Abstract
This paper develops type assignment systems with intersection and union types, and type quantifiers. We show that the known system for these types is not semantically complete. However, the following two hold for a certain class of typing statements, called stable statements, which include all statements without type quantifier: (1) the validity of stable statements for Kripke models is equivalent to that for standard models, (2) if we add two axioms expressing the distributive laws of intersection over union and existential-type quantifier, then the resulting system is complete for Kripke models. As a consequence, the known system with the axioms for distributive laws is complete for standard models if we restrict statements to stable ones. All the results are obtained in a systematic way with sequent-style formulations of type assignment and the cut-elimination property for them.
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