Logic: Lecture 2, 06 August 2015 -------------------------------- Review of syntax, valuations, validity, satisfiability, truth tables Notation: v |= a for v(a) = tt |= a for a is valid, i.e. v |= a for all v Axiomatizations System H (A1) A -> (B -> A) (A2) (A -> (B -> C)) -> ((A -> B) -> (A -> C)) (A3) (~B -> ~A) -> ((~B -> A) -> B) A, A -> B (MP) --------- B Presentation: Axiom schemes vs concrete axioms and substitutions A,B will denote schemes, a,b will denote concrete formulas Definition: "Derivation" or "proof" of a in system H Finite sequence of formulas ending with a such that each formula is an instance of an axiom or is obtained from a pair of earlier formulas in the sequence by applying MP. Notation: |- a for a is derivable (or "a has a proof", or "a is a theorem") Example: Derivation of p -> p |- (p -> ((p -> p) -> p)) -> ((p -> (p -> p)) -> (p -> p)) (A2) |- p -> ((p -> p) -> p) (A1) |- (p -> (p -> p)) -> (p -> p) (MP) |- p -> (p -> p) (A1) |- p -> p (MP) Theorem to prove: |- a iff |= a Soundness : If |- a then |= a Completeness : If |= a then |- a Proof of soundness By induction on the length of deriviation ------------------ Adding constraints May constrain valuations to reflect connections between propositions. e.g. p = "Sky is blue", q = "It is raining", insist that p -> ~q always holds Under this assumption, ~~q -> ~p (contrapositive), or q -> ~p will also always hold. How can we derive this formally? Assume X, prove a Formulas in X are concrete, not axiom schemes X |- a Redefine deriviations: A sequence where each formula is an instance of an axiom, obtained using MP, or a member of X. ======================================================================