Disjunction with Contradiction

Theorem

A disjunction with a contradiction:

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

Proof

Proof by Natural deduction

This is proved by the Tableau method:

$p \lor \bot \vdash p$

Line		Pool	Formula	Rule	Depends upon
1		1	$p \lor \bot$	P	(None)
2		2	$p$	A	(None)
3		3	$\bot$	A	(None)
4		3	$p$	$\bot \mathcal E$	3
5		1	$p$	$\lor \mathcal E$	1, 2-2, 3-4

$\blacksquare$

$p \vdash p \lor \bot$

Line		Pool	Formula	Rule	Depends upon
1		1	$p$	P	(None)
2		1	$p \lor \bot$	$\lor \mathcal I_1$	1

$\blacksquare$

Proof by Truth Table

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 models.

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

$\blacksquare$