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:
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: $\neg \EE$ | 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 $p$.
$\blacksquare$
Sources
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.3.3$