thodg/slides/modal-logic

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• Author : Thomas de Grivel
  Date : 2020-05-22 11:42:51
  Hash : ccd5ed5c
  Message : update modal-logic

• index.md

• # 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