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22C/55:181 Propositional Logic Definition: the (well formed) formulas (wffs) over a set of variables V consist of: (i) T, F, and each XŒV, (ii) for each pair of wffs a and b • (ÿa), • (aŸb), • (a⁄b), • (afib), • (aÛb), To avoid excessive parenthesizes in wffs we adopt the following precedence and take fi to be rightassociative highest precedence ÿ Ÿ ⁄ fi, Û lowest precedence The definitions of the logical connectives are given by their “truth-tables” P Q ÿP PŸQ P⁄Q PfiQ PÛQ T T F F T F T F F F T T T F F F T T T F T F T T T F F T Definition: an assignment to a set V of variables is a function s: V Æ {T,F}. Each assignment is inductively extended to apply to wffs. For wffs a and b • s(ÿa) = ÿs(a), • s(aŸb) = s(a) Ÿ s(b), • s(a⁄b) = s(a) ⁄ s(b), • s(afib) = s(a) fi s(b), • s(aÛb) = s(a) Û s(b), and • s(T) = T, s(F) = F, Definition: wffs a and b are logically equivalent, a ≡ b, if s(a) = s(b) for each assignment s. Definition: Let a and b be wffs. Then a is satisfiable if there is an assignment s so that s(a) = T; an unsatisfiable wff is also called a contradiction. If s(a) = T for every assignment s, then a is a tautology. Definition: a set of wffs S logically implies a wff a, S |= a, provided that for each assignment s such that s(b) = T for each bŒS, s(a) = T (if S = ∅, write |= a and a is a tautology). page 1 of 2 22C/55:181 Definition: a proof system consists of the following constituents • a subset of wffs called axioms — we expect there is a decision procedure to effectively determine whether or not a wff is an axiom. • a finite collection {R1 , … , Rn } of inference rules, where each rule Ri allows us to decide for wffs a1 , a1 … am i … , a m , b whether or not b is a direct consequence of a1 , … , a m , written . b i i • proofs which are sequences a1 , … , a k of wffs so that each i (1≤i≤k), either ai is an axiom or ai is a direct consequence of some preceding wffs in the sequence; the last wff of a proof is called a theorem. Two important rules of inference are: a afib b a⁄b ÿa⁄g • resolution — for any wffs a, b, and g, . b⁄g • modus ponens — for any wffs a and b, Definition: in a proof system with axioms A, a wff a is a consequence of a set of wffs G if it is a theorem in the proof system with axioms A » G. The elements of G are called hypotheses or premises and we write G |— a; if G = ∅, write |— a (i.e., with no premises, the consequences are just the theorems). a1 … am i Definition: a rule of inference is sound provided that whenever each aj (1≤j≤mi ) is a b tautology, b is also a tautology. A proof system is sound if each theorem is a tautology. A proof system is complete if each tautology is a theorem. page 2 of 2