Equation form expr-005518057817a68d

$\mModel{M} = \tuple{W,R,V}$

Read as: model M is the ordered triple W, R, V

Means: model M is the ordered triple W, R, V

1 occurrence in this chapter

• Occurrence projected-formula-0024854

Equation form expr-00ddf8f5cda0d1e1

$\mSat/{M}{p}$

Read as: p is not true in model M

Means: p is not true in model M

2 occurrences in this chapter

• Occurrence projected-formula-0024710
• Occurrence projected-formula-0024724

Equation form expr-01208e159d2aec8c

$\land$

Read as: conjunction

Means: conjunction

1 occurrence in this chapter

• Occurrence projected-formula-0024494

Equation form expr-03215cd7410368f9

$\mSat/{M}{!A \lif !B}$

Read as: the conditional from A to B is not true in model M

Means: the conditional from A to B is not true in model M

1 occurrence in this chapter

• Occurrence projected-formula-0024723

Equation form expr-034a69bf5d96f8a8

$i = 1$

Read as: i equals one

Means: i equals one

1 occurrence in this chapter

• Occurrence projected-formula-0024540

Equation form expr-043a718774c572bd

$s$

Read as: s

Means: s

1 occurrence in this chapter

• Occurrence projected-formula-0024477

Equation form expr-0588ffe69d967fd7

$\mSat{M}{\Box !B}[w]$

Read as: model M satisfies necessity of B at world w

Means: model M satisfies necessity of B at world w

2 occurrences in this chapter

• Occurrence projected-formula-0024858
• Occurrence projected-formula-0024865

Equation form expr-09314a8cbcb65295

$\mSat{M'}{!A}[w]$

Read as: model M prime satisfies A at world w

Means: model M prime satisfies A at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024878

Equation form expr-09ad6ae190e9283e

$\Box !B$

Read as: necessity of B

Means: necessity of B

3 occurrences in this chapter

• Occurrence projected-formula-0024623
• Occurrence projected-formula-0024627
• Occurrence projected-formula-0024629

Equation form expr-0a014a2b252cd898

$\mSat{M}{\Diamond p}[w_1]$

Read as: model M satisfies possibility of p at world w subscript one

Means: model M satisfies possibility of p at world w subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0024970

Equation form expr-0ab83730c263f0b7

$\Entails !A$

Read as: A is valid in all models

Means: A is valid in all models

2 occurrences in this chapter

• Occurrence projected-formula-0024769
• Occurrence projected-formula-0024776

Equation form expr-0b39637627e56c83

$\mSat{M}{!A}[w_2]$

Read as: model M satisfies A at world w subscript two

Means: model M satisfies A at world w subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0024684

Equation form expr-0de5af79a9cb012f

$\Box \Diamond !A \lif !B$

Read as: if necessarily possibly A, then B

Means: if necessarily possibly A, then B

1 occurrence in this chapter

• Occurrence projected-formula-0024917

Equation form expr-0ea1e3755dad0c75

$!D_2$

Read as: formula D subscript two

Means: formula D subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0024564

Equation form expr-0ef817c3f243f4b9

$\mSat{M}{!B}$

Read as: B is true at every world in model M

Means: B is true at every world in model M

4 occurrences in this chapter

• Occurrence projected-formula-0024697
• Occurrence projected-formula-0024713
• Occurrence projected-formula-0024719
• Occurrence projected-formula-0024722

Equation form expr-1047bed0f592edef

$\mSat{M}{\Box(!A \lif !B)}[w]$

Read as: model M satisfies necessity of the conditional from A to B at world w

Means: model M satisfies necessity of the conditional from A to B at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024856

Equation form expr-1106189e319461bd

$\mSat{M}{\lnot p}[w']$

Read as: model M satisfies not p at world w prime

Means: model M satisfies not p at world w prime

1 occurrence in this chapter

• Occurrence projected-formula-0024962

Equation form expr-1329e0c7d7026f0f

$R = \{\tuple{w_1, w_2},\tuple{w_1, w_3}\}$

Read as: R is the set containing the ordered pair w subscript one, w subscript two, and the ordered pair w subscript one, w subscript three

Means: R is the set containing the ordered pair w subscript one, w subscript two, and the ordered pair w subscript one, w subscript three

1 occurrence in this chapter

• Occurrence projected-formula-0024588

Equation form expr-132b1ed834aca791

$q \not\ident p_i$

Read as: q is not syntactically identical to p subscript i

Means: q is not syntactically identical to p subscript i

1 occurrence in this chapter

• Occurrence projected-formula-0024539

Equation form expr-1373dafa6c90ec37

$p\lif \Diamond p$

Read as: if p, then possibly p

Means: if p, then possibly p

1 occurrence in this chapter

• Occurrence projected-formula-0024740

Equation form expr-141a788a845573ec

$\mSat{M}{\indfrm}[w]$

Read as: model M satisfies the formula in the current induction case at world w

Means: model M satisfies the formula in the current induction case at world w

6 occurrences in this chapter

• Occurrence projected-formula-0024604
• Occurrence projected-formula-0024606
• Occurrence projected-formula-0024609
• Occurrence projected-formula-0024612
• Occurrence projected-formula-0024615
• Occurrence projected-formula-0024619

Equation form expr-148de9c5a7a44d19

$p$

Read as: p

Means: p

12 occurrences in this chapter

• Occurrence projected-formula-0024573
• Occurrence projected-formula-0024579
• Occurrence projected-formula-0024678
• Occurrence projected-formula-0024728
• Occurrence projected-formula-0024741
• Occurrence projected-formula-0024745
• Occurrence projected-formula-0024747
• Occurrence projected-formula-0024751
• Occurrence projected-formula-0024849
• Occurrence projected-formula-0024897
• Occurrence projected-formula-0024900
• Occurrence projected-formula-0024903

Equation form expr-149a5cde6dd9e884

$!B \lif \Box !A$

Read as: if B, then necessarily A

Means: if B, then necessarily A

1 occurrence in this chapter

• Occurrence projected-formula-0024794

Equation form expr-170468ad2237edad

$p \lif \Box q \Entails/\Box (p \lif q)$

Read as: if p then necessarily q does not entail necessity of the conditional from p to q

Means: if p then necessarily q does not entail necessity of the conditional from p to q

1 occurrence in this chapter

• Occurrence projected-formula-0024990

Equation form expr-172717ef1e584d16

$!A \to !B$

Read as: A implies B

Means: A implies B

1 occurrence in this chapter

• Occurrence projected-formula-0024464

Equation form expr-172a6d6592e8e4f4

$\mClass{C} \Entails !A$

Read as: A is valid in the class C of models

Means: A is valid in the class C of models

1 occurrence in this chapter

• Occurrence projected-formula-0024768

Equation form expr-17fa59edf213cfb9

$\mSat{M}{!A \lif !B}$

Read as: the conditional from A to B is true at every world in model M

Means: the conditional from A to B is true at every world in model M

2 occurrences in this chapter

• Occurrence projected-formula-0024695
• Occurrence projected-formula-0024711

Equation form expr-1880909af91a07f0

$\mSat{M}{p}[w']$

Read as: model M satisfies p at world w prime

Means: model M satisfies p at world w prime

1 occurrence in this chapter

• Occurrence projected-formula-0024960

Equation form expr-1a53000974926187

$V(p)$

Read as: V of p

Means: V of p

1 occurrence in this chapter

• Occurrence projected-formula-0024574

Equation form expr-1b16b1df538ba12d

$n$

Read as: n

Means: n

1 occurrence in this chapter

• Occurrence projected-formula-0024541

Equation form expr-1e2cf0a95163ca0a

$\mSat/{M}{\SSubst{!A}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}}[w]$

Read as: model M does not satisfy the result of simultaneously substituting D subscript one through D subscript n for p subscript one through p subscript n in A, at world w

Means: model M does not satisfy the result of simultaneously substituting D subscript one through D subscript n for p subscript one through p subscript n in A, at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024829

Equation form expr-2054805cec45318d

$\mSat{M}{\lnot\Diamond !A}[w]$

Read as: model M satisfies not possibly A at world w

Means: model M satisfies not possibly A at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024686

Equation form expr-22457bba9cc63931

$\mSat{M}{\Box\lnot p \lif \lnot p}[w]$

Read as: model M satisfies the conditional from necessarily not p to not p at world w

Means: model M satisfies the conditional from necessarily not p to not p at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024950

Equation form expr-22fcce974c4bdb33

$\mModel M$

Read as: model M

Means: model M

1 occurrence in this chapter

• Occurrence projected-formula-0024599

Equation form expr-234ef32b30394a54

$(!A \lor !B)$

Read as: the disjunction of A and B, enclosed in parentheses

Means: the disjunction of A and B, enclosed in parentheses

1 occurrence in this chapter

• Occurrence projected-formula-0024508

Equation form expr-23fac266c08cc34b

$\Obj p_i$

Read as: propositional variable p subscript i

Means: propositional variable p subscript i

1 occurrence in this chapter

• Occurrence projected-formula-0024500

Equation form expr-240313d5e995741c

$\pAssign{v}$

Read as: truth-value assignment v

Means: truth-value assignment v

3 occurrences in this chapter

• Occurrence projected-formula-0024813
• Occurrence projected-formula-0024832
• Occurrence projected-formula-0024835

Equation form expr-240381337f804e9f

$!A \lif !B$

Read as: if A, then B

Means: if A, then B

2 occurrences in this chapter

• Occurrence projected-formula-0024467
• Occurrence projected-formula-0024868

Equation form expr-241307f534cace2c

$V(p) = \{w_1, w_2\}$

Read as: V of p is the set containing w subscript one and w subscript two

Means: V of p is the set containing w subscript one and w subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0024589

Equation form expr-251d0df9e1d4a680

$\mSat/{M}{\Box p \lif p}[w_1]$

Read as: model M does not satisfy the conditional from necessarily p to p at world w subscript one

Means: model M does not satisfy the conditional from necessarily p to p at world w subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0024974

Equation form expr-25e12cee4c403162

$\mTrue{p}$

Read as: p is printed true at this world

Means: p is printed true at this world

1 occurrence in this chapter

• Occurrence projected-formula-0024585

Equation form expr-271202c561ada888

$\mSat{M}{\lnot !A}[w']$

Read as: model M satisfies not A at world w prime

Means: model M satisfies not A at world w prime

1 occurrence in this chapter

• Occurrence projected-formula-0024661

Equation form expr-284c47be48fb0c66

$\lnot$

Read as: negation

Means: negation

1 occurrence in this chapter

• Occurrence projected-formula-0024493

Equation form expr-28bfdd7dc1292f25

$p \lif \Box p$

Read as: if p, then necessarily p

Means: if p, then necessarily p

1 occurrence in this chapter

• Occurrence projected-formula-0024850

Equation form expr-2da171389d41722d

$\Box (!A \land !B) \Entails \Box !A$

Read as: necessity of the conjunction of A and B entails necessity of A

Means: necessity of the conjunction of A and B entails necessity of A

1 occurrence in this chapter

• Occurrence projected-formula-0024988

Equation form expr-2db5e5ed1e3f1510

$(\SSubst{!B}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}} \liff \SSubst{!C}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}).$

Read as: the biconditional between the simultaneous substitution instance of B and the corresponding simultaneous substitution instance of C

Means: the biconditional between the simultaneous substitution instance of B and the corresponding simultaneous substitution instance of C

1 occurrence in this chapter

• Occurrence projected-formula-0024553

Equation form expr-2e23407a47cc8eb9

$!B \ident \SSubst{!A}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}$

Read as: B is syntactically identical to the result of simultaneously substituting D subscript one through D subscript n for p subscript one through p subscript n in A

Means: B is syntactically identical to the result of simultaneously substituting D subscript one through D subscript n for p subscript one through p subscript n in A

1 occurrence in this chapter

• Occurrence projected-formula-0024807

Equation form expr-2f2773f485404693

$!B \Entails !A$

Read as: B entails A

Means: B entails A

1 occurrence in this chapter

• Occurrence projected-formula-0024944

Equation form expr-303d52c8d005ab48

$!B \in \Gamma$

Read as: B belongs to Gamma

Means: B belongs to Gamma

1 occurrence in this chapter

• Occurrence projected-formula-0024940

Equation form expr-3106fac3e3c8f992

$w_2$

Read as: w subscript two

Means: w subscript two

3 occurrences in this chapter

• Occurrence projected-formula-0024592
• Occurrence projected-formula-0024736
• Occurrence projected-formula-0024976

Equation form expr-318cbe8dda2d2c2b

$\mSat{M}{ p}[w]$

Read as: model M satisfies p at world w

Means: model M satisfies p at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024954

Equation form expr-318f93b8c3835cd2

$\Obj p_1$

Read as: propositional variable p subscript one

Means: propositional variable p subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0024491

Equation form expr-339707771b150f03

$\mSat{M}{!A \lif !B}[w]$

Read as: model M satisfies the conditional from A to B at world w

Means: model M satisfies the conditional from A to B at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024715

Equation form expr-346cf456a77b1ce5

$p_n$

Read as: p subscript n

Means: p subscript n

3 occurrences in this chapter

• Occurrence projected-formula-0024527
• Occurrence projected-formula-0024804
• Occurrence projected-formula-0024810

Equation form expr-34c867dbe53a287a

$\mSat{M}{\Diamond !A}[w]$

Read as: model M satisfies possibility of A at world w

Means: model M satisfies possibility of A at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024653

Equation form expr-367663e572248060

$\mSat/{M}{p}[w_1]$

Read as: model M does not satisfy p at world w subscript one

Means: model M does not satisfy p at world w subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0024973

Equation form expr-376d5037f2a5b8b0

$\Obj p_0$

Read as: propositional variable p subscript zero

Means: propositional variable p subscript zero

1 occurrence in this chapter

• Occurrence projected-formula-0024490

Equation form expr-3791ff36591fd55f

$\pSat{v}{\lnot !B} \Leftrightarrow {} & \pSat/{v}{!B}\\ &\qquad \text{by definition of $\pSat{v}{}$};\\ \Leftrightarrow {} & \mSat/{M}{\SSubst{!B}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}}[w]\\ &\qquad \text{by induction hypothesis}\\ \Leftrightarrow {} & \mSat{M}{\SSubst{\lnot !B}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}}[w]\\ &\qquad \text{by definition of $\pSat{v}{}$}.$

Read as: Induction chain for negation. Assignment v satisfies not B if and only if v does not satisfy B. By the induction hypothesis, this holds if and only if model M does not satisfy at w the simultaneous substitution instance of B. This holds if and only if M satisfies at w the simultaneous substitution instance of not B. The source attributes the last step to propositional satisfaction, although the displayed relation is modal satisfaction; that attribution is preserved and noted. End chain.

Means: Induction chain for negation. Assignment v satisfies not B if and only if v does not satisfy B. By the induction hypothesis, this holds if and only if model M does not satisfy at w the simultaneous substitution instance of B. This holds if and only if M satisfies at w the simultaneous substitution instance of not B. The source attributes the last step to propositional satisfaction, although the displayed relation is modal satisfaction; that attribution is preserved and noted. End chain.

1 occurrence in this chapter

• Occurrence projected-formula-0024824

Equation form expr-385ce03182cf3a0b

$Rw_1w$

Read as: world w is accessible from w subscript one

Means: world w is accessible from w subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0024680

Equation form expr-38bc0230ec229809

$(\SSubst{!B}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}} \land \SSubst{!C}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}).$

Read as: the conjunction of the simultaneous substitution instance of B and the corresponding simultaneous substitution instance of C

Means: the conjunction of the simultaneous substitution instance of B and the corresponding simultaneous substitution instance of C

1 occurrence in this chapter

• Occurrence projected-formula-0024547

Equation form expr-3b49e1a3f6189bbd

$!A \liff !B$

Read as: A if and only if B

Means: A if and only if B

1 occurrence in this chapter

• Occurrence projected-formula-0024518

Equation form expr-3b524d8b1a1063d9

$\mSat{M}{!C}[w]$

Read as: model M satisfies C at world w

Means: model M satisfies C at world w

3 occurrences in this chapter

• Occurrence projected-formula-0024608
• Occurrence projected-formula-0024611
• Occurrence projected-formula-0024614

Equation form expr-3bc7c48edc33ed52

$\Diamond(p_2 \lif p_3) & \lif \Box(\Diamond(p_2 \lif p_3) \land \lnot\Box p_1) \intertext{while $\SSubst{!A}{\subst{!D_2}{p_1}, \subst{!D_1}{p_2}}$ is} \lnot\Box p_1 & \lif \Box(\lnot\Box p_1 \land \Diamond(p_2 \lif p_3)) \intertext{Note that simultaneous substitution is in general not the same as iterated substitution, e.g., compare $\SSubst{!A}{\subst{!D_1}{p_1}, \subst{!D_2}{p_2}}$ above with $\Subst{(\Subst{!A}{!D_1}{p_1})}{!D_2}{p_2}$, which is:} \Diamond(p_2 \lif p_3) & \Subst{\lif \Box(\Diamond(p_2 \lif p_3) \land p_2)}{\lnot\Box p_1}{p_2}, \text{ i.e.,}\\ \Diamond(\lnot\Box p_1 \lif p_3) & \lif \Box(\Diamond(\lnot\Box p_1 \lif p_3) \land \lnot\Box p_1)\\ \intertext{and with $\Subst{(\Subst{!A}{!D_2}{p_2})}{!D_1}{p_1}$:} p_1 & \lif \Subst{\Box(p_1 \land \lnot\Box p_1)}{\Diamond(p_2 \lif p_3)}{p_1}, \text{ i.e.,}\\ \Diamond(p_2 \lif p_3) & \lif \Box(\Diamond(p_2 \lif p_3) \land \lnot\Box\Diamond(p_2 \lif p_3)).$

Read as: Worked simultaneous and iterated substitutions. First: if possibly the conditional from p two to p three, then necessarily its conjunction with not necessarily p one. Reversing the two simultaneous replacements gives: if not necessarily p one, then necessarily its conjunction with possibly the conditional from p two to p three. Iterating D one then D two instead yields: if possibly the conditional from not necessarily p one to p three, then necessarily its conjunction with not necessarily p one. Iterating D two then D one yields: if possibly the conditional from p two to p three, then necessarily its conjunction with not necessarily possibly the conditional from p two to p three. End worked substitutions.

Means: Worked simultaneous and iterated substitutions. First: if possibly the conditional from p two to p three, then necessarily its conjunction with not necessarily p one. Reversing the two simultaneous replacements gives: if not necessarily p one, then necessarily its conjunction with possibly the conditional from p two to p three. Iterating D one then D two instead yields: if possibly the conditional from not necessarily p one to p three, then necessarily its conjunction with not necessarily p one. Iterating D two then D one yields: if possibly the conditional from p two to p three, then necessarily its conjunction with not necessarily possibly the conditional from p two to p three. End worked substitutions.

1 occurrence in this chapter

• Occurrence projected-formula-0024567

Equation form expr-414f84ca9ba2b5e0

$\mModel{M'} = \{W', R', V'\}$

Read as: the source writes model M prime equals the set containing W prime, R prime, and V prime

Means: the source writes model M prime equals the set containing W prime, R prime, and V prime

1 occurrence in this chapter

• Occurrence projected-formula-0024979

Equation form expr-41e82746afe25cb2

$w_1 \in V(p)$

Read as: w subscript one belongs to V of p

Means: w subscript one belongs to V of p

1 occurrence in this chapter

• Occurrence projected-formula-0024676

Equation form expr-438757f12dd8c3fc

$w_1$

Read as: w subscript one

Means: w subscript one

4 occurrences in this chapter

• Occurrence projected-formula-0024591
• Occurrence projected-formula-0024730
• Occurrence projected-formula-0024735
• Occurrence projected-formula-0024975

Equation form expr-43e803d5632f4cc0

$\mSat{M}{p \lor q}[w_1]$

Read as: model M satisfies p or q at world w subscript one

Means: model M satisfies p or q at world w subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0024644

Equation form expr-442967b989492313

$V(p) = \{w\}$

Read as: V of p is the singleton set containing w

Means: V of p is the singleton set containing w

1 occurrence in this chapter

• Occurrence projected-formula-0024899

Equation form expr-468c5d976e16a9c7

$\mSat{M}{\Box\lnot!A}[w]$

Read as: model M satisfies necessarily not A at world w

Means: model M satisfies necessarily not A at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024687

Equation form expr-46b7a5140fe90304

$!A \lif \Box !A$

Read as: if A, then necessarily A

Means: if A, then necessarily A

1 occurrence in this chapter

• Occurrence projected-formula-0024915

Equation form expr-48075a486ff44dfc

$\mSat{M}{\Diamond q}[w_1]$

Read as: model M satisfies possibility of q at world w subscript one

Means: model M satisfies possibility of q at world w subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0024648

Equation form expr-4c0f80705a953246

$p \lif \Diamond\Diamond p$

Read as: if p, then possibly possibly p

Means: if p, then possibly possibly p

1 occurrence in this chapter

• Occurrence projected-formula-0024926

Equation form expr-4d758f3eb8936a2a

$\pSat{v}{!B \land !C} \Leftrightarrow {} & \pSat{v}{!B} \text{ and } \pSat{v}{!C}\\ &\qquad \text{by definition of $\pSat{v}{}$}\\ \Leftrightarrow {} & \mSat{M}{\SSubst{!B}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}}[w] \text{ and } \\ & \mSat{M}{\SSubst{!C}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}}[w]\\ &\qquad \text{by induction hypothesis}\\ \Leftrightarrow{} & \mSat{M}{\SSubst{(!B \land !C)}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}}[w]\\ &\qquad \text{by definition of $\mSat{M}{}[w]$}.$

Read as: Induction chain for conjunction. Assignment v satisfies B and C if and only if it satisfies B and it satisfies C. By the induction hypotheses, this holds if and only if model M at w satisfies both corresponding simultaneous substitution instances. By modal satisfaction, this holds if and only if M at w satisfies the simultaneous substitution instance of the conjunction of B and C. End chain.

Means: Induction chain for conjunction. Assignment v satisfies B and C if and only if it satisfies B and it satisfies C. By the induction hypotheses, this holds if and only if model M at w satisfies both corresponding simultaneous substitution instances. By modal satisfaction, this holds if and only if M at w satisfies the simultaneous substitution instance of the conjunction of B and C. End chain.

1 occurrence in this chapter

• Occurrence projected-formula-0024825

Equation form expr-4ef14388a47d369b

$w_2 \in V(p)$

Read as: w subscript two belongs to V of p

Means: w subscript two belongs to V of p

1 occurrence in this chapter

• Occurrence projected-formula-0024677

Equation form expr-4f86984ef1f09207

$\mSat{M}{p \lif \Diamond p}[w]$

Read as: model M satisfies the conditional from p to possibly p at world w

Means: model M satisfies the conditional from p to possibly p at world w

2 occurrences in this chapter

• Occurrence projected-formula-0024949
• Occurrence projected-formula-0024955

Equation form expr-5045ba03c70ef68c

$(!A \lif !B) \land (!B \lif !A)$

Read as: the conjunction of the conditional from A to B and the conditional from B to A

Means: the conjunction of the conditional from A to B and the conditional from B to A

1 occurrence in this chapter

• Occurrence projected-formula-0024519

Equation form expr-5093eaa4ffa5bb9b

$p \lif \Diamond p \Entails \Box p \lif p$

Read as: if p then possibly p entails if necessarily p then p

Means: if p then possibly p entails if necessarily p then p

1 occurrence in this chapter

• Occurrence projected-formula-0024978

Equation form expr-50e721e49c013f00

$w$

Read as: w

Means: w

22 occurrences in this chapter

• Occurrence projected-formula-0024476
• Occurrence projected-formula-0024478
• Occurrence projected-formula-0024481
• Occurrence projected-formula-0024484
• Occurrence projected-formula-0024577
• Occurrence projected-formula-0024580
• Occurrence projected-formula-0024582
• Occurrence projected-formula-0024583
• Occurrence projected-formula-0024596
• Occurrence projected-formula-0024598
• Occurrence projected-formula-0024624
• Occurrence projected-formula-0024628
• Occurrence projected-formula-0024630
• Occurrence projected-formula-0024633
• Occurrence projected-formula-0024635
• Occurrence projected-formula-0024718
• Occurrence projected-formula-0024785
• Occurrence projected-formula-0024831
• Occurrence projected-formula-0024898
• Occurrence projected-formula-0024901
• Occurrence projected-formula-0024904
• Occurrence projected-formula-0024985

Equation form expr-5136fc4246e7d497

$\Diamond$

Read as: possibility

Means: possibility

5 occurrences in this chapter

• Occurrence projected-formula-0024498
• Occurrence projected-formula-0024522
• Occurrence projected-formula-0024753
• Occurrence projected-formula-0024758
• Occurrence projected-formula-0024799

Equation form expr-51b864bb63453dd2

$\pSat/{v}{\lfalse}$

Read as: assignment v does not satisfy falsity

Means: assignment v does not satisfy falsity

1 occurrence in this chapter

• Occurrence projected-formula-0024821

Equation form expr-525a1d0e8bb940a9

$w' \in W$

Read as: w prime belongs to W

Means: w prime belongs to W

2 occurrences in this chapter

• Occurrence projected-formula-0024617
• Occurrence projected-formula-0024621

Equation form expr-5280c8f9e471df51

$\ltrue$

Read as: truth

Means: truth

1 occurrence in this chapter

• Occurrence projected-formula-0024516

Equation form expr-52884be383e0a550

$\mSat{M}{\Box \lnot p}[w]$

Read as: model M satisfies necessarily not p at world w

Means: model M satisfies necessarily not p at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024961

Equation form expr-529ad2daacd7efcb

$\lor$

Read as: disjunction

Means: disjunction

1 occurrence in this chapter

• Occurrence projected-formula-0024495

Equation form expr-52d7448aebc51481

$\mSat{M}{\lfalse}[w]$

Read as: model M satisfies falsity at world w

Means: model M satisfies falsity at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024601

Equation form expr-5323aa1f0bf11e2c

$(\SSubst{!B}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}} \lor \SSubst{!C}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}).$

Read as: the disjunction of the simultaneous substitution instance of B and the corresponding simultaneous substitution instance of C

Means: the disjunction of the simultaneous substitution instance of B and the corresponding simultaneous substitution instance of C

1 occurrence in this chapter

• Occurrence projected-formula-0024549

Equation form expr-537f7b515e650db7

$\SSubst{!A}{\subst{!D_1}{p_1}, \subst{!D_2}{p_2}}$

Read as: the result of simultaneously substituting D subscript one for p subscript one and D subscript two for p subscript two in A

Means: the result of simultaneously substituting D subscript one for p subscript one and D subscript two for p subscript two in A

1 occurrence in this chapter

• Occurrence projected-formula-0024566

Equation form expr-563087af17ac1bdc

$\lnot \SSubst{!B}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}$

Read as: the negation of the simultaneous substitution instance of B

Means: the negation of the simultaneous substitution instance of B

1 occurrence in this chapter

• Occurrence projected-formula-0024545

Equation form expr-563993a0b64357c6

$M = \tuple{W, R, V}$

Read as: M is the ordered triple W, R, V

Means: M is the ordered triple W, R, V

1 occurrence in this chapter

• Occurrence projected-formula-0024689

Equation form expr-57885e4c75965b23

$\Gamma$

Read as: Gamma

Means: Gamma

3 occurrences in this chapter

• Occurrence projected-formula-0024932
• Occurrence projected-formula-0024934
• Occurrence projected-formula-0024942

Equation form expr-58eb88f3d8608c45

$\Entails \Box!A$

Read as: necessarily A is valid in all models

Means: necessarily A is valid in all models

2 occurrences in this chapter

• Occurrence projected-formula-0024777
• Occurrence projected-formula-0024786

Equation form expr-5dafb363af77c7f8

$\mSat{M}{q}[w_1]$

Read as: model M satisfies q at world w subscript one

Means: model M satisfies q at world w subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0024642

Equation form expr-6064bd03f7272b46

$\mSat{M}{q}$

Read as: q is true at every world in model M

Means: q is true at every world in model M

1 occurrence in this chapter

• Occurrence projected-formula-0024726

Equation form expr-6158f77c3833268f

$p_i$

Read as: p subscript i

Means: p subscript i

1 occurrence in this chapter

• Occurrence projected-formula-0024532

Equation form expr-618b8e285848f0ab

$\lnot p \lif \Diamond \Box p$

Read as: if not p, then possibly necessarily p

Means: if not p, then possibly necessarily p

1 occurrence in this chapter

• Occurrence projected-formula-0024746

Equation form expr-62019e757a9cfb48

$\mSat/{M}{\Diamond !A}[w]$

Read as: model M does not satisfy possibility of A at world w

Means: model M does not satisfy possibility of A at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024640

Equation form expr-623fc0387a93ef39

$\mSat/{M}{\lfalse}[w]$

Read as: model M does not satisfy falsity at world w

Means: model M does not satisfy falsity at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024822

Equation form expr-62d8bfbf711cee05

$\Diamond (!A \lor !B) \liff (\Diamond !A \lor \Diamond !B)$

Read as: the disjunction of A and B is possible if and only if either A is possible or B is possible

Means: the disjunction of A and B is possible if and only if either A is possible or B is possible

1 occurrence in this chapter

• Occurrence projected-formula-0024920

Equation form expr-63713d1f6d4dc4ed

$\mSat{M}{!A}$

Read as: A is true at every world in model M

Means: A is true at every world in model M

6 occurrences in this chapter

• Occurrence projected-formula-0024690
• Occurrence projected-formula-0024693
• Occurrence projected-formula-0024696
• Occurrence projected-formula-0024698
• Occurrence projected-formula-0024712
• Occurrence projected-formula-0024721

Equation form expr-64a8722b9aebd40c

$\Box (p \lif q) \Entails/ p \lif \Box q$

Read as: necessity of the conditional from p to q does not entail the conditional from p to necessarily q

Means: necessity of the conditional from p to q does not entail the conditional from p to necessarily q

1 occurrence in this chapter

• Occurrence projected-formula-0024989

Equation form expr-6569be2ecbc7d7d4

$\mSat{M}{\lnot!A}$

Read as: not A is true at every world in model M

Means: not A is true at every world in model M

1 occurrence in this chapter

• Occurrence projected-formula-0024702

Equation form expr-65edae0ed060878a

$V'(p_i) = \Setabs{w}{\mSat{M}{!D_i}[w]}$

Read as: V prime of p subscript i is the set of worlds w at which model M satisfies D subscript i

Means: V prime of p subscript i is the set of worlds w at which model M satisfies D subscript i

1 occurrence in this chapter

• Occurrence projected-formula-0024876

Equation form expr-65fc0961f49725f0

$\pSat{v}{!A}$

Read as: assignment v satisfies A

Means: assignment v satisfies A

1 occurrence in this chapter

• Occurrence projected-formula-0024818

Equation form expr-669451e62a7d20f1

$\mSat/{M}{\lnot p}$

Read as: not p is not true at every world in model M

Means: not p is not true at every world in model M

1 occurrence in this chapter

• Occurrence projected-formula-0024709

Equation form expr-672eb44f56c4f5e1

$!A \strictif !B$

Read as: necessarily, if A then B

Means: necessarily, if A then B

1 occurrence in this chapter

• Occurrence projected-formula-0024472

Equation form expr-6941e31c1142a156

$\Setabs{!B}{\lexists[!D_1], \dots, \lexists[!D_n] \left(!B = \SSubst{!C}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}} \right) }.$

Read as: the set of formulas B such that there exist formulas D subscript one through D subscript n for which B equals the simultaneous substitution instance of C

Means: the set of formulas B such that there exist formulas D subscript one through D subscript n for which B equals the simultaneous substitution instance of C

1 occurrence in this chapter

• Occurrence projected-formula-0024840

Equation form expr-69ef9ee2c45eb3b7

$\mSat{M}{!A}[w']$

Read as: model M satisfies A at world w prime

Means: model M satisfies A at world w prime

4 occurrences in this chapter

• Occurrence projected-formula-0024667
• Occurrence projected-formula-0024671
• Occurrence projected-formula-0024781
• Occurrence projected-formula-0024862

Equation form expr-6ad5362532f7d650

$\Diamond \Box !A$

Read as: possibly necessarily A

Means: possibly necessarily A

1 occurrence in this chapter

• Occurrence projected-formula-0024748

Equation form expr-6b3e498919635337

$\mSat{M}{p \lif \Diamond p}[w_1]$

Read as: model M satisfies the conditional from p to possibly p at world w subscript one

Means: model M satisfies the conditional from p to possibly p at world w subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0024971

Equation form expr-6b78c5f6c2948fb7

$\mSat{M}{\Box (p \lor q)}[w_1]$

Read as: model M satisfies necessity of p or q at world w subscript one

Means: model M satisfies necessity of p or q at world w subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0024645

Equation form expr-6ef942abd8cdd431

$\Box \Box !A \lif \Box !A$

Read as: if necessarily necessarily A, then necessarily A

Means: if necessarily necessarily A, then necessarily A

1 occurrence in this chapter

• Occurrence projected-formula-0024919

Equation form expr-6f9da282bc96125e

$q \notin \{p_1, \dots, p_n \}$

Read as: q does not belong to the set containing p subscript one through p subscript n

Means: q does not belong to the set containing p subscript one through p subscript n

1 occurrence in this chapter

• Occurrence projected-formula-0024836

Equation form expr-7023b10755c9b26e

$p \lif \Box\Diamond p$

Read as: if p, then necessarily possibly p

Means: if p, then necessarily possibly p

1 occurrence in this chapter

• Occurrence projected-formula-0024907

Equation form expr-717a14ca6bc50ec7

$\Box p \lor \lnot \Box p$

Read as: either necessarily p or not necessarily p

Means: either necessarily p or not necessarily p

1 occurrence in this chapter

• Occurrence projected-formula-0024800

Equation form expr-71f2b4f16853c08a

$\mSat{M}{!D_i}[w]$

Read as: model M satisfies D subscript i at world w

Means: model M satisfies D subscript i at world w

2 occurrences in this chapter

• Occurrence projected-formula-0024817
• Occurrence projected-formula-0024834

Equation form expr-73b48c6c1d281bc8

$\Diamond(p_2 \lif p_3)$

Read as: possibility of the conditional from p subscript two to p subscript three

Means: possibility of the conditional from p subscript two to p subscript three

1 occurrence in this chapter

• Occurrence projected-formula-0024563

Equation form expr-741046a1e88bd478

$\lnot\Box p_1$

Read as: not necessarily p subscript one

Means: not necessarily p subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0024565

Equation form expr-7819f162040043dd

$\mSat/{M}{\lnot !A}[w']$

Read as: model M does not satisfy not A at world w prime

Means: model M does not satisfy not A at world w prime

2 occurrences in this chapter

• Occurrence projected-formula-0024665
• Occurrence projected-formula-0024666

Equation form expr-786aa0b01b2874ad

$Rww'$

Read as: world w prime is accessible from world w

Means: world w prime is accessible from world w

11 occurrences in this chapter

• Occurrence projected-formula-0024575
• Occurrence projected-formula-0024618
• Occurrence projected-formula-0024622
• Occurrence projected-formula-0024626
• Occurrence projected-formula-0024639
• Occurrence projected-formula-0024660
• Occurrence projected-formula-0024664
• Occurrence projected-formula-0024670
• Occurrence projected-formula-0024780
• Occurrence projected-formula-0024860
• Occurrence projected-formula-0024959

Equation form expr-78aaeb244101d779

$\SSubst{\indfrm}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}$

Read as: the result of simultaneously substituting D subscript one through D subscript n for p subscript one through p subscript n in the formula of the current induction case

Means: the result of simultaneously substituting D subscript one through D subscript n for p subscript one through p subscript n in the formula of the current induction case

10 occurrences in this chapter

• Occurrence projected-formula-0024535
• Occurrence projected-formula-0024537
• Occurrence projected-formula-0024542
• Occurrence projected-formula-0024544
• Occurrence projected-formula-0024546
• Occurrence projected-formula-0024548
• Occurrence projected-formula-0024550
• Occurrence projected-formula-0024552
• Occurrence projected-formula-0024554
• Occurrence projected-formula-0024556

Equation form expr-79e7bf33ddc100a2

$\Box p \lif \Box (q \lif p)$

Read as: if necessarily p, then necessarily if q then p

Means: if necessarily p, then necessarily if q then p

1 occurrence in this chapter

• Occurrence projected-formula-0024787

Equation form expr-7b50bc0a16e147b9

$\mSat{M}{}[w]$

Read as: satisfaction in model M at world w

Means: satisfaction in model M at world w

3 occurrences in this chapter

• Occurrence projected-formula-0024657
• Occurrence projected-formula-0024672
• Occurrence projected-formula-0024945

Equation form expr-7b8dff15e7d0e9be

$!D_n$

Read as: formula D subscript n

Means: formula D subscript n

3 occurrences in this chapter

• Occurrence projected-formula-0024529
• Occurrence projected-formula-0024806
• Occurrence projected-formula-0024812

Equation form expr-7c2443102c181d52

$p_3$

Read as: p subscript three

Means: p subscript three

1 occurrence in this chapter

• Occurrence projected-formula-0024734

Equation form expr-7cc21a3bda17828d

$\mSat{M}{!B}[w]$

Read as: model M satisfies B at world w

Means: model M satisfies B at world w

6 occurrences in this chapter

• Occurrence projected-formula-0024607
• Occurrence projected-formula-0024610
• Occurrence projected-formula-0024717
• Occurrence projected-formula-0024877
• Occurrence projected-formula-0024939
• Occurrence projected-formula-0024967

Equation form expr-7cdcc210af5bfaf1

$\mSat{M}{\lnot\Diamond\lnot !A}[w]$

Read as: model M satisfies not possibly not A at world w

Means: model M satisfies not possibly not A at world w

3 occurrences in this chapter

• Occurrence projected-formula-0024652
• Occurrence projected-formula-0024655
• Occurrence projected-formula-0024668

Equation form expr-7dedcc56da009b6c

$\Box \Diamond !A \lif \Diamond \Box !A$

Read as: if necessarily possibly A, then possibly necessarily A

Means: if necessarily possibly A, then possibly necessarily A

1 occurrence in this chapter

• Occurrence projected-formula-0024921

Equation form expr-7eeba8a02b6b43f3

$\pSat{v}{!B \lor !C} \Leftrightarrow{} & \pSat{v}{!B} \text{ or } \pSat{v}{!C}\\ &\qquad \text{by definition of $\pSat{v}{}$};\\ \Leftrightarrow{} & \mSat{M}{\SSubst{!B}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}}[w] \text{ or }\\ & \mSat{M}{\SSubst{!C}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}}[w]\\ &\qquad \text{by induction hypothesis}\\ \Leftrightarrow{} & \mSat{M}{\SSubst{(!B \lor !C)}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}}[w]\\ &\qquad \text{by definition of $\mSat{M}{}[w]$}.$

Read as: Induction chain for disjunction. Assignment v satisfies B or C if and only if it satisfies at least one of B and C. By the induction hypotheses, this holds if and only if model M at w satisfies at least one corresponding substitution instance. By modal satisfaction, this holds if and only if M at w satisfies the substitution instance of B or C. End chain.

Means: Induction chain for disjunction. Assignment v satisfies B or C if and only if it satisfies at least one of B and C. By the induction hypotheses, this holds if and only if model M at w satisfies at least one corresponding substitution instance. By modal satisfaction, this holds if and only if M at w satisfies the substitution instance of B or C. End chain.

1 occurrence in this chapter

• Occurrence projected-formula-0024826

Equation form expr-7faf9b0bc6d8753e

$\Diamond (!A \lif !B) \lif (\Box !A \lif \Diamond !B)$

Read as: if the conditional from A to B is possible, then if A is necessary, B is possible

Means: if the conditional from A to B is possible, then if A is necessary, B is possible

1 occurrence in this chapter

• Occurrence projected-formula-0024912

Equation form expr-7fb7ecb60a6350a2

$\Diamond !A \lif !B$

Read as: if possibly A, then B

Means: if possibly A, then B

1 occurrence in this chapter

• Occurrence projected-formula-0024795

Equation form expr-8034f48a3ceeebf5

$V'(p) = \emptyset$

Read as: V prime of p is empty

Means: V prime of p is empty

1 occurrence in this chapter

• Occurrence projected-formula-0024982

Equation form expr-8096a23ff2d820db

$\mSat{M}{!A \lif !B}[w']$

Read as: model M satisfies the conditional from A to B at world w prime

Means: model M satisfies the conditional from A to B at world w prime

1 occurrence in this chapter

• Occurrence projected-formula-0024861

Equation form expr-8238c028f61fc0f7

$!A$

Read as: formula A

Means: formula A

73 occurrences in this chapter

• Occurrence projected-formula-0024455
• Occurrence projected-formula-0024456
• Occurrence projected-formula-0024457
• Occurrence projected-formula-0024458
• Occurrence projected-formula-0024459
• Occurrence projected-formula-0024460
• Occurrence projected-formula-0024462
• Occurrence projected-formula-0024465
• Occurrence projected-formula-0024468
• Occurrence projected-formula-0024470
• Occurrence projected-formula-0024473
• Occurrence projected-formula-0024474
• Occurrence projected-formula-0024475
• Occurrence projected-formula-0024480
• Occurrence projected-formula-0024482
• Occurrence projected-formula-0024487
• Occurrence projected-formula-0024488
• Occurrence projected-formula-0024501
• Occurrence projected-formula-0024503
• Occurrence projected-formula-0024506
• Occurrence projected-formula-0024509
• Occurrence projected-formula-0024512
• Occurrence projected-formula-0024514
• Occurrence projected-formula-0024520
• Occurrence projected-formula-0024523
• Occurrence projected-formula-0024524
• Occurrence projected-formula-0024525
• Occurrence projected-formula-0024533
• Occurrence projected-formula-0024534
• Occurrence projected-formula-0024559
• Occurrence projected-formula-0024560
• Occurrence projected-formula-0024595
• Occurrence projected-formula-0024597
• Occurrence projected-formula-0024641
• Occurrence projected-formula-0024682
• Occurrence projected-formula-0024688
• Occurrence projected-formula-0024699
• Occurrence projected-formula-0024703
• Occurrence projected-formula-0024705
• Occurrence projected-formula-0024743
• Occurrence projected-formula-0024749
• Occurrence projected-formula-0024762
• Occurrence projected-formula-0024766
• Occurrence projected-formula-0024770
• Occurrence projected-formula-0024771
• Occurrence projected-formula-0024774
• Occurrence projected-formula-0024782
• Occurrence projected-formula-0024802
• Occurrence projected-formula-0024808
• Occurrence projected-formula-0024820
• Occurrence projected-formula-0024828
• Occurrence projected-formula-0024838
• Occurrence projected-formula-0024842
• Occurrence projected-formula-0024843
• Occurrence projected-formula-0024846
• Occurrence projected-formula-0024852
• Occurrence projected-formula-0024867
• Occurrence projected-formula-0024870
• Occurrence projected-formula-0024871
• Occurrence projected-formula-0024874
• Occurrence projected-formula-0024880
• Occurrence projected-formula-0024882
• Occurrence projected-formula-0024887
• Occurrence projected-formula-0024888
• Occurrence projected-formula-0024889
• Occurrence projected-formula-0024891
• Occurrence projected-formula-0024893
• Occurrence projected-formula-0024895
• Occurrence projected-formula-0024929
• Occurrence projected-formula-0024931
• Occurrence projected-formula-0024933
• Occurrence projected-formula-0024935
• Occurrence projected-formula-0024964

Equation form expr-82482b4b82458641

$\Diamond \SSubst{!B}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}.$

Read as: possibility of the simultaneous substitution instance of B

Means: possibility of the simultaneous substitution instance of B

1 occurrence in this chapter

• Occurrence projected-formula-0024557

Equation form expr-85c3ce29e3a4dc40

$\mClass{C}$

Read as: the class C of models

Means: the class C of models

6 occurrences in this chapter

• Occurrence projected-formula-0024763
• Occurrence projected-formula-0024764
• Occurrence projected-formula-0024765
• Occurrence projected-formula-0024767
• Occurrence projected-formula-0024772
• Occurrence projected-formula-0024791

Equation form expr-89390fc45c7bbfe2

$\mSat{M}{\Box p}[w_1]$

Read as: model M satisfies necessity of p at world w subscript one

Means: model M satisfies necessity of p at world w subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0024972

Equation form expr-8947db7b8d020f79

$\Box !A \lif \Box (!B \lif !A)$

Read as: if A is necessary, then the conditional from B to A is necessary

Means: if A is necessary, then the conditional from B to A is necessary

1 occurrence in this chapter

• Occurrence projected-formula-0024916

Equation form expr-8b57c1f938ff39f9

$W \neq \emptyset$

Read as: W is nonempty

Means: W is nonempty

1 occurrence in this chapter

• Occurrence projected-formula-0024701

Equation form expr-8bd2005830815bd2

$\Box!A$

Read as: necessity of A

Means: necessity of A

1 occurrence in this chapter

• Occurrence projected-formula-0024775

Equation form expr-8c2574892063f995

$R$

Read as: R

Means: R

6 occurrences in this chapter

• Occurrence projected-formula-0024479
• Occurrence projected-formula-0024485
• Occurrence projected-formula-0024486
• Occurrence projected-formula-0024570
• Occurrence projected-formula-0024637
• Occurrence projected-formula-0024760

Equation form expr-8c7566c2fdcbadf8

$\mSat{M}{p}$

Read as: p is true at every world in model M

Means: p is true at every world in model M

1 occurrence in this chapter

• Occurrence projected-formula-0024725

Equation form expr-8e35c2cd3bf6641b

$q$

Read as: q

Means: q

2 occurrences in this chapter

• Occurrence projected-formula-0024538
• Occurrence projected-formula-0024729

Equation form expr-8e8237e560bad76b

$\Box(!A \lif !B) \lif (\Diamond !A \lif \Diamond !B)$

Read as: if the conditional from A to B is necessary, then if A is possible, B is possible

Means: if the conditional from A to B is necessary, then if A is possible, B is possible

1 occurrence in this chapter

• Occurrence projected-formula-0024910

Equation form expr-8f8932956e0e561e

$W' = \{w\}$

Read as: W prime is the singleton set containing w

Means: W prime is the singleton set containing w

1 occurrence in this chapter

• Occurrence projected-formula-0024980

Equation form expr-900aeb8b11807edf

$\mSat{M}{\Diamond p}[w]$

Read as: model M satisfies possibility of p at world w

Means: model M satisfies possibility of p at world w

2 occurrences in this chapter

• Occurrence projected-formula-0024956
• Occurrence projected-formula-0024957

Equation form expr-911dea1526a594a6

$\mSat/{M'}{!A}[w]$

Read as: model M prime does not satisfy A at world w

Means: model M prime does not satisfy A at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024886

Equation form expr-93aa632d054e9ffb

$!A \lif (!B \lif !A)$

Read as: if A, then if B then A

Means: if A, then if B then A

1 occurrence in this chapter

• Occurrence projected-formula-0024848

Equation form expr-9539983363a76211

$R = \emptyset$

Read as: R is empty

Means: R is empty

1 occurrence in this chapter

• Occurrence projected-formula-0024797

Equation form expr-96310c0704bf132b

$\pSat{v}{p_i} \Leftrightarrow {} & \pAssign{v}(p_i) = \True \\ & \qquad \text{by definition of $\pSat{v}{p_i}$}\\ \Leftrightarrow {} & \mSat{M}{!D_i}[w] \\ &\qquad \text{by assumption}\\ \Leftrightarrow {} & \mSat{M}{\SSubst{p_i}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}}[w]\\ &\qquad \text{since $\SSubst{p_i}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}} \ident !D_i$}.$

Read as: Induction chain for p subscript i. Assignment v satisfies p subscript i if and only if v assigns it true, by propositional satisfaction. This holds if and only if model M satisfies D subscript i at w, by assumption. This holds if and only if M satisfies at w the simultaneous substitution instance obtained from p subscript i, because that instance is syntactically identical to D subscript i. End chain.

Means: Induction chain for p subscript i. Assignment v satisfies p subscript i if and only if v assigns it true, by propositional satisfaction. This holds if and only if model M satisfies D subscript i at w, by assumption. This holds if and only if M satisfies at w the simultaneous substitution instance obtained from p subscript i, because that instance is syntactically identical to D subscript i. End chain.

1 occurrence in this chapter

• Occurrence projected-formula-0024823

Equation form expr-96f180f41f91ce63

$(\Diamond !A \lif \Box !B) \lif (\Box !A \lif \Box !B)$

Read as: if the conditional from possibly A to necessarily B holds, then if A is necessary, B is necessary

Means: if the conditional from possibly A to necessarily B holds, then if A is necessary, B is necessary

1 occurrence in this chapter

• Occurrence projected-formula-0024922

Equation form expr-96f8c8771552fb97

$!B \ident \SSubst{!A}{\subst{!D_1}{p_1},\dots,\subst{!D_n}{p_n}}$

Read as: B is syntactically identical to the simultaneous substitution instance of A replacing p subscript one through p subscript n by D subscript one through D subscript n

Means: B is syntactically identical to the simultaneous substitution instance of A replacing p subscript one through p subscript n by D subscript one through D subscript n

1 occurrence in this chapter

• Occurrence projected-formula-0024873

Equation form expr-9763a163e7ff6591

$w_3$

Read as: w subscript three

Means: w subscript three

3 occurrences in this chapter

• Occurrence projected-formula-0024593
• Occurrence projected-formula-0024737
• Occurrence projected-formula-0024977

Equation form expr-992accb9917efeb5

$!C$

Read as: C

Means: C

3 occurrences in this chapter

• Occurrence projected-formula-0024839
• Occurrence projected-formula-0024841
• Occurrence projected-formula-0024845

Equation form expr-99adc3223e0a4ae7

$p \lif (q \lif p)$

Read as: if p, then if q then p

Means: if p, then if q then p

1 occurrence in this chapter

• Occurrence projected-formula-0024851

Equation form expr-9dd4e4b646218a71

$\mSat{M}{!B}[w']$

Read as: model M satisfies B at world w prime

Means: model M satisfies B at world w prime

3 occurrences in this chapter

• Occurrence projected-formula-0024616
• Occurrence projected-formula-0024620
• Occurrence projected-formula-0024863

Equation form expr-9ff421c81040cbb0

$\pAssign{v}(p_i) = \True$

Read as: assignment v gives p subscript i the value true

Means: assignment v gives p subscript i the value true

2 occurrences in this chapter

• Occurrence projected-formula-0024816
• Occurrence projected-formula-0024833

Equation form expr-a24f02133530e49d

$\mSat/{M}{p \lif q}$

Read as: the conditional from p to q is not true at every world in model M

Means: the conditional from p to q is not true at every world in model M

1 occurrence in this chapter

• Occurrence projected-formula-0024727

Equation form expr-a294d9efdc5319f9

$\Box (!A \lor !B) \lif (\Box !A \lor \Box !B)$

Read as: if the disjunction of A and B is necessary, then either A is necessary or B is necessary

Means: if the disjunction of A and B is necessary, then either A is necessary or B is necessary

1 occurrence in this chapter

• Occurrence projected-formula-0024911

Equation form expr-a39d2ca036172648

$p_1$

Read as: p subscript one

Means: p subscript one

4 occurrences in this chapter

• Occurrence projected-formula-0024526
• Occurrence projected-formula-0024732
• Occurrence projected-formula-0024803
• Occurrence projected-formula-0024809

Equation form expr-a471f5a7deca1730

$\mSat{M}{\lnot \Box\Box \lnot q}[w_1]$

Read as: model M satisfies not necessarily necessarily not q at world w subscript one

Means: model M satisfies not necessarily necessarily not q at world w subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0024650

Equation form expr-a4a51fbde8d39ce5

$\Obj p_2$

Read as: propositional variable p subscript two

Means: propositional variable p subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0024492

Equation form expr-a8f9c4ec8ae3bd39

$\mSat/{M'}{p}[w]$

Read as: model M prime does not satisfy p at world w

Means: model M prime does not satisfy p at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024983

Equation form expr-a94a8ddf251c366b

$\mSat{M}{\lnot\Box\lnot !A}[w]$

Read as: model M satisfies not necessarily not A at world w

Means: model M satisfies not necessarily not A at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024654

Equation form expr-a97c38e84f241d67

$\mSat{M'}{\Box p}[w]$

Read as: model M prime satisfies necessity of p at world w

Means: model M prime satisfies necessity of p at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024986

Equation form expr-aa816ef1a7dad1da

$\Diamond !A \liff \lnot\Box\lnot !A. \tag{\Dual}$

Read as: Dual schema: possibly A if and only if not necessarily not A

Means: Dual schema: possibly A if and only if not necessarily not A

1 occurrence in this chapter

• Occurrence projected-formula-0024866

Equation form expr-ab733e02bed54e75

$\Gamma \Entails !A$

Read as: Gamma entails A

Means: Gamma entails A

1 occurrence in this chapter

• Occurrence projected-formula-0024936

Equation form expr-ac5162902de4efd2

$\mSat{M}{\SSubst{!A}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}}[w]$

Read as: model M satisfies the simultaneous substitution instance of A at world w

Means: model M satisfies the simultaneous substitution instance of A at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024819

Equation form expr-ad9ead843445ff0d

$\mSat/{M}{\lnot p}[w]$

Read as: model M does not satisfy not p at world w

Means: model M does not satisfy not p at world w

2 occurrences in this chapter

• Occurrence projected-formula-0024952
• Occurrence projected-formula-0024953

Equation form expr-ae691208d80475a7

$\mSat{M}{!A}[w_1]$

Read as: model M satisfies A at world w subscript one

Means: model M satisfies A at world w subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0024683

Equation form expr-ae7d61ce04a31447

$\Box (!A \land !B) \liff (\Box !A \land \Box !B)$

Read as: the conjunction of A and B is necessary if and only if both A and B are necessary

Means: the conjunction of A and B is necessary if and only if both A and B are necessary

1 occurrence in this chapter

• Occurrence projected-formula-0024914

Equation form expr-b01634352703cd63

$\mSat/{M}{\lnot !A}$

Read as: not A is not true at every world in model M

Means: not A is not true at every world in model M

2 occurrences in this chapter

• Occurrence projected-formula-0024694
• Occurrence projected-formula-0024704

Equation form expr-b318c8360463bdb3

$\Box \Diamond p$

Read as: necessarily possibly p

Means: necessarily possibly p

1 occurrence in this chapter

• Occurrence projected-formula-0024750

Equation form expr-b59c0ca457682eca

$\mSat{M}{p}[w]$

Read as: model M satisfies p at world w

Means: model M satisfies p at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024602

Equation form expr-b5b446abb4d0323b

$\mSat{M}{!A}[w]$

Read as: model M satisfies A at world w

Means: model M satisfies A at world w

5 occurrences in this chapter

• Occurrence projected-formula-0024600
• Occurrence projected-formula-0024691
• Occurrence projected-formula-0024707
• Occurrence projected-formula-0024716
• Occurrence projected-formula-0024941

Equation form expr-b7051a8441b8596e

$p \lif \Diamond p \Entails/ \Box p \lif p$

Read as: if p then possibly p does not entail if necessarily p then p

Means: if p then possibly p does not entail if necessarily p then p

1 occurrence in this chapter

• Occurrence projected-formula-0024969

Equation form expr-b7e49009c4bf4d26

$\mSat{M}{\Box q}[w_1]$

Read as: model M satisfies necessity of q at world w subscript one

Means: model M satisfies necessity of q at world w subscript one

1 occurrence in this chapter

• Occurrence projected-formula-0024649

Equation form expr-b83087dda7cf3716

$\mSat{M}{\Box!A}[w]$

Read as: model M satisfies necessity of A at world w

Means: model M satisfies necessity of A at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024783

Equation form expr-b95d5fdc7f8374c2

$\mSat{M}{\lnot q}[w_3]$

Read as: model M satisfies not q at world w subscript three

Means: model M satisfies not q at world w subscript three

1 occurrence in this chapter

• Occurrence projected-formula-0024643

Equation form expr-b99d2e5f37ae7e17

$w'$

Read as: w prime

Means: w prime

12 occurrences in this chapter

• Occurrence projected-formula-0024483
• Occurrence projected-formula-0024576
• Occurrence projected-formula-0024581
• Occurrence projected-formula-0024584
• Occurrence projected-formula-0024625
• Occurrence projected-formula-0024638
• Occurrence projected-formula-0024659
• Occurrence projected-formula-0024663
• Occurrence projected-formula-0024669
• Occurrence projected-formula-0024859
• Occurrence projected-formula-0024864
• Occurrence projected-formula-0024958

Equation form expr-baacfd9d189243cd

$\Diamond !A$

Read as: possibility of A

Means: possibility of A

1 occurrence in this chapter

• Occurrence projected-formula-0024515

Equation form expr-bb4af7f6f6021855

$\Diamond p \lif \Box \Diamond p$

Read as: if possibly p, then necessarily possibly p

Means: if possibly p, then necessarily possibly p

1 occurrence in this chapter

• Occurrence projected-formula-0024909

Equation form expr-bc5b3c46de9c2f57

$\mSat{M}{\Diamond\lnot !A}[w]$

Read as: model M satisfies possibly not A at world w

Means: model M satisfies possibly not A at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024658

Equation form expr-bf1df883a744abb3

$!A \land !B$

Read as: A and B

Means: A and B

1 occurrence in this chapter

• Occurrence projected-formula-0024461

Equation form expr-c0ff42fe2d6e4445

$\lnot\lfalse$

Read as: not falsity

Means: not falsity

1 occurrence in this chapter

• Occurrence projected-formula-0024517

Equation form expr-c229d8c18cd80a5c

$W = \{w\}$

Read as: W is the singleton set containing w

Means: W is the singleton set containing w

1 occurrence in this chapter

• Occurrence projected-formula-0024793

Equation form expr-c257c20c8ce26d2e

$\mModel{M'} = \tuple{W, R, V'}$

Read as: model M prime is the ordered triple W, R, V prime

Means: model M prime is the ordered triple W, R, V prime

1 occurrence in this chapter

• Occurrence projected-formula-0024875

Equation form expr-c3eb983cbd345929

$\mClass{C}' \subseteq \mClass{C}$

Read as: the class C prime is a subclass of the class C

Means: the class C prime is a subclass of the class C

1 occurrence in this chapter

• Occurrence projected-formula-0024773

Equation form expr-c40e112d9ad379d0

$Rww$

Read as: world w is accessible from itself

Means: world w is accessible from itself

1 occurrence in this chapter

• Occurrence projected-formula-0024636

Equation form expr-c4ddea87b1451ed2

$(\Diamond !A \land \Diamond !B) \lif \Diamond (!A \land !B)$

Read as: if A is possible and B is possible, then the conjunction of A and B is possible

Means: if A is possible and B is possible, then the conjunction of A and B is possible

1 occurrence in this chapter

• Occurrence projected-formula-0024913

Equation form expr-c63f9557f464c93a

$\lnot !A$

Read as: not A

Means: not A

1 occurrence in this chapter

• Occurrence projected-formula-0024502

Equation form expr-c6528118ba158042

$\pSat{v}{!B \lif !C} \Leftrightarrow{} & \pSat/{v}{!B} \text{ or } \pSat{v}{!C}\\ &\qquad \text{by definition of $\pSat{v}{}$}\\ \Leftrightarrow{} & \mSat/{M}{\SSubst{!B}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}}[w] \text{ or }\\ & \mSat{M}{\SSubst{!C}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}}[w]\\ &\qquad \text{by induction hypothesis}\\ \Leftrightarrow{} & \mSat{M}{\SSubst{(!B \lif !C)}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}}[w]\\ &\qquad \text{by definition of $\mSat{M}{}[w]$}.$

Read as: Induction chain for the conditional. Assignment v satisfies if B then C if and only if v does not satisfy B or v satisfies C. By the induction hypotheses, this holds if and only if model M at w does not satisfy the substitution instance of B or does satisfy the substitution instance of C. By modal satisfaction, this holds if and only if M at w satisfies the substitution instance of if B then C. End chain.

Means: Induction chain for the conditional. Assignment v satisfies if B then C if and only if v does not satisfy B or v satisfies C. By the induction hypotheses, this holds if and only if model M at w does not satisfy the substitution instance of B or does satisfy the substitution instance of C. By modal satisfaction, this holds if and only if M at w satisfies the substitution instance of if B then C. End chain.

1 occurrence in this chapter

• Occurrence projected-formula-0024827

Equation form expr-c865789bc525e11a

$!D_i$

Read as: formula D subscript i

Means: formula D subscript i

2 occurrences in this chapter

• Occurrence projected-formula-0024531
• Occurrence projected-formula-0024543

Equation form expr-cb1ce2373aab389a

$Rw_2w$

Read as: world w is accessible from w subscript two

Means: world w is accessible from w subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0024681

Equation form expr-ccb87154d21d3d96

$2+2=4$

Read as: two plus two equals four

Means: two plus two equals four

1 occurrence in this chapter

• Occurrence projected-formula-0024454

Equation form expr-cdc2ed7d3b3d72c2

$\lfalse$

Read as: falsity

Means: falsity

4 occurrences in this chapter

• Occurrence projected-formula-0024489
• Occurrence projected-formula-0024499
• Occurrence projected-formula-0024536
• Occurrence projected-formula-0024902

Equation form expr-ce36a9166e75bce6

$\Box \lnot \lfalse$

Read as: necessarily not falsity

Means: necessarily not falsity

1 occurrence in this chapter

• Occurrence projected-formula-0024788

Equation form expr-cf26fe5bd7e6ea18

$(!A \lif !B)$

Read as: the conditional from A to B, enclosed in parentheses

Means: the conditional from A to B, enclosed in parentheses

1 occurrence in this chapter

• Occurrence projected-formula-0024511

Equation form expr-d04ff80d9f6dc462

$\lif$

Read as: conditional

Means: conditional

1 occurrence in this chapter

• Occurrence projected-formula-0024496

Equation form expr-d055ee4dbcdd0c8b

$!B$

Read as: formula B

Means: formula B

17 occurrences in this chapter

• Occurrence projected-formula-0024463
• Occurrence projected-formula-0024466
• Occurrence projected-formula-0024469
• Occurrence projected-formula-0024471
• Occurrence projected-formula-0024504
• Occurrence projected-formula-0024507
• Occurrence projected-formula-0024510
• Occurrence projected-formula-0024631
• Occurrence projected-formula-0024632
• Occurrence projected-formula-0024801
• Occurrence projected-formula-0024844
• Occurrence projected-formula-0024869
• Occurrence projected-formula-0024881
• Occurrence projected-formula-0024928
• Occurrence projected-formula-0024930
• Occurrence projected-formula-0024943
• Occurrence projected-formula-0024963

Equation form expr-d0a2b90b3d18abd7

$\mModel{M}$

Read as: model M

Means: model M

13 occurrences in this chapter

• Occurrence projected-formula-0024594
• Occurrence projected-formula-0024720
• Occurrence projected-formula-0024731
• Occurrence projected-formula-0024738
• Occurrence projected-formula-0024739
• Occurrence projected-formula-0024784
• Occurrence projected-formula-0024830
• Occurrence projected-formula-0024890
• Occurrence projected-formula-0024892
• Occurrence projected-formula-0024894
• Occurrence projected-formula-0024896
• Occurrence projected-formula-0024924
• Occurrence projected-formula-0024925

Equation form expr-d0b530525cf3b93c

$\mSat/{M}{\Diamond\lnot !A}[w]$

Read as: model M does not satisfy possibly not A at world w

Means: model M does not satisfy possibly not A at world w

2 occurrences in this chapter

• Occurrence projected-formula-0024656
• Occurrence projected-formula-0024662

Equation form expr-d148fb86b1972296

$\mSat/{M}{!A}[w]$

Read as: model M does not satisfy A at world w

Means: model M does not satisfy A at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024968

Equation form expr-d16b73ce79fc2dfd

$\mSat/{M}{!B}[w]$

Read as: model M does not satisfy B at world w

Means: model M does not satisfy B at world w

3 occurrences in this chapter

• Occurrence projected-formula-0024605
• Occurrence projected-formula-0024613
• Occurrence projected-formula-0024885

Equation form expr-d2da161508eebc66

$w \in W$

Read as: w belongs to W

Means: w belongs to W

13 occurrences in this chapter

• Occurrence projected-formula-0024679
• Occurrence projected-formula-0024692
• Occurrence projected-formula-0024706
• Occurrence projected-formula-0024708
• Occurrence projected-formula-0024714
• Occurrence projected-formula-0024779
• Occurrence projected-formula-0024815
• Occurrence projected-formula-0024855
• Occurrence projected-formula-0024879
• Occurrence projected-formula-0024884
• Occurrence projected-formula-0024938
• Occurrence projected-formula-0024948
• Occurrence projected-formula-0024966

Equation form expr-d5c8ee73677dc4ea

$p_2$

Read as: p subscript two

Means: p subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0024733

Equation form expr-d6d00853b0416049

$\mSat{M}{\Box \bot}[w_3]$

Read as: model M satisfies necessity of falsity at world w subscript three

Means: model M satisfies necessity of falsity at world w subscript three

1 occurrence in this chapter

• Occurrence projected-formula-0024647

Equation form expr-dcfed30b9104243f

$\Diamond(!A \lif !B) \lor \Box(!B \lif !A)$

Read as: either the conditional from A to B is possible, or the conditional from B to A is necessary

Means: either the conditional from A to B is possible, or the conditional from B to A is necessary

1 occurrence in this chapter

• Occurrence projected-formula-0024923

Equation form expr-dd3023768fff60cc

$\mSat{M}{\Box\lnot p}[w]$

Read as: model M satisfies necessarily not p at world w

Means: model M satisfies necessarily not p at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024951

Equation form expr-dd3ddedffb9c5a13

$\Box p \lif (\Box q \lif \Box p)$

Read as: if necessarily p, then if necessarily q then necessarily p

Means: if necessarily p, then if necessarily q then necessarily p

1 occurrence in this chapter

• Occurrence projected-formula-0024789

Equation form expr-dda6488d6b6a1fca

$p \lif \Diamond p \Entails \Box\lnot p \lif \lnot p$

Read as: if p then possibly p entails if necessarily not p then not p

Means: if p then possibly p entails if necessarily not p then not p

1 occurrence in this chapter

• Occurrence projected-formula-0024946

Equation form expr-de5a6f78116eca62

$V$

Read as: V

Means: V

1 occurrence in this chapter

• Occurrence projected-formula-0024572

Equation form expr-df1c6f7455cf6ff3

$!A \lif \Box!A$

Read as: if A, then necessarily A

Means: if A, then necessarily A

2 occurrences in this chapter

• Occurrence projected-formula-0024790
• Occurrence projected-formula-0024847

Equation form expr-df70b62ace0c6812

$\Box p \lif \Box \Box p$

Read as: if necessarily p, then necessarily necessarily p

Means: if necessarily p, then necessarily necessarily p

1 occurrence in this chapter

• Occurrence projected-formula-0024908

Equation form expr-e0b0fc0401e776a4

$\mSat/{M'}{\Box p \lif p}[w]$

Read as: model M prime does not satisfy the conditional from necessarily p to p at world w

Means: model M prime does not satisfy the conditional from necessarily p to p at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024987

Equation form expr-e1f8ee83625bb238

$\SSubst{!A}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}$

Read as: the result of simultaneously substituting D subscript one through D subscript n for p subscript one through p subscript n in A

Means: the result of simultaneously substituting D subscript one through D subscript n for p subscript one through p subscript n in A

2 occurrences in this chapter

• Occurrence projected-formula-0024530
• Occurrence projected-formula-0024558

Equation form expr-eac72d1087008c7b

$!A\lif \Diamond !A$

Read as: if A, then possibly A

Means: if A, then possibly A

1 occurrence in this chapter

• Occurrence projected-formula-0024742

Equation form expr-eb1a74600fe3e5c5

$\Diamond !B$

Read as: possibility of B

Means: possibility of B

1 occurrence in this chapter

• Occurrence projected-formula-0024634

Equation form expr-ec103117ee0c4cda

$(\SSubst{!B}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}} \lif \SSubst{!C}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}).$

Read as: the conditional from the simultaneous substitution instance of B to the corresponding simultaneous substitution instance of C

Means: the conditional from the simultaneous substitution instance of B to the corresponding simultaneous substitution instance of C

1 occurrence in this chapter

• Occurrence projected-formula-0024551

Equation form expr-ec7567117fa65b44

$\mSat{M'}{p \lif \Diamond p}[w]$

Read as: model M prime satisfies the conditional from p to possibly p at world w

Means: model M prime satisfies the conditional from p to possibly p at world w

1 occurrence in this chapter

• Occurrence projected-formula-0024984

Equation form expr-ef636ba8b6892981

$V(q) = \{w_2\}$

Read as: V of q is the singleton set containing w subscript two

Means: V of q is the singleton set containing w subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0024590

Equation form expr-efeb43a2f191e913

$\Box p \lif \Diamond p$

Read as: if necessarily p, then possibly p

Means: if necessarily p, then possibly p

1 occurrence in this chapter

• Occurrence projected-formula-0024905

Equation form expr-f10c0ace0b647bfa

$R' = \emptyset$

Read as: R prime is empty

Means: R prime is empty

1 occurrence in this chapter

• Occurrence projected-formula-0024981

Equation form expr-f157f71e0163a897

$\Box$

Read as: necessity

Means: necessity

6 occurrences in this chapter

• Occurrence projected-formula-0024497
• Occurrence projected-formula-0024521
• Occurrence projected-formula-0024752
• Occurrence projected-formula-0024755
• Occurrence projected-formula-0024757
• Occurrence projected-formula-0024798

Equation form expr-f192680bfea27cb4

$\Box(p \land q) \lif \Box p$

Read as: if the conjunction of p and q is necessary, then p is necessary

Means: if the conjunction of p and q is necessary, then p is necessary

1 occurrence in this chapter

• Occurrence projected-formula-0024754

Equation form expr-f213f863105aabf3

$\Box \SSubst{!B}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}}.$

Read as: necessity of the simultaneous substitution instance of B

Means: necessity of the simultaneous substitution instance of B

1 occurrence in this chapter

• Occurrence projected-formula-0024555

Equation form expr-f25337353d18e5f2

$\Box(!A \lif !B) \lif (\Box !A \lif \Box !B). \tag{\Ax{K}}$

Read as: K schema: if the conditional from A to B is necessary, then if A is necessary, B is necessary

Means: K schema: if the conditional from A to B is necessary, then if A is necessary, B is necessary

1 occurrence in this chapter

• Occurrence projected-formula-0024853

Equation form expr-f4dec0cece1e3f6b

$\Diamond p \lif \Box p$

Read as: if possibly p, then necessarily p

Means: if possibly p, then necessarily p

1 occurrence in this chapter

• Occurrence projected-formula-0024927

Equation form expr-f64fdd85fd924a20

$\mSat{M}{\Box q}[w_3]$

Read as: model M satisfies necessity of q at world w subscript three

Means: model M satisfies necessity of q at world w subscript three

1 occurrence in this chapter

• Occurrence projected-formula-0024646

Equation form expr-f6a654ac829398d2

$\lnot \Diamond !A \lif \Box (!A \lif !B)$

Read as: if A is not possible, then the conditional from A to B is necessary

Means: if A is not possible, then the conditional from A to B is necessary

1 occurrence in this chapter

• Occurrence projected-formula-0024918

Equation form expr-f80eece1214c4a79

$!D_1$

Read as: formula D subscript one

Means: formula D subscript one

4 occurrences in this chapter

• Occurrence projected-formula-0024528
• Occurrence projected-formula-0024562
• Occurrence projected-formula-0024805
• Occurrence projected-formula-0024811

Equation form expr-f87a549f62a7a792

$\Box !A$

Read as: necessity of A

Means: necessity of A

1 occurrence in this chapter

• Occurrence projected-formula-0024513

Equation form expr-f8dcd48388f1ed21

$w \in V(p)$

Read as: w belongs to V of p

Means: w belongs to V of p

3 occurrences in this chapter

• Occurrence projected-formula-0024578
• Occurrence projected-formula-0024586
• Occurrence projected-formula-0024603

Equation form expr-f9c6cbd6245f9917

$\mModel{M} = \tuple{W, R, V}$

Read as: model M is the ordered triple W, R, V

Means: model M is the ordered triple W, R, V

12 occurrences in this chapter

• Occurrence projected-formula-0024568
• Occurrence projected-formula-0024674
• Occurrence projected-formula-0024685
• Occurrence projected-formula-0024778
• Occurrence projected-formula-0024792
• Occurrence projected-formula-0024796
• Occurrence projected-formula-0024814
• Occurrence projected-formula-0024872
• Occurrence projected-formula-0024883
• Occurrence projected-formula-0024937
• Occurrence projected-formula-0024947
• Occurrence projected-formula-0024965

Equation form expr-f9e469296b427af3

$\Box p \lif p$

Read as: if necessarily p, then p

Means: if necessarily p, then p

5 occurrences in this chapter

• Occurrence projected-formula-0024744
• Occurrence projected-formula-0024756
• Occurrence projected-formula-0024759
• Occurrence projected-formula-0024761
• Occurrence projected-formula-0024906

Equation form expr-fbfc39c6e797a6a3

$W = \{w_1, w_2, w_3\}$

Read as: W is the set containing w subscript one, w subscript two, and w subscript three

Means: W is the set containing w subscript one, w subscript two, and w subscript three

1 occurrence in this chapter

• Occurrence projected-formula-0024587

Equation form expr-fc66cc046baf3fab

$w_1, w_2 \in W$

Read as: w subscript one and w subscript two belong to W

Means: w subscript one and w subscript two belong to W

1 occurrence in this chapter

• Occurrence projected-formula-0024675

Equation form expr-fcb5f40df9be6bae

$W$

Read as: W

Means: W

3 occurrences in this chapter

• Occurrence projected-formula-0024569
• Occurrence projected-formula-0024571
• Occurrence projected-formula-0024700

Equation form expr-febc2d0a7ade256c

$\pSat/{v}{!A}$

Read as: assignment v does not satisfy A

Means: assignment v does not satisfy A

1 occurrence in this chapter

• Occurrence projected-formula-0024837

Equation form expr-ff252dc637121a1d

$p_1 \lif \Box(p_1 \land p_2)$

Read as: if p subscript one, then necessarily the conjunction of p subscript one and p subscript two

Means: if p subscript one, then necessarily the conjunction of p subscript one and p subscript two

1 occurrence in this chapter

• Occurrence projected-formula-0024561

Equation form expr-ff3dd1e8af63e51f

$\mSat{M}{\Box !A}[w]$

Read as: model M satisfies necessity of A at world w

Means: model M satisfies necessity of A at world w

3 occurrences in this chapter

• Occurrence projected-formula-0024651
• Occurrence projected-formula-0024673
• Occurrence projected-formula-0024857

Equation form expr-ff9ba27416d63e9a

$(!A \land !B)$

Read as: the conjunction of A and B, enclosed in parentheses

Means: the conjunction of A and B, enclosed in parentheses

1 occurrence in this chapter

• Occurrence projected-formula-0024505

Primitive symbols of the basic modal language

The projected language has falsity, the denumerable propositional variables p subscript zero, p subscript one, and so on, negation, conjunction, disjunction, the conditional, and the necessity and possibility operators. The source lists each symbol and its syntactic role.

Source

Inductive definition of basic modal formulas

Falsity and every propositional variable are atomic formulas. Negation takes one formula; conjunction, disjunction, and the conditional take two formulas; necessity and possibility each take one formula. Nothing else is a formula. Parentheses and construction order are retained.

Source

Projected abbreviations for defined modal operators

In the selected source profile, truth abbreviates not falsity and the biconditional between A and B abbreviates the conjunction of the conditional from A to B with the conditional from B to A. Unselected source branches are not silently inserted.

Source

Definition of simultaneous substitution

Simultaneously replace p subscript one through p subscript n in formula A by D subscript one through D subscript n. The source gives the atomic, negation, conjunction, disjunction, conditional, biconditional, necessity, and possibility cases in order. Each case applies the same replacement list, and no replacement is performed sequentially.

Source

Example contrasting simultaneous and iterated substitution

A is the conditional from p one to necessarily p one and p two. D one is possibly if p two then p three, and D two is not necessarily p one. The display gives both orders of simultaneous replacement, then both orders of iterated replacement, preserving the different resulting formulas.

Source

Four source substitution results

The alignment first gives the simultaneous D one, D two instance and then the reversed simultaneous instance. It next gives the result of substituting D one before D two, and finally D two before D one. Alignment columns are layout; each spoken result is a complete conditional.

Source

Definition of a relational modal model

A model M is an ordered triple W, R, V. W is a nonempty set of worlds, R is a binary accessibility relation on W, and V assigns to every propositional variable the set of worlds where it is true. R w w prime means w prime is accessible from w.

Source

Figure containing the simple three-world model

The figure contains the source model graph with worlds w one, w two, and w three, printed truth labels for p and q at every world, and exactly two arrows, both from w one. Its inner TikZ object supplies the ordered structural reading.

Source

Simple three-world relational model

World w one prints p true and q false; w two prints p true and q true; w three prints p false and q false. The only directed accessibility edges are from w one to w two and from w one to w three. No loop or other edge is drawn.

Source

Inductive definition of truth at a world

Model M satisfies A at w according to the atomic and Boolean clauses, followed by the modal clauses. Necessity of B holds at w exactly when B holds at every world accessible from w. Possibility of B holds at w exactly when B holds at at least one accessible world.

Source

Exercise evaluating nine formulas in the simple model

For the earlier three-world model, decide whether each of nine displayed satisfaction claims holds, including vacuous necessities at w three and a nested necessity-negation claim at w one. The source supplies questions only; no answers are added.

Source

Duality of necessity and possibility

At any world, necessarily A is equivalent to not possibly not A, and possibly A is equivalent to not necessarily not A. The selected proof establishes the first equivalence; the second is assigned separately as an exercise.

Source

Exercise completing the modal duality proof

Complete the omitted second part of the preceding duality proposition. The requested proof is not supplied in the source and remains unsolved.

Source

Exercise on worlds with matching atoms and successors

Assume w one and w two agree on every propositional variable and have exactly the same accessible worlds. Prove by induction that they agree on every modal formula. The source states the conditions and goal but gives no solution.

Source

Exercise on not-possible and necessary-not

For a model M, prove that not possibly A holds at w exactly when necessarily not A holds there. This source exercise remains unsolved.

Source

Definition of truth throughout a model

A is true in model M exactly when model M satisfies A at every world w in W. This global notion is distinct from satisfaction at one specified world.

Source

Two facts about truth throughout a model

If A is true throughout a nonempty model, not A is not true throughout it; the converse fails. If the conditional from A to B and A are both true throughout a model, then B is too; the converse relationship between the global statements fails. The proof uses the earlier simple model for counterexamples.

Source

Exercise evaluating formulas throughout a three-world model

The graph fixes the printed values of p one, p two, and p three and four directed edges. Decide whether six listed formulas or schemas are true at every world. The source requests explanations but provides no solutions.

Source

Three-world model for the global-truth exercise

World w one prints p one true and p two and p three false. World w two prints p one and p two true and p three false. World w three prints all three true. Arrows go from w one to w two, w two to w three, and w one to w three; w three also has a loop.

Source

Definition of validity relative to a class of models

A is valid in class C exactly when it is true at every world in every model in C. C semantically entails A records class-relative validity; an entailment sign without C records validity in all models.

Source

Validity is inherited by subclasses

If A is valid throughout class C, it is valid throughout every subclass C prime of C.

Source

Validity is closed under necessitation

If A is valid in all models, then necessarily A is valid in all models. The proof takes an arbitrary model and world and applies validity at every accessible world.

Source

Exercise proving three valid modal formulas

Show validity of the three listed formulas involving necessity, falsity, and nested conditionals. The source gives no solutions.

Source

Exercise on validity in singleton and edgeless models

Prove one schema valid when the model has one world, and two further schemas valid when the accessibility relation is empty. The formulas and model classes remain exactly as printed; no proof is supplied.

Source

Definition of a tautological instance

A modal formula B is a tautological instance when it is obtained by simultaneously substituting modal formulas D one through D n for the variables of a modal-free tautology A.

Source

Lemma transferring a modal-free valuation into a model

For modal-free A, choose assignment v so that v of p subscript i is true exactly when model M satisfies D subscript i at w. Then v satisfies A exactly when M at w satisfies the simultaneous substitution instance of A. The proof proceeds by the projected induction cases.

Source

Atomic-variable induction chain

The three equivalences connect propositional satisfaction of p subscript i, the assigned value true, modal satisfaction of D subscript i at w, and satisfaction of the corresponding substitution instance.

Source

Negation induction chain

The chain moves from propositional satisfaction of not B to failure of B, uses the induction hypothesis for the substitution instance of B, and concludes modal satisfaction of the substitution instance of not B. The source's final justification names propositional rather than modal satisfaction; that attribution is preserved with a note.

Source

Conjunction induction chain

The chain expands propositional satisfaction of B and C, applies both induction hypotheses in source order, and recombines the two modal claims as satisfaction of the substitution instance of the conjunction.

Source

Disjunction induction chain

The chain expands propositional satisfaction of B or C, applies the two induction hypotheses, and recombines the alternatives as satisfaction of the substitution instance of the disjunction.

Source

Conditional induction chain

The chain expands the conditional as failure of B or satisfaction of C, applies the two induction hypotheses with the negative antecedent retained, and recombines the alternatives as satisfaction of the substituted conditional.

Source

Every tautological instance is valid

The contraposition proof turns a modal countermodel to a substitution instance into a propositional counterassignment to the underlying modal-free formula, using the preceding lemma.

Source

Definition of a modal schema

A schema is exactly the set of simultaneous substitution instances of a characteristic modal formula C. Characteristic formulas are unique up to renaming propositional variables; membership makes a formula an instance of the schema.

Source

Truth and validity of a schema

A schema is true in a model when all its instances are true there, and valid when it is true in every model. This is stronger than truth of only its characteristic formula in one model.

Source

Validity of the K schema

K says that necessarily if A then B implies that necessarily A implies necessarily B. The proof fixes an arbitrary accessible world and applies both boxed assumptions there.

Source

Displayed K schema

If necessarily if A then B, then if necessarily A, then necessarily B. The source tags this display K.

Source

Validity of the Dual schema

The proposition states that possibly A is equivalent to not necessarily not A. Its proof body is the source placeholder Exercise and is not expanded.

Source

Displayed Dual schema

Possibly A if and only if not necessarily not A. The source tags this display Dual.

Source

Exercise proving the Dual schema

Prove the preceding proposition that the Dual schema is valid. The source provides no proof here.

Source

Semantic modus ponens

If A and the conditional from A to B are true at a world, B is true there. Consequently the valid formulas are closed under modus ponens.

Source

Validity is equivalent to validity of every substitution instance

A is valid exactly when every simultaneous substitution instance of A is valid. The only-if proof changes the valuation component to make each p subscript i true exactly where D subscript i is true and leaves the supporting induction claim as an exercise.

Source

Exercise proving the substitution-model claim

Prove by induction that the original model satisfies substitution instance B at w exactly when the modified model satisfies A at w. The proof is not supplied.

Source

Exercise showing five familiar modal schemas are not generally valid

Give countermodels to the schemas D, T, B, four, and five as printed. They respectively involve seriality-like, reflexive, symmetric, transitive, and Euclidean patterns studied later; no answers are inserted here.

Source

Outer table of valid and invalid modal schemas

This TeX table wrapper contains one two-column tabular object with six source rows. Its caption literally says valid and open-parenthesis or question mark close-parenthesis invalid schemas; that source uncertainty is retained. The inner table supplies the nonduplicated listener structure.

Source

Six paired valid and invalid modal schemas

The first column contains six schemas identified by the source as valid, and the second contains six identified as invalid. Each row is a pair, read left then right; formulas are not regrouped by operator or inferred beyond the source labels.

Source

Exercise proving the table classifications

Prove every schema in the table's first column valid and every schema in its second column invalid. The source supplies no proofs or countermodels.

Source

Exercise classifying two modal schemas

Decide whether each of two displayed compound schemas is valid or invalid. The source does not reveal either classification.

Source

Exercise finding models that validate every instance

For each of two characteristic formulas, find one model in which every substitution instance is true. The requested models and justification are not supplied.

Source

Definition of modal entailment

Gamma entails A exactly when, at every world of every model, satisfaction of every B in Gamma guarantees satisfaction of A. With one premise B, the notation is B entails A.

Source

Worked entailment and countermodel example

The first argument proves that if p then possibly p entails if necessarily not p then not p. The second refutes entailment of if necessarily p then p using the pictured three-world model and then gives a simpler one-world edgeless countermodel. The source's set braces around the latter model triple are retained and disclosed.

Source

Figure containing the modal countermodel

The figure contains the source graph with p false at w one, p true at w two and w three, and exactly the arrows from w one to each of w two and w three. Its inner TikZ object supplies the ordered structural reading.

Source

Three-world countermodel graph

World w one prints p false. Worlds w two and w three each print p true. Directed accessibility edges go from w one to w two and from w one to w three. No loop or other edge is printed.

Source

Exercise proving a modal entailment

Show that necessity of A and B entails necessity of A. The source gives the statement only and the exercise remains unsolved.

Source

Exercise proving two non-entailments

Give countermodels showing that necessarily if p then q does not entail if p then necessarily q, and the converse entailment also fails. No countermodels are supplied.

Source

Cross-reference reference-000938

the simple three-world model figure

Source occurrence

Cross-reference reference-000939

the simple three-world model figure

Source occurrence

Cross-reference reference-000940

the necessity clause in the definition of truth at a world

Source occurrence

Cross-reference reference-000941

the simple three-world model figure

Source occurrence

Cross-reference reference-000942

the proposition on duality of necessity and possibility

Source occurrence

Cross-reference reference-000943

the simple three-world model figure

Source occurrence

Cross-reference reference-000944

the simple three-world model figure

Source occurrence

Cross-reference reference-000945

the lemma transferring modal-free satisfaction to a substitution instance

Source occurrence

Cross-reference reference-000946

the proposition that the Dual schema is valid

Source occurrence

Cross-reference reference-000947

the proposition equating validity with validity of every substitution instance

Source occurrence

Cross-reference reference-000948

the table of six paired valid and invalid schemas

Source occurrence

Cross-reference reference-000949

the three-world countermodel figure

Source occurrence

Source disclosures

• TR050-SAR-002: Source caveat. The negation induction case is routed by the source tag named prvFalse rather than prvNot. Its visible selected mathematical content is preserved; the routing name is not reinterpreted as a falsity case. source
• TR050-SAR-001: Source caveat. The negation induction chain attributes its last modal-satisfaction equivalence to the definition of propositional satisfaction. The displayed formulas and printed attribution are retained. source
• TR050-SAR-003: Source caveat. The table caption literally questions whether the second class is joined by and or or. The displayed column labels Valid Schemas and Invalid Schemas remain the operative source labels; the caption uncertainty is not silently edited away. source
• TR050-SAR-004: Source caveat. The simpler countermodel is introduced with set braces around W prime, R prime, and V prime, although models were defined as ordered triples. The printed braces are preserved and spoken as a source notation difference. source

Source-generated case expression tr050-source-macro-0001

$!A \ident \lfalse$

Read as: Case: A is the falsity constant.

Read in context source

Source-generated case expression tr050-source-macro-0002

$!A \ident q$

Read as: Case: A is the propositional variable q.

Read in context source

Source-generated case expression tr050-source-macro-0003

$!A \ident p_i$

Read as: Case: A is the propositional variable p subscript i.

Read in context source

Source-generated case expression tr050-source-macro-0004

$!A \ident \lnot !B$

Read as: Case: A is the negation of B.

Read in context source

Source-generated case expression tr050-source-macro-0005

$!A \ident (!B \land !C)$

Read as: Case: A is the conjunction of B and C.

Read in context source

Source-generated case expression tr050-source-macro-0006

$!A \ident (!B \lor !C)$

Read as: Case: A is the disjunction of B and C.

Read in context source

Source-generated case expression tr050-source-macro-0007

$!A \ident (!B \lif !C)$

Read as: Case: A is the conditional from B to C.

Read in context source

Source-generated case expression tr050-source-macro-0008

$!A \ident (!B \liff !C)$

Read as: Case: A is the biconditional between B and C.

Read in context source

Source-generated case expression tr050-source-macro-0009

$!A \ident \Box !B$

Read as: Case: A is necessarily B.

Read in context source

Source-generated case expression tr050-source-macro-0010

$!A \ident \Diamond !B$

Read as: Case: A is possibly B.

Read in context source

Source-generated case expression tr050-source-macro-0024

$\mFalse{p}$

Read as: p is false

Read in context source

Source-generated case expression tr050-source-macro-0025

$\mTrue{p}$

Read as: p is true

Read in context source

Source-generated case expression tr050-source-macro-0026

$\mFalse{q}$

Read as: q is false

Read in context source

Source-generated case expression tr050-source-macro-0027

$\mTrue{p}$

Read as: p is true

Read in context source

Source-generated case expression tr050-source-macro-0028

$\mTrue{q}$

Read as: q is true

Read in context source

Source-generated case expression tr050-source-macro-0029

$\mFalse{p}$

Read as: p is false

Read in context source

Source-generated case expression tr050-source-macro-0030

$\mFalse{q}$

Read as: q is false

Read in context source

Source-generated case expression tr050-source-macro-0011

$!A \ident \lfalse$

Read as: Case: A is the falsity constant.

Read in context source

Source-generated case expression tr050-source-macro-0012

$!A \ident \lnot !B$

Read as: Case: A is the negation of B.

Read in context source

Source-generated case expression tr050-source-macro-0013

$!A \ident (!B \land !C)$

Read as: Case: A is the conjunction of B and C.

Read in context source

Source-generated case expression tr050-source-macro-0014

$!A \ident (!B \lor !C)$

Read as: Case: A is the disjunction of B and C.

Read in context source

Source-generated case expression tr050-source-macro-0015

$!A \ident (!B \lif !C)$

Read as: Case: A is the conditional from B to C.

Read in context source

Source-generated case expression tr050-source-macro-0016

$!A \ident \Box !B$

Read as: Case: A is necessarily B.

Read in context source

Source-generated case expression tr050-source-macro-0017

$!A \ident \Diamond !B$

Read as: Case: A is possibly B.

Read in context source

Source-generated case expression tr050-source-macro-0031

$\mTrue{p_1}$

Read as: p subscript one is true

Read in context source

Source-generated case expression tr050-source-macro-0032

$\mFalse{p_2}$

Read as: p subscript two is false

Read in context source

Source-generated case expression tr050-source-macro-0033

$\mFalse{p_3}$

Read as: p subscript three is false

Read in context source

Source-generated case expression tr050-source-macro-0034

$\mTrue{p_1}$

Read as: p subscript one is true

Read in context source

Source-generated case expression tr050-source-macro-0035

$\mTrue{p_2}$

Read as: p subscript two is true

Read in context source

Source-generated case expression tr050-source-macro-0036

$\mFalse{p_3}$

Read as: p subscript three is false

Read in context source

Source-generated case expression tr050-source-macro-0037

$\mTrue{p_1}$

Read as: p subscript one is true

Read in context source

Source-generated case expression tr050-source-macro-0038

$\mTrue{p_2}$

Read as: p subscript two is true

Read in context source

Source-generated case expression tr050-source-macro-0039

$\mTrue{p_3}$

Read as: p subscript three is true

Read in context source

Source-generated case expression tr050-source-macro-0018

$!A \ident \lfalse$

Read as: Case: A is the falsity constant.

Read in context source

Source-generated case expression tr050-source-macro-0019

$!A \ident p_i$

Read as: Case: A is the propositional variable p subscript i.

Read in context source

Source-generated case expression tr050-source-macro-0020

$!A \ident \lnot !B$

Read as: Case: A is the negation of B.

Read in context source

Source-generated case expression tr050-source-macro-0021

$!A \ident (!B \land !C)$

Read as: Case: A is the conjunction of B and C.

Read in context source

Source-generated case expression tr050-source-macro-0022

$!A \ident (!B \lor !C)$

Read as: Case: A is the disjunction of B and C.

Read in context source

Source-generated case expression tr050-source-macro-0023

$!A \ident (!B \lif !C)$

Read as: Case: A is the conditional from B to C.

Read in context source

Source-generated case expression tr050-source-macro-0040

$\mFalse{p}$

Read as: p is false

Read in context source

Source-generated case expression tr050-source-macro-0041

$\mTrue{p}$

Read as: p is true

Read in context source

Source-generated case expression tr050-source-macro-0042

$\mTrue{p}$

Read as: p is true

Read in context source

Ordered structures

Simple three-world relational model

Structure: diagram tikz.

Model graph. Node one is w subscript one. Its printed valuations are, in source order, p is true and q is false. Node two is w subscript two. Its printed valuations are p is true and q is true. Node three is w subscript three. Its printed valuations are p is false and q is false. Directed accessibility edges, in source order: from w subscript one to w subscript two; then from w subscript one to w subscript three. No loop or further edge is printed. End model graph.

Read the source-bound structure in context

Three-world model for the global-truth exercise

Structure: diagram tikz.

Exercise model graph. Node one is w subscript one. Its printed valuations are p subscript one is true, p subscript two is false, and p subscript three is false. Node two is w subscript two. Its printed valuations are p subscript one is true, p subscript two is true, and p subscript three is false. Node three is w subscript three. Its printed valuations are p subscript one is true, p subscript two is true, and p subscript three is true. Directed accessibility edges, in source order: a loop at w subscript three; from w subscript one to w subscript two; from w subscript two to w subscript three; and from w subscript one to w subscript three. End exercise model graph.

Read the source-bound structure in context

Six paired valid and invalid modal schemas

Structure: table.

Two-column schema table. Headers: Valid Schemas; Invalid Schemas. Row one: valid schema if the conditional from A to B is necessary, then if A is possible, B is possible; invalid schema if the disjunction of A and B is necessary, then either A is necessary or B is necessary. Row two: valid schema if the conditional from A to B is possible, then if A is necessary, B is possible; invalid schema if A is possible and B is possible, then the conjunction of A and B is possible. Row three: valid schema the conjunction of A and B is necessary if and only if both A and B are necessary; invalid schema if A, then necessarily A. Row four: valid schema if A is necessary, then the conditional from B to A is necessary; invalid schema if necessarily possibly A, then B. Row five: valid schema if A is not possible, then the conditional from A to B is necessary; invalid schema if necessarily necessarily A, then necessarily A. Row six: valid schema the disjunction of A and B is possible if and only if either A is possible or B is possible; invalid schema if necessarily possibly A, then possibly necessarily A. End schema table.

Read the source-bound structure in context

Three-world countermodel graph

Structure: diagram tikz.

Countermodel graph. Node one is w subscript one, with printed valuation p is false. Node two is w subscript two, with printed valuation p is true. Node three is w subscript three, with printed valuation p is true. Directed accessibility edges, in source order: from w subscript one to w subscript two; then from w subscript one to w subscript three. Only p is printed in this diagram; no value for any other atom is inferred. End countermodel graph.

Read the source-bound structure in context