False Statement implies Every Statement/Formulation 1

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Theorem

$\neg p \vdash p \implies q$


Proof 1

By the tableau method of natural deduction:

$\neg p \vdash p \implies q$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg p$ Premise (None)
2 1 $\neg p \lor q$ Rule of Addition: $\lor \mathcal I_1$ 1
3 1 $p \implies q$ Sequent Introduction 2 Rule of Material Implication

$\blacksquare$


Proof 2

We apply the Method of Truth Tables.

As can be seen by inspection, where the truth value in the relevant column on the left hand side is $T$, that under the one on the right hand side is also $T$:

$\begin{array}{|cc||ccc|} \hline \neg & p & p & \implies & q \\ \hline T & F & F & T & F \\ T & F & F & T & T \\ F & F & T & F & F \\ F & F & T & T & T \\ \hline \end{array}$

$\blacksquare$


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