Double negation  

From The Art and Popular Culture Encyclopedia

Jump to: navigation, search

Related e

Wikipedia
Wiktionary
Shop


Featured:

Kunstformen der Natur (1904) by Ernst Haeckel
Enlarge
Kunstformen der Natur (1904) by Ernst Haeckel

In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition A is logically equivalent to not (not-A), or by the formula A ≡ ~(~A) where the sign ≡ expresses logical equivalence and the sign ~ expresses negation.

Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic, (Hamilton is discussing Hegel in the following: "In the more recent systems of philosophy, the universality and necessity of the axiom of Reason has, with other logical laws, been controverted and rejected by speculators on the absolute.[On principle of Double Negation as another law of Thought, see Fries, Logik, §41, p. 190; Calker, Denkiehre odor Logic und Dialecktik, §165, p. 453; Beneke, Lehrbuch der Logic, §64, p. 41.]" (Hamilton 1860:68)</ref> but it is disallowed by intuitionistic logic.) The o of Kleene's formula *49o indicates "the demonstration is not valid for both systems [classical system and intuitionistic system]", Kleene 1952:101.</ref> The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:

<math>\mathbf{*4\cdot13}. \ \ \vdash.\ p \ \equiv \ \thicksim(\thicksim p)</math>
"This is the principle of double negation, i.e. a proposition is equivalent of the falsehood of its negation."

See also




Unless indicated otherwise, the text in this article is either based on Wikipedia article "Double negation" or another language Wikipedia page thereof used under the terms of the GNU Free Documentation License; or on original research by Jahsonic and friends. See Art and Popular Culture's copyright notice.

Personal tools