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6slides(2) - Chapter 6 Definability of Connectives,Equivalence of Languages Definition of Logical equivalence For any formulas A,B A â‰ B iff | = A

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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 sub-formula 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.

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6slides(2) - Chapter 6 Definability of Connectives,Equivalence of Languages Definition of Logical equivalence For any formulas A,B A â‰ B iff | = A

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