Rule of Transposition/Formulation 1

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 $\textrm{TP}$.

It is also known as the Rule of Contraposition.

Law of Excluded Middle
Note that the proof of the reverse implication 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
We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, the truth values under the main connectives match for all models.

$\begin{array}{|ccc||ccccc|} \hline p & \implies & q & \neg & q & \implies & \neg & p \\ \hline F & T & F & T & F & T & T & F \\ F & T & T & F & T & T & T & F \\ T & F & F & T & F & F & F & T \\ T & T & T & F & T & T & F & T \\ \hline \end{array}$