Many-valued logics

Three-valued Logics

Equation form expr-01208e159d2aec8c

∧\land

Read as: conjunction

Means: conjunction

12 occurrences in this chapter

Equation form expr-043b130790fae68b

¬A∨B\lnot !A \lor !B

Read as: not A, or B

Means: not A, or B

1 occurrence in this chapter

Equation form expr-069ac2aed6bc166f

→~Ł3\tf{\lif}[\LogLuk[3]]

Read as: the conditional truth function in three valued Lukasiewicz logic

Means: the conditional truth function in three valued Lukasiewicz logic

1 occurrence in this chapter

Equation form expr-0d4cbaecc5a93859

⊥~=F\tf{\lfalse} = \False

Read as: the truth function of falsity equal to false

Means: the truth function of falsity equal to false

1 occurrence in this chapter

Equation form expr-0e2c28d3cdde75db

Kw\LogKw

Read as: K subscript w

Means: K subscript w

3 occurrences in this chapter

Equation form expr-0ee8cf6488bca418

¬~(U)=∨~(U,U)=∧~(U,U)=→~(U,U)=U\tf{\lnot}(\Undef) = \tf{\lor}(\Undef, \Undef) = \tf{\land}(\Undef, \Undef) = \tf{\lif}(\Undef, \Undef) = \Undef

Read as: negation of U, disjunction of U and U, conjunction of U and U, and the conditional from U to U all have value U

Means: negation of U, disjunction of U and U, conjunction of U and U, and the conditional from U to U all have value U

1 occurrence in this chapter

Equation form expr-123ddce011c7f73e

v′\pAssign {v'}

Read as: the assignment v prime

Means: the assignment v prime

1 occurrence in this chapter

Equation form expr-148de9c5a7a44d19

pp

Read as: p

Means: p

13 occurrences in this chapter

Equation form expr-14baaa8c23dd4738

v′¯C(C)=T\pValue {v'}(!C)[\LogCL] = \True

Read as: the classical value of C under v prime is true

Means: the classical value of C under v prime is true

1 occurrence in this chapter

Equation form expr-15371170798ae4ac

C\LogCL

Read as: classical logic

Means: classical logic

1 occurrence in this chapter

Equation form expr-160f609692c14c04

¬(p∨q)↔(¬p∧¬q)\lnot(p \lor q) \liff (\lnot p \land \lnot q)

Read as: not, open scope, p or q, close scope, if and only if, open scope, not p and not q, close scope

Means: not, open scope, p or q, close scope, if and only if, open scope, not p and not q, close scope

1 occurrence in this chapter

Equation form expr-16f38629e1f81ade

Ł3\LogLuk[3]

Read as: three valued Lukasiewicz logic

Means: three valued Lukasiewicz logic

5 occurrences in this chapter

Equation form expr-17e0a63336f3eac5

LP\LogLP

Read as: L P

Means: L P

8 occurrences in this chapter

Equation form expr-1b16b1df538ba12d

nn

Read as: n

Means: n

1 occurrence in this chapter

Equation form expr-1fb0fae2b2e81a2d

G\LogGod

Read as: G

Means: G

1 occurrence in this chapter

Equation form expr-20ddfc618734c4ad

∨~G\tf{\lor}[\LogGod]

Read as: the disjunction truth function in Goedel logic

Means: the disjunction truth function in Goedel logic

1 occurrence in this chapter

Equation form expr-215dbf551bc6bc67

{T}\{\True\}

Read as: the singleton set containing true

Means: the singleton set containing true

1 occurrence in this chapter

Equation form expr-22fc4a7f60284f42

(¬p∧p)→q)(\lnot p \land p) \lif q)

Read as: if, open scope, not p and p, close scope, then q, followed in the source by an unmatched closing parenthesis

Means: if, open scope, not p and p, close scope, then q, followed in the source by an unmatched closing parenthesis

1 occurrence in this chapter

Equation form expr-24c7b022f83d2803

V+={T,U}V^+=\{\True,\Undef\}

Read as: V plus is the set containing true and U

Means: V plus is the set containing true and U

1 occurrence in this chapter

Equation form expr-284c47be48fb0c66

¬\lnot

Read as: negation

Means: negation

14 occurrences in this chapter

Equation form expr-28c682f88e571975

◇~\tf{\Diamond}

Read as: the possibility truth function

Means: the possibility truth function

1 occurrence in this chapter

Equation form expr-2a8f097676febe82

V+={T,U}V^+ = \{\True, \Undef\}

Read as: V plus is the set containing true and U

Means: V plus is the set containing true and U

5 occurrences in this chapter

Equation form expr-2ef86478f3d3c710

V={F,U,T}V = \{\False, \Undef,\True\}

Read as: V is the set containing false, U, and true

Means: V is the set containing false, U, and true

1 occurrence in this chapter

Equation form expr-3163c74b28295918

∨~Ł3\tf{\lor}[\LogLuk[3]]

Read as: the disjunction truth function in three valued Lukasiewicz logic

Means: the disjunction truth function in three valued Lukasiewicz logic

1 occurrence in this chapter

Equation form expr-33175b010415711c

¬q,p→q⊨¬p\lnot q, p \lif q \Entails \lnot p

Read as: not q, together with if p then q, entails not p

Means: not q, together with if p then q, entails not p

1 occurrence in this chapter

Equation form expr-352daf9d368686e5

v¯Ks(B)=T\pValue v(!B)[\LogKs] = \True

Read as: the strong Kleene value of B under v is true

Means: the strong Kleene value of B under v is true

3 occurrences in this chapter

Equation form expr-382ee137aedcf4de

A≡p!A \ident p

Read as: A is syntactically identical to p

Means: A is syntactically identical to p

1 occurrence in this chapter

Equation form expr-3b49e1a3f6189bbd

A↔B!A \liff !B

Read as: A if and only if B

Means: A if and only if B

1 occurrence in this chapter

Equation form expr-3f6855a646122c85

v¯Ks(¬B)=F\pValue v(\lnot!B)[\LogKs] = \False

Read as: the strong Kleene value of not B under v is false

Means: the strong Kleene value of not B under v is false

1 occurrence in this chapter

Equation form expr-3f864991b716ea11

⊭LPA\Entails/[\LogLP] !A

Read as: A is not valid in the logic of paradox

Means: A is not valid in the logic of paradox

1 occurrence in this chapter

Equation form expr-428908baa70543da

p⊨p∧pp \Entails p \land p

Read as: p entails the conjunction of p with p

Means: p entails the conjunction of p with p

3 occurrences in this chapter

Equation form expr-47bb5ad953a876ee

RM3\LogRM[3]

Read as: R M subscript three

Means: R M subscript three

2 occurrences in this chapter

Equation form expr-4b001c00ec881fed

∧~(T,U)=∧~(U,T)=U.\tf{\land}(\True, \Undef) = \tf{\land}(\Undef, \True) = \Undef.

Read as: conjunction of true and U, and conjunction of U and true, both have value U

Means: conjunction of true and U, and conjunction of U and true, both have value U

1 occurrence in this chapter

Equation form expr-4d4239bf8a56d4e4

v¯Ks(A)=F\pValue v(!A)[\LogKs] = \False

Read as: the strong Kleene value of A under v is false

Means: the strong Kleene value of A under v is false

2 occurrences in this chapter

Equation form expr-4e07408562bedb8b

33

Read as: three

Means: three

2 occurrences in this chapter

Equation form expr-4ed18699be22b927

v¯(A)\pValue v(!A)

Read as: the value of A under v

Means: the value of A under v

1 occurrence in this chapter

Equation form expr-4f6fd48d045ea8d2

∼p\mathord{\sim}p

Read as: external negation of p

Means: external negation of p

1 occurrence in this chapter

Equation form expr-5045ba03c70ef68c

(A→B)∧(B→A)(!A \lif !B) \land (!B \lif !A)

Read as: the conjunction of if A then B, with if B then A

Means: the conjunction of if A then B, with if B then A

1 occurrence in this chapter

Equation form expr-5090e2573ea7884e

v¯Ks(B∧C)=T\pValue v(!B \land !C)[\LogKs] = \True

Read as: the strong Kleene value of the conjunction of B and C under v is true

Means: the strong Kleene value of the conjunction of B and C under v is true

1 occurrence in this chapter

Equation form expr-5136fc4246e7d497

◇\Diamond

Read as: the possibility operator

Means: the possibility operator

1 occurrence in this chapter

Equation form expr-52237fac4b282473

∧~Ks\tf{\land}[\LogKs]

Read as: the conjunction truth function in strong Kleene logic

Means: the conjunction truth function in strong Kleene logic

3 occurrences in this chapter

Equation form expr-529ad2daacd7efcb

∨\lor

Read as: disjunction

Means: disjunction

11 occurrences in this chapter

Equation form expr-534731714098afd2

◇p↔¬□¬p\Diamond p \liff \lnot \Box \lnot p

Read as: the biconditional with possibility of p on the left and the negation of the necessity of the negation of p on the right

Means: the biconditional with possibility of p on the left and the negation of the necessity of the negation of p on the right

1 occurrence in this chapter

Equation form expr-55b13714e82ad56c

v′:At0→{F,T}\pAssign {v'}\colon \PVar \to \{\False, \True\}

Read as: v prime, mapping the propositional variables to the set containing false and true,

Means: v prime, mapping the propositional variables to the set containing false and true,

1 occurrence in this chapter

Equation form expr-563cb0d9ba9e242b

v′¯C(B)=F\pValue {v'}(!B)[\LogCL] = \False

Read as: the classical value of B under v prime is false

Means: the classical value of B under v prime is false

2 occurrences in this chapter

Equation form expr-57524682e75835c3

∨~Kw\tf{\lor}[\LogKw]

Read as: the disjunction truth function in weak Kleene logic

Means: the disjunction truth function in weak Kleene logic

1 occurrence in this chapter

Equation form expr-5768da80750ca2ee

v¯(B→C)=v¯(¬(B∧¬C))\pValue v(!B \lif !C) = \pValue v(\lnot(!B \land \lnot !C))

Read as: the value under v of if B then C equals the value under v of not, open scope, B and not C, close scope

Means: the value under v of if B then C equals the value under v of not, open scope, B and not C, close scope

1 occurrence in this chapter

Equation form expr-58197ff8ca27696b

¬~\tf{\lnot}

Read as: the negation truth function

Means: the negation truth function

3 occurrences in this chapter

Equation form expr-59cf7cd4fdb9ff5b

T\True

Read as: true

Means: true

103 occurrences in this chapter

Equation form expr-59eb94792e9fcc1b

p→q,q→r⊨p→rp \lif q, q\lif r \Entails p \lif r

Read as: if p then q, together with if q then r, entails if p then r

Means: if p then q, together with if q then r, entails if p then r

1 occurrence in this chapter

Equation form expr-5a6380c1b37b78f5

p⊨p∨qp \Entails p \lor q

Read as: p entails the disjunction of p and q

Means: p entails the disjunction of p and q

5 occurrences in this chapter

Equation form expr-5a67d1b997e85a2a

V={T,U,F}V = \{\True, \Undef, \False\}

Read as: V is the set containing true, U, and false

Means: V is the set containing true, U, and false

7 occurrences in this chapter

Equation form expr-5d94be88a66c614f

→~G\tf{\lif}[\LogGod]

Read as: the conditional truth function in Goedel logic

Means: the conditional truth function in Goedel logic

1 occurrence in this chapter

Equation form expr-616522c60d7b122c

∼\sim

Read as: the external negation operator

Means: the external negation operator

2 occurrences in this chapter

Equation form expr-62100fd7f0329cf4

v(p)∈{T,F}\pAssign v(p) \in \{\True, \False\}

Read as: assignment v gives p a value in the set containing true and false

Means: assignment v gives p a value in the set containing true and false

1 occurrence in this chapter

Equation form expr-64dd9ec086ad4f5b

U\Undef

Read as: U

Means: U

93 occurrences in this chapter

Equation form expr-6897e6e945c61fe1

v¯(B∨C)=v¯(¬(¬B∧¬C))\pValue v(!B \lor !C) = \pValue v(\lnot(\lnot!B \land \lnot !C))

Read as: the value under v of B or C equals the value under v of not, open scope, not B and not C, close scope

Means: the value under v of B or C equals the value under v of not, open scope, not B and not C, close scope

1 occurrence in this chapter

Equation form expr-696037095e038a87

□~\tf{\Box}

Read as: the necessity truth function

Means: the necessity truth function

1 occurrence in this chapter

Equation form expr-6b4edfc5fe38d079

G3\LogGod[3]

Read as: three valued Goedel logic

Means: three valued Goedel logic

6 occurrences in this chapter

Equation form expr-6b86b273ff34fce1

11

Read as: one

Means: one

1 occurrence in this chapter

Equation form expr-6da11151a32fddba

Γ⊭CB\Gamma \Entails/[\LogCL] !B

Read as: Gamma does not entail B in classical logic

Means: Gamma does not entail B in classical logic

1 occurrence in this chapter

Equation form expr-6ff6952079b48b47

v′¯C(¬B)=T\pValue {v'}(\lnot !B)[\LogCL] = \True

Read as: the classical value of not B under v prime is true

Means: the classical value of not B under v prime is true

1 occurrence in this chapter

Equation form expr-720edbf3808e1a0a

Γ⊭LB\Gamma \Entails/[\Log L] !B

Read as: Gamma does not entail B in logic L

Means: Gamma does not entail B in logic L

1 occurrence in this chapter

Equation form expr-72f83412bc8d55ea

¬p,p⊭q\lnot p, p \Entails/ q

Read as: not p, together with p, does not entail q

Means: not p, together with p, does not entail q

1 occurrence in this chapter

Equation form expr-75dbdf1fbb537d75

q)q\smash{)}

Read as: q, then close parenthesis

Means: q, then close parenthesis

1 occurrence in this chapter

Equation form expr-78a09349ed42e07b

p∧¬pp \land \lnot p

Read as: p and not p

Means: p and not p

1 occurrence in this chapter

Equation form expr-78dd27eeb74ae588

v¯(A↔B)=v¯((A→B)∧(B→A))\pValue v(!A \liff !B) = \pValue v((!A \lif !B) \land (!B \lif !A))

Read as: the value under v of A if and only if B equals the value under v of the conjunction of if A then B, with if B then A

Means: the value under v of A if and only if B equals the value under v of the conjunction of if A then B, with if B then A

1 occurrence in this chapter

Equation form expr-7a39e3222506cef4

V+={T,U}V^+=\{\True, \Undef\}

Read as: V plus is the set containing true and U

Means: V plus is the set containing true and U

1 occurrence in this chapter

Equation form expr-7c36cd49875e64f5

¬¬p∨¬p(p→q)∨(q→p)¬(p∧q)→(¬p∨¬q)(p→q)∨(q→r)∨(r→s)& \lnot\lnot p \lor \lnot p\\ & (p \lif q) \lor (q \lif p) \\ & \lnot(p \land q) \lif (\lnot p \lor \lnot q) \\ & (p \lif q) \lor (q \lif r) \lor (r \lif s)

Read as: Four formulas, in source row order. First, not not p, or not p. Second, either if p then q, or if q then p. Third, if not, open scope, p and q, close scope, then, open scope, not p or not q, close scope. Fourth, the disjunction of if p then q, if q then r, and if r then s.

Means: Four formulas, in source row order. First, not not p, or not p. Second, either if p then q, or if q then p. Third, if not, open scope, p and q, close scope, then, open scope, not p or not q, close scope. Fourth, the disjunction of if p then q, if q then r, and if r then s.

1 occurrence in this chapter

Equation form expr-8124eb9a8372e942

∧~G\tf{\land}[\LogGod]

Read as: the conjunction truth function in Goedel logic

Means: the conjunction truth function in Goedel logic

1 occurrence in this chapter

Equation form expr-81eb243294171ac5

∧~Kw\tf{\land}[\LogKw]

Read as: the conjunction truth function in weak Kleene logic

Means: the conjunction truth function in weak Kleene logic

1 occurrence in this chapter

Equation form expr-8238c028f61fc0f7

A!A

Read as: A

Means: A

13 occurrences in this chapter

Equation form expr-82623f345da734fd

∧~Ł3\tf{\land}[\LogLuk[3]]

Read as: the conjunction truth function in three valued Lukasiewicz logic

Means: the conjunction truth function in three valued Lukasiewicz logic

1 occurrence in this chapter

Equation form expr-8375313438d45878

(p→q)→(¬p∨q)¬(¬p∧¬q)→(p∨q)((p→q)→p)→p¬(p→q)→(p∧¬q)& (p \lif q) \lif (\lnot p \lor q) \\ & \lnot(\lnot p \land \lnot q) \lif (p \lor q) \\ & ((p \lif q) \lif p) \lif p \\ & \lnot(p \lif q) \lif (p \land \lnot q)

Read as: Four formulas, in source row order. First, if, open scope, if p then q, close scope, then, open scope, not p or q, close scope. Second, if not, open scope, not p and not q, close scope, then, open scope, p or q, close scope. Third, if, open scope, if, open scope, if p then q, close scope, then p, close scope, then p. Fourth, if not, open scope, if p then q, close scope, then, open scope, p and not q, close scope.

Means: Four formulas, in source row order. First, if, open scope, if p then q, close scope, then, open scope, not p or q, close scope. Second, if not, open scope, not p and not q, close scope, then, open scope, p or q, close scope. Third, if, open scope, if, open scope, if p then q, close scope, then p, close scope, then p. Fourth, if not, open scope, if p then q, close scope, then, open scope, p and not q, close scope.

1 occurrence in this chapter

Equation form expr-8547b80de065fac2

∼~\tf{\sim}

Read as: the external negation truth function

Means: the external negation truth function

1 occurrence in this chapter

Equation form expr-85f60ccf031ac51b

→~Kw\tf{\lif}[\LogKw]

Read as: the conditional truth function in weak Kleene logic

Means: the conditional truth function in weak Kleene logic

1 occurrence in this chapter

Equation form expr-886721686ccc2e5b

A≡(B∨C)!A \ident (!B \lor !C)

Read as: A is syntactically identical to the disjunction of B and C

Means: A is syntactically identical to the disjunction of B and C

1 occurrence in this chapter

Equation form expr-889336cffefbfda1

V+={T}V^+ = \{\True\}

Read as: V plus is the singleton set containing true

Means: V plus is the singleton set containing true

5 occurrences in this chapter

Equation form expr-89121db8e9897dc6

¬¬p∨¬p\lnot\lnot p \lor \lnot p

Read as: the disjunction of the double negation of p with the negation of p

Means: the disjunction of the double negation of p with the negation of p

1 occurrence in this chapter

Equation form expr-8dcea9becec4732c

v(p)=U\pAssign v(p) = \Undef

Read as: assignment v gives p the value U

Means: assignment v gives p the value U

3 occurrences in this chapter

Equation form expr-8e35c2cd3bf6641b

qq

Read as: q

Means: q

5 occurrences in this chapter

Equation form expr-8e557f58a3cc9f7c

→~(T,U)=U\tf{\lif}(\True, \Undef) = \Undef

Read as: the conditional with true antecedent and U consequent has value U

Means: the conditional with true antecedent and U consequent has value U

1 occurrence in this chapter

Equation form expr-8e62c3a6292cd2ea

L\Log L

Read as: L

Means: L

1 occurrence in this chapter

Equation form expr-90f52b7465bd0323

v¯(¬◇(p∧¬p))=U\pValue v(\lnot \Diamond(p \land \lnot p)) = \Undef

Read as: the value under v of not possibly, open scope, p and not p, close scope, equals U

Means: the value under v of not possibly, open scope, p and not p, close scope, equals U

1 occurrence in this chapter

Equation form expr-9115802679501615

→~(U,F)=F\tf{\lif}(\Undef, \False) = \False

Read as: the conditional with U antecedent and false consequent has value false

Means: the conditional with U antecedent and false consequent has value false

1 occurrence in this chapter

Equation form expr-921d834a8a5dc250

v(p)=T\pAssign v(p) = \True

Read as: assignment v gives p the value true

Means: assignment v gives p the value true

1 occurrence in this chapter

Equation form expr-9244c801c50f58fd

(p→q)∨(q→p)(p \lif q) \lor (q \lif p)

Read as: the disjunction of the conditional from p to q with the conditional from q to p

Means: the disjunction of the conditional from p to q with the conditional from q to p

1 occurrence in this chapter

Equation form expr-942d22889fe96b82

¬p→(p→q)\lnot p \lif (p \lif q)

Read as: if not p, then, open scope, if p then q, close scope

Means: if not p, then, open scope, if p then q, close scope

1 occurrence in this chapter

Equation form expr-96362a0075907dc5

A→A!A \lif !A

Read as: the conditional from A to A

Means: the conditional from A to A

2 occurrences in this chapter

Equation form expr-983b33490f9bc3d6

(p\smash{(}p

Read as: open parenthesis, then p

Means: open parenthesis, then p

1 occurrence in this chapter

Equation form expr-99adc3223e0a4ae7

p→(q→p)p \lif (q \lif p)

Read as: if p, then, open scope, if q then p, close scope

Means: if p, then, open scope, if q then p, close scope

1 occurrence in this chapter

Equation form expr-9a3cf63b37dacd6b

¬◇(p∧¬p)\lnot \Diamond(p \land \lnot p)

Read as: not possibly, open scope, p and not p, close scope

Means: not possibly, open scope, p and not p, close scope

1 occurrence in this chapter

Equation form expr-9e002426f28e8210

v¯Ks(¬B)=T\pValue v(\lnot!B)[\LogKs] = \True

Read as: the strong Kleene value of not B under v is true

Means: the strong Kleene value of not B under v is true

1 occurrence in this chapter

Equation form expr-9eb5c6a4d0ba7fc3

v¯LP(A)=v¯Ks(A)=F\pValue v(!A)[\LogLP] = \pValue v[\LogKs](!A) = \False

Read as: the value of A under v in the logic of paradox equals its strong Kleene value under v, and both equal false

Means: the value of A under v in the logic of paradox equals its strong Kleene value under v, and both equal false

1 occurrence in this chapter

Equation form expr-9efe52b1fc4300ee

⊨CA\Entails[\LogCL] !A

Read as: A is valid in classical logic

Means: A is valid in classical logic

1 occurrence in this chapter

Equation form expr-a318c24216defe20

++

Read as: the plus operator

Means: the plus operator

2 occurrences in this chapter

Equation form expr-a4f9b3a61fd77bdc

v′¯C(B∧C)=T\pValue {v'}(!B \land !C)[\LogCL] = \True

Read as: the classical value of the conjunction of B and C under v prime is true

Means: the classical value of the conjunction of B and C under v prime is true

1 occurrence in this chapter

Equation form expr-a8a7c6373dcc5e37

→~(U,F)=U\tf{\lif}(\Undef, \False) = \Undef

Read as: the conditional with U antecedent and false consequent has value U

Means: the conditional with U antecedent and false consequent has value U

1 occurrence in this chapter

Equation form expr-a923dcf28617bc76

q→pq \lif p

Read as: the conditional from q to p

Means: the conditional from q to p

2 occurrences in this chapter

Equation form expr-aa122470d608ed04

v¯(A)=U\pValue{v}(!A) = \Undef

Read as: the value of A under v is U

Means: the value of A under v is U

1 occurrence in this chapter

Equation form expr-aae570b52fd91ba2

v′¯C(A)=F\pValue{v'}[\LogCL](!A) = \False

Read as: the classical value of A under v prime is false

Means: the classical value of A under v prime is false

1 occurrence in this chapter

Equation form expr-ac5b11ed7afd3d9c

v′¯C(C)=F\pValue {v'}(!C)[\LogCL] = \False

Read as: the classical value of C under v prime is false

Means: the classical value of C under v prime is false

1 occurrence in this chapter

Equation form expr-ac7698cc0feb251a

→~(F,U)=T\tf{\lif}(\False, \Undef) = \True

Read as: the conditional with false antecedent and U consequent to true

Means: the conditional with false antecedent and U consequent to true

1 occurrence in this chapter

Equation form expr-b24c2720089cdd3e

↔\liff

Read as: the biconditional

Means: the biconditional

1 occurrence in this chapter

Equation form expr-b5b0910c652c270d

v′¯C(A)=T\pValue {v'}(!A)[\LogCL] = \True

Read as: the classical value of A under v prime is true

Means: the classical value of A under v prime is true

2 occurrences in this chapter

Equation form expr-b7459cd0c0f0525d

v′¯C(B)=T\pValue {v'}(!B)[\LogCL] = \True

Read as: the classical value of B under v prime is true

Means: the classical value of B under v prime is true

2 occurrences in this chapter

Equation form expr-baacfd9d189243cd

◇A\Diamond !A

Read as: possibly A

Means: possibly A

1 occurrence in this chapter

Equation form expr-bcf72d9d8c6df720

¬p,p⊨q\lnot p, p \Entails q

Read as: not p, together with p, entails q

Means: not p, together with p, entails q

2 occurrences in this chapter

Equation form expr-bda7750408b17618

¬(p∧¬p)\lnot(p \land \lnot p)

Read as: not, open scope, p and not p, close scope

Means: not, open scope, p and not p, close scope

1 occurrence in this chapter

Equation form expr-bdcb7192fe841cb0

v\pAssign v

Read as: v

Means: v

2 occurrences in this chapter

Equation form expr-bf085b80deb44241

⊭CA\Entails/[\LogCL] !A

Read as: A is not valid in classical logic

Means: A is not valid in classical logic

1 occurrence in this chapter

Equation form expr-bf1df883a744abb3

A∧B!A \land !B

Read as: A and B

Means: A and B

1 occurrence in this chapter

Equation form expr-bfafd1f337f5011a

v′(p)={Tif v(p)∈{T,U}Fotherwise\pAssign {v'}(p) = \begin{cases} \True & \text{if } \pAssign {v}(p) \in \{\True, \Undef\}\\ \False & \text{otherwise} \end{cases}

Read as: Assignment v prime gives p the value true if assignment v gives p either true or U, and gives p the value false otherwise.

Means: Assignment v prime gives p the value true if assignment v gives p either true or U, and gives p the value false otherwise.

1 occurrence in this chapter

Equation form expr-c12bac278f87ea66

((p→q)→p)→p((p \lif q) \lif p) \lif p

Read as: if, open scope, if, open scope, if p then q, close scope, then p, close scope, then p

Means: if, open scope, if, open scope, if p then q, close scope, then p, close scope, then p

1 occurrence in this chapter

Equation form expr-c131adcc98170384

¬¬p⊨p\lnot\lnot p \Entails p

Read as: not not p entails p

Means: not not p entails p

1 occurrence in this chapter

Equation form expr-c3173025e5d9bcd9

A≡(B∧C)!A \ident (!B \land !C)

Read as: A is syntactically identical to the conjunction of B and C

Means: A is syntactically identical to the conjunction of B and C

1 occurrence in this chapter

Equation form expr-c5a46ff9bd27357b

v¯Ks(B)=F\pValue v(!B)[\LogKs] = \False

Read as: the strong Kleene value of B under v is false

Means: the strong Kleene value of B under v is false

3 occurrences in this chapter

Equation form expr-c5bf73cb03229b8b

p→qp \lif q

Read as: the conditional from p to q

Means: the conditional from p to q

2 occurrences in this chapter

Equation form expr-c5c418e07c87e5db

¬~Ks\tf{\lnot}[\LogKs]

Read as: the negation truth function in strong Kleene logic

Means: the negation truth function in strong Kleene logic

2 occurrences in this chapter

Equation form expr-c62679b2d772dc9f

(p→(p→q))→(p→q)(p \lif (p \lif q)) \lif (p \lif q)

Read as: if, open scope, if p then, open scope, if p then q, close scope, close scope, then, open scope, if p then q, close scope

Means: if, open scope, if p then, open scope, if p then q, close scope, close scope, then, open scope, if p then q, close scope

1 occurrence in this chapter

Equation form expr-c6fc77970f067e46

¬~G\tf{\lnot}[\LogGod]

Read as: the negation truth function in Goedel logic

Means: the negation truth function in Goedel logic

1 occurrence in this chapter

Equation form expr-c80c7957cf601f0b

→~(U,U)=T\tf{\lif}(\Undef, \Undef) = \True

Read as: that the conditional with U antecedent and U consequent has value true

Means: that the conditional with U antecedent and U consequent has value true

2 occurrences in this chapter

Equation form expr-c95a1ecf73da207a

∧~(F,U)=∧~(F,U)=F.\tf{\land}(\False, \Undef) = \tf{\land}(\False, \Undef) = \False.

Read as: conjunction of false and U equals conjunction of false and U, and both equal false

Means: conjunction of false and U equals conjunction of false and U, and both equal false

1 occurrence in this chapter

Equation form expr-caf372f8d210da27

+~\tf{+}

Read as: the is undefined truth function

Means: the is undefined truth function

2 occurrences in this chapter

Equation form expr-cb264a86414cc186

v′¯C(¬B)=F\pValue {v'}(\lnot !B)[\LogCL] = \False

Read as: the classical value of not B under v prime is false

Means: the classical value of not B under v prime is false

1 occurrence in this chapter

Equation form expr-cc31c2b2d7bcdb68

v¯Ks(A)=T\pValue v(!A)[\LogKs] = \True

Read as: the strong Kleene value of A under v is true

Means: the strong Kleene value of A under v is true

1 occurrence in this chapter

Equation form expr-cc458b36c641e059

¬~(U)=U\tf{\lnot}(\Undef) = \Undef

Read as: negation of U has value U

Means: negation of U has value U

1 occurrence in this chapter

Equation form expr-cd24150fd65a2673

v¯G(A)∈{T,U}\pValue v(!A)[\LogGod] \in \{\True,\Undef\}

Read as: the Goedel value of A under v belongs to the set containing true and U

Means: the Goedel value of A under v belongs to the set containing true and U

1 occurrence in this chapter

Equation form expr-cd37871db1ba340f

A∨¬A!A \lor \lnot !A

Read as: A or not A

Means: A or not A

1 occurrence in this chapter

Equation form expr-cdc2ed7d3b3d72c2

⊥\lfalse

Read as: the falsity constant

Means: the falsity constant

3 occurrences in this chapter

Equation form expr-ced13cbfd149c473

¬~(U)=F\tf{\lnot}(\Undef) = \False

Read as: negation of U has value false

Means: negation of U has value false

1 occurrence in this chapter

Equation form expr-d04ff80d9f6dc462

→\lif

Read as: the conditional

Means: the conditional

11 occurrences in this chapter

Equation form expr-d055ee4dbcdd0c8b

B!B

Read as: B

Means: B

4 occurrences in this chapter

Equation form expr-d5c4445b1703277c

v:At0→{F,T,U}\pAssign v\colon \PVar \to \{\False, \True, \Undef\}

Read as: v, mapping the propositional variables to the set containing false, true, and U,

Means: v, mapping the propositional variables to the set containing false, true, and U,

1 occurrence in this chapter

Equation form expr-d6582f2718df65db

v¯Ł3(A)=v¯C(A)\pValue v(!A)[\LogLuk[3]] = \pValue v(!A)[\LogCL]

Read as: the value of A under v in three valued Lukasiewicz logic equals its classical value under v

Means: the value of A under v in three valued Lukasiewicz logic equals its classical value under v

1 occurrence in this chapter

Equation form expr-d72c2cedf15e5fa4

L0\Lang L_0

Read as: L subscript zero

Means: L subscript zero

7 occurrences in this chapter

Equation form expr-d750201fde947d53

v′¯C(A)=F\pValue {v'}(!A)[\LogCL] = \False

Read as: the classical value of A under v prime is false

Means: the classical value of A under v prime is false

3 occurrences in this chapter

Equation form expr-d80bf60f39e339f6

p∨q,¬p⊨qp \lor q, \lnot p \Entails q

Read as: p or q, together with not p, entails q

Means: p or q, together with not p, entails q

4 occurrences in this chapter

Equation form expr-da85e41722636295

p∧q⊨pp \land q \Entails p

Read as: the conjunction of p and q entails p

Means: the conjunction of p and q entails p

3 occurrences in this chapter

Equation form expr-ddded48f62774e02

□p↔¬◇¬p\Box p \liff \lnot\Diamond \lnot p

Read as: the biconditional with necessity of p on the left and the negation of the possibility of the negation of p on the right

Means: the biconditional with necessity of p on the left and the negation of the possibility of the negation of p on the right

1 occurrence in this chapter

Equation form expr-df881ce6109ce666

Ks\LogKs

Read as: K subscript s

Means: K subscript s

6 occurrences in this chapter

Equation form expr-e275166f9d1c8f2b

¬(p→¬p)∨¬(¬p→p)\lnot(p \lif \lnot p) \lor \lnot(\lnot p \lif p)

Read as: either not, open scope, if p then not p, close scope, or not, open scope, if not p then p, close scope

Means: either not, open scope, if p then not p, close scope, or not, open scope, if not p then p, close scope

1 occurrence in this chapter

Equation form expr-e5229bfc9118a1b1

Hal\LogHal

Read as: Hal

Means: Hal

3 occurrences in this chapter

Equation form expr-e54931a36854f35f

p∨¬p(p→q)→(¬p∨q)¬¬p→p¬(¬p∧¬q)→(p∨q)((p→q)→p)→p¬(p→q)→(p∧¬q)& p \lor \lnot p && (p \lif q) \lif (\lnot p \lor q) \\ & \lnot\lnot p \lif p && \lnot(\lnot p \land \lnot q) \lif (p \lor q) \\ & ((p \lif q) \lif p) \lif p && \lnot(p \lif q) \lif (p \land \lnot q)

Read as: Six formulas in three source rows, with two formulas per row. Row one, left: p or not p. Row one, right: if, open scope, if p then q, close scope, then, open scope, not p or q, close scope. Row two, left: if not not p then p. Row two, right: if not, open scope, not p and not q, close scope, then, open scope, p or q, close scope. Row three, left: if, open scope, if, open scope, if p then q, close scope, then p, close scope, then p. Row three, right: if not, open scope, if p then q, close scope, then, open scope, p and not q, close scope.

Means: Six formulas in three source rows, with two formulas per row. Row one, left: p or not p. Row one, right: if, open scope, if p then q, close scope, then, open scope, not p or q, close scope. Row two, left: if not not p then p. Row two, right: if not, open scope, not p and not q, close scope, then, open scope, p or q, close scope. Row three, left: if, open scope, if, open scope, if p then q, close scope, then p, close scope, then p. Row three, right: if not, open scope, if p then q, close scope, then, open scope, p and not q, close scope.

1 occurrence in this chapter

Equation form expr-e560be96e4317388

⊨LPA\Entails[\LogLP] !A

Read as: A is valid in the logic of paradox

Means: A is valid in the logic of paradox

1 occurrence in this chapter

Equation form expr-e7e9536310cdba7f

12\frac{1}{2}

Read as: one half

Means: one half

1 occurrence in this chapter

Equation form expr-eb1f040769d24f8a

v¯Ks(B∧C)=F\pValue v(!B \land !C)[\LogKs] = \False

Read as: the strong Kleene value of the conjunction of B and C under v is false

Means: the strong Kleene value of the conjunction of B and C under v is false

1 occurrence in this chapter

Equation form expr-ecc160d7823fbc6c

¬(p∧q)↔(¬p∨¬q)\lnot(p \land q) \liff (\lnot p \lor \lnot q)

Read as: not, open scope, p and q, close scope, if and only if, open scope, not p or not q, close scope

Means: not, open scope, p and q, close scope, if and only if, open scope, not p or not q, close scope

1 occurrence in this chapter

Equation form expr-ed74bab370186601

→~Ks\tf{\lif}[\LogKs]

Read as: the conditional truth function in strong Kleene logic

Means: the conditional truth function in strong Kleene logic

1 occurrence in this chapter

Equation form expr-edd47f1916d0f3d2

v¯Ks(A)=v(p)=v′¯C(A)\pValue v(!A)[\LogKs] = \pAssign v(p) = \pValue {v'}(!A)[\LogCL]

Read as: the strong Kleene value of A under v equals the value assigned to p by v, which equals the classical value of A under v prime

Means: the strong Kleene value of A under v equals the value assigned to p by v, which equals the classical value of A under v prime

1 occurrence in this chapter

Equation form expr-ee1432360a41ee7c

p,p→q⊨qp, p \lif q \Entails q

Read as: p, together with if p then q, entails q

Means: p, together with if p then q, entails q

5 occurrences in this chapter

Equation form expr-ee5d070389487c34

U∉V+\Undef \notin V^+

Read as: U does not belong to V plus

Means: U does not belong to V plus

1 occurrence in this chapter

Equation form expr-ef0bc6a09eb9081d

¬(A∧¬A)\lnot(!A \land \lnot !A)

Read as: not, open scope, A and not A, close scope

Means: not, open scope, A and not A, close scope

1 occurrence in this chapter

Equation form expr-ef79b06f8bd2b48d

p∨¬pp \lor \lnot p

Read as: p or not p

Means: p or not p

1 occurrence in this chapter

Equation form expr-f041b60bcf71f586

¬A,A⊨B\lnot !A, !A \Entails !B

Read as: not A, together with A, entails B

Means: not A, together with A, entails B

1 occurrence in this chapter

Equation form expr-f157f71e0163a897

□\Box

Read as: the necessity operator

Means: the necessity operator

1 occurrence in this chapter

Equation form expr-f23982583aa961f1

A≡¬B!A \ident \lnot !B

Read as: A is syntactically identical to not B

Means: A is syntactically identical to not B

1 occurrence in this chapter

Equation form expr-f6df7f33c29d5c12

∨~Ks\tf{\lor}[\LogKs]

Read as: the disjunction truth function in strong Kleene logic

Means: the disjunction truth function in strong Kleene logic

1 occurrence in this chapter

Equation form expr-f6dff6a3ed0cca73

→~(U,T)=T\tf{\lif}(\Undef, \True) = \True

Read as: the conditional with U antecedent and true consequent has value true

Means: the conditional with U antecedent and true consequent has value true

1 occurrence in this chapter

Equation form expr-f7bd8f50661f608e

A≡(B→C)!A \ident (!B \lif !C)

Read as: A is syntactically identical to the conditional from B to C

Means: A is syntactically identical to the conditional from B to C

1 occurrence in this chapter

Equation form expr-f82586e5f6e57f67

v¯(A)=U\pValue v(!A) = \Undef

Read as: U

Means: U

1 occurrence in this chapter

Equation form expr-f87a549f62a7a792

□A\Box !A

Read as: necessarily A

Means: necessarily A

1 occurrence in this chapter

Equation form expr-f995bc65e98b9ff5

v′¯C(B∧C)=F\pValue {v'}(!B \land !C)[\LogCL] = \False

Read as: the classical value of the conjunction of B and C under v prime is false

Means: the classical value of the conjunction of B and C under v prime is false

1 occurrence in this chapter

Equation form expr-f9baaf9f77711629

F\False

Read as: false

Means: false

82 occurrences in this chapter

Equation form expr-fa7267cbcd62197e

v¯G(A)=F\pValue v(!A)[\LogGod] = \False

Read as: the Goedel value of A under v is false

Means: the Goedel value of A under v is false

1 occurrence in this chapter

Definition of three valued Lukasiewicz logic

The language has negation, conjunction, disjunction, and the conditional. Its values are true, U, and false; only true is designated. The four tables fix every input case. In particular the conditional from U to U is true, unlike strong Kleene logic.

Source

Lukasiewicz negation truth table

Lukasiewicz negation truth table. Input rows are true, U, and false, in that order. Each row gives input then output.

Source

Lukasiewicz conjunction truth table

Lukasiewicz conjunction truth table. Rows are first inputs true, U, and false, in that order. Columns are second inputs true, U, and false, in that order. Each row gives the first input followed by all three outputs.

Source

Lukasiewicz disjunction truth table

Lukasiewicz disjunction truth table. Rows are first inputs true, U, and false, in that order. Columns are second inputs true, U, and false, in that order. Each row gives the first input followed by all three outputs.

Source

Lukasiewicz conditional truth table

Lukasiewicz conditional truth table. Rows are first inputs true, U, and false, in that order. Columns are second inputs true, U, and false, in that order. Each row gives the first input followed by all three outputs.

Source

Agreement with classical values on settled assignments

If every propositional variable occurring in A receives true or false, its value in three valued Lukasiewicz logic equals its classical value under the same assignment.

Source

Exercise on the Lukasiewicz biconditional

Define the biconditional as the conjunction of the two directed conditionals. Determine its truth table in three valued Lukasiewicz logic. The source leaves the table to the reader; no solution is supplied.

Source

Nine row Lukasiewicz tautology verification table

Eight columns preserve the source decomposition of if not p then, if p then q. The first two columns enumerate all nine ordered assignments to p and q in true, U, false order. Columns three to eight are the six symbol positions in the displayed formula. Column five is the main conditional and is true in all nine source rows. The repeated occurrences of p and q are retained, not merged.

Source

Exercise on three Lukasiewicz tautologies

Show by truth tables that the three listed formulas are tautologies. The second and third use the biconditional defined as the conjunction of both conditionals, with explicit links to those items and the preceding biconditional exercise. The exercise remains unsolved.

Source

Exercise on classical tautologies that fail in Lukasiewicz logic

Show that each of the three listed classical tautologies fails to be a Lukasiewicz tautology. The source has an extra closing parenthesis after q in the first formula; that notation defect is disclosed without changing its logical connectives. No countervaluation or solution is supplied.

Source

Exercise on five Lukasiewicz consequence relations

Determine whether each of the five displayed premise and conclusion patterns holds in Lukasiewicz logic and give a truth table for each. Premises before the entailment symbol are jointly assumed. The exercise remains unsolved.

Source

Possibility truth table

Possibility truth table. Input rows are true, U, and false, in that order. Each row gives input then output.

Source

Necessity truth table

Necessity truth table. Input rows are true, U, and false, in that order. Each row gives input then output.

Source

Exercise on possibility and necessity duality

Using the supplied unary tables, show that necessity of p is equivalent to not possibly not p, and possibility of p to not necessarily not p. These are exercises in the stated three valued truth functional extension; no solution is supplied.

Source

Definition of strong Kleene logic

The values are true, U, and false and only true is designated. Unknown input can sometimes be overridden: a false conjunct makes conjunction false and a true disjunct makes disjunction true. The conditional is the disjunction of the negated antecedent with the consequent. The four source tables specify every case.

Source

Strong Kleene negation truth table

Strong Kleene negation truth table. Input rows are true, U, and false, in that order. Each row gives input then output.

Source

Strong Kleene conjunction truth table

Strong Kleene conjunction truth table. Rows are first inputs true, U, and false, in that order. Columns are second inputs true, U, and false, in that order. Each row gives the first input followed by all three outputs.

Source

Strong Kleene disjunction truth table

Strong Kleene disjunction truth table. Rows are first inputs true, U, and false, in that order. Columns are second inputs true, U, and false, in that order. Each row gives the first input followed by all three outputs.

Source

Strong Kleene conditional truth table

Strong Kleene conditional truth table. Rows are first inputs true, U, and false, in that order. Columns are second inputs true, U, and false, in that order. Each row gives the first input followed by all three outputs.

Source

Definition of weak Kleene logic

The values are true, U, and false and only true is designated. For every binary connective in the matrix, an input U forces output U; settled inputs follow classical truth functions. The four source tables specify every case.

Source

Weak Kleene negation truth table

Weak Kleene negation truth table. Input rows are true, U, and false, in that order. Each row gives input then output.

Source

Weak Kleene conjunction truth table

Weak Kleene conjunction truth table. Rows are first inputs true, U, and false, in that order. Columns are second inputs true, U, and false, in that order. Each row gives the first input followed by all three outputs.

Source

Weak Kleene disjunction truth table

Weak Kleene disjunction truth table. Rows are first inputs true, U, and false, in that order. Columns are second inputs true, U, and false, in that order. Each row gives the first input followed by all three outputs.

Source

Weak Kleene conditional truth table

Weak Kleene conditional truth table. Rows are first inputs true, U, and false, in that order. Columns are second inputs true, U, and false, in that order. Each row gives the first input followed by all three outputs.

Source

Neither Kleene logic has a tautology

Assign U to every propositional variable. Every formula in either stated language then has value U, since all the given truth functions preserve U when every argument is U. U is undesignated in both logics, so no formula is valid. This is a claim about the languages in the two preceding definitions, not their later extensions by extra operators.

Source

Exercise comparing strong and weak Kleene consequence

For each of the five listed relations, decide separately for strong Kleene and weak Kleene logic whether it holds, and supply a truth table. No exercise answers are included.

Source

Bochvar external negation truth table

Bochvar external negation truth table. Input rows are true, U, and false, in that order. Each row gives input then output.

Source

Bochvar is undefined truth table

Bochvar is undefined truth table. Input rows are true, U, and false, in that order. Each row gives input then output.

Source

Exercise defining Bochvar external negation

Find a formula in the single variable p, using ordinary negation and the plus operator but no external negation, that always has the value of external negation of p. Justify it by a truth table. The required formula and table remain unsupplied.

Source

Definition of three valued Goedel logic

The language also includes a falsity constant with value false. Its values are true, U, and false and only true is designated. Negation of U is false. Its conditional gives true on U and U, but false on U antecedent and false consequent. The four tables retain all rows and columns.

Source

Goedel negation truth table

Goedel negation truth table. Input rows are true, U, and false, in that order. Each row gives input then output.

Source

Goedel conjunction truth table

Goedel conjunction truth table. Rows are first inputs true, U, and false, in that order. Columns are second inputs true, U, and false, in that order. Each row gives the first input followed by all three outputs.

Source

Goedel disjunction truth table

Goedel disjunction truth table. Rows are first inputs true, U, and false, in that order. Columns are second inputs true, U, and false, in that order. Each row gives the first input followed by all three outputs.

Source

Goedel conditional truth table

Goedel conditional truth table. Rows are first inputs true, U, and false, in that order. Columns are second inputs true, U, and false, in that order. Each row gives the first input followed by all three outputs.

Source

Six classical formulas that are not Goedel tautologies

The source display has three rows and two formula columns. Read each row left then right. The surrounding text identifies all six formulas as not tautologies of three valued Goedel logic; it does not supply valuations in this display.

Source

Exercise on four Goedel tautologies

Give truth tables showing that the following four formulas are tautologies of three valued Goedel logic. Their order follows the four rows of the display. No truth tables or solutions are supplied.

Source

Four formulas to prove valid in Goedel logic

The four source rows give, in order, weak excluded middle, comparability of the two conditionals, a negated conjunction implication, and a disjunction of three linked conditionals. These are the formulas for the enclosing exercise; their truth tables are not supplied.

Source

Exercise on four failures of Goedel validity

Give truth tables showing that the four displayed formulas are not tautologies of three valued Goedel logic. Keep their four row order. The source provides no solutions.

Source

Four formulas to show not Goedel tautologies

The four rows are the displayed tasks for the enclosing exercise: a conditional implying a disjunction, a negated conjunction implying a disjunction, the nested conditional known as Peirce law, and a negated conditional implying a conjunction. Their exact nesting is retained in the formula reading. No countervaluations are supplied.

Source

Exercise on five Goedel consequence relations

Determine which of the five listed relations hold in Goedel logic, giving a truth table for each. These are tests of consequence from the stated premises, not just tests of whether the conclusion is a tautology. The exercise remains unsolved.

Source

Definition of the logic of paradox

The logic of paradox uses the same language, three values, and truth functions as strong Kleene logic, but designates both true and U. Changing designation, not changing a table cell, produces the logic presented here.

Source

Definition of Hallden logic of nonsense

Use the weak Kleene truth functions and add a one place plus operator meaning is meaningless. Its values are true, U, and false with true and U both designated. The plus table returns true exactly on U, and false on the other two inputs.

Source

Hallden is meaningless truth table

Hallden is meaningless truth table. Input rows are true, U, and false, in that order. Each row gives input then output.

Source

The logic of paradox has the classical tautologies

The source proves both directions. For the converse it converts a three valued assignment into a classical assignment by replacing U with true, and proves preservation of settled false and settled true by simultaneous induction. The proof prints an unrestricted equality in the atomic case and repeats B where the conjunction cases need C; these source slips are disclosed. The two remaining connective cases are left as exercises.

Source

Exercise completing the logic of paradox proof

Complete both preservation claims in the preceding proof for disjunction and conditional formulas. The negation and conjunction cases were presented in the source; the requested remaining cases are not solved here.

Source

Exercise on Hallden tautologies

Prove that every classical tautology is a tautology in Hallden logic. The source gives no proof in this exercise, and no solution is added.

Source

Exercise comparing consequence in paradox and nonsense logics

For each of the six listed relations, decide separately for the logic of paradox and Hallden logic whether it holds, and give a truth table. The exercise includes reasoning from a contradictory pair of premises. No truth tables or answers are supplied.

Source

Source definition named three valued R Mingle

The source names a matrix three valued R Mingle. It takes true, U, and false as values, designates true and U, and refers back to the Lukasiewicz truth functions. It also adds the falsity symbol to the listed language, although the referenced Lukasiewicz definition did not give its value. This missing clause is disclosed, not silently supplied.

Source

Exercise on the matrix named R Mingle

Determine which of the four listed consequence relations hold for the matrix defined in the immediately preceding source definition. Use that source matrix and its designation choice. The exercise remains unsolved.

Source

Designating true and U in Goedel logic gives classical logic

The source states that the Goedel truth functions on false, U, and true, with both true and U designated, define classical consequence. Its proof is explicitly the word Exercise. That placeholder is preserved without a solution.

Source

Exercise proving classical consequence after changing Goedel designation

Prove the preceding proposition by turning a countervaluation in the modified Goedel matrix into a classical countervaluation. Establish that a formula has value false exactly when its transformed classical value is false; equivalently, that true or U corresponds to classical true. Explain why this proves the proposition. The proof is not supplied.

Source

Cross-reference reference-000920

the preceding exercise item on the negation of a conjunction

Source occurrence

Cross-reference reference-000921

the preceding exercise item on the negation of a disjunction

Source occurrence

Cross-reference reference-000922

the exercise defining the Lukasiewicz biconditional

Source occurrence

Cross-reference reference-000923

the earlier proposition comparing many valued and classical validity

Source occurrence

Cross-reference reference-000924

the definition of the value of a formula under an assignment

Source occurrence

Cross-reference reference-000925

the proposition that the logic of paradox has exactly the classical tautologies

Source occurrence

Cross-reference reference-000926

the proposition that designating true and U in Goedel logic gives classical consequence

Source occurrence

Cross-reference reference-000927

the proposition that the logic of paradox has exactly the classical tautologies

Source occurrence

Source disclosures

Source-generated mathematical component tr047-source-macro-0001

Ł3\LogLuk[3]

Read as: three valued Lukasiewicz logic

Read in context source

Source-generated mathematical component tr047-source-macro-0002

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0003

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0004

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0005

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0006

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0007

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0008

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0009

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0010

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0011

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0012

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0013

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0014

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0015

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0016

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0017

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0018

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0019

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0020

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0021

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0022

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0023

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0024

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0025

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0026

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0027

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0028

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0029

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0030

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0031

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0032

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0033

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0034

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0035

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0036

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0037

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0038

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0039

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0040

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0041

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0042

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0043

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0044

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0045

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0046

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0047

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0048

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0049

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0050

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0051

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0052

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0053

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0054

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0055

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0056

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0057

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0058

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0059

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0060

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0061

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0062

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0063

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0064

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0065

U\Undef

Read as: U

Read in context source

Source-generated mathematical component tr047-source-macro-0066

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0067

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0068

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0069

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0070

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0071

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0072

T\True

Read as: true

Read in context source

Source-generated mathematical component tr047-source-macro-0073

F\False

Read as: false

Read in context source

Source-generated mathematical component tr047-source-macro-0074

Ł3\LogLuk[3]

Read as: three valued Lukasiewicz logic

Read in context source

Source-generated mathematical component tr047-source-macro-0075

Ł3\LogLuk[3]

Read as: three valued Lukasiewicz logic

Read in context source

Source-generated mathematical component tr047-source-macro-0076

Ł3\LogLuk[3]

Read as: three valued Lukasiewicz logic

Read in context source