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

a	b	$a\iff b$
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,

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

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.

Title	biconditional
Canonical name	Biconditional
Date of creation	2013-03-22 11:53:06
Last modified on	2013-03-22 11:53:06
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	17
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 03-00
Synonym	iff
Related topic	PropositionalLogic
Related topic	Equivalent3