Negation of Excluded Middle is False/Form 2
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Theorem
$\vdash \neg \neg (p \lor \neg p)$
Proof
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $\neg (p \lor \neg p)$ | Assumption | (None) | ||
| 2 | 1 | $\bot$ | Sequent Introduction | 1 | Negation of Excluded Middle is False: Form 1 | |
| 3 | $\neg (p \lor \neg p) \implies \bot$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged | ||
| 4 | $\neg \neg (p \lor \neg p)$ | Sequent Introduction | 3 | Negation as Implication of Bottom |
$\blacksquare$