Abstract.In this paper we re-examine the semantics of classical higher-order logic with the purpose of clarifying the role of extensionality. To reach this goal, we distinguish nine classes of higher-order models with respect to various combinations of Boolean extensionality and three forms of functional extensionality. Furthermore, we develop a methodology of abstract consistency methods (by providing the necessary model existence theorems) needed to analyze completeness of (machine-oriented) higher-order calculi with respect to these model classes.

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Abstract. We investigate cut-elimination and cut-simulation in impredicative (higherorder) logics. We illustrate that adding simple axioms such as Leibniz equations to a calculus for an impredicative logic -in our case a sequent calculus for classical type theory -is like adding cut. The phenomenon equally applies to prominent axioms like Boolean-and functional extensionality, induction, choice, and description. This calls for the development of calculi where these principles are built-in instead of being treated axiomatically.

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While many higher-order interactive theorem provers include a choice operator, higher-order automated theorem provers so far have not. In order to support automated reasoning in the presence of a choice operator, we present a cut-free ground tableau calculus for Church's simple type theory with choice. The tableau calculus is designed with automated search in mind. In particular, the rules only operate on the top level structure of formulas. Additionally, we restrict the instantiation terms for quantifiers to a universe that depends on the current branch. At base types the universe of instantiations is finite. Both of these restrictions are intended to minimize the number of rules a corresponding search procedure is obligated to consider. We prove completeness of the tableau calculus relative to Henkin models.