Originally Posted by

**coach2uf** 1. A

Therefore, B ⊃ (~ A ⊃ C)

Prove valid using only the rule of inference and the rules of replacement (cannot use Conditional or Indirect Proofs)

Here’s how I have it done right now:

1. A

2. A v (C v ~B) 1 Add

3. (C v ~B) v A 2 Comm

4. C v ( ~B v A) 3 Assoc

5. ( ~B v A) v C 4. Comm

6. (~B v ~ ~A) v C 5. DN

7. ~ (B • ~A) v C 6. DeM

8. (B • ~A) ⊃ C 7. Cont

9. B ⊃ (~A ⊃ C) 8. Exp

I ended up working backwards on this proof which explains why each justification derives from the preceding line. I don't see anything wrong with my justifications (though if there is please correct me), but I'm wondering if there is a more eloquent way to do this proof. Also, say I didn't want to work backwards. Obviously I would have to start with an Addition, but how would I go about choosing what to add to A?

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1. A v B

2. C ⊃ ~ A

3. D ⊃ E

4. ~ D ⊃ C

5. E ⊃ ~ A

Therefore, B

Prove valid using any of the rules of deductive logic

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~(~ T ⊃ ~ R) , ~ S v T , R ≡ S

Using Natural Deduction demonstrate that the following set of statements are inconsistent.