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: $\neg \II$	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: $\neg \EE$	1, 2
4		1	$p \implies \bot$	Rule of Implication: $\implies \II$	2 – 3	Assumption 2 has been discharged

$\blacksquare$