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)

Example

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)

* A*↔*B
*∴

*B*→

*A*

*(then we infer that*

*B*→

*A*is true)

Example

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.

Examples

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’.)