Peirce's Law implies Law of Excluded Middle

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From Peirce's Law:

$\left({p \implies q}\right) \implies p \vdash p$

follows the Law of Excluded Middle:

$\vdash p \lor \neg p$


By the tableau method of natural deduction:

$\vdash p \lor \neg p$
Line Pool Formula Rule Depends upon Notes
1 1 $(p \lor \neg p) \implies \bot$ Assumption (None)
2 1 $\neg(p \lor \neg p)$ Sequent Introduction 1 Negation as Implication of Bottom
3 1 $\bot$ Sequent Introduction 2 Negation of Excluded Middle is False
4 1 $p \lor \neg p$ Rule of Explosion: $\bot \EE$ 3
5 $((p \lor \neg p) \implies \bot) \implies p \lor \neg p$ Rule of Implication: $\implies \II$ 1 – 4 Assumption 1 has been discharged
6 $p \lor \neg p$ Sequent Introduction 5 Peirce's Law