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:
“The History of the world is not the theatre of happiness. Periods of happiness are blank pages in it, for they are periods of harmony—periods when the antithesis is in abeyance.”
—Georg Wilhelm Friedrich Hegel (1770–1831)
“The custard is setting; meanwhile
I not only have my own history to worry about
But am forced to fret over insufficient details related to large
Unfinished concepts that can never bring themselves to the point
Of being, with or without my help, if any were forthcoming.”
—John Ashbery (b. 1927)
“It’s nice to be a part of history but people should get it right. I may not be perfect, but I’m bloody close.”
—John Lydon (formerly Johnny Rotten)