Theorem Proving Semantic Tableaux CIS 548 November 15

Theorem Proving Semantic Tableaux CIS 548 November 15, 2006 Kutztown University

Review of Symbolic Logic n All sentences fall in one of 3 categories Ø Tautology – true under all truth values of variables Ø Inconsistency – false under all truth values Ø Contingency – true under some; false under others n Notation Ø Logical connectives » » » ^ or & : : and v : : or ~ or ¬ : : not → : : implication ↔ : : biconditional Ø Quantifiers (standard symbols unavailable in ppt) » Ă : : universal quantifier » Ë : : existential quantifier 2006 Kutztown University 2

Review of Symbolic Logic (cont. ) n Implication can be replaced Ø P → Q ≡ ~P v Q Ø Useful for tableaux method n Other substitutions that can be made Ø Ø Ø 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: http: //www. niu. edu/phil/~kapitan/Sheffer. pdf Ref: http: //mathworld. wolfram. com/NAND. html Ref: http: //en. wikipedia. org/wiki/Logical_NAND Ø Not immediately useful for tableaux 2006 Kutztown University 3

The Tableaux Method n Start with negation of sentence Ø E. g. , to prove P v ~P » start with ~ (P v ~P) » let that be the root of a tree n Simplify Ø Ø 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 Ø If branch has only atoms or negations of atoms » Stop; branch cannot be closed » Sentence is not a tautology 2006 Kutztown University 4

The Tableaux Method Propositional Logic Rules n Action determined by main connective Ø p ^ q : : p & q on the same branch Ø p v q : : p & q on two separate branches Ø ~(p ^ q) : : ~p & ~q on two branches Ø ~(p v q) : : ~p & ~q on the same branch Ø p → q : : ~p & q on two branches Ø ~(p → q) : : p & ~q on same branch Ø p ↔ q : : p → q and q → p on same branch Ø ~(p ↔ q) : : p ^ ~q and q ^ ~p on two branches 2006 Kutztown University 5

The Tableaux Method Propositional Logic Example n ((p → q) ^ (q → r)) → (p → r) Ø Place ~((p → q) ^ (q → r)) → (p → r)) at root of tree. Ø Rule 6 : : ((p → q) ^ (q → r)) & ~(p → r) on branch 1. Ø Rule 6 : : p & ~r on branch 1. Ø Rule 1 : : (p → q) & (q → r) on branch 1. Ø Rule 5 : : ~p on branch 1 a; q branch 1 b. Ø Close off branch 1 a. Ø Rule 5 : : ~q on branch 1 b 1; r branch 1 b 2. Ø Close off branch 1 b 1. Ø Close off branch 1 b 2. Ø All branches are closed a tautology. 2006 Kutztown University 6

The Tableaux Method Quantificational Logic Rules n To come. . . 2006 Kutztown University 7