Słupecki's Generalized Mereology and Its Flaws

Abstract

One of the streams in the early development of set theory was an attempt to use mereology, a formal theory of parthood, as a foundational tool. The first such attempt is due to a Polish logician, Stanisław Leśniewski (1886–1939). The attempt failed, but there is another, prima facie more promising attempt by Jerzy Słupecki (1904–1987), who employed his generalized mereology to build mereological foundations for type theory. In this paper I (1) situate Leśniewski's attempt in the development of set theory, (2) describe and evaluate Leśniewski's approach, (3) describe Słupecki's strategy without unnecessary technical details, and (4) evaluate it with a rather negative outcome. The issues discussed go beyond merely historical interests due to the current popularity of mereology and because they are related to nominalistic attempts to understand mathematics in general. The introduction describes very briefly the situation in which mereology entered the scene of foundations of mathematics — it can be safely skipped by anyone familiar with the early development of set theory. Section 2 describes and evaluates Leśniewski's attempt to use mereology as a foundational tool. In Section 3, I describe an attempt by Słupecki to improve on Leśniewski's work, which resulted in a system called generalized mereology. In Section 4, I point out the reasons why this attempt is still not successful. Section 5 contains an explanation of why Leśniewski's use of Ontology in developing arithmetic also is not nominalistically satisfactory.