# Duality principle

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The duality principle in mathematical logic is a theorem on the acceptability of mutual substitution (in a certain sense) of logical operations in the formulas of formal logical and logical-objective languages. Let be a formula in the language of propositional or predicate logic not containing the implication symbol ; a formula is said to be dual to a formula if it may be obtained from by replacing in each occurrence (cf. Imbedded word) of the symbols , , , by their dual operations, i.e. by the symbols , , , and , respectively. The duality principle states that if is true, then is true as well. In particular, if two formulas and are equivalent, their dual formulas and are equivalent too. The duality principle is valid for classical systems, and the equivalence and the truth of the formulas involved in its formulation may be understood both in terms of interpretations and in the sense of being deducible in the corresponding classical calculus. The duality principle is no longer valid if the formulas are understood in their constructive sense. For instance, in the language of propositional logic the implication is constructively true, and is even deducible in a Heyting formal system, but the converse implication of the dual formula is constructively untrue (it is not Kleene-realizable).
The following theorem is closely connected with the duality principle: If is a formula dual to a propositional or predicate formula constructed without making use of implications from the elementary propositions , then the formula is equivalent to the formula in the classical propositional or predicate calculus, respectively.