T h e logic CD is an intermediate logic (stronger than intuitionistic logic and weaker than classical logic) which exactly corresponds t o the Kripke models with constant domains. It is known that the logic CD has a Gentzen-type formulation called LD (which is same as LK except that ( -3 ) and (-+ 7 ) rules are replaced by the corresponding intuitionistic rules) and that the cut-elimination theorem does not hold for LD. In this paper we present a modification of LD and prove the cut-elimination theorem for it. Moreover we prove a "weak" version of cut-elimination theorem for LD, saying that all "cuts" except some special forms can be eliminated from a proof in LD. From these cut-elimination theorems we obtain some corollaries on syntactical properties of CD: fragments collapsing into intuitionistic logic, Harrop disjunction and existence properties, and a fact on the number of logical symbols in the axiom of CD.Mathematics Subject Classiflcation: 03B55, 03F05.In this paper we present a Gentzen-type cut-eliminable system for a natural variant of intuitionistic logic, called CD following [2]. CD is characterized by Kripke models with constant domains, from which the name CD comes. Syntactically, CD is obtained from intuitionistic logic by adding the axiomThe importance of CD is well-established (e. g., by its close relation to the notion of forcing in set theory -see [l], [4]), and so this intermediate logic is extensively studied in [l] -[8], [lo], [ll], and many other papers.CD also has a quite natural Gentzen-type formulation LD, that is LK with (+3) and ( 4 7 ) rules replaced by the corresponding intuitionistic (LJ) rules. But unfortunately, the cut-elimination theorem does not hold for LD (see [6], [7]). Moreover, it is known that even if we add any finite number of inference rules to LD, we can not get a cut-free system for CD (see [S]).

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