## ::Double negation

### ::concepts

Logic::negation    ''A''::''double    Negative::logical    Formula::judgment    Kleene::proof    ''B''::vdash

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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.<ref>Or alternate symbolism such as A ↔ ¬(¬A) or Kleene's *49o: A ∾ ¬¬A (Kleene 1952:119; in the original Kleene uses an elongated tilde ∾ for logical equivalence, approximated here with a "lazy S".)</ref>

Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic,<ref>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.<ref>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:

$\mathbf{*4\cdot13}. \ \ \vdash.\ p \ \equiv \ \thicksim(\thicksim p)$<ref>PM 1952 reprint of 2nd edition 1927 pages 101-102, page 117.</ref>
"This is the principle of double negation, i.e. a proposition is equivalent of the falsehood of its negation."

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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.<ref>Sigwart 1895:142-143</ref>

Double negation sections