Theorem Proving Semantic Tableaux CIS548 November 15, 2006

2006Kutztown University2 Review of Symbolic Logic All sentences fall in one of 3 categories All sentences fall in one of 3 categories  Tautology – true under all truth values of variables  Inconsistency – false under all truth values  Contingency – true under some; false under others Notation Notation  Logical connectives » ^ or & :: and » v :: or » ~ or ¬ :: not »→ :: implication » ↔ :: biconditional  Quantifiers (standard symbols unavailable in ppt) » Ă :: universal quantifier » Ë :: existential quantifier

2006Kutztown University3 Review of Symbolic Logic (cont.) Implication can be replaced Implication can be replaced  P → Q ≡ ~P v Q  Useful for tableaux method Other substitutions that can be made Other substitutions that can be made  p v q ≡ ~(~p ^ ~q) {eliminates v}  p ^ q ≡ ~(~p v ~q) {eliminates ^}  Sheffer stroke – the ultimate (or extreme) » neither-nor ≡ nand {eliminates all connectives} »Ref: » Ref: » Ref:  Not immediately useful for tableaux

2006Kutztown University4 The Tableaux Method Start with negation of sentence Start with negation of sentence  E.g., to prove P v ~P » start with ~ (P v ~P) » let that be the root of a tree Simplify Simplify  Choose any sentence on the branch  Use rules of logic to break it apart  Place pieces onto tree  If atom and its negation appear on a branch, close the branch  If branch has only atoms or negations of atoms » Stop; branch cannot be closed » Sentence is not a tautology

2006Kutztown University5 The Tableaux Method Propositional Logic Rules Action determined by main connective Action determined by main connective  p ^ q :: p & q on the same branch  p v q :: p & q on two separate branches  ~(p ^ q) :: ~p & ~q on two branches  ~(p v q) :: ~p & ~q on the same branch  p → q :: ~p & q on two branches  ~(p → q) :: p & ~q on same branch  p ↔ q :: p → q and q → p on same branch  ~(p ↔ q) :: p ^ ~q and q ^ ~p on two branches

2006Kutztown University6 The Tableaux Method Propositional Logic Example ((p → q) ^ (q → r)) → (p → r) ((p → q) ^ (q → r)) → (p → r)  Place ~((p → q) ^ (q → r)) → (p → r)) at root of tree.  Rule 6 :: ((p → q) ^ (q → r)) & ~(p → r) on branch 1.  Rule 6 :: p & ~r on branch 1.  Rule 1 :: (p → q) & (q → r) on branch 1.  Rule 5 :: ~p on branch 1a; q branch 1b.  Close off branch 1a.  Rule 5 :: ~q on branch 1b1; r branch 1b2.  Close off branch 1b1.  Close off branch 1b2.  All branches are closed  a tautology.

2006Kutztown University7 The Tableaux Method Quantificational Logic Rules To come... To come...