Behavioral Mereology (Proofs and Properties)

TL;DR

This paper introduces a general, topos-theoretic behavioral approach to mereology, focusing on how parts of systems relate through behavioral constraints, and develops an inter-modal logic to formalize these relationships.

Contribution

It presents a novel, abstract framework for mereology based on system behaviors and formalizes it using an inter-modal logic, extending traditional mereological concepts.

Findings

Develops a topos-theoretic, behavior-based mereology framework

Formalizes part relationships using an inter-modal logic

Provides a general, elementary set-theoretic perspective

Abstract

Mereology is the study of parts and the relationships that hold between them. We introduce a behavioral approach to mereology, in which systems and their parts are known only by the types of behavior they can exhibit. Our discussion is formally topos-theoretic, and agnostic to the topos, providing maximal generality; however, by using only its internal logic we can hide the details and readers may assume a completely elementary set-theoretic discussion. We consider the relationship between various parts of a whole in terms of how behavioral constraints are passed between them, and give an inter-modal logic that generalizes the usual alethic modalities in the setting of symmetric accessibility.