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

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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: $\land \EE_1$ 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

$\blacksquare$


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