Definition:Interderivable

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Definition

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

then $p$ and $q$ are said to be interderivable.

That is:

$p \dashv \vdash q$

means:

$p \vdash q \ \text {and} \ q \vdash p$.

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.

Also note that by the Extended Rule of Implication, we can also define:

$p \dashv \vdash q$

to mean:

$\vdash \left({p \implies q}\right) \land \left({q \implies p}\right)$

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

Let $A$ and $B$ be statement forms.

Then $A$ is logically equivalent to $B$ iff $\left({A \iff B}\right)$ is a tautology.

Two interderivable statements can be referred to as logically equivalent or provably equivalent.

This is compatible with the definition of equivalence from the result Equivalences are Interderivable.