Principle of Non-Contradiction/Sequent Form/Formulation 2

Theorem

$\vdash \neg \paren {p \land \neg p}$

$\vdash \neg \left({p \land \neg p}\right)$
Line	Pool	Formula	Rule	Depends upon	Notes
1	1	$p \land \neg p$	Assumption	(None)
2	1	$p$	Rule of Simplification: $\land \EE_1$	1
3	1	$\neg p$	Rule of Simplification: $\land \EE_2$	1
4	1	$\bot$	Principle of Non-Contradiction: $\neg \EE$	2, 3
5		$\neg \left({p \land \neg p}\right)$	Proof by Contradiction: $\neg \II$	1 – 4	Assumption 1 has been discharged

$\blacksquare$

We apply the Method of Truth Tables to the proposition $\neg \left({p \land \neg p}\right)$.

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

$\begin{array}{|ccccc|} \hline \neg & (p & \land & \neg & p)\\ \hline T & F & F & T & F \\ T & T & F & F & T \\ \hline \end{array}$

$\blacksquare$