Contradictory Antecedent

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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: $\top \mathcal I$ (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: $\bot \mathcal E$ 1
4 $\bot \implies p$ Rule of Implication: $\implies \mathcal I$ 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


Sources