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Completeness of intermediate logics with doubly negated axioms



Let math formula denote a first-order logic in a language that contains infinitely many constant symbols and also containing intuitionistic logic math formula. By math formula, we mean the associated logic axiomatized by the double negation of the universal closure of the axioms of math formula plus math formula. We shall show that if math formula is strongly complete for a class of Kripke models math formula, then math formula is strongly complete for the class of Kripke models that are ultimately in math formula.