Mereological Theory Research Articles

From slot mereology to a mereology of slots

In 2013, Bennett proposed a mereological theory in which the parthood relation is defined on the basis of two primitive relations: a is a part of b iff a fills a slot owned by b. However, this theory has issues counting how many parts an entity has. We explore the various counting problems and propose a new theory to solve them. Keeping the core idea of Bennett’s slots, this theory introduces mereological relations between slots. This theory enables us to solve all known counting problems and to go beyond the limits of Bennett’s theory by theorising expected features of mereological theories: supplementation principles and mereological sum and fusion. The theory is illustrated on ontological issues on the nature of structural universals and informational entities.

Boolean Mereology

Most ordinary objects - cats, humans, mountains, ships, tables, etc. - have indeterminate mereological boundaries. If the theory of mereology is meant to include ordinary objects at all, we need it to have some space for mereological indeterminacy. In this paper, we present a novel degree-theoretic semantics - Boolean semantics - and argue that it is the best degree-theoretic semantics for modeling mereological indeterminacy, for three main reasons: (a) it allows for incomparable degrees of parthood, (b) it enforces classical logic, and (c) it is compatible with all the axioms of classical mereology. Using Boolean semantics, we will also investigate the connection between vagueness in parthood and vagueness in existence/identity. We show that, contrary to what many have argued, the connection takes neither the form of entailment nor the form of exclusion.

MEREOLOGICAL BIMODAL LOGICS

AbstractIn this paper, using a propositional modal language extended with the window modality, we capture the first-order properties of various mereological theories. In this setting, $\Box \varphi $ reads all the parts (of the current object) are $\varphi $ , interpreted on the models with a whole-part binary relation under various constraints. We show that all the usual mereological theories can be captured by modal formulas in our language via frame correspondence. We also correct a mistake in the existing completeness proof for a basic system of mereology by providing a new construction of the canonical model.

The Denotation of Copredicative Nouns

Copredication is the phenomenon whereby two or more predicates seem to require that their argument denotes different things. The denotation of words that copredicate has been broadly discussed. In this paper, I investigate the metaphysics behind this question. Thus, mereological theories of dot objects claim that these nouns denote complex entities; Asher (Lexicalmeaning in context, Cambridge University Press, 2011, https://doi.org/10.1017/CBO9780511793936) thinks that they denote bare particulars; and the Activation Package Theory contends that they stand for multiple denotations. According to the Activation Package Theory, copredicative nouns stand for complex knowledge structures that offer a range of multiple potential denotations. In this paper, I claim that the Activation Package Theory contributes to solve some of the metaphysical questions that arise from copredication.

On the effective universality of mereological theories

AbstractMereological theories are based on the binary relation “being a part of”. The systematic investigations of mereology were initiated by Leśniewski. More recent authors (including Simons, Casati and Varzi, Hovda) formulated a series of first‐order mereological axioms. These axioms give rise to a plenitude of theories, which are of great philosophical interest. The paper considers first‐order mereological theories from the point of view of computable (or effective) algebra. Following the approach of Hirschfeldt, Khoussainov, Shore, and Slinko, we isolate two important computability‐theoretic properties P (namely, degree spectra of structures, and effective dimensions), and consider the following problem: for a given mereological theory T, is it true that its models can realize every possible variant of the property P? If the answer is positive, then we say that the theory T is ‐universal. We obtain the following results about known mereological theories. Any theory T which is weaker than Extensional Closure Mereology (CEM) is ‐universal. A similar fact is true for the theory GM2. On the other hand, any theory stronger that CEM + (C) + (G) is not ‐universal. In particular, General Extensional Mereology is not ‐universal.

Hylomorphism and Complex Properties

Abstract Hylomorphism is the Aristotelian theory according to which objects are composites of form and matter. Form is what unifies the various parts of an object – the matter – into a cohesive whole. Some contemporary hylomorphists argue their theory applies beyond the realm of concreta, and that it explains the unity of various abstract entities. Not everyone agrees. Recent criticism alleges that hylomorphism fails to explain the unity of certain abstract entities, namely, complex properties – properties with other properties as proper parts. Here, I both respond to this criticism and show that and how hylomorphism extends to the domain of complex properties. By discussing hylomorphism’s applicability to the hitherto unchartered domain of complex properties, I hope to defend its credentials as a general mereological theory.

Intuitionistic mereology

Two mereological theories are presented based on a primitive apartness relation along with binary relations of mereological excess and weak excess, respectively. It is shown that both theories are acceptable from the standpoint of constructive reasoning while remaining faithful to the spirit of classical mereology. The two theories are then compared and assessed with regard to their extensional import.

Mereology and Infinity

This paper deals with the treatment of infinity and finiteness in mereology. After an overview of some first-order mereological theories, finiteness axioms are introduced along with a mereological definition of “x is finite” in terms of which the axioms themselves are derivable in each of those theories. The finiteness axioms also provide the background for definitions of “(mereological theory) T makes an assumption of infinity”. In addition, extensions of mereological theories by the axioms are investigated for their own sake. In the final part, a definition of “x is finite” stated in a second-order language is also presented, followed by some concluding remarks on the motivation for the study of the (first-order) extensions of mereological theories dealt with in the paper.

Sequents for non-wellfounded mereology

The paper explores the proof theory of non-wellfounded mereology with binary fusions and provides a cut-free sequent calculus equivalent to the standard axiomatic system.

‘Identity’ as a mereological term

The mereological predicate ‘is part of’ can be used to define the predicate ‘is identical with’. I argue that this entails that mereological theories can be ideologically simpler than nihilistic theories that do not use the notion of parthood—contrary to what has been argued by Ted Sider. Moreover, if one accepts an extensional mereology, there are good philosophical reasons apart from ideological simplicity to give a mereological definition of identity.

Rough – Granular Computing knowledge discovery models for medical classification

Medical domain has become one of the most important areas of research in order to richness huge amounts of medical information about the symptoms of diseases and how to distinguish between them to diagnose it correctly. Knowledge discovery models play vital role in refinement and mining of medical indicators to help medical experts to settle treatment decisions. This paper introduces four hybrid Rough – Granular Computing knowledge discovery models based on Rough Sets Theory, Artificial Neural Networks, Genetic Algorithm and Rough Mereology Theory. A comparative analysis of various knowledge discovery models that use different knowledge discovery techniques for data pre-processing, reduction, and data mining supports medical experts to extract the main medical indicators, to reduce the misdiagnosis rates and to improve decision-making for medical diagnosis and treatment. The proposed models utilized two medical datasets: Coronary Heart Disease dataset and Hepatitis C Virus dataset. The main purpose of this paper was to explore and evaluate the proposed models based on Granular Computing methodology for knowledge extraction according to different evaluation criteria for classification of medical datasets. Another purpose is to make enhancement in the frame of KDD processes for supervised learning using Granular Computing methodology.

An addendum to: “Notes on models of first-order mereological theories”

Andrzej Pietruszczak has made a comment on my Claim 1 in his paper entitled “Classical mereology is not elementarily axiomatizable”. His paper is a wonderful exposition of mereological structures and I think his comment is fair. However, the following are some remarks inspired by his comment.

The mereology of structural universals

This paper explores the mereology of structural universals, using the structural richness of a non-classical mereology without unique fusions. The paper focuses on a problem posed by David Lewis, who using the example of methane, and assuming classical mereology, argues against any purely mereological theory of structural universals. The problem is that being a methane molecule would have to contain being a hydrogen atom four times over, but mereology does not have the concept of the same part occurring several times. This paper takes up the challenge by providing mereological analysis of three operations sufficient for a theory of structural universals: (1) Reflexive binding, i.e. identifying two of the places of a universal; (2) Existential binding, i.e. the language-independent correlate of an existential quantification; and (3) Conjunction.

Notes on models of first-order mereological theories

This paper will consider some interesting mereological models and, by looking into them carefully, will clarify some important metalogical issues, such as definability, atomicity and decidability. More precisely, this paper will inquire into what kind of subsets can be defined in certain mereological models, what kind of axioms can guarantee that any member is composed of atoms and what kind of axioms are crucial, by regulating the models in a certain way, for an axiomatized mereological theory to be decidable.

More on the decidability of mereological theories

Quite a few results concerning the decidability of mereological theories have been given in my previous paper. But many mereological theories are still left unaccounted for. In this paper I will refine a general method for proving the undecidability of a theory and then by making use of it, I will show that most mereological theories that are strictly weaker than CEM are finitely inseparable and hence undecidable. The same results might be carried over to some extensions of those weak theories by adding the fusion axiom schema. Most of the proofs to be presented in this paper take finite lattices as the base models when applying the refined method. However, I shall also point out the limitation of this kind of reduction and make some observations and conjectures concerning the decidability of stronger mereological theories.

Exemplification and Parthood

Consider the things that exist—the entities—and let us suppose they are mereologically structured, that is, some are parts of others. The project of ontology within the bounds of bare mereology use this structure to say which of these entities belong to various ontological kinds, such as properties and particulars. My purpose in this paper is to defend the most radical section of the project, the mereological theory of the exemplification of universals. Along the way I help myself to several hypotheses: the existence of merely possible worlds; that particulars have thisnesses; and that mereology is far from classical. Moreover, the way I characterize instantiation might be judged too complicated to be plausible. At the end of the paper, I reply to these objections based on complexity.

A Comprehensive Picture of the Decidability of Mereological Theories

The signature of the formal language of mereology contains only one binary predicate which stands for the relation "being a part of" and it has been strongly suggested that such a predicate must at least define a partial ordering. Mereological theories owe their origin to Leśniewski. However, some more recent authors, such as Simons as well as Casati and Varzi, have reformulated mereology in a way most logicians today are familiar with. It turns out that any theory which can be formed by using the reformulated mereological axioms or axiom schemas is in a sense a subtheory of the elementary theory of Boolean algebras or of the theory of infinite atomic Boolean algebras. It is known that the theory of partial orderings is undecidable while the elementary theory of Boolean algebras and the theory of infinite atomic Boolean algebras are decidable. In this paper, I will look into the behaviors in terms of decidability of those mereological theories located in between. More precisely, I will give a comprehensive picture of the said issue by offering solutions to the open problems which I have raised in some of my papers published previously.

The Limits of Hylomorphism

Aristotle’s theory of physical objects, hylomorphism, has resurfaced in contemporary metaphysics. In its current version, hylomorphism is proposed as a general theory of mereology, its purview extending beyond material objects to chemical composites, events, and non-physical mathematical, linguistic, and musical objects. While I agree that hylomorphism works well in all of the newly proposed applications, it fails as a theory of properties and their parts. I show that this is the case and then theorize about why this is so.

Granular computing in the frame of rough mereology. A case study: Classification of data into decision categories by means of granular reflections of data

This work extends the authors' contribution to the International Conference on Rough Sets and Current Trends in Computing (RSCTC 2008) held at the University of Akron in October 2008. It is dedicated to the topic of granular computing, formalized within the theory of rough mereology, as proposed by Polkowski; as an application of the idea of a granular reflection of data and of classifiers induced from it (Polkowski, 2005), we give an account of recent results in this area. A scheme for classifier construction based on factoring a classifier through a granular reflection of data; voting by granules of training objects; voting by granules of decision rules induced from the training set; voting by granules induced from a granular reflection of data, is presented here. In voting cases, voting is based on weights computed by means of rough inclusions induced from residual implications of continuous t-norms. The results show a high effectiveness of this approach as witnessed by the reported tests with some well--known data sets from University of California, at Irvine (UCI) repository whose results are compared against the standard rough set exhaustive classifier whose accuracy is indicated under the radius of “nil.” © 2011 Wiley Periodicals, Inc.

Decidability of mereological theories

Mereological theories are theories based on a binary predicate ‘being a part of’. It is believed that such a predicate must at least define a partial ordering. A mereological theory can be obtained by adding on top of the basic axioms of partial orderings some of the other axioms posited based on pertinent philosophical insights. Though mereological theories have aroused quite a few philosophers’ interest recently, not much has been said about their meta-logical properties. In this paper, I will look into whether those theories are decidable or not. Besides, since theories of Boolean algebras are in some sense upper bounds of mereological theories which can be found in the literature, I shall also make some observations about the possibility of getting mereological theories beyond Boolean algebras.