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Homelogically equivalent

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# 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 $A$):

$\models A\leftrightarrow B.$ |

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

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

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

$\Gamma\models A$ |

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

$B\models A.$ |

Using this, one sees that

$\models A\leftrightarrow B\qquad\mbox{iff}\qquad A\models B\mbox{ and }B% \models A.$ |

To see this: if $\models A\leftrightarrow B$, 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 $A\models B$ and $B\models A$. 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.

## Mathematics Subject Classification

03B05*no label found*

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