thodg/slides/modal-logic/index.md
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Hash : 7849e27b
Author :
Thomas de Grivel
Date : 2020-05-22T11:45:17
modal-logic: simplify definitions
Modal logic
1. Notation
1.1 ¬A
“Not A”
1.2 □A
“It is necessary that A”
1.3 ◊A
“It is possible that A”
1.4 A → B
“If A then B”
1.5 A ∧ B
“A and B”
1.6 A ∨ B
“A or B”
1.7 A ⊕ B
“A xor B”
1.8 A ↔ B
“If A then B and if B then A”
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
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modal-logic/index.md
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# Modal logic ## 1. Notation ### 1.1 ¬A "Not A" ### 1.2 □A "It is necessary that A" ### 1.3 ◊A "It is possible that A" ### 1.4 A → B "If A then B" ### 1.5 A ∧ B "A and B" ### 1.6 A ∨ B "A or B" ### 1.7 A ⊕ B "A xor B" ### 1.8 A ↔ B "If A then B and if B then A" ## 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