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In a proof by contradiction, what if both the proposition and its negation lead to contradictions? [duplicate]

I'm learning math. I've recently thought more about the proof by contradiction technique, and I have a question that I would like cleared up. Let me set the stage. Suppose I am trying to prove a ...
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Proof by contradiction, status of initial assumption after the proof is complete. [duplicate]

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 ...
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Does mathematics become circular at the bottom? What is at the bottom of mathematics? [duplicate]

I am trying to understand what mathematics is really built up of. I thought mathematical logic was the foundation of everything. But from reading a book in mathematical logic, they use "="(equals-sign)...
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I am familiar with the mechanism of proof by contradiction: we want to prove $P$, so we assume $¬P$ and prove that this is false; hence $P$ must be true. I have the following devil's advocate ...
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In a proof by contradiction, how do we know the assumption is the cause of the contradiction?

In a proof by contradiction, how do we know the assumption is the cause of the contradiction? And not just the result of some other property more fundamental to numbers? In other words, how can we ...
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Are there important situations where we study false statements as if they were true?

I know of two situations resulting from asserting that a false mathematical statement is true (by this we assume that the statement has been made to be a mathematical axiom and that it must be true ...
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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: ...
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Prove that $\lnot (p \implies q)$ is equivalent to $p \land \lnot q$?

By equivalent I mean the biconditional, as in $$\lnot (p \implies q) \iff p \land \lnot q$$ Given the definition of implication, I understand why this is true, but I need a bit of help showing this ...
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Lack of implication and logical quantifiers

After some botched attempts at formulating my question correctly, I stopped trying to simplify it and just ask straight out. Here's the situation. We have a statement that we're trying to prove: ...