Definition: A biconditional is a compound statement formed by combining 2 conditionals under “and.” Biconditionals are true when both statements have the exact same truth value.
A biconditional actually contains 2 conditional statements. A↔B means (A→B)˄(B→A). Either both are true or neither are true.
How can this we see this?
From 2 conditional statements we can infer a biconditional.
A→B (if A→B is true)
B→A (if B→A is true)
∴ A↔B (then we infer that A↔B is true)
If I’m breathing, then I’m alive
If I’m alive, then I’m breathing
Therefore I’m breathing if and only if I’m alive. Or also: I’m alive if and only if I’m breathing.
And the other way around is also possible: from a biconditional we can infer 2 conditionals.
A↔B (iff A↔B is true)
∴ A→B (then we infer that A→B is true)
∴ B→A (then we infer that B→A is true)
if it’s true that I’m breathing if and only if I’m alive,
then it’s true that if I’m breathing, I’m alive;
likewise, it’s true that if I’m alive, I’m breathing.
Different interpretations in language:
A biconditional is of the form (A→B)˄(B→A), expressed as A if B and B if A, or A↔B, A if and only if B.
Or we say, A implies B and B implies A.
Sometimes ‘if’ is used as a biconditional, depending on the context.
I’ll buy you a new wallet if you need one.
If the speaker means with ‘if’, whether or not the wallet is needed, then ‘if’ is meant as a biconditional.
It is cloudy if it is raining.
Since it can be cloudy while it’s not raining, ‘if’ here is not meant as a biconditional. (And also not as a conditional ‘conjunction’, but as a time ‘conjunction’. So correct would be ‘when’.)