Rule of Transposition/Formulation 1/Proof by Truth Table

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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

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 boolean interpretations.


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

$\blacksquare$