Rule of Transposition

Theorem
A statement and its contrapositive have the same truth value:


 * $$p \implies q \dashv \vdash \neg q \implies \neg p$$

Its abbreviation in a tableau proof is $$\textrm{TP}$$.

Proof by Natural deduction
These are proved by the Tableau method.

This follows directly from Modus Tollendo Tollens:

Comment
Note that the second part of this proof requires the use of double negation elimination, which depends on the Law of the Excluded Middle. This axiom is not accepted by the intuitionist school.

Proof by Truth Table
Let $$v: \left\{{p, q}\right\} \to \left\{{T, F}\right\}$$ be an interpretation for a logical formula $$\phi$$ of two variables $$p, q$$.

We see that $$v \left({p \implies q}\right) = v \left({\neg q \implies \neg p}\right)$$ for all interpretations $$v$$.

Hence the result by the definition of interderivable.