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First of all I'd like to say that I have looked for the answers to my specific question and have not found it in the existing topics.

The question is fairly simple. Say, we need to prove statement P by the method of contradiction. Assuming that $\lnot P$ holds, using the list of statements proven earlier to hold or derived by us during the proof, we arrive to P being $true$.

$$\lnot P \to A_1 \to\ ... \ \to A_n \to P$$ $$\lnot P \to P \iff \lnot(\lnot P) \lor P \iff P $$

We can therefore add P to the list of our proven statements, because it was derived. Most of the proofs contain something in the lines of "the obtained contradiction proves that our initial assumption ($\lnot P$) was wrong and so $P$ holds".

What I don't understand is, if the initial assumption ($\lnot P$) is thus proven to be false, then why can we be sure that anything derived from it holds (in particular, that P holds)? On the other hand, if it cannot be derived then the assumption ($\lnot P$) can in fact be true.

Can someone explain why this type of argument cannot be used?

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    No, you proved, from no assumptions, the implication $(\lnot P)\to P$. And it is a tautology that $(\lnot P)\to P$ implies (in fact, it is equivalent to) $P$. So, from no assumptions, you proved $P$. Same reasoning applies if you are using some additional assumptions to conclude from $\lnot P$ that $P$ holds. In that case, what you get is that those assumptions give you $P$. – Andrés E. Caicedo Apr 22 '19 at 13:00

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We are not deriving $P$ from the assumption $\lnot P$. What we are doing is that we are deriving a contradiction from the assumption $\lnot P$. The contradiction can be something like $P \land \lnot P$, or it could be something else which contradicts an already proven statement.

The fact that we have arrived at a contradiction in turn implies that our initial assumptions have to be false. This is how we derive that $\lnot P$ must be false and hence $P$ must be true.

To reiterate, we are not proving $P$ with $\lnot P$ as an assumption. We are using the assumption to arrive at a contradiction which then implies $P$.

Ekanshdeep Gupta
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The basic assumption is that we reason in a consistent logical system. What we want to derive is not necessarily that $P$ holds, we just want to show that the assumption $\neg P$ leads to a contradiction and, as we assume the logical system to be consistent, it cannot have contradictions. This shows that $\neg P$ is wrong.

Then, if we further assume that tertium non datur holds, i.e. that for any given statement $P$ the statement $P\lor \neg P$ holds, it follows that $P$ must be true.

Again, just to reiterate: Proof by contradiction just has to yield any contradiction under the assumption of $\neg P$ to prove $P$. However, such reasoning is only sound if the logical calculus in which you're reasoning has something like TND or a contradiction rule and is consistent.

J. Becker
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  • Do you therefore mean that for proof by contradiction to work, "assumption ¬P leads to a contradiction and, as we assume the logical system to be consistent, it cannot have contradictions. This shows that ¬P is wrong" statement has to be a part of the axiomatic system that we use in our logic to perform the proofs? –  Apr 22 '19 at 13:05
  • Well, we need to know that our theory is consistent for such a reasoning to work. If our theory is not consistent, then being able to derive a contradiction from $\neg P$ would not prove the falsity of $\neg P$. However, the consistency of our theory does not need to be formulated as an axiom, see for example Zermelo Frenkel axiom system for sets. It does not include an axiom that states its concistency.

    In fact, including such an axiom is basically unnecessary. If the axiom set including the axiom is satisfiable, then there exists logically concistent models for it sans consistency axiom.

     – J. Becker Apr 22 '19 at 13:15
  • Ok. I see now. So, this kind of reasoning is possible when using "non-constructive" classical mathematics and would fail otherwise. Its a matter of belief. –  Apr 22 '19 at 14:50
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What I don't understand is, if the initial assumption ($\lnot P$) is thus proven to be false, then why can we be sure that anything derived from it holds (in particular, that P holds)? On the other hand, if it cannot be derived then the assumption ($\lnot P$) can in fact be true.

When you introduce an assumption $P$ in a proof, you do not need to first prove that it is true. You simply want see what conclusions could be drawn from $P$ if it were true.

Having introduced the assumption $P$, if you can then derive $Q$ without introducing any other assumptions, you can then conclude that $P \implies Q$

On the other hand, if you can then derive a contradiction of the form $Q\land \neg Q$ without introducing any other assumptions, you can then conclude $\neg P$.

As to the status of the initial assumption (your initial question here), in both cases, after you have written down your conclusion, you may refer to the conclusion in subsequent lines of your proof. You may not, however, refer to the initial assumption ($P$ in this case) or any subsequent lines prior to the resulting conclusion. In a sense, those lines will have been deactivated after forming your conclusion. We say that the initial assumption has then been discharged.

Such assumption-conclusion blocks, may, of course, be nested within one another as sub-proofs. The above rules will apply for each of these blocks individually.

Dan Christensen
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  • Is deriving from a true statement the same thing as deriving from a presumably true statement? Also, is it something that holds only in classical mathematics (based on classical logic)? –  Apr 24 '19 at 21:02
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    @Eval No. Not sure. The only "true statement" in a mathematical proof is an axiom or a theorem. A proof results in a theorem if all assumptions introduced in that proof have been discharged. Assumptions (i.e. presumably true statements) are not derived from other statements. They may even contradict true statements. So deriving a statement from an assumption will not necessarily result in a true statement. Deriving a statement from a true statement without any active assumptions (all have been discharged) will result in a true statement. Not sure whether this holds in all non-classical logics. – Dan Christensen Apr 24 '19 at 23:14