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Unformatted text preview: Chapter 6: Definability of Connectives,Equivalence of Languages Definition of Logical equivalence : For any formulas A,B , A â‰¡ B iff  = ( A â‡” B ) . Property: A â‰¡ B iff  = ( A â‡’ B ) and  = ( B â‡’ A ) . 1 Substitution Theorem Let B 1 be obtained from A 1 by substitution of a formula B for one or more occurrences of a subformula A of A 1 . We denote it as B 1 = A 1 ( A/B ) . Then the following holds. If A â‰¡ B, then A 1 â‰¡ B 1 , 2 The next set of equivalences, or correspond ing tautologies, deals with what is called a definability of connectives in classical se mantics. For example, a tautology  = (( A â‡’ B ) â‡” ( Â¬ A âˆª B )) makes it possible to define implication in terms of disjunction and negation. We state it in a form of logical equivalence as follows. Definability of Implication in terms of nega tion and disjunction: ( A â‡’ B ) â‰¡ ( Â¬ A âˆª B ) 3 We use logical equivalence notion, instead of the tautology notion, as it makes the ma nipulation of formulas much easier. Definability of Implication equivalence allows us, by the force of Substitution Theo rem to replace any formula of the form ( A â‡’ B ) placed anywhere in another for mula by a formula ( Â¬ A âˆª B ). Hence we transform a given formula contain ing implication into an logically equivalent formula that does contain implication (but contains negation and disjunction). 4 Example 1 We transform (via Substitution Theorem) a formula (( C â‡’ Â¬ B ) â‡’ ( B âˆª C )) into its logically equivalent form not con taining â‡’ as follows. (( C â‡’ Â¬ B ) â‡’ ( B âˆª C )) â‰¡ ( Â¬ ( C â‡’ Â¬ B ) âˆª ( B âˆª C ))) â‰¡ ( Â¬ ( Â¬ C âˆª B ) âˆª ( B âˆª C ))) ....
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This note was uploaded on 02/12/2011 for the course CSE 541 taught by Professor Bachmair,l during the Spring '08 term at SUNY Stony Brook.
 Spring '08
 Bachmair,L
 Computer Science

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