# 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 |