# 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.

 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