Last time: , Formula ::= P | | | | where P - - PowerPoint PPT Presentation

Last time: ◮ ϕ, ψ ∈ Formula ::= P | ϕ ∧ ψ | ϕ ∨ ψ | ϕ → ψ | ¬ϕ where P ∈ Prop ◮ (ϕ ψ) “ϕ entails ψ” means that for all I that satisfy ϕ, I also satisfies ψ ◮ Claim: ϕ ψ if and only if ϕ → ψ ◮ A sequent ϕ1, ϕ2, . . . , ϕn ⊢ ψ represents a place in the middle of a proof

◮ ϕ1, ϕ2, . . . are the assumptions (what you already know) ◮ ψ is the goal

◮ A proof rule transforms premises (sequents) into a conclusion (another sequent) ◮ A proof is a collection of proof steps where each premise has its own proof

◮ each statement follows from what came before. . .

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· · · ⊢ ϕ ∧ ψ · · · ⊢ ϕ (∧ elimination) · · · ⊢ ϕ ∧ ψ · · · ⊢ ψ (∧ elimination) · · · ⊢ ϕ · · · ⊢ ψ · · · ⊢ ϕ ∧ ψ (∧ introduction) · · · ⊢ ϕ1 ∨ ϕ2 · · · , ϕ1 ⊢ ψ · · · , ϕ2 ⊢ ψ · · · ⊢ ψ (∨ elimination) · · · ⊢ ϕ · · · ⊢ ϕ ∨ ψ (∨ introduction) · · · ⊢ ψ · · · ⊢ ϕ ∨ ψ (∨ introduction) · · · ⊢ ϕ · · · ⊢ ϕ → ψ · · · ⊢ ψ (→ elimination) · · · , ϕ ⊢ ψ · · · ⊢ ϕ → ψ (→ introduction) · · · , ϕ ⊢ ϕ (assumption) · · · ⊢ ϕ ∨ ¬ϕ (law of excluded middle) · · · ⊢ ϕ · · · ⊢ ¬ϕ · · · ⊢ ψ (absurd)

◮ Defn: ϕ1, . . . , ϕn ⊢ ψ means that there exists a proof tree for ϕ1, . . . , ϕn ⊢ ψ. ◮ Defn: ϕ1, . . . , ϕn ψ means that every interp. satisfying all ϕi also satisfies ψ ◮ Claim: if (ϕi) ⊢ ψ then (ϕi) ψ (⊢ is sound) ◮ Claim: if (ϕi) ψ then (ϕi) ⊢ ψ (⊢ is complete) ◮ Propositional logic: ϕ, ψ ∈ Formula ::= P | ϕ ∧ ψ | ϕ ∨ ψ | ϕ → ψ | ¬ϕ ◮ Predicate (first order) logic: ϕ, ψ ∈ Formula ::= P | ϕ ∧ ψ | ϕ ∨ ψ | ϕ → ψ | ¬ϕ | P(x) | ∀x, P | ∃x, P ◮ Claim: there exists a sound and complete proof system for first-order logic ◮ Claim (G¨

• del’s theorem): any system that is powerful enough to describe N

cannot be both sound and complete