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:
| {(x → y) → 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 (x → y) → x is true. If this is true, either its consequent, x, is true, when the whole formula would be true, or its antecedent x → y is false. But in the last case the antecedent of x → y, 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:
| {(x → y) → a} → x, |
- where the a is used in such a sense that (x → y) → a means that from (x → 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. (Peirce, the Collected Papers 3.384).
Warning: ((x→y)→a)→x is not a tautology. However, → is a tautology.
Read more about this topic: Peirce's Law
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