biconditional

1 Biconditional

A biconditional is a truth function that is true only in the case that both parameters are true or both are false.

Symbolically, the biconditional is written as

$$a\iff b$$

or

$$a\leftrightarrow b$$

with the latter being rare outside of formal logic. The truth table for the biconditional is

The biconditional function is often written as “iff,” meaning “if and only if.”

It gets its name from the fact that it is really two conditionals in conjunction,

$$(a\to b)\wedge (b\to a)$$

This fact is important to recognize when writing a mathematical proof, as both conditionals must be proven independently.

2 Colloquial Usage

The only unambiguous way of stating a biconditional in plain English is of the form “$b$ if $a$ and $a$ if $b$.” Slightly more formal, one would say “$b$ implies $a$ and $a$ implies $b$.” The plain English “if” may sometimes be used as a biconditional. One must weigh context heavily.

For example, “I’ll buy you an ice cream if you pass the exam” is meant as a biconditional, since the speaker doesn’t intend a valid outcome to be buying the ice cream whether or not you pass the exam (as in a conditional). However, “it is cloudy if it is raining” is not meant as a biconditional, since it can obviously be cloudy while not raining.