Peirce's Law - History

History

Here is Peirce's own statement of the 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:
{(xy) → x} → 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 (xy) → x is true. If this is true, either its consequent, x, is true, when the whole formula would be true, or its antecedent xy is false. But in the last case the antecedent of xy, that is x, must be true. (Peirce, the Collected Papers 3.384).

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

From the formula just given, we at once get:
{(xy) → a} → x,
where the a is used in such a sense that (xy) → a means that from (xy) 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. (Peirce, the Collected Papers 3.384).

Warning: ((xy)→a)→x is not a tautology. However, → is a tautology.

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