assuming the conclusion A natural deduction proof goes from premmisses to conclusion, and under normal circumstances you will not assume the conclusion. Sometimes you may assume the negation of the conclusion and do some reductio ad absurdum and that ilk, but never assume the conclusion unnegated. But recently I saw a wrong proof where a student did assume the conclusion. So (internally) I was ranting "never, never , never in a proof assume the conclusion, it is a crime against logic, prooftheory . Where is my broad red marker, and all that." But then I became reflective: "is this really always the case? Are there really no valid proofs where assuming the conclusion is needed or where assuming the conclusion speeds up or shortens the proof?" Any proofs why assuming the conclusion cannot help, or examples where it is useful welcome.

You can't assume the conclusion. If you could, you could prove any statement P as follows:

1. Assume P.
2. Then P.
3. Done.



It's the mathematical equivalent of saying that something is true "because it is".

That said, in a formal system, you might have to do a subproof premised upon a statement that happens to be the conclusion of the overall proof; for example, to prove $P$ from $P \vee ((P \rightarrow Q) \rightarrow P)$, you might have to use "assume $P$" and "assume $((P \rightarrow Q) \rightarrow P)$" cases and prove $P$ in both cases. However, the "assume $P$" case is trivial. In a proof with words, we'd say something like "we only need to consider the second case", or perhaps devote one sentence to "If P, then we're done." This is quite different from unconditionally assuming the desired result.