Disjunction with Contradiction

Theorem

$p \lor \bot \dashv \vdash p$

$p \lor \bot \vdash p$
Line	Pool	Formula	Rule	Depends upon	Notes
1	1	$p \lor \bot$	Premise	(None)
2	2	$p$	Assumption	(None)
3	3	$\bot$	Assumption	(None)
4	3	$p$	Rule of Explosion: $\bot \EE$	3
5	1	$p$	Proof by Cases: $\text{PBC}$	1, 2 – 2, 3 – 4	Assumptions 2 and 3 have been discharged

$\Box$

$p \vdash p \lor \bot$
Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p$	Premise	(None)
2		1	$p \lor \bot$	Rule of Addition: $\lor \II_1$	1

$\blacksquare$

We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, in each case, the truth values in the appropriate columns match for all boolean interpretations.

$\begin{array}{|c|ccc||c|ccc|} \hline \bot & p & p & \lor & \bot \\ \hline \F & \F & \F & \F & \F \\ \F & \T & \T & \T & \F \\ \hline \end{array}$

$\blacksquare$