The paper focuses on extending to the ﬁrst order case the semantical program for modalities ﬁrst introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we oﬀer a series of new completeness results for salient classical systems of ﬁrst order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula in terms of neighborhood frames with constant domains. The ﬁrst order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either diﬃcult of impossible to build in terms of relational Kripkean semantics. We conclude by introducing general ﬁrst order neighborhood frames and we oﬀer a general completeness result in terms of them which circumvents some well-known problems of propositional and ﬁrst order neighborhood semantics. We argue that the semantical program that thus arises surpasses both in expressivity and adequacy the standard Kripkean approach, even when it comes to the study of ﬁrst order normal systems
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