Peirce's Law/Historical Note

Historical Note
'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:


 * $\left({\left({x \mathop {-\!\!\!<} y}\right) \mathop {-\!\!\!<} x}\right) \mathop {-\!\!\!<} x$


 * 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 $\left({x \mathop {-\!\!\!<} y}\right) \mathop {-\!\!\!<} x$ 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:


 * $\left({\left({x \mathop {-\!\!\!<} y}\right) \mathop {-\!\!\!<} a}\right) \mathop {-\!\!\!<} x$


 * where the $a$ is used in such a sense that $\left({x \mathop {-\!\!\!<} y}\right) \mathop {-\!\!\!<} a$ means that from $\left({x \mathop {-\!\!\!<} y}\right)$ 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 of the sign of illation $-\!\!\!<$ for implication.