Atomism Axiomatised Using Mereological Composition as a Primitive Notion

TL;DR

This paper introduces a new axiomatic system for atomism based on primitive mereological composition, defining atoms as indivisible objects and parts as inclusion, without relying on traditional part or part-whole notions.

Contribution

It presents the first formal theory of atomism using primitive composition, with only two axioms, and proves its equivalence to atomistic extensional mereology.

Findings

The theory is definitionally equivalent to atomistic extensional mereology.

It requires no full comprehension schema or specific composition existence.

Models can contain only atoms, supporting atomistic concepts without part notions.

Abstract

Atomism is the view that everything is composed of atoms. The view within the framework of the contemporary formal approach is expressed on the ground of mereology with the use of the primitive notion of being a part as every object has at least one atomic part [2, 48], [3, 145], [17, 42], or using mereological fusion [16, 24] which is defined by being a part. We will briefly present a discussion between A. Varzi and A. Shiver concerning the two approaches and propose a new intuitive axiomatic characterization of atomism. We build a system with a primitive notion of composition that holds between individuals and pluralities. We assume only two specific axioms: each object is a unique composition of unique atoms, and being a composition of some objects is equivalent to being the composition of all atoms of these objects. In our approach, notions of part, and atom, are secondary to composition: atom is defined as an object that cannot be a composition of two or more objects, and part is defined as inclusion between atoms. We will show that the theory, with only these two specific axioms, is sufficient to adequately express atomism as we prove that the theory is definitionally equivalent to atomistic extensional mereology with plural quantification allowed and mereological fusion defined. Our theory requires neither full comprehension schema nor the existence of any specific compositions (there are models with only atoms). Therefore, it may constitute the basis on which atomistic concepts of reality are further strengthened, specifically those in which the notion of being a part is not used. To the best of our knowledge, our proposal is the only formal theory of atomism using the primitive notion of composition.