Logic: Construct formal proofs for the following

Question

1. A ⊃ (~B ⊃ ~A)

⊢ A ⊃ B

2. ~C

⊢ (D ∧ C) ⊃ E

3. F ⊃ G

F ∨ (H ∧ G)

⊢G

4. ɪ ≡ J

K ∨ ~J

⊢ɪ ⊃ K

Answer 1

There are many ways to construct these proofs--it'll depend on your logical system and which inference rules you have at your disposal. Feel free to reach out to me if you're still looking for some direction on these!

Answer 2

1) A --> (A --> B) by contrapositive

A --> (~A or B) by implication identity

~A or (~A or B) by implication identity

~A or ~A or (~A or B) by distributive

(~A or ~A or ~A) or B by associative

~A or B

A -->B by implication identity

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2) Given ~C,

~C = ~C or ~D must hold

= ~(C and D) must hold

This implies ~(C and D) or E must hold

(C and D) -> E

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3) If F then F or (H and G)

then F or (H and G) => G

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4) if K, then ~I or K ---> I-->K

if ~J, then ~I ---> ~I or K ---> I --> K