Rule of Implication/Sequent Form/Proof by Truth Table
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Theorem
The Rule of Implication can be symbolised by the sequent:
\(\ds \paren {p \vdash q}\) | \(\) | \(\ds \) | ||||||||||||
\(\ds \vdash \ \ \) | \(\ds p \implies q\) | \(\) | \(\ds \) |
Proof
We apply the Method of Truth Tables.
$\begin{array}{|c|c||ccc|} \hline p & q & p & \implies & q\\ \hline \F & \F & \F & \T & \F \\ \F & \T & \F & \T & \T \\ \T & \F & \T & \F & \F \\ \T & \T & \T & \T & \T \\ \hline \end{array}$
As can be seen by inspection, only when $p$ is true and $q$ is false, then so is $p \implies q$ also false.
$\blacksquare$