How Can I Prove De Morgan's Law Using Basic Logical Rules?

[LT123]

Homework Statement

I am trying to prove (~P & ~Q) => ~(P V Q) using only MP, negation elimination, &I conjunction, &E simplification, addition, constructive dilemma, biconditional introduction and elimination, conditional proof, and reductio ad absurdum.

Homework Equations

The Attempt at a Solution

It seems to me that the only way to do the prove is reductio ad absurdum. But,
I am stuck after hypothesizing (P V Q). I looked at embedding other hypothesis to get a useful negation but could only come up with ~(P & Q). I don't think it will be useful.

Any hints on the hypothesis I need to use?

 

[mighty2000]

Hi there! As you mentioned, the best approach to prove this statement is through reductio ad absurdum. Here's one possible way to do it:

1. Assume ~(P V Q) [Hypothesis]
2. Assume P [Hypothesis]
3. From (2), we can use &I to get P & P.
4. Using &E, we can get P.
5. From (2) and (5), we can use Constructive Dilemma to get P V Q.
6. But this contradicts our original assumption in (1).
7. Therefore, we can use Reductio Ad Absurdum to conclude ~(P V Q) => ~P.
8. Similarly, we can also prove ~(P V Q) => ~Q.
9. Using &I, we can get ~(P V Q) & ~P & ~Q.
10. By Biconditional Introduction, we can get (~P & ~Q) => ~(P V Q).

I hope this helps! Let me know if you have any further questions. Good luck!