Absorption / modus tollens / modus ponendo tollens
Negation introduction
Rules of replacement
Associativity
Commutativity
Distributivity
Double negation
De Morgan's laws
Transposition
Material implication
Exportation
Tautology
Predicate logic
Rules of inference
Universal generalization / instantiation
Existential generalization / instantiation
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".[citation needed] 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.[1]
Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic,[2] but it is disallowed by intuitionistic logic.[3] The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:
[4]
"This is the principle of double negation, i.e. a proposition is equivalent of the falsehood of its negation."
^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".)
^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)
^The o of Kleene's formula *49o indicates "the demonstration is not valid for both systems [classical system and intuitionistic system]", Kleene 1952:101.
^PM 1952 reprint of 2nd edition 1927 pp. 101–02, 117.
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doubled negatives intensify the negation. Languages where multiple negatives affirm each other are said to have negative concord or emphatic negation...
operators. Within a system of classical logic, doublenegation, that is, the negation of the negation of a proposition P {\displaystyle P} , is logically...
inconsistent. Having proven such a mere double-negation also still aids in negating other statements through negation introduction, as then ( ϕ → ¬ ψ ) →...
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for negation are given below. A desideratum is always the negation introduction law, discussed next. A quick analysis of the valid rules for negation gives...
called negation – the grammatical rules for negation vary from language to language, and a given language may have multiple methods of negation. Affirmative...
negation and one of the two other operations are basic because of the following identities that allow one to define conjunction in terms of negation and...
In Rapa Nui, doublenegation is more frequent than single negation (with the negator ꞌina often co-occurring with another clause negator most of the time)...
the tilde represents negation: "~p" means "not p", where "p" is a proposition. Modern use often replaces the tilde with the negation symbol (¬) for this...
and negation (as Russell, Whitehead, and Hilbert did), or using only implication and negation (as Frege did), or using only conjunction and negation, or...
definition cites Hilbert's two axioms of negation A → (~A → B) (A → B) → { (~A → B) → B} Hilbert's first axiom of negation, "anything follows from the false"...
In mathematical logic, a formula is in negation normal form (NNF) if the negation operator ( ¬ {\displaystyle \lnot } , not) is only applied to variables...
non-standard dialects. The doublenegation follows the idea of two different morphemes, one that causes the doublenegation, and one that is used for the...
language for classical propositional logic can be expressed using just negation (¬), implication (→) and propositional symbols. A well-known axiomatization...
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the negation of a propositional function's existential quantification is a universal quantification of that propositional function's negation; symbolically...
inconsistency of its negation. It is thus related to reductio ad absurdum, but it can prove a proposition using just its own negation and the concept of...
include de Morgan's laws, commutation, association, distribution, doublenegation, transposition, material implication, logical equivalence, exportation...
class shares characteristic properties: Law of excluded middle and doublenegation elimination Law of noncontradiction, and the principle of explosion...
by doublenegation, concluding that x {\displaystyle x} exists. Constructive mathematics does not allow the last step of removing the doublenegation to...
Negation introduction is a rule of inference, or transformation rule, in the field of propositional calculus. Negation introduction states that if a given...
are derivable from each other via the rules of contraposition and doublenegation. Semantically, (1) and (2) are true in exactly the same models (interpretations...
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