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In logic, negation, also called the logical not or logical complement, is an operation that takes a proposition to another proposition "not ", written , , ′ [1] or ¯. [citation needed] It is interpreted intuitively as being true when is false, and false when is true.
In the following rules, (/) is exactly like except for having the term wherever has the free variable . Universal Generalization (or Universal Introduction) (/) _Restriction 1: is a variable which does not occur in .
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 ...
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.
Negation: the symbol appeared in Heyting in 1930 [3] [4] (compare to Frege's symbol ⫟ in his Begriffsschrift [5]); the symbol appeared in Russell in 1908; [6] an alternative notation is to add a horizontal line on top of the formula, as in ¯; another alternative notation is to use a prime symbol as in ′.
For example, x 1 is a positive literal, ¬x 2 is a negative literal, and x 1 ∨ ¬x 2 is a clause. The formula (x 1 ∨ ¬x 2) ∧ (¬x 1 ∨ x 2 ∨ x 3) ∧ ¬x 1 is in conjunctive normal form; its first and third clauses are Horn clauses, but its second clause is not.
The symbol is used to denote negation. For example, if P ( x ) is the predicate " x is greater than 0 and less than 1", then, for a domain of discourse X of all natural numbers, the existential quantification "There exists a natural number x which is greater than 0 and less than 1" can be symbolically stated as:
In Boolean logic, logical NOR, [1] non-disjunction, or joint denial [1] is a truth-functional operator which produces a result that is the negation of logical or.That is, a sentence of the form (p NOR q) is true precisely when neither p nor q is true—i.e. when both p and q are false.