# Peirce's Law/Formulation 2

## Theorem

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

## Proof 1

By the tableau method of natural deduction:

$\vdash \paren {\paren {p \implies q} \implies p} \implies p$
Line Pool Formula Rule Depends upon Notes
1 1 $\paren {p \implies q} \implies p$ Assumption (None)
2 1 $p$ Sequent Introduction 1 Peirce's Law: Formulation 1: $\paren {p \implies q} \implies p \vdash p$
3 $\paren {\paren {p \implies q} \implies p} \implies p$ Rule of Implication: $\implies \mathcal I$ 1 – 2 Assumption 1 has been discharged

$\blacksquare$

## Proof 2

We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, the truth values under the main connective are $T$ for all boolean interpretations.

$\begin{array}{|ccccc|c|c|}\hline ((p & \implies & q) & \implies & p) & \implies & p \\ \hline F & T & F & F & F & T & F \\ F & T & T & F & F & T & F \\ T & F & F & T & T & T & T \\ T & T & T & T & T & T & T \\ \hline \end{array}$

$\blacksquare$

## Source of Name

This entry was named for Charles Sanders Peirce.