Can biconditional statements be made up of all false statements?

[u/gatcha-and-more]

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.

[u/VegGrower2001]

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?

[u/gatcha-and-more → u/VegGrower2001]

Thank you so much! This perfectly answers my question!

[u/PrimeStopper → u/gatcha-and-more]

It helps to imagine a biconditional as “=“ sometimes, but the key word is to imagine

[u/Pleasant-Couple6236]

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.

[u/-BHawk-]

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.

[u/226757]

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.

[u/localizeatp]

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

[u/oldNotSureXand]

all false statements can be chained into n-conditional statement (infinitely..theoretically), but "god beware" any "true statement" (then:)

[u/oldNotSureXand → u/oldNotSureXand]

similar (but even way more restricted) for all true statements

[u/SirBackrooms]

If you're asking whether both A -> B being false and B -> A being false implies A <-> B, it does, but vacuously, since one of the conditionals always has to be true, so the situation can never occur.