lecture-1.md (970B)
1 +++ 2 title = 'Lecture 1' 3 +++ 4 5 # Intro 6 7 Basic model logic operators: 8 - □: necessary, known, provable 9 - ◇: possible, considered possible 10 11 - ◇ φ ⇔ ¬□ ¬φ 12 - □ φ ⇔ ¬◇ ¬φ 13 14 # First-order propositional logic 15 Includes variables, T, ⊥, not, and, or, implication. 16 Proofs are given by structural induction. 17 Precedence is ¬, then ∧∨, then →. 18 19 a valuation v : Var → {0,1} maps propositional variables to truth values. 20 21 the semantics of a formula under a valuation is defined with ⟦p⟧ᵥ = v(p), with p ∈ Var 22 23 if ⟦φ⟧ᵥ = 1, we write v ⊨ φ (read "v models φ") 24 - then, φ has a model, so φ is satisfiable 25 26 If every model of all φᵢ is a model of ψ, we write φ₁,...,φn ⊨ ψ 27 - then ψ is a semantic consequence of φ₁,...,φn 28 29 If v ⊨ φ for all valuations of v, then ⊨ φ (φ is a tautology) 30 31 Soundness: ⊢ implies ⊨, proved by induction on length of proof 32 33 Completeness: ⊨ implies ⊢, can be proven using consistency