Categories: How I Learned to Stop Worrying and Love Two Sorts

TL;DR

This paper introduces an epistemic interpretation of RS-frames for modal logic, linking them to categorization systems and social interactions, with implications for decision-making and belief modeling.

Contribution

It proposes a novel epistemic interpretation of RS-frames in modal logic, connecting them to categorization and social belief systems.

Findings

RS-frames can model agents' beliefs about categories.

Galois-stable sets represent potential categories.

Fixed points of belief modalities model socially constructed categories.

Abstract

RS-frames were introduced by Gehrke as relational semantics for substructural logics. They are two-sorted structures, based on RS-polarities with additional relations used to interpret modalities. We propose an intuitive, epistemic interpretation of RS-frames for modal logic, in terms of categorization systems and agents' subjective interpretations of these systems. Categorization systems are a key to any decision-making process and are widely studied in the social and management sciences. A set of objects together with a set of properties and an incidence relation connecting objects with their properties forms a polarity which can be `pruned' into an RS-polarity. Potential categories emerge as the Galois-stable sets of this polarity, just like the concepts of Formal Concept Analysis. An agent's beliefs about objects and their properties (which might be partial) is modelled by a relation which gives rise to a normal modal operator expressing the agent's beliefs about category membership. Fixed-points of the iterations of the belief modalities of all agents are used to model categories constructed through social interaction.