Logical biconditional

As it turns out, these two statements are only the same—when zero or two arguments are involved. In fact, the following truth tables only show the same bit pattern in the line with no argument and in the lines with two arguments:

The left Venn diagram below, and the lines (AB    ) in these matrices represent the same operation.

Like all connectives in first-order logic, the biconditional has rules of inference that govern its use in formal proofs.

Biconditional introduction allows one to infer that if B follows from A and A follows from B, then A if and only if B.

For example, from the statements "if I'm breathing, then I'm alive" and "if I'm alive, then I'm breathing", it can be inferred that "I'm breathing if and only if I'm alive" or equivalently, "I'm alive if and only if I'm breathing." Or more schematically:

For example, the statement "I'll buy you a new wallet if you need one" may be interpreted as a biconditional, since the speaker doesn't intend a valid outcome to be buying the wallet whether or not the wallet is needed (as in a conditional). However, "it is cloudy if it is raining" is generally not meant as a biconditional, since it can still be cloudy even if it is not raining.