If supposing that a statement is false gives rise to a paradox, does this prove that the statement is true?
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?
Edit: Define paradox as follows: a situation is a paradox if and only if it presents a logical inconsistency somehow "equivalent" to the one presented in Russell's paradox.