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
Famous quotes containing the word history:
“There is nothing truer than myth: history, in its attempt to realize myth, distorts it, stops halfway; when history claims to have succeeded this is nothing but humbug and mystification. Everything we dream is realizable. Reality does not have to be: it is simply what it is.”
—Eugène Ionesco (b. 1912)
“The history of every country begins in the heart of a man or a woman.”
—Willa Cather (18761947)
“The steps toward the emancipation of women are first intellectual, then industrial, lastly legal and political. Great strides in the first two of these stages already have been made of millions of women who do not yet perceive that it is surely carrying them towards the last.”
—Ellen Battelle Dietrick, U.S. suffragist. As quoted in History of Woman Suffrage, vol. 4, ch. 13, by Susan B. Anthony and Ida Husted Harper (1902)