# logically equivalent

Two formulas^{} $A$ and $B$ are said to be *logically equivalent* (typically shortened to *equivalent ^{}*) when $A$ is true if and only if $B$ is true (that is, $A$ implies $B$ and $B$ implies $A$):

$$\vDash A\leftrightarrow B.$$ |

This is sometimes abbreviated as $A\iff B$.

For example, for any integer $z$, the statement “$z$ is positive” is equivalent to “$z$ is not negative and $z\ne 0$”.

More generally, one says that a formula $A$ is a logical consequence of a set $\mathrm{\Gamma}$ of formulas, written

$$\mathrm{\Gamma}\vDash A$$ |

if whenever every formula in $\mathrm{\Gamma}$ is true, so is $A$. If $\mathrm{\Gamma}$ is a singleton consisting of formula $B$, we also write

$$B\vDash A.$$ |

Using this, one sees that

$$\vDash A\leftrightarrow B\mathit{\hspace{1em}\hspace{1em}}\text{iff}\mathit{\hspace{1em}\hspace{1em}}A\vDash B\text{and}B\vDash A.$$ |

To see this: if $\vDash A\leftrightarrow B$, then $A\to B$ and $B\to A$ are both true, which means that if $A$ is true so is $B$ and that if $B$ is true so is $A$, or $A\vDash B$ and $B\vDash A$. The argument can be reversed.

Remark. Some authors call the above notion semantical equivalence or tautological equivalence, rather than logical equivalence. In their view, logical equivalence is a syntactic notion: $A$ and $B$ are logically equivalent whenever $A$ is deducible^{} from $B$ and $B$ is deducible from $A$ in some deductive system.

Title | logically equivalent |

Canonical name | LogicallyEquivalent |

Date of creation | 2013-03-22 13:17:00 |

Last modified on | 2013-03-22 13:17:00 |

Owner | sleske (997) |

Last modified by | sleske (997) |

Numerical id | 10 |

Author | sleske (997) |

Entry type | Definition |

Classification | msc 03B05 |

Synonym | tautologically equivalent |

Synonym | semantically equivalent |

Synonym | tautological equivalence |

Synonym | semantical equivalence |

Synonym | tautological consequence |

Synonym | semantical consequence |

Related topic | Biconditional^{} |

Defines | logical equivalence |

Defines | logical consequence |