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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.
Accordingly, negation in classical logic satisfies the law of double negation: ¬¬A is equivalent to A. Generally in non-classical logics, negation that satisfies the law of double negation is called involutive. In algebraic semantics, such a negation is realized as an involution on the algebra of truth values.
Within a system of classical logic, double negation, that is, the negation of the negation of a proposition , is logically equivalent to . Expressed in symbolic terms, . In intuitionistic logic, a proposition implies its double negation, but not conversely. This marks one important difference between classical and intuitionistic negation.
A double negation does not affirm the law of the excluded middle ; while it is not necessarily the case that PEM is upheld in any context, no counterexample can be given either. Such a counterexample would be an inference (inferring the negation of the law for a certain proposition) disallowed under classical logic and thus PEM is not allowed ...
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 logic, a rule of replacement [1] [2] [3] is a transformation rule that may be applied to only a particular segment of an expression.A logical system may be constructed so that it uses either axioms, rules of inference, or both as transformation rules for logical expressions in the system.
In mathematics and mathematical logic, Boolean algebra is a branch of algebra.It differs from elementary algebra in two ways. First, the values of the variables are the truth values true and false, usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers.
In proof theory, a discipline within mathematical logic, double-negation translation, sometimes called negative translation, is a general approach for embedding classical logic into intuitionistic logic. Typically it is done by translating formulas to formulas that are classically equivalent but intuitionistically inequivalent.