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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.
The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics. Additionally, the subsequent columns contains an informal explanation, a short example, the Unicode location, the name for use in HTML documents, [1] and the LaTeX symbol.
In mathematics, a negative number is the opposite of a positive real number. [1] Equivalently, a negative number is a real number that is less than zero . Negative numbers are often used to represent the magnitude of a loss or deficiency.
A negative literal is the negation of an atom (e.g., ). The polarity of a literal is positive or negative depending on whether it is a positive or negative literal. In logics with double negation elimination (where ¬ ¬ x ≡ x {\displaystyle \lnot \lnot x\equiv x} ) the complementary literal or complement of a literal l {\displaystyle l} can ...
In mathematics and physics, the phrase "change of sign" is associated with exchanging an object for its additive inverse (multiplication with −1, negation), an operation which is not restricted to real numbers. It applies among other objects to vectors, matrices, and complex numbers, which are not prescribed to be only either positive ...
In most programming languages, subtraction and negation are indicated with the ASCII hyphen-minus character, -. In APL a raised minus sign (here written using U+00AF ¯ MACRON) is used to denote a negative number, as in ¯3. While in J a negative number is denoted by an underscore, as in _5.
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 ′.
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.