Principle of Non-Contradiction/Sequent Form/Formulation 1

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Theorem

The Principle of Non-Contradiction can be symbolised by the sequent:

$p, \neg p \vdash \bot$


Proof 1

By the tableau method of natural deduction:

$p, \neg p \vdash \bot$
Line Pool Formula Rule Depends upon Notes
1 1 $p$ Premise (None)
2 2 $\neg p$ Premise (None)
3 1, 2 $\bot$ Principle of Non-Contradiction: $\neg \mathcal E$ 1, 2

$\blacksquare$


Proof 2

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 $p$.

$\blacksquare$


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