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Together with double negation elimination one may infer our originally formulated rule, namely that anything follows from an absurdity. Typically the intuitionistic negation of is defined as . Then negation introduction and elimination are just special cases of implication introduction (conditional proof) and elimination (modus ponens).
Since biconditionality is an equivalence relation, any instance of ¬¬A in a well-formed formula can be replaced by A, leaving unchanged the truth-value of the well-formed formula. Double negative elimination is a theorem of classical logic, but not of weaker logics such as intuitionistic logic and minimal logic.
Propositions for which double-negation elimination is possible are also called stable. Intuitionistic logic proves stability only for restricted types of propositions. A formula for which excluded middle holds can be proven stable using the disjunctive syllogism, which is discussed more thoroughly below. The converse does however not hold in ...
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.
Minimal logic proves double-negation elimination for negated formulas, () (). More generally, Heyting arithmetic proves this classical equivalence for any Harrop formula . And Σ 1 0 {\displaystyle \Sigma _{1}^{0}} -results are well behaved as well: Markov's rule at the lowest level of the arithmetical hierarchy is an admissible rule of ...
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.
A de Morgan negation is a simple negation satisfying double negation elimination: ... disjunction, and negation (such as a formula in disjunctive normal form), ...
The case = now directly shows how double-negation elimination implies consequentia mirabilis over minimal logic. In intuitionistic logic, explosion can be used for ⊥ → ( P ∧ ⊥ ) {\displaystyle \bot \to (P\land \bot )} , and so here the law's special case consequentia mirabilis also implies double-negation elimination.