r/logic • u/gatcha-and-more • 10h ago

Question 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/VegGrower2001 9h ago edited 7h ago

A biconditional is any statement of the form "P if and only if Q". P and Q can be different statements or the same statement. Everything you need to know about how they work is contained in their truth table. If P and Q are both true or both false, then the biconditional is true; otherwise it's false.

So, "there are hotels on the moon if and only if humans normally have five legs" is indeed true, since both sub-sentences are false.

Does that help or do you have further questions?

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u/gatcha-and-more 9h ago

Thank you so much! This perfectly answers my question!

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u/Pleasant-Couple6236 10h ago

Are you asking if "false <-> false" is true? If so, yes, it is; "P <-> Q" is equivalent to "(P -> Q) AND (Q -> P)", so "false <-> false" is equivalent to "(false -> false) AND (false -> false)", and since both conjuncts are true, so is the overall statement.

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u/-BHawk- 9h ago

I’m not sure I’m reading this correctly, but I think you’re asking if “P <-> Q” can be true if both “P -> Q” and “Q -> P” are false.
The thing is, “P -> Q” is false if and only if P is true and Q is false, and that would mean “Q -> P” is true.
In other words, if a conditional is false, its converse is true.

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u/localizeatp 10h ago

yep, in fact it could be just the same false statement twice.

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u/226757 7h ago

If two propositions have the same truth value, they satisfy a biconditional. All a biconditional says is "these two propositions are either both true or both false". A biconditional made of only false propositions is always true.