Conjunction with Contradiction

Theorem
A conjunction with a contradiction:


 * $p \land \bot \dashv \vdash \bot$

Proof

 * align="right" | 2 ||
 * align="right" | 1
 * $p \land \bot$
 * $\bot \mathcal E$
 * 1
 * From a bottom, we can prove what we like.
 * From a bottom, we can prove what we like.

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.

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