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# 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 |

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.

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## Comments

## if and only if

I would suggest adding "if and only if" to the "Other Names" section.