Theory Of Parthood Research Articles

Słupecki's Generalized Mereology and Its Flaws

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.

On the formal impossibility of analysing subfunctions as parts of functions in design methodology

In this paper, a proof is given that in design methods, the relation between technical functions and their subfunctions in functional descriptions of technical products cannot be analysed as a formal relation of parthood. This result holds for design methods in which transformations of flows of energy, material and signals are accepted as functions. First, two specific categories of such technical functions are modelled. Second, the composition relation by which ordered sets of these functions define other functions is characterised. Third, it is shown that this composition relation for technical functions does not meet the basic postulates of parthood relations as given by mereology, the theory of parthood. It still may be beneficial to designing to take subfunctions informally as the parts of the functions they compose. Yet, the proof shows that when functional descriptions are formalised for, for instance, the development of automated design reasoning tools or for incorporation in engineering ontologies, the composition relation for technical functions cannot unconditionally be taken as a parthood relation.

Structural Universals as Structural Parts: Toward a General Theory of Parthood and Composition

David Lewis famously argued against structural universals since they allegedly required what he called a composition “sui generis” that differed from standard mereological composition. In this paper it is shown that, although traditional Boolean mereology does not describe parthood and composition in its full generality, a better and more comprehensive theory is provided by the foundational theory of categories. In this category-theoretical framework a theory of structural universals can be formulated that overcomes the conceptual difficulties that Lewis and his followers regarded as unsurmountable. As a concrete example of structural universals groups are considered in some detail.

Ontological foundations for conceptual modelling

Etymologically, the term “ontology” means the study of existence. In philosophy, ontology is the branch of metaphysics concerned with the fundamental nature of being, addressing deep questions such as “Do nonphysical things exist?”, “Does an object remain identical to itself when it undergoes change?”, and so on. Ontology in this sense is a mature discipline, with a history of systematic development in western philosophy at least since Aristotle. The business of ontology “. . . is to study the most general features of reality” (Peirce, 1935), as opposed to the several specific scientific disciplines (e.g., physics, chemistry, biology), which deal only with entities that fall within their respective domain. However, there are many ontological principles that are utilized in scientific research, for instance, in the selection of concepts and hypotheses, in the axiomatic reconstruction of scientific theories, in the design of techniques, and in the evaluation of scientific results (Bunge, 1977, p. 19). Thus, to quote the physicist and philosopher of science Mario Bunge, “every science presupposes some metaphysics”. In the beginning of the 20th century, the philosopher Edmund Husserl coined the term “formal ontology” as analogous to formal logic. Whilst formal logic deals with formal logical structures (e.g., truth, validity, consistency) independently of their veracity, formal ontology deals with formal ontological structures (e.g., theory of parthood, types and instantiation, identity, dependence, unity), i.e., with formal aspects of entities irrespective of their particular nature. The unfolding of formal ontology as a philosophical discipline aims at developing a system of general categories that can be used in the development of scientific theories and domain-specific common sense theories of reality. According to Smith (2004), the term “ontology” in the computer and information science literature appeared for the first time in 1967, in a work on the foundations of data modeling by S.H. Mealy, in a passage where he distinguishes three distinct realms in the field of data processing, namely: (i) “the real world itself ”; (ii) “ideas about it existing in the minds of men”; (iii) “symbols on paper or some other storage medium”. This view bears obvious resemblance to the famous semiotic triangle of Ogden & Richards (1923). Mealy concludes the passage arguing about the existence of things in the world

A framework in prolog for computing structural relationships

We present a low-level Prolog framework for computing structural relationships in finite models of General Extensional Mereology (GEM), which is probably the most prominent predicate logic theory of parthood. Based on a minimal amount of structural input knowledge about a domain the framework generates a consistent model of parthood and other structural relations and functions on the entities of that domain. Although some of its algorithms have exponential time complexity, the framework may contribute to the generation of consistent models of structural knowledge in various domains.

Parts, wholes, and part-whole relations: The prospects of mereotopology

We can see mereology as a theory of parthood and topology as a theory of wholeness. How can the two be combined to obtain a unified theory of parts and wholes'? This paper examines three main ways of answering this question. On the first account, mereology and topology form two independent (though mutually related) domains. The second account grants priority to topology and characterizes mereology derivatively, by defining parthood in terms of wholeness (more specifically: connectedness). The third approach reverses the order, exploiting the idea that wholeness (connectedness) can be explained in terms of parthood along with other predicates or relations. The analysis and comparison of these strategies is mostly formal (and within the confines of standard first-order theories), but their relevance to spatio-temporal reasoning and representation is emphasized. Some more speculative strategies and directions for further research, such as the development of a unified framework based on a single mereotopological primitive of connected parthood, are also briefly considered.