# If supposing that a statement is false gives rise to a paradox, does this prove that the statement is true?

The title pretty much says it all:

If supposing that a statement is false gives rise to a paradox, does this prove that the statement is true?

Edit: Let me attempt to be a little more precise:

Suppose you have a proposition. Furthermore, suppose that assuming the proposition is false leads to a paradox. Does this imply the proposition is true? In other words, can I replace the "contradiction" in "proof by contradiction" with "paradox."

This question might still be somewhat ambiguous; I'm reluctant to attempt to precisely define "paradox" here. As a (somewhat loose) example however, consider some proposition whose negation leads to, for example, Russell's paradox. Would this prove that the proposition is true?

• except if you like the set of all sets THAT much ;) – DoctorJAM Sep 18 '14 at 21:17

It depends on the statement. Some statements e.g.

This statement is false.

lead to a contradiction whether you assume them true or false, so don't have an assignable truth value.

You also need to know or prove that your statement has a truth value (i.e. is either true or false) before you can conclude your argument.

Certainly not. Suppose you have a red box and a green box, exactly one of which contains a treasure, and the following two statements about the boxes:

1. The treasure is in the green box.
2. Exactly one of these statements is true.

Assuming that the treasure is in the green box results in an easy paradox: statement 2 can't be either true or false. Many people erroneously conclude from this that the treasure must be in the red box. However, this conclusion is invalid; the treasure could be in either box.

Yes. This is what is known as a proof by contradiction. When you want to prove a statement $P$ implies a statement $Q$ (i.e., you want to prove $P \implies Q$ is true), you always start by assuming $P$ is true.

Then, if you want to proceed by contradiction, you suppose $Q$ is false. Usually, if $P \implies Q$ is a true statement, then assuming that $Q$ is false will lead to a result that contradicts something about the assumption $P$.

Note that sometimes the contradiction you find doesn't contradict any assumptions in $P$ directly, but may contradict any background assumptions you have, such as assumptions about the space you are in in general.

• While your answer would be valid if $P$ has a specific truth value, Mark Bennet given an example of a statement which is neither true nor false so it cannot be proved by contradiction. I.e. OP is not correct in general. – Jam Sep 18 '14 at 21:48
• @EulCan Ouch. I did say "if $P \implies Q$ is a true statement." If you were the one that downvoted me, I think that's sad because I don't think my answer was "not useful." – layman Sep 18 '14 at 23:49

The whole question boils down to which sentences have a truth value. See this answer for a detailed analysis of this notion. All modern mathematics is done in a manner that can in fact be formalized in a formal system that does not allow self-reference or meta-reference. So one cannot even assign to a propositional variable $P$ the liar sentence or the Quine sentence. Therefore one cannot suppose that such a paradoxical sentence is true or false to derive a contradiction or anything whatsoever. We also have the law of excluded middle in classical logic, so there you have the valid deduction method of proof by contradiction, which works regardless of what form the contradiction takes. The term "paradox" is quite vague, but if by it you mean something that you can prove cannot possibly be true, then deriving any paradox is equivalent to deriving a contradiction, and implies that at least one of the currently active assumptions is false.