# indirect proofs in natural deduction

Choose one of the proofs below and use one of the indirect proof techniques (reductio ad absurdum or conditional proof) presented in Chapter 8 to demonstrate the validity of the argument. The proofs below may use any of the rules of inference or replacement rules given in Chapter 8.

1.(G â€¢ P) â†’ K, E â†’ Z, ~P â†’ ~ Z, G â†’ (E v L), therefore, (G â€¢ ~L) â†’ K

2.(S v T) â†” ~E, S â†’ (F â€¢ ~G), A â†’ W, T â†’ ~W, therefore, (~E â€¢ A) â†’ ~G

3.(S v T) v (U v W), therefore, (U v T) v (S v W)

4.~Q â†’ (L â†’ F), Q â†’ ~A, F â†’ B, L, therefore, ~A v B

5.~S â†’ (F â†’ L), F â†’ (L â†’ P), therefore, ~S â†’ (F â†’ P)

In mathematics, it is very common for there to be multiple ways to solve a given a problem; the same can be said of logic. There is often a variety of ways to perform a natural deduction. Now, construct an alternate proof. In other words, if the proof was done using RAA, now use CP; if you used CP, now use RAA. Consider the following questions, as well, in your journal response:

â€¢Will a direct proof work for any of these?

â€¢Can the proof be performed more efficiently by using different equivalence rules? 