# Law of Excluded Middle implies Peirce's Law

## Theorem

From the Law of Excluded Middle follows Peirce's Law:

$\paren {p \lor \neg p} \vdash \paren {\paren {p \implies q} \implies p} \implies p$

## Proof

By the tableau method of natural deduction:

$\paren {p \lor \neg p} \vdash \paren {\paren {p \implies q} \implies p} \implies p$
Line Pool Formula Rule Depends upon Notes
1 1 $p \lor \neg p$ Premise (None)
2 2 $p$ Assumption (None)
3 2 $\paren {\paren {p \implies q} \implies p} \implies p$ Sequent Introduction 2 True Statement is implied by Every Statement
4 4 $\neg p$ Assumption (None)
5 4 $p \implies q$ Sequent Introduction 4 False Statement implies Every Statement
6 6 $\paren {p \implies q} \implies p$ Assumption (None)
7 6, 5 $p$ Modus Ponendo Ponens: $\implies \mathcal E$ 6, 5
8 5 $\paren {\paren {p \implies q} \implies p} \implies p$ Rule of Implication: $\implies \II$ 6 – 7 Assumption 6 has been discharged
9 1 $\paren {\paren {p \implies q} \implies p} \implies p$ Proof by Cases: $\text{PBC}$ 1, 2 – 3, 4 – 8 Assumptions 2 and 4 have been discharged

$\blacksquare$