03B05-LogicallyEquivalent.tex

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Two formulas $A$ and $B$ are said to be \emph{logically equivalent} (typically shortened to \emph{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.

\textbf{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.
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