Abstract

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 in the first part of the paper 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 (like FOL + K)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 [40]. We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics.

Highlights

  • Dana Scott and Richard Montague proposed in 1970 a new semantic framework for the study of modalities, which today tends to be known as neighborhood semantics.A neighborhood frame is a pair W, N, where W is a set of states, or worlds, and N : W → 22W is a neighborhood function which associates a set of neighborhoods with each world

  • 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

  • More recently one of us (Arlo-Costa) presented in [?] preliminary results in this area showing that the role of the Barcan schemas in this context is quite different from the corresponding role of these schemas in the relational case.† the use of neighborhood semantics permits the development of models with constant domains where neither the Barcan (BF) nor the Converse Barcan formulas (CBF) are valid

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Summary

Introduction

Dana Scott and Richard Montague proposed in 1970 (independently, in [?] and [?] respectively) a new semantic framework for the study of modalities, which today tends to be known as neighborhood semantics. More recently one of us (Arlo-Costa) presented in [?] preliminary results in this area showing that the role of the Barcan schemas in this context is quite different from the corresponding role of these schemas in the relational case.† the use of neighborhood semantics permits the development of models with constant domains where neither the Barcan (BF) nor the Converse Barcan formulas (CBF) are valid. Question.§ But, again, there are many interesting applications, ranging from the modeling of contextual modals in linguistics [?] to the logic of finitely additive conditional probability where the use of varying domains is not immediately motivated and where the asymmetry between the CBF (as valid) and the BF (as invalid) holds. We conclude by discussing some examples and suggesting topics for future research

Classical systems of propositional modal logic
Classical systems of first order modal logic
First-order neighborhood frames
General Frames
Conclusion and Further Work
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