Propositional logic
- Declarative sentence: true or false
- Argument abstraction: If p and not g, then r. Not r. p. Therefore q.
- Argument formalisation: [ ( (p ∧ ¬ q) ➝ r ) ∧ (¬ r ∧ p) ) ] ➝ q
- Symbols: ∧ (and), ∨ (or), ⨁ (xor), ¬ (not), ➝ (implication)
Constructing formulae
- every propositional variable is a formula
- so is its negation
- so are constructors wit operators
Symbol priority: negation, then conjunction/disjunction, then implication
Types of proposition:
- Tautology (p ∨ ¬ p) is always true
- Contradiction (p ∧ ¬ p) is always false
- Contingency is neither a tautology nor a contradiction
Rules of propositional logic
- Implication ϕ ➝ Ψ is
- false if ϕ true and Ψ false
- true otherwise
- Bi-implication ϕ ⟷ Ψ (“ϕ if and only if Ψ”) is
- true if ϕ and Ψ have same truth value
- false otherwise
- conjunction/disjunction (with conjunction as example)
- p ∧ q ⟷ q ∧ p
- p ∧ (q ∧ r) ⟷ (p ∧ q) ∧ r
- p ∧ (q ∨ r) ⟷ (p ∧ q) ∨ (p ∧ r)
- p ∧ p ⟷ p
- p ∧ (p ∨ q) ⟷ p
- negation
- p ∧ ¬ p ⟷ F
- p ∨ ¬ p ⟷ T
- ¬ ¬ p ⟷ p
- demorgan
- ¬ p ∧ ¬ q ⟷ ¬ (p ∨ q)
- ¬ (p ∧ q) ⟷ ¬ p ∨ ¬ q
- identity
- implication