6slides(2) - Chapter 6 Definability of Connectives,Equivalence of Languages Definition of Logical equivalence For any formulas A,B A 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 ≡ B iff | = ( A ⇔ B ) . Property: A ≡ B iff | = ( A ⇒ B ) and | = ( B ⇒ 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 ≡ B, then A 1 ≡ 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 ⇒ B ) ⇔ ( ¬ A ∪ 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 ⇒ B ) ≡ ( ¬ A ∪ 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 ⇒ B ) placed anywhere in another for- mula by a formula ( ¬ A ∪ 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 ⇒ ¬ B ) ⇒ ( B ∪ C )) into its logically equivalent form not con- taining ⇒ as follows. (( C ⇒ ¬ B ) ⇒ ( B ∪ C )) ≡ ( ¬ ( C ⇒ ¬ B ) ∪ ( B ∪ C ))) ≡ ( ¬ ( ¬ C ∪ B ) ∪ ( B ∪ 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 B iff | = A

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