Definition:Interderivable

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 interderivable.

That is:
 * $$p \dashv \vdash q$$

means:
 * $$p \vdash q \ \mathbf {and} \ q \vdash p$$.

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:
 * $$\vdash \left({p \implies q}\right) \and \left({q \implies p}\right)$$

See the entry on Material Equivalence.

Alternative Definition
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.

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

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

This is justified by Equivalences are Interderivable.

Logical Equivalence
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.