# Peirce's Law

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## Theorem

### Formulation 1

- $\paren {p \implies q} \implies p \vdash p$

### Formulation 2

- $\vdash \paren {\paren {p \implies q} \implies p} \implies p$

## Strong Form

- $\paren {\paren {p \implies q} \implies p} \dashv \vdash p$

## Also see

## Source of Name

This entry was named for Charles Sanders Peirce.

## Historical Note

Peirce's own statement and proof of the Peirce's Law:

*A*fifth icon*is required for the principle of excluded middle and other propositions connected with it. One of the simplest formulae of this kind is:*

- $\paren {\paren {x \mathop {-\!\!\!<} y} \mathop {-\!\!\!<} x} \mathop {-\!\!\!<} x$

*This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent $x$ being false while its antecedent $\paren {x \mathop {-\!\!\!<} y} \mathop {-\!\!\!<} x$ is true. If this is true, either its consequent $x$ is true, when the whole formula would be true, or its antecedent $x \mathop{-\!\!\!<} y$ is false. But in the last case the antecedent of $x \mathop{-\!\!\!<} y$, that is $x$, must be true.*

Peirce goes on to point out an immediate application of the law:

*From the formula just given, we at once get:*

- $\paren {\paren {x \mathop {-\!\!\!<} y} \mathop {-\!\!\!<} a} \mathop {-\!\!\!<} x$

*where the $a$ is used in such a sense that $\paren {x \mathop {-\!\!\!<} y} \mathop {-\!\!\!<} a$ means that from $\paren {x \mathop {-\!\!\!<} y}$ every proposition follows. With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of $x$ follows the truth of $x$.*

Note the use by Peirce of the sign of illation $-\!\!\!<$ for implication.

## Sources

- 1885: Charles Sanders Peirce:
*On the Algebra of Logic: A Contribution to the Philosophy of Notation*(*Amer. J. Math.***Vol. 7**: pp. 180 – 202) www.jstor.org/stable/2369451