Definition:Logical Equivalence

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


 * $p \vdash q$, that is: $p$ therefore $q$
 * $q \vdash p$, that is: $q$ 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
 * coimplicant

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

Others use $p \equiv q$.

On, during the course of development of general proofs of logical equivalence, the notation $p \leadstoandfrom q$ is used as a matter of course.

Note that in Distinction between Logical Implication and Conditional, the distinction between $\implies$ and $\leadsto$ is explained.

In the same way, $\leadstoandfrom$ and $\iff$ are not the same -- it makes no sense to write:
 * $A \iff B \iff C$

when what should be written is:
 * $A \leadstoandfrom B \leadstoandfrom C$

Also see

 * Definition:Logical Implication
 * Definition:Biconditional