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