Lecture_9 - Lecture 9 Lecture 9 Counterexamples...

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Unformatted text preview: Lecture 9 Lecture 9 Counterexamples Counterexamples Counterexample Equivalence: A truth­value assignment to the sentential variables is a counterexample to the left­hand side [of a law] iff it is a counterexample to at least one of the implications on the right­hand side. (p.128) Elementary Implications Elementary Implications I. II. Definition Claims If an elementary implication is valid, then it is self­evident. If an elementary implication is not self­ evident then there is a unique assignment to its sentential variables that constitutes a counterexample. Fool­Proof Method Fool­Proof Method Two kinds of self-evident implications Laws for conjunction: (∧╞) and (╞,∧ , ) Laws for disjunction: (∨ and (╞,∨ ,╞) ) Laws for conditional: (→,╞) and (╞,→) Laws for bi-conditional: (↔,╞) and (╞,↔) Laws for negated compounds. If you know (∧╞) ,(╞,∧ (∨ and (╞,→), then you , ), ,╞) If ), shouldn’t have much difficulty in recalling the others. Example Problems Example Problems [A → (B∨ ∧ ╞ ¬C → A C)] B A∧ , ¬B → ¬ C╞ C B B, ¬C → A ╞ [A → (B∨ ∧ C)] B Proofs by Contradiction Proofs by Contradiction Take ⊥ to be your favorite contradiction. I’ll assume that you have one and that it is unique. Observe that Γ ╞ A if and only if Γ,¬A ╞ ⊥ This suggests the following proof strategy: Replace initial goal Γ ╞ A with equivalent A with equivalent goal Γ,¬A ╞ ⊥ . Examples Examples ¬A, A → B╞ ¬B A∧ , ¬B → ¬ C╞ C B B, A →B, B → C╞ ¬C → ¬A ...
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This note was uploaded on 02/27/2011 for the course PHILOSOPHY 101 taught by Professor H during the Spring '11 term at Columbia College.

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