Conjunction with Contradiction

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: $\land \EE_2$	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: $\bot \EE$	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.

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

$\blacksquare$

Sources

• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.3.3$