Hash :
ccd5ed5c
Author :
Thomas de Grivel
Date :
2020-05-22T11:42:51
update modal-logic
“Not A”
¬A
“It is necessary that A”
□A
“It is possible that A”
◊A
“If A then B”
A → B
“A and B”
A ∧ B
“A or B”
A ∨ B
“A xor B”
A ⊕ B
“If A then B and if B then A”
A ↔ B
K is a weak logic (Saul Kripke)
A is a theorem of K → □A is a theorem of K
□(A → B) → (□A → □B)
◊A = ¬□¬A
□(A ∧ B) ↔ □A ∧ □B
◻A ∨ ◻B → □(A ∨ B)
T is K plus the following axiom :
□A → A
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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