Basic Logic: Presuming Truth

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

3 Answers
3

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.

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.

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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