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This paper investigates the quantifier-free definability of specific connectives—such as those of Taranovsky and Kreisel–Połacik type—in intuitionistic second-order logic. Addressing limitations of conventional model-theoretic approaches in characterizing quantifier-free definable mappings between structures, it establishes, for the first time, a deep correspondence between such definability and cut elimination as well as local provable equivalence. The analysis employs proof-theoretic methods—including natural deduction and sequent calculus, local conservativity arguments, counterexample construction, and type-elimination techniques. Main contributions include: (i) complete criteria for quantifier-free definability in classical theories such as linear and discrete orders; (ii) correction and generalization of Tarski’s seminal result on real closed fields; and (iii) the first systematic proof-theoretic framework for definability theory in intuitionistic logic.