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Given my belief that X is false, is it still possible for me to agree that if X were true, then Y?

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    Could you maybe say a little more about why you think it might and/or might not be the case? – Paul Ross Sep 9 '18 at 12:56
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Yes, absolutely.

This is a common technique in both informal and formal logic.

In informal logic, this is called "ex hypothesi"

In more formal treatment, this is exactly how a conditional proof works. i.e, we make an assumption and then wind up with a "If this assumption is true, then ..." as our conclusion.

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Given my belief that X is false, is it still possible for me to agree that if X were true, then Y?

It would unavoidable using classical logic. ~X => (X => Y) is a tautology. You can prove it with a truth table. In fact, it corresponds to the last two lines of the truth table where X is false. It doesn't matter whether Y is true or false.

Knowing nothing about Y, not even if it is true or false, we can infer from X being false that X => Y will always be true. Y seems to come from out of nowhere and yet this is a legitimate, commonly used method of proof (the so-called Principle of Explosion -- from a falsehood, all things follow). I often use it myself.

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    +1 For the link to the truth table software and the statement of the proposition. – Frank Hubeny Sep 10 '18 at 16:15
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This is a part of the mathematical and philosophical proof concept of reductio ad absurdam, which consists of proving that A implies B and that B is false.

For example, the Wikipedia article uses the concept of the smallest positive rational number. If there was one, we could divide it by 2 and still have a rational number which will be smaller than the smallest one.

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    RAA is given by [~A => [B & ~B]] => A. – Dan Christensen Sep 10 '18 at 21:32

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