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# HW2 - Homework 2 PHIL 151 Due Wed Jan 19th in class at 10...

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Homework 2 PHIL 151 Due Wed. Jan 19th in class at 10 am 1. (2 points) Problem 1b on page 20. 2. (8 points)Prove or find a counter-example to (a) If Γ | = ϕ ψ then Γ | = ϕ and Γ | = ψ (b) If Γ | = ϕ ψ then Γ | = ϕ or Γ | = ψ where Γ is a set of propositional formulas (or in other words a subset of PROP ) and ϕ and ψ are individual propositional formulas (elements of PROP ). 3. Suppose that ϕ 1 , ..., ϕ k and ψ are in PROP . (a) (5 points) Show that [ | ϕ 1 ... ϕ k | ] v = 1 if and only if [ | ϕ i | ] v = 1 for each i k , where ϕ 1 ... ϕ k abbreviates ( .. ( ϕ 1 ϕ 2 ) ϕ 3 ) ... ) ϕ k ). (What has to be shown, specifically, is that the bicondtional holds for any natural number k .) (b) (5 points) Using 3(a), prove that { ϕ 1 , ..., ϕ k } | = ψ if and only if | = ( ϕ 1 ... ϕ k ) ψ . 4. Define the simultaneous substitution of ϕ 1 , ..., ϕ k for propositional sym- bols q 1 , ..., q k in ψ recursively as follows For atomic ψ , ψ [ ϕ 1 , ..., ϕ k /q 1 , ..., q k ] = ϕ i if ψ = q i , and ψ [ ϕ 1 , ..., ϕ k /q 1 , ..., q k ] = ψ otherwise.

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HW2 - Homework 2 PHIL 151 Due Wed Jan 19th in class at 10...

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