Truth-Tables

1. Truth-Tables • Three ways of using truth-tables to answer if • KB |= Q : • a) by the definition of entailment: • KB |= Q iff for every interpretation I, • if I satisfies KB then I satisfies Q. • b) by transforming into a unsatisfiability problem: • KB |= Q iff KB U { Q} is unsatisfiable • c) by transforming into a validity problem: • {w1, …, wn} |= Q iff |= ((w1  …  wn)  Q)

2. Deductive System • Language • Inference Rules (R) • Logical axioms (AX)

3. Inference Rules • Inference rules allow us to deduce new wffs from • known ones • Notation • <given wffs that match these patterns> • --------------------------------------------------- • <we can deduce this>

4. Modus Ponens • If we believe a rule, and we believe that its antecedent is true, we can believe that its conclusion is true. • Let A, B be wffs. A , A  B --------------------------------------------------- B

5. Unit resolution • If at least one of two wffs is true (A or B) & we know one is false, then the other must be true • Let A, B be wffs. A , A  B --------------------------------------------------- B • Really, just a variant of modus ponens

6. Resolution • Case analysis on the possible values of B: Because B cannot be both true and false, one of the other disjuncts must be true in one of the premisses: B  C , A  B --------------------------------------------------- C  A • Alternatively (implication is transitive): • Given: A  B, and also B  C • A  C

7. Logical Axioms • Valid wffs • Examples: • A  A • (A  B)  B • (A  B)  A • A  (B  A)

8. Proof Systems KB |- Q iff there is a sequence of wffs D1, ..., Dn such that Dn is Q and for each Di in the sequence: a) either Di is in KB or b) Di can be inferred from a wff (or wffs) earlier in the sequence by using one of the rules of inference in R, or c) Di is an instance of a logical axiom in AX The sequence (if exists) D1, ..., Dn is called a proof or a deduction of Q from KB. Q is said to be a theorem of KB. KB |- Q : a) by the definition of entailment: