indirect proofs in natural deduction

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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 • P) → K, E → Z, ~P → ~ Z, G → (E v L), therefore, (G • ~L) → K

2.(S v T) ↔ ~E, S → (F • ~G), A → W, T → ~W, therefore, (~E • A) → ~G

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

4.~Q → (L → F), Q → ~A, F → B, L, therefore, ~A v B

5.~S → (F → L), F → (L → P), therefore, ~S → (F → 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:

•Will a direct proof work for any of these?

•Can the proof be performed more efficiently by using different equivalence rules?

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