Contradictory Antecedent

Theorem

An implication with a contradiction as antecedent:

$\bot \implies p \dashv \vdash \top$

Proof by Natural Deduction

By the tableau method of natural deduction:

$\bot \implies p \vdash \top$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$\bot \implies p$	Premise	(None)
2			$\top$	Rule of Top-Introduction: $\top \II$	(None)

$\Box$

By the tableau method of natural deduction:

$\top \vdash \bot \implies p$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$\bot$	Assumption	(None)
2		2	$\top$	Premise	(None)
3		1	$p$	Rule of Explosion: $\bot \EE$	1
4			$\bot \implies p$	Rule of Implication: $\implies \II$	1 – 3	Assumption 1 has been discharged

$\blacksquare$

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.

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

$\blacksquare$

Also see

• Rule of Explosion

Sources

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