- Double negation
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.
- 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.
- ^ 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)
- ^ Sigwart 1895:142-143
- 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.
- Double negation elimination
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