Double negation

This article is about the logical concept. For the linguistic concept, see Double negative.

In the theory of logic, double negation is expressed by saying that a proposition A is identical to (equivalent to) not (not-A), or by the formula A = ~~A. Like the Law of Excluded Middle, this principle when extended to an infinite collection of individuals is disallowed by Intuitionistic logic. Some writers grant this "law" the status of a Law of thought.^[1]

The principium contradictiones of modern logicians (particularly Leibnitz and Kant) in the formula A is not not-A, differs entirely in meaning and application from the Aristotelian proposition [ i.e. Law of Contradiction: not (A and not-A) i.e. ~(A & ~A), or not (( B is A) and (B is not-A))]. This latter refers to the relation between an affirmative and a negative judgment. According to Aristotle, one judgment [B is judged to be an A] contradicts another [B is judged to be a not-A]. The later proposition [ A is not not-A ] refers to the relation between subject and predicate in a single judgment; the predicate contradicts the subject. Aristotle states that one judgment is false when another is true; the later writers [Leibniz and Kant] state that a judgment is in itself and absolutely false, because the predicate contradicts the subject. What the later writers desire is a principle from which it can be known whether certain propositions are in themselves true. From the Aristotelian proposition we cannot immediately infer the truth or falsehood of any particular proposition, but only the impossibility of believing both affirmation and negation at the same time.^[2]

Footnotes

1. ^ 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)
2. ^ Sigwart 1895:142-143

References

• William Hamilton, 1860, Lectures on Metaphysics and Logic, Vol. II. Logic; Edited by Henry Mansel and John Veitch, Boston, Gould and Lincoln. Available online from googlebooks.
• Christoph Sigwart, 1895, Logic: The Judgment, Concept, and Inference; Second Edition, Translated by Helen Dendy, Macmillan & Co. New York. Available online from googlebooks.

See also

Media related to Double negation at Wikimedia Commons

• Negation
• Double negation elimination

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This mathematical logic-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Mathematical logic Logic Logic stubs Mathematical logic stubs Wikimedia Foundation. 2010. Игры ⚽ Поможем сделать НИР Double name Double or Nothing (1971 novel) Look at other dictionaries: Double négation — Négation (linguistique) Pour les articles homonymes, voir négation. En linguistique, la négation (du latin negare, nier)[1] est une opération qui consiste à désigner comme fausse une proposition préalablement exprimée ou non ; elle s’oppose… …   Wikipédia en Français double negation — noun The principle that the negation of a negation of a proposition means the same as the proposition • • • Main Entry: ↑double …   Useful english dictionary Double négation — ● Double négation présence à l intérieur de la même phrase de deux négations qui s annulent plus ou moins complètement (par exemple Vous n êtes pas sans savoir que…) …   Encyclopédie Universelle double negation — The result of negating the negation of a proposition. In classical propositional calculus, ¬¬ p is equivalent to p . In intuitionistic logic this is not so; we cannot infer p from ¬¬ p …   Philosophy dictionary Principe de double négation — ● Principe de double négation principe selon lequel ¬ ¬ p ≡ p. (Ce principe est rejeté par la logique intuitionniste.) …   Encyclopédie Universelle Negation (linguistique) — Négation (linguistique) Pour les articles homonymes, voir négation. En linguistique, la négation (du latin negare, nier)[1] est une opération qui consiste à désigner comme fausse une proposition préalablement exprimée ou non ; elle s’oppose… …   Wikipédia en Français Négation linguistique — Négation (linguistique) Pour les articles homonymes, voir négation. En linguistique, la négation (du latin negare, nier)[1] est une opération qui consiste à désigner comme fausse une proposition préalablement exprimée ou non ; elle s’oppose… …   Wikipédia en Français négation — [ negasjɔ̃ ] n. f. • negatiun XII e; lat. negatio, de negare « nier » 1 ♦ Acte de l esprit qui consiste à nier, à rejeter un rapport, une proposition, une existence; expression de cet acte. Négation de Dieu. Négation de la vérité, des valeurs. ⇒… …   Encyclopédie Universelle Negation en francais — Négation en français Article principal : Négation (linguistique). Sommaire 1 Syntaxe de la négation 1.1 Règle générale 1.2 Le « ne » explétif et comparatif …   Wikipédia en Français Negation (algebra) — Negation is the mathematical operation that reverses the sign of a number. Thus the negation of a positive number is negative, and the negation of a negative number is positive. The negation of zero is zero. The negation of a number a, usually… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”