Concept
Double negation
Overview
In propositional logic, the double negation of a statement states that "it is not the case that the statement is not true". In classical logic, every statement is logically equivalent to its double negation, but this is not true in intuitionistic logic; this can be expressed 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, but it is disallowed by intuitionistic logic. The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:
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{\displaystyle \mathbf {*4\cdot 13} .\ \ \vdash .\ p\ \equiv \ \thicksim (\thicksim p)}
"This is the principle of double negation, i.e. a proposition is equivalent of the falsehood of its negation."
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.
Alfred North Whitehead, Science and the modern world
States what Double negation is; the author is tied to Double negation in the knowledge graph.
The mere absence of negation is not enough; it must be present, but present with its fangs drawn. What you posit as A must already have cancelled the alternative or made it innocuous, by having negated it in advance. Double negation is the only form of affirmation that fully plays into the hands of the dogmatic ideal. Simply and innocently affirmative statements are good enough for empiricists, but unfit for…
William James, A Pluralistic Universe
States what Double negation is; names Double negation.
It is by this means that they are understood and demonstrated. In common life, and it is the same in the finite sciences, this reflective form is often employed as a key to the secret of the real condition of the objects under investigation. So long as we deal with what may be termed the household needs of knowledge, nothing can be urged against this method of study. But it can never afford definitive satisfaction…
Georg Wilhelm Friedrich Hegel, The Logic of Hegel
States what Double negation is; the author is tied to Double negation in the knowledge graph.
Related ideas
- Georg Wilhelm Friedrich Hegel
- Negation
- Truth value
- Intuitionistic logic
- Logical equivalence
- Law of excluded middle
- Propositional logic
- Validity (logic)
- Principia Mathematica
- Rule of inference
- Minimal logic
- Jan Łukasiewicz
- Inference
- Well-formed formula
- Bertrand Russell
- Alfred North Whitehead
- Tautology (logic)
- Classical logic
- Equivalence relation
- Hypothetical syllogism
Opposing perspectives
Place in the tradition
Part of Classical logic.