Propositional logic.md (1499B)
1 +++ 2 title = 'Propositional logic' 3 +++ 4 5 # Propositional logic 6 - Declarative sentence: true or false 7 - Argument abstraction: If p and not g, then r. Not r. p. Therefore q. 8 - Argument formalisation: [ ( (p ∧ ¬ q) ➝ r ) ∧ (¬ r ∧ p) ) ] ➝ q 9 - Symbols: ∧ (and), ∨ (or), ⨁ (xor), ¬ (not), ➝ (implication) 10 11 Constructing formulae 12 13 - every propositional variable is a formula 14 - so is its negation 15 - so are constructors wit operators 16 17 Symbol priority: negation, then conjunction/disjunction, then implication 18 19 Types of proposition: 20 21 - Tautology (p ∨ ¬ p) is always true 22 - Contradiction (p ∧ ¬ p) is always false 23 - Contingency is neither a tautology nor a contradiction 24 25 Rules of propositional logic 26 27 - Implication ϕ ➝ Ψ is 28 - false if ϕ true and Ψ false 29 - true otherwise 30 - Bi-implication ϕ ⟷ Ψ (“ϕ if and only if Ψ”) is 31 - true if ϕ and Ψ have same truth value 32 - false otherwise 33 - conjunction/disjunction (with conjunction as example) 34 - p ∧ q ⟷ q ∧ p 35 - p ∧ (q ∧ r) ⟷ (p ∧ q) ∧ r 36 - p ∧ (q ∨ r) ⟷ (p ∧ q) ∨ (p ∧ r) 37 - p ∧ p ⟷ p 38 - p ∧ (p ∨ q) ⟷ p 39 - negation 40 - p ∧ ¬ p ⟷ F 41 - p ∨ ¬ p ⟷ T 42 - ¬ ¬ p ⟷ p 43 - demorgan 44 - ¬ p ∧ ¬ q ⟷ ¬ (p ∨ q) 45 - ¬ (p ∧ q) ⟷ ¬ p ∨ ¬ q 46 - identity 47 - disjunction 48 - p ∨ T ⟷ T 49 - p ∨ F ⟷ p 50 - conjunction 51 - p ∧ T ⟷ p 52 - p ∧ F ⟷ F 53 - implication 54 - p ➝ q == ¬ p ∨ q