Rule of Addition/Sequent Form/Proof by Truth Table
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Theorem
The Rule of Addition can be symbolised by the sequents:
| \(\text {(1)}: \quad\) | \(\ds p\) | \(\) | \(\ds \) | |||||||||||
| \(\ds \vdash \ \ \) | \(\ds p \lor q\) | \(\) | \(\ds \) |
| \(\text {(2)}: \quad\) | \(\ds q\) | \(\) | \(\ds \) | |||||||||||
| \(\ds \vdash \ \ \) | \(\ds p \lor q\) | \(\) | \(\ds \) |
Proof
We apply the Method of Truth Tables.
$\begin{array}{|c|c||ccc|} \hline p & q & p & \lor & q\\ \hline \F & \F & \F & \F & \F \\ \F & \T & \F & \T & \T \\ \T & \F & \T & \T & \F \\ \T & \T & \T & \T & \T \\ \hline \end{array}$
As can be seen, whenever either $p$ or $q$ (or both) are true, then so is $p \lor q$.
$\blacksquare$