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A double negative is a construction occurring when two forms of grammatical negation are used in the same sentence. This is typically used to convey a different shade of meaning from a strictly positive sentence ("You're not unattractive" vs "You're attractive").
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.
In C (and some other languages descended from C), double negation (!!x) is used as an idiom to convert x to a canonical Boolean, ie. an integer with a value of either 0 or 1 and no other. Although any integer other than 0 is logically true in C and 1 is not special in this regard, it is sometimes important to ensure that a canonical value is ...
double turnstile: semantically entails: ... (U+2308 and U+2309) or by using a negation symbol and a reversed negation symbol ⌐ ¬ in superscript mode.)
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 ...
The double negation law can be seen by complementing the shading in the third diagram for ¬x, which shades the x circle. To visualize the first De Morgan's law, (¬ x ) ∧ (¬ y ) = ¬( x ∨ y ) , start with the middle diagram for x ∨ y and complement its shading so that only the region outside both circles is shaded, which is what the ...
Syntactically, (1) and (2) are derivable from each other via the rules of contraposition and double negation. Semantically, (1) and (2) are true in exactly the same models (interpretations, valuations); namely, those in which either Lisa is in Denmark is false or Lisa is in Europe is true. (Note that in this example, classical logic is assumed.
The double-negation translation was used by Gödel (1933) to study the relationship between classical and intuitionistic theories of the natural numbers ("arithmetic"). He obtains the following result: If a formula φ is provable from the axioms of Peano arithmetic then φ N is provable from the axioms of Heyting arithmetic.