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One usual way to formulate classical negation in a natural deduction setting is to take as primitive rules of inference negation introduction (from a derivation of to both and , infer ; this rule also being called reductio ad absurdum), negation elimination (from and infer ; this rule also being called ex falso quodlibet), and double negation ...
Double negation elimination occurs in classical logics but not in intuitionistic logic. In the context of a formula in the conjunctive normal form, a literal is pure if the literal's complement does not appear in the formula. In Boolean functions, each separate occurrence of a variable, either in inverse or uncomplemented form, is a literal.
Each logic operator can be used in an assertion about variables and operations, showing a basic rule of inference. Examples: The column-14 operator (OR), shows Addition rule: when p=T (the hypothesis selects the first two lines of the table), we see (at column-14) that p∨q=T.
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.
The use of constructivist logics in general has been a controversial topic among mathematicians and philosophers (see, for example, the Brouwer–Hilbert controversy). A common objection to their use is the above-cited lack of two central rules of classical logic, the law of excluded middle and double negation elimination.
Classical logic is the standard logic of mathematics. Many mathematical theorems rely on classical rules of inference such as disjunctive syllogism and the double negation elimination. The adjective "classical" in logic is not related to the use of the adjective "classical" in physics, which has another meaning.
A discussion of the introduction and elimination forms for higher-order logic is beyond the scope of this article. It is possible to be in-between first-order and higher-order logics. For example, second-order logic has two kinds of propositions, one kind quantifying over terms, and the second kind quantifying over propositions of the first kind.
Negation normal form is not a canonical form: for example, () and () are equivalent, and are both in negation normal form. In classical logic and many modal logics , every formula can be brought into this form by replacing implications and equivalences by their definitions, using De Morgan's laws to push negation inwards, and eliminating double ...