Modal Semantics for Reasoning with Probability and Uncertainty

TL;DR

This paper compares Kripke and neighbourhood semantics in probabilistic modal logic, showing neighbourhood semantics' greater expressiveness in representing belief functions and analyzing modal equivalence between models.

Contribution

It provides a detailed analysis of the differences between Kripke and neighbourhood semantics, highlighting neighbourhood semantics' ability to express belief functions and establishing conditions for modal equivalence.

Findings

Neighbourhood semantics can express belief functions (lower probabilities).

Kripke semantics is less expressive for probabilistic beliefs.

Modal equivalence between probabilistic Kripke models and additive belief neighbourhood models is characterized.

Abstract

This paper belongs to the field of probabilistic modal logic, focusing on a comparative analysis of two distinct semantics: one rooted in Kripke semantics and the other in neighbourhood semantics. The primary distinction lies in the following: The latter allows us to adequately express belief functions (lower probabilities) over propositions, whereas the former does not. Thus, neighbourhood semantics is more expressive. The main part of the work is a section in which we study the modal equivalence between probabilistic Kripke models and a subclass of belief neighbourhood models, namely additive ones. We study how to obtain modally equivalent structures.