Conjunction with Contradiction

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Theorem

A conjunction with a contradiction:

$p \land \bot \dashv \vdash \bot$


Proof by Natural Deduction

By the tableau method of natural deduction:

$p \land \bot \vdash \bot$
Line Pool Formula Rule Depends upon Notes
1 1 $p \land \bot$ Premise (None)
2 1 $\bot$ Rule of Simplification: $\land \mathcal E_2$ 1

$\Box$


By the tableau method of natural deduction:

$\bot \vdash p \land \bot$
Line Pool Formula Rule Depends upon Notes
1 1 $\bot$ Premise (None)
2 1 $p \land \bot$ Rule of Explosion: $\bot \mathcal E$ 1 From a bottom, we can prove what we like

$\blacksquare$


Proof by Truth Table

We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, in each case, the truth values in the appropriate columns match for all boolean interpretations.

$\begin{array}{|c|ccc||c|ccc|} \hline \bot & p & \land & \bot & p \\ \hline F & F & F & F & F \\ F & T & F & F & T \\ \hline \end{array}$

$\blacksquare$


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