thodg/slides/modal-logic/index.md
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Hash : ccd5ed5c
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Thomas de Grivel
Date : 2020-05-22T11:42:51
update modal-logic
Modal logic
1. Notation
1.1 Not
“Not A”
¬A
1.2 Necessarilly
“It is necessary that A”
□A
1.3 Possibly
“It is possible that A”
◊A
1.4 Implies
“If A then B”
A → B
1.5 Conjunction
“A and B”
A ∧ B
1.6 Disjunction
“A or B”
A ∨ B
1.7 Exclusive disjunction
“A xor B”
A ⊕ B
1.8 Mutual implication
“If A then B and if B then A”
A ↔ B
2. Contruction
K is a weak logic (Saul Kripke)
2.1 Necessitation rule
A is a theorem of K → □A is a theorem of K
2.2 Distribution axiom
□(A → B) → (□A → □B)
2.3 Operator ◊
◊A = ¬□¬A
3. Lemmas
3.1 Necessary conjunction
□(A ∧ B) ↔ □A ∧ □B
3.2 Disjonction of necessities
◻A ∨ ◻B → □(A ∨ B)
4. T
T is K plus the following axiom :
□A → A
Source
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modal-logic/index.md
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# Modal logic ## 1. Notation ### 1.1 Not "Not A" ¬A ### 1.2 Necessarilly "It is necessary that A" □A ### 1.3 Possibly "It is possible that A" ◊A ### 1.4 Implies "If A then B" A → B ### 1.5 Conjunction "A and B" A ∧ B ### 1.6 Disjunction "A or B" A ∨ B ### 1.7 Exclusive disjunction "A xor B" A ⊕ B ### 1.8 Mutual implication "If A then B and if B then A" A ↔ B ## 2. Contruction K is a weak logic (Saul Kripke) ### 2.1 Necessitation rule A is a theorem of K → □A is a theorem of K ### 2.2 Distribution axiom □(A → B) → (□A → □B) ### 2.3 Operator ◊ ◊A = ¬□¬A ## 3. Lemmas ### 3.1 Necessary conjunction □(A ∧ B) ↔ □A ∧ □B ### 3.2 Disjonction of necessities ◻A ∨ ◻B → □(A ∨ B) ## 4. T T is K plus the following axiom : □A → A