6slides(1) - Chapter 6: Examples of Propositional...

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Unformatted text preview: Chapter 6: Examples of Propositional Tautologies, Logical Equivalences Implication Modus Ponens known to the Stoics (3rd cen- tury B.C) | = (( A ∩ ( A ⇒ B )) ⇒ B ) Detachment | = (( A ∩ ( A ⇔ B )) ⇒ B ) | = (( B ∩ ( A ⇔ B )) ⇒ A ) 1 Sufficient Given an implication ( A ⇒ B ) , A is called a sufficient condition for B to hold. Necessary Given an implication ( A ⇒ B ) , B is called a necessary condition for A to hold. 2 Implication Names Simple ( A ⇒ B ) is called a simple impli- cation . Converse ( B ⇒ A ) is called a converse implication to ( A ⇒ B ). Opposite ( ¬ B ⇒ ¬ A ) is called an opposite implication to ( A ⇒ B ). Contrary ( ¬ A ⇒ ¬ B ) is called a contrary implication to ( A ⇒ B ). 3 Laws of contraposition | = (( A ⇒ B ) ⇔ ( ¬ B ⇒ ¬ A )) , | = (( B ⇒ A ) ⇔ ( ¬ A ⇒ ¬ B )) . The laws of contraposition make it possible to replace, in any deductive argument, a sentence of the form ( A ⇒ B ) by ¬ B ⇒ ¬ A ), and conversely....
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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(1) - Chapter 6: Examples of Propositional...

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