Principle of Non-Contradiction/Sequent Form/Formulation 1/Proof 2

Theorem

$p, \neg p \vdash \bot$

Proof

We apply the Method of Truth Tables.

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

As can be seen by inspection, the truth value of the main connective, that is $\land$, is $F$ for each boolean interpretation for $p$.

$\blacksquare$