## How do I prove: 1. A v (B & C) 2. (A v C) > ~(G & O) / ~G v ~O

1

This is a question for my philosophy.

Prove this valid using any of the rules we've studied so far:

1. A v (B & C)

2. (A v C) > ~(G & O) / ~G v ~O

1What rules did you study so far? More importantly, what did you try yourself so far? – Bram28 – 2018-05-16T19:38:00.123

1You may have the following rules to work with: distribution, commutation, simplification, modus ponens & DeMorgans theorem. Many of these terms fall under Rules of Replacement in most texts. Following these rules if you have them available will help you solve the problem in 5 steps. That is your fifth step would be 5. ~G V ~O. – Logikal – 2018-05-16T19:41:12.817

Actually I made a mistake the final line would not be line 5. I meant to say within five steps after the original two premises you would reach the goal conclusion. – Logikal – 2018-05-16T20:17:26.893

It seems like a homework to me. – lukuss – 2018-05-17T06:35:27.503

1

I am not an expert, but here is my solution (corrections are welcome):

1. A v (B^C)
2. (A v C) > ~(G ^ O) / ~G v ~O
3. (A v B) ^ (A v C) 1, Dist.
4. (A v C) > (~G v ~O) 2, De M.
5. (A v C) 3, Simp.
6. ~G v ~O 4, 5, M.P.

0

The following proof uses these rules: disjunction introduction (∨I), conjunction elimination (∧E), disjunction elimination (∨E), conditional elimination (→E) and the DeMorgan Rule (DeM).

Of the rules used, the DeMorgan Rule may not be in the "rules we've studied so far".

Here is another proof not using the DeMorgan Rule. It is based on the derivation for the DeMorgan Rule in forall x: Calgary Remix, page 142. It uses the following additional rules: contradiction introduction (⊥I), negation introduction (¬I) and the law of excluded middle (LEM).

References

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/