Arguments with Indirect Proofs

Indirect proofs (aka reductio ad absurdum proofs) make assumptions. Based on the assumption of the proposition Φ, we might be able to derive, based on Φ and our given premises P1, P2, …, Pn, the proposition of the form ψ& ~ψ. Since ψ& ~ψ is a contradiction, it cannot be true. What led to it, also, cannot be true because it entails a contradiction. Therefore, we conclude ~Φ.

Note that since ψ&~ ψ is a contradiction, Φ cannot be true and, therefore, this must mean that ~Φ is true. By the Law of Excluded Middle, a proposition must either be true or false (exclusively). This means that if Φ is false, then ~Φ is true.

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