Negation as Implication of Bottom

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Theorem

$p \implies \bot \dashv\vdash \neg p$


Proof

By the tableau method of natural deduction:

$p \implies \bot \vdash \neg p$
Line Pool Formula Rule Depends upon Notes
1 1 $p \implies \bot$ Premise (None)
2 2 $p$ Assumption (None)
3 1,2 $\bot$ Modus Ponendo Ponens: $\implies \mathcal E$ 1, 2
4 1 $\neg p$ Proof by Contradiction: $\neg \mathcal I$ 2 – 3 Assumption 2 has been discharged

$\Box$

By the tableau method of natural deduction:

$\neg p \vdash p \implies \bot$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg p$ Premise (None)
2 2 $p$ Assumption (None)
3 1,2 $\bot$ Principle of Non-Contradiction: $\neg \mathcal E$ 1, 2
4 1 $p \implies \bot$ Rule of Implication: $\implies \mathcal I$ 2 – 3 Assumption 2 has been discharged

$\blacksquare$