose that the changed side is the antecedent and that left exchange appears intended. source line 147.
  96. The next inference is labeled right conditional rule. source line 149.
  97. An upper premise slot is intentionally blank in this staged diagram. source line 161.
  98. An upper premise slot is intentionally blank in this staged diagram. source line 170.
  99. The next inference is labeled right negation rule. source line 172.
  100. An upper premise slot is intentionally blank in this staged diagram. source line 182.
  101. The next inference is labeled left conjunction rule. source line 184.
  102. The next inference is labeled right negation rule. source line 186.
  103. The next inference is labeled left negation rule. source line 192.
  104. An upper premise slot is intentionally blank in this staged diagram. source line 194.
  105. The next inference is labeled question mark placeholder for an inference rule not yet chosen. source line 195.
  106. The next inference is labeled left negation rule. source line 197.
  107. The next inference is labeled left disjunction rule. source line 199.
  108. The next inference is labeled left exchange rule. source line 201.
  109. The next inference is labeled left conjunction rule. source line 203.
  110. The next inference is labeled right negation rule. source line 205.
  111. An upper premise slot is intentionally blank in this staged diagram. source line 214.
  112. An upper premise slot is intentionally blank in this staged diagram. source line 217.
  113. The next inference is labeled left disjunction rule. source line 220.
  114. The next inference is labeled left exchange rule. source line 222.
  115. The next inference is labeled right negation rule. source line 224.
  116. The next inference is labeled left conjunction rule. source line 230.
  117. The next inference is labeled left negation rule. source line 232.
  118. The next inference is labeled left conjunction rule. source line 236.
  119. The next inference is labeled left negation rule. source line 238.
  120. The next inference is labeled left disjunction rule. source line 241.
  121. The next inference is labeled left exchange rule. source line 243.
  122. The next inference is labeled right negation rule. source line 245.
  123. An upper premise slot is intentionally blank in this staged diagram. source line 258.
  124. The next inference is labeled right disjunction rule. source line 260.
  125. The source ends this displayed proof segment here. source line 262.
  126. An upper premise slot is intentionally blank in this staged diagram. source line 263.
  127. The next inference is labeled right negation rule. source line 265.
  128. The next inference is labeled right disjunction rule. source line 267.
  129. An upper premise slot is intentionally blank in this staged diagram. source line 276.
  130. The next inference is labeled right disjunction rule. source line 278.
  131. The next inference is labeled right contraction rule. source line 280.
  132. The next inference is labeled right negation rule. source line 287.
  133. The next inference is labeled right disjunction rule. source line 289.
  134. The next inference is labeled right exchange rule. source line 291.
  135. The next inference is labeled right disjunction rule. source line 293.
  136. The next inference is labeled right contraction rule. source line 295.
  137. An upper premise slot is intentionally blank in this staged diagram. source line 33.
  138. An upper premise slot is intentionally blank in this staged diagram. source line 41.
  139. The next inference is labeled left existential quantifier rule. source line 43.
  140. An upper premise slot is intentionally blank in this staged diagram. source line 48.
  141. The next inference is labeled left negation rule. source line 50.
  142. The next inference is labeled left exchange rule. source line 52.
  143. The next inference is labeled right negation rule. source line 54.
  144. The next inference is labeled left existential quantifier rule. source line 56.
  145. The next inference is labeled left universal quantifier rule. source line 66.
  146. The next inference is labeled left negation rule. source line 68.
  147. The next inference is labeled left exchange rule. source line 70.
  148. The next inference is labeled right negation rule. source line 72.
  149. The next inference is labeled left existential quantifier rule. source line 74.
  150. An upper premise slot is intentionally blank in this staged diagram. source line 55.
  151. The next inference is labeled left contraction rule. source line 57.
  152. The next inference is labeled left exchange rule. source line 59.
  153. The next inference is labeled left weakening rule. source line 61.
  154. An upper premise slot is intentionally blank in this staged diagram. source line 115.
  155. The next inference is labeled derivation pi sub zero. source line 116.
  156. An upper premise slot is intentionally blank in this staged diagram. source line 118.
  157. The next inference is labeled derivation pi sub one. source line 119.
  158. The next inference is labeled cut rule. source line 121.
  159. An upper premise slot is intentionally blank in this staged diagram. source line 31.
  160. The next inference is labeled derivation pi sub zero. source line 32.
  161. An upper premise slot is intentionally blank in this staged diagram. source line 34.
  162. The next inference is labeled derivation pi sub one. source line 35.
  163. The next inference is labeled cut rule. source line 37.
  164. The next inference is labeled right negation rule. source line 61.
  165. An upper premise slot is intentionally blank in this staged diagram. source line 63.
  166. The next inference is labeled derivation pi sub one. source line 64.
  167. The next inference is labeled cut rule. source line 66.
  168. An upper premise slot is intentionally blank in this staged diagram. source line 85.
  169. The next inference is labeled derivation pi. source line 86.
  170. The next inference is labeled left negation rule. source line 89.
  171. The next inference is labeled left exchange rule. source line 91.
  172. The next inference is labeled cut rule. source line 93.
  173. An upper premise slot is intentionally blank in this staged diagram. source line 111.
  174. The next inference is labeled derivation pi sub zero. source line 112.
  175. The next inference is labeled right negation rule. source line 114.
  176. An upper premise slot is intentionally blank in this staged diagram. source line 116.
  177. The next inference is labeled derivation pi sub one. source line 117.
  178. The next inference is labeled cut rule. source line 119.
  179. The next inference is labeled left conjunction rule. source line 41.
  180. The source ends this displayed proof segment here. source line 43.
  181. The next inference is labeled left conjunction rule. source line 45.
  182. The next inference is labeled right conjunction rule. source line 52.
  183. The next inference is labeled left negation rule. source line 71.
  184. A double inference line abbreviates one or more weakening, contraction, or exchange steps. source line 73.
  185. The next inference is labeled left negation rule. source line 76.
  186. A double inference line abbreviates one or more weakening, contraction, or exchange steps. source line 78.
  187. The next inference is labeled left disjunction rule. source line 80.
  188. The next inference is labeled right disjunction rule. source line 89.
  189. The source ends this displayed proof segment here. source line 91.
  190. The next inference is labeled right disjunction rule. source line 93.
  191. The next inference is labeled left conditional rule. source line 113.
  192. The next inference is labeled left negation rule. source line 120.
  193. The next inference is labeled left exchange rule. source line 122.
  194. The next inference is labeled right weakening rule. source line 124.
  195. The next inference is labeled right conditional rule. source line 126.
  196. The source ends this displayed proof segment here. source line 128.
  197. The next inference is labeled left weakening rule. source line 130.
  198. The next inference is labeled right conditional rule. source line 132.
  199. The next inference is labeled right existential quantifier rule. source line 46.
  200. The next inference is labeled left universal quantifier rule. source line 53.
  201. An upper premise slot is intentionally blank in this staged diagram. source line 80.
  202. The next inference is labeled left weakening rule. source line 82.
  203. The source ends this displayed proof segment here. source line 84.
  204. An upper premise slot is intentionally blank in this staged diagram. source line 85.
  205. The next inference is labeled right weakening rule. source line 87.
  206. An upper premise slot is intentionally blank in this staged diagram. source line 107.
  207. The next inference is labeled left negation rule. source line 109.
  208. An upper premise slot is intentionally blank in this staged diagram. source line 136.
  209. The next inference is labeled left conjunction rule. source line 138.
  210. An upper premise slot is intentionally blank in this staged diagram. source line 164.
  211. The next inference is labeled right disjunction rule. source line 166.
  212. An upper premise slot is intentionally blank in this staged diagram. source line 186.
  213. The next inference is labeled right conditional rule. source line 188.
  214. An upper premise slot is intentionally blank in this staged diagram. source line 211.
  215. The next inference is labeled left universal quantifier rule. source line 213.
  216. An upper premise slot is intentionally blank in this staged diagram. source line 231.
  217. The next inference is labeled right universal quantifier rule. source line 233.
  218. An upper premise slot is intentionally blank in this staged diagram. source line 272.
  219. An upper premise slot is intentionally blank in this staged diagram. source line 274.
  220. The next inference is labeled cut rule. source line 276.
  221. An upper premise slot is intentionally blank in this staged diagram. source line 292.
  222. An upper premise slot is intentionally blank in this staged diagram. source line 294.
  223. The next inference is labeled right conjunction rule. source line 296.
  224. An upper premise slot is intentionally blank in this staged diagram. source line 311.
  225. An upper premise slot is intentionally blank in this staged diagram. source line 313.
  226. The next inference is labeled left conditional rule. source line 315.
  227. The next inference is labeled identity rule. source line 24.
  228. The source ends this displayed proof segment here. source line 26.
  229. The next inference is labeled identity rule. source line 29.
  230. The source ends this displayed proof segment here. source line 31.
  231. The next inference is labeled left weakening rule. source line 39.
  232. The next inference is labeled identity rule. source line 41.
  233. The next inference is labeled left weakening rule. source line 50.
  234. The next inference is labeled identity rule. source line 52.
  235. The source ends this displayed proof segment here. source line 54.
  236. The next inference is labeled left weakening rule. source line 56.
  237. The next inference is labeled identity rule. source line 58.
  238. The next inference is labeled left exchange rule. source line 60.