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

## Theorem

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

## Proof

By the tableau method of natural deduction:

$\vdash \neg \left({p \land \neg p}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $p \land \neg p$ Assumption (None)
2 1 $p$ Rule of Simplification: $\land \mathcal E_1$ 1
3 1 $\neg p$ Rule of Simplification: $\land \mathcal E_2$ 1
4 1 $\bot$ Principle of Non-Contradiction: $\neg \mathcal E$ 2, 3
5 $\neg \left({p \land \neg p}\right)$ Proof by Contradiction: $\neg \mathcal I$ 1 – 4 Assumption 1 has been discharged

$\blacksquare$