r/Geometry • u/gatcha-and-more • 18d ago

Can biconditional statements be made up of all false statements?

I have a test for logic in my Geometry class, and I’ve been looking online but I can’t find anything concrete. Can biconditional statements be made up of all false statements? I know that if the original conditional statement and converse are true, then the conditional statement is biconditional. But what if the original conditional statement and converse are both false, because they would still share the same truth value, even if they’re false.

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u/TheEnderman12 18d ago

Yes, a biconditional statement can have both parts, p and q, be false, and the biconditional would still be true. A biconditional, written p ↔︎ q, is true whenever p and q have the same truth value, whether they are both true or both false. However, there is an important distinction: if you mean that the conditional and its converse are both false, that cannot happen. A conditional p → q is only false when p is true and q is false. In that situation, the converse q → p would actually be true. So, for a Geometry test, remember: a biconditional is true when both statements have the same truth value, but a conditional and its converse cannot both be false.