# Lecture_8 - Lecture 8 Lecture 8 Basic Implication Laws...

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Unformatted text preview: Lecture 8 Lecture 8 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, 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, (╞,∨ ). Give a derivation of the following: A ∨B ╞ (A → B) → [(B → A) → (A ∧B)] Other Rules and Negated Other Rules and Negated Compounds 1.. 1 2.. 2 (→,╞) (↔,╞) (╞ ,↔) Negated Compounds: ¬¬ cases ¬β cases cases Exercise 4.6 Exercise 4.6 1. 2. 3. ╞ ¬(A → B) → A (A ╞ (A ∧B) → (A ↔ B) A ∨B, A → C ╞ ¬B → C Logically Inconsistent Premises Logically Inconsistent Premises Comments about inconsistent premises Self­evident implications, termination points Disjoining as a derived rule Exercise 4.7 Exercise 4.7 ╞ (A → ¬A) → ¬A A) ╞ (A → B) → (¬B → ¬A) (A ∨B) → C╞ (A → C) ∧(B → C) (A ...
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