Definition:Logical Equivalence

Definition
If two statements $p$ and $q$ are such that:


 * $p \vdash q$, i.e. $p$ therefore $q$
 * $q \vdash p$, i.e. $q$ therefore $p$

then $p$ and $q$ are said to be logically equivalent.

That is:


 * $p \dashv \vdash q$

means:


 * $p \vdash q$ and $q \vdash p$.

Note that because the conclusion of an argument is a single statement, there can be only one statement on either side of the $\dashv \vdash$ sign.

In symbolic logic, the notion of logical equivalence occurs in the form of provable equivalence and semantic equivalence.

Also known as
Two logically equivalent statements are also referred to as:


 * interderivable
 * equivalent

Also denoted as
Some sources denote $p \dashv \vdash q$ by $p \leftrightarrow q$.

Also see

 * Definition:Logical Implication
 * Definition:Biconditional

Boolean Interpretation
Two statements $p$ and $q$ are interderivable if $v \left({p}\right) = v \left({q}\right)$ for all boolean interpretations $v$.

This follows from Equivalences are Interderivable.