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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).
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.
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.
Implication introduction / elimination (modus ponens) Biconditional introduction / elimination; Conjunction introduction / elimination; Disjunction introduction / elimination; Disjunctive / hypothetical syllogism; Constructive / destructive dilemma; Absorption / modus tollens / modus ponendo tollens; Negation introduction; Rules of replacement
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 ...
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 ...
Classical logic is the standard logic of mathematics. Many mathematical theorems rely on classical rules of inference such as disjunctive syllogism and the double negation elimination. The adjective "classical" in logic is not related to the use of the adjective "classical" in physics, which has another meaning.
Indirect Proof (IP), [17] negation introduction (−I), [17] negation elimination (−E) [17] m, n RAA (k) [17] The union of the assumption sets at lines m and n, excluding k (the denied assumption). [17] From a sentence and its denial [b] at lines m and n, infer the denial of any assumption appearing in the proof (at line k). [17] Double arrow ...