Nonclassical mereology and its application to sets
type: Item
instance of: scholarly article
title: Nonclassical mereology and its application to sets
zbMATH Open document ID: 1049.03003
author: Peter Forrest
published in: Notre Dame Journal of Formal Logic
publication date: 2004-05-27T00:00:00Z
review text: The paper consists of two parts. The author himself qualifies the first part as ``the case against classical mereology''; it contains some objections to the latter and some motivation to what is called there Heyting mereology. The second part shows how this alternative system of mereology provides us with ``all the sets we need''. The systems of mereology discussed in the paper are based on first-order logic with identity and the definite description operator; the non-logical primitives are the proper part relation and a name for the ``fictious null thing''. (However, the null thing is admitted in the scope of quantified variables and thus, formally, is not fictious.) Therefore, the classical mereology is reduced in the paper to the elementary part of Leśniewski's original system. The author argues that the axiom of the classical mereology which declares uniqueness of mereological classes (called fusions in the paper) is a careless extrapolation from the finite to the infinite case. However, he does not reject the principle that things have a unique sum. The position of the author can be explained more precisely as follows. A model of a mereology is a complete lattice with zero deleted. In such a lattice, the fusion of a subset \(F\) is an upper bound \(z\) of \(F\) such that any \(x\) disjoint from every element of \(F\) is disjoint from \(z\). The sum of \(F\) is a fusion which happens to be the join of \(F\). In classical mereology, \(F\) has to have a unique fusion; hence the notions of fusion, join and sum coincide there. In the author's nonclassical mereology, uniqueness of fusions is not required, but, in order to obtain a mathematically well-behaved mereology, it is assumed that meet distributes over arbitrary joins (the latter condition is known to be equivalent to the one that the lattice is Heyting). By the way, this assumption provides that the join of \(F\) is always a fusion. In contrast to classical mereology, the sum of proper parts of a thing \(x\) …
reviewed by: Jānis Cīrulis
Mathematics Subject Classification ID: 03A05
Mathematics Subject Classification ID: 03E70
zbMATH DE Number: 2068394
zbMATH Keywords: class
zbMATH Keywords: fusion
zbMATH Keywords: mereology
zbMATH Keywords: unique sum
zbMATH Keywords: model of a mereology
zbMATH Keywords: Heyting lattice
zbMATH Keywords: pseudosets
zbMATH Keywords: set theory
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1071509430
OpenAlex ID: W2025422824
cites work: How innocent is mereology?
cites work: Q3965241
cites work: Q4382420
cites work: On Understanding Leśniewski
cites work: Q3240239
DOI: 10.1305/NDJFL/1071509430
description: scientific article; zbMATH DE number 2068394