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Abstract
FAQs
Introduction
Conclusion and Further Work
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Philosophy
Logic

First-order classical modal logic

Profile image of Eric PacuitEric Pacuit

2006, Studia Logica - An International Journal for Symbolic Logic

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 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.

FAQs

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AI

What are neighborhood frames and how do they function?add

Neighborhood frames consist of a set of worlds and a neighborhood function associating sets of propositions to each world. They serve to model modalities by interpreting necessity and possibility through the neighborhoods attached to each state.

How does first-order neighborhood semantics address Barcan formulas?add

First-order neighborhood semantics allows the development of models where the Barcan and Converse Barcan formulas can fail, offering necessary conditions for their validity. Specifically, frameworks can be constructed where the Converse Barcan formula holds while the Barcan formula does not, demonstrating nuanced control over these axioms.

What advancements have been made in modeling non-normal classical systems?add

Recent research has shown that many non-normal classical modalities can be represented elegantly using neighborhood models. This represents a shift from earlier views that dismissed these logics due to their lack of clear applications or underlying theoretical frameworks.

How do general first-order frames improve modal logic completeness?add

General first-order frames with constant domains provide a robust framework that addresses many incompleteness issues found in standard relational models. This offers a more coherent interpretation of first-order modal logics, leading to strong completeness results for entire modal families.

What practical implications arise from applying first-order neighborhood semantics?add

The application of first-order neighborhood semantics has significant implications in fields such as Artificial Intelligence and Game Theory, particularly in modeling multi-agent belief states and probabilistic reasoning. These models facilitate understanding complex interactions among agents within uncertain environments.

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Related topics

  • Modal Logic
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