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Lecture_7_09 - Lecture 7 Lecture 7 Tautological Implication...

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Unformatted text preview: Lecture 7 Lecture 7 Tautological Implication Tautological Implication Definition of implication Equivalence and implication Conjunction and implication Conditional and implication Reflexivity and transitivity Substitution and implication Show that the following can fail Show that the following can fail 1. 2. 3. 4. 5. 6. A ╞ A ∧B A ∨ ╞ A B A → B╞ B A → B╞ ¬A A ↔ B╞ A ∧B A ↔ B╞ ¬A ∧¬B Basic Implication Laws Basic Implication Laws The conclusion­conditional law, (╞,→). Γ,A╞ B iff Γ╞ A → B ,A╞ Monotonicity Equivalent premise lists Disjoining Γ, A, A → B╞ C iff Γ,A,B╞ C A, Provide “top­down” derivations Provide “top­down” derivations ╞ [A → (B → C)] → [B → (A → C)] A → B, ¬A → C╞ B ∨C A → (B ∨C), ¬B╞ A → C B╞ Additional Implication Laws Additional Implication Laws The conjunction premise law, (∧╞). , Γ, A ∧B ╞ C iff Γ, A , B╞ C The conjunction conclusion law, (╞,∧ ) Γ╞ A ∧B iff Γ╞ A and Γ╞ B and Give a derivation of the following: C → A╞ (C → B) → (C →(A ∧B)) (A Additional Implication Laws Additional Implication Laws The disjunction premise law, (∨╞). , The disjunction conclusion law, (╞,∨ ). to be continued … ...
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