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

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.

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.

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.

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

Also see

 * Therefore
 * Because
 * Conditional
 * Logical Implication
 * Biconditional