# 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

 $a\Leftrightarrow b$

or

 $a\leftrightarrow b$

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

a b $a\Leftrightarrow b$
F F T
F T F
T F F
T T T

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,

 $(a\rightarrow b)\land(b\rightarrow a)$

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.

Title biconditional Biconditional 2013-03-22 11:53:06 2013-03-22 11:53:06 Mathprof (13753) Mathprof (13753) 17 Mathprof (13753) Definition msc 03-00 iff PropositionalLogic Equivalent3