Principle of Non-Contradiction/Sequent Form/Formulation 2

From ProofWiki
Jump to navigation Jump to search


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

Proof 1

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


Proof 2

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 $p$.

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