This paper examines whether classical extensional mereology is adequate for formalizing the whole–parts relation in quantum chemical systems. Although other philosophers have argued that classical extensional and summative mereology does not adequately formalize whole–parts relation within organic wholes and social wholes, such critiques often assume that summative mereology is appropriate for formalizing the whole–parts relation in inorganic wholes such as atoms and molecules. However, my discussion of atoms and molecules as they are conceptualized in quantum chemistry will (...) establish that standard mereology cannot adequately fulfill this task, since the properties and behavior of such wholes are context-dependent and cannot simply be reduced to the summative properties of their parts. To the extent that philosophers of chemistry have called for the development of an alternative mereology for quantum chemical systems, this paper ends by proposing behavioral mereology as a promising step in that direction. According to behavioral mereology, considerations of what constitutes a part of a whole is dependent upon the observable behavior displayed by these entities. Thus, relationality and context-dependence are stipulated from the outset and this makes behavioral mereology particularly well-suited as a mereology of quantum chemical wholes. The question of which mereology is appropriate for formalizing the whole–parts relation in quantum chemical systems is relevant to contemporary philosophy of chemistry, since this issue is related to the more general questions of the reducibility of chemical wholes to their parts and of the reducibility of chemistry to physics, which have been of central importance within the philosophy of chemistry for several decades. More generally, this paper puts contemporary discussions of mereology within the philosophy of chemistry into a broader historical and philosophical context. In doing so, this paper also bridges the gap between formal mereology, conceived as a branch of formal ontology, and “applied” mereology, conceived as a branch of philosophy of science. (shrink)

Mereological principles are sometimes used to support general claims about the structure and arrangement of objects in the world. I focus initially on one such mereological principle, the weak supplementation principle (WSP). It is not obvious that (WSP) is prescribed by ordinary thinking about parthood. Further, (WSP) is not needed for a fairly strong formal characterization of the part–whole relation. For these reasons, some arguments relying on (WSP) might be countered by simply denying (WSP). I argue more generally that (...) there is no reason to assume that one core mereology functions as a common basis for all plausible metaphysical theories. (shrink)

This volume is the first systematic and thorough attempt to investigate the relation and the possible applications of mereology to contemporary science. It gathers contributions from leading scholars in the field and covers a wide range of scientific theories and practices such as physics, mathematics, chemistry, biology, computer science and engineering. Throughout the volume, a variety of foundational issues are investigated both from the formal and the empirical point of view. The first section looks at the topic as (...) it applies to physics. The section addresses questions of persistence and composition within quantum and relativistic physics and concludes by scrutinizing the possibility to capture continuity of motion as described by our best physical theories within gunky space times. The second part tackles mathematics and shows how to provide a foundation for point-free geometry of space switching to fuzzy-logic. The relation between mereological sums and set-theoretic suprema is investigated and issues about different mereological perspectives such as classical and natural Mereology are thoroughly discussed. The third section in the volume looks at natural science. Several questions from biology, medicine and chemistry are investigated. From the perspective of biology, there is an attempt to provide axioms for inferring statements about part hood between two biological entities from statements about their spatial relation. From the perspective of chemistry, it is argued that classical mereological frameworks are not adequate to capture the practices of chemistry in that they consider neither temporal nor modal parameters. The final part introduces computer science and engineering. A new formal mereological framework in which an indeterminate relation of part hood is taken as a primitive notion is constructed and then applied to a wide variety of disciplines from robotics to knowledge engineering. A formal framework for discrete mereotopology and its applications is developed and finally, the importance of mereology for the relatively new science of domain engineering is also discussed. (shrink)

Is a whole something more than the sum of its parts? Are there things composed of the same parts? If you divide an object into parts, and divide those parts into smaller parts, will this process ever come to an end? Can something lose parts or gain new ones without ceasing to be the thing it is? Does any multitude of things (including disparate things such as you, this book, and the tail of a cat) compose a whole of some (...) sort? Questions such as these have occupied us for at least as long as philosophy has existed. They define the field that has come to be known as mereology-the study of all relations of part to whole and of part to part within a whole-and have deep and far-reaching ramifications in metaphysics as well as in logic, the foundations of mathematics, the philosophy of language, the philosophy of science, and beyond. In Mereology, A. J. Cotnoir and Achille C. Varzi have compiled decades of advanced research into a comprehensive, up-to-date, and formally rigorous picture. The early chapters cover the more classical aspects of mereology; the rest of the book deals with variants and extensions. Whether you are an established professional philosopher, an interested student, or a newcomer, inside you will find all the tools you need to join this ever-evolving field of inquiry and theorize about all things mereological. (shrink)

The peculiar legacy of Husserl’s mereology, chiefly studied by analytic philosophers interested in ontology, has led to a partial understanding of the III. LU, which is too often reduced to a chapter of “formal ontology”. Yet, the power of this Investigation goes far beyond: it enabled Husserl to deal, in the framework of a unified theory, with a vast range of particular problems. The paper focuses on one of these issues, namely abstraction, so as to expose how Husserl (...) instrumentalizes his formal tools in order to tackle material issues. The existence of an up and down pattern is uncovered: Husserl first reinterprets the psychological problem of abstraction in ontological terms, before coming back to the original problem with new insights. The second, correlative aim of the paper is to emphasize the key role played by Friedrich Schumann, a forgotten yet crucial character for Husserl’s conception of abstraction. (shrink)

Classical mereology is a formal theory of the part-whole relation, essentially involving a notion of mereological fusion, or sum. There are various different definitions of fusion in the literature, and various axiomatizations for classical mereology. Though the equivalence of the definitions of fusion is provable from axiom sets, the definitions are not logically equivalent, and, hence, are not inter-changeable when laying down the axioms. We examine the relations between the main definitions of fusion and correct some technical (...) errors in prominent discussions of the axiomatization of mereology. We show the equivalence of four different ways to axiomatize classical mereology, using three different notions of fusion. We also clarify the connection between classical mereology and complete Boolean algebra by giving two "neutral" axiom sets which can be supplemented by one or the other of two simple axioms to yield the full theories; one of these uses a notion of "strong complement" that helps explicate the connections between the theories. (shrink)

In his book, 'History as a Science and the System of the Sciences', Thomas Seebohm articulates the view that history can serve to mediate between the sciences of explanation and the sciences of interpretation, that is, between the natural sciences and the human sciences. Among other things, Seebohm analyzes history from a phenomenological perspective to reveal the material foundations of the historical human sciences in the lifeworld. As a preliminary to his analyses, Seebohm examines the formal and material presuppositions (...) of phenomenological epistemology, as well as the emergence of the human sciences and the traditional distinctions and divisions that are made between the natural and the human sciences. -/- As part of this examination, Seebohm devotes a section to discussing Husserl’s formal mereology because he understands that a reflective analysis of the foundations of the historical sciences requires a reflective analysis of the objects of the historical sciences, that is, of concrete organic wholes (i.e., social groups) and of their parts. Seebohm concludes that Husserl’s mereological ontology needs to be altered with regard to the historical sciences because the relations between organic wholes and their parts are not summative relations. Seebohm’s conclusion is relevant for the issue of the reducibility of organic wholes such as social groups to their parts and for the issue of the reducibility of the historical sciences to the lower-order sciences, that is, to the sciences concerned with lower-order ontologies. -/- In this paper, I propose to extend Seebohm’s conclusion to the ontology of chemical wholes as object of quantum chemistry and to argue that Husserl’s formal mereology is descriptively inadequate for this regional ontology as well. This may seem surprising at first, since the objects studied by quantum chemists are not organic wholes. However, my discussion of atoms and molecules as they are understood in quantum chemistry will show that Husserl’s classical summative and extensional mereology does not accurately capture the relations between chemical wholes and their parts. This conclusion is relevant for the question of the reducibility of chemical wholes to their parts and of the reducibility of chemistry to physics, issues that have been of central importance within the philosophy of chemistry for the past several decades. (shrink)

How best to think about quantum systems under permutation invariance is a question that has received a great deal of attention in the literature. But very little attention has been paid to taking seriously the proposal that permutation invariance reflects a representational redundancy in the formalism. Under such a proposal, it is far from obvious how a constituent quantum system is represented. Consequently, it is also far from obvious how quantum systems compose to form assemblies, i.e. what is the (...) class='Hi'>formal structure of their relations of parthood, overlap and fusion. In this paper, I explore one proposal for the case of fermions and their assemblies. According to this proposal, fermionic assemblies which are not entangled—in some heterodox, but natural sense of ‘entangled’—provide a prima facie counterexample to classical mereology. This result is puzzling; but, I argue, no more intolerable than any other available interpretative option. (shrink)

ABSTRACTHow did the traditional doctrine of parts and wholes evolve into contemporary formal mereology? This paper argues that a crucial missing link may lie in the early modern and especially Wolf...

Mereological universalists and nihilists disagree on the conditions for composition. In this paper, we show how this debate is a function of one’s chosen semantics for plural quantifiers. Debating mereologists have failed to appreciate this point because of the complexity of the debate and extraneous theoretical commitments. We eliminate this by framing the debate between universalists and nihilists in a formal model where these two theses about composition are contradictory. The examination of the two theories in the model brings (...) clarity to a debate in which opponents frequently talk past one another. With the two views stated precisely, our investigation reveals the dependence of the mereologists’ ontological commitments on the semantics of plural quantifiers. Though we discuss the debate with respect to a simplified and idealized model, the insights provided will make more complex debates on composition more productive and deflationist criticisms of the debate less substantial. (shrink)

The interactive theorem prover Isabelle/HOL is used to verify elementary theorems of classical extensional mereology.

Part One of this paper is a case against classical mereology and for Heyting mereology. This case proceeds by first undermining the appeal of classical mereology and then showing how it fails to cohere with our intuitions about a measure of quantity. Part Two shows how Heyting mereology provides an account of sets and classes without resort to any nonmereological primitive.

One purely formal discipline, mathematics, permeates modern natural, social, and medical sciences as well as technology and engineering. Another purely formal discipline, formal mereology, has the last decades rapidly grown within philosophy. The obvious question now is: of what use can formal mereology be in the sciences and in engineering? In philosophy, this question has seldom so far been posed, but now there has appeared an anthology in which the editors C. Calosi and P. (...) Graziani rightly state: “It should be clear that the present volume fills indeed the gap between philosophical reflections on parts and wholes and the contemporary scientific context. Or better, it begins to fill this gap” . The book is truly to be welcomed. The editors have divided the contributions into four parts: I “Physics” , II “Mathematics” , III “Natural Sciences” , and IV “Computer Sciences and Engineering” . Each part is preceded by a brief two pages introduc .. (shrink)

Metaphysicians of various stripes claim that a single object can have more than one exact location in space or time – e.g. endurantists claim that an object persists by being ‘all there’ at different moments in time. Antony Eagle has developed a formal theory of location which is prima facie consistent with multi-location, but Damiano Costa and Claudio Calosi argue that the theory is unattractive to multi-location theorists on other grounds. I examine their charge that Eagle’s theory won’t allow (...) an endurantist to account for certain cases of mereological change. I argue that the charge sticks, but not for the reasons Costa and Calosi think. Along the way, I explore an issue which is underexplored in their paper, namely, how an endurantist might modify Eagle’s theory to incorporate a parthood relation which obtains, not absolutely, but only relative to times. (shrink)

Many mereological propositions are true contingently, so we are entitled to ask why they are true. One frequently given type of answer to such questions evokes truth-makers, that is, entities in virtue of whose existence the propositions in question are true. However, even without endorsing the extreme view that all contingent propositions have truth-makers, it turns out to be puzzlingly hard to provide intuitively convincing candidate truth-makers for even a core class of basic mereological propositions. Part of the problem is (...) that the relation of part to whole is ontologically intimate in a way reminiscent of identity. Such intimacy bespeaks a formal or internal relation, which typically requires no truth-makers beyond its terms. But truth-makers are held to necessitate their truths, so whence the contingency when A is part of B but need not be, or B need not have A as part? This paper addresses and attempts to disentangle the conundrum. (shrink)

The special composition question, which asks under which conditions objects compose a further object, establishes a central debate in modern metaphysics. Recent successes of inductive metaphysics, which studies the implications of the natural sciences for metaphysical problems, suggest that insights into the SCQ can be gained by investigating the physics of composite systems. In this work, I show that the minus first law of thermodynamics, which is concerned with the approach to equilibrium, leads to a new approach to the SCQ, (...) the thermodynamic composition principle : Multiple systems in thermal contact compose a single system. This principle, which is justified based on a systematic classification of possible mereological models for thermodynamic systems, might form the basis of an inductive argument for universalism. A formal analysis of the TCP is provided on the basis of mereotopology, which is a combination of mereology and topology. Here, “thermal contact” can be analyzed using the mereotopological predicate “self-connectedness”. Self-connectedness has to be defined in terms of mereological sums to ensure that scattered objects cannot be self-connected. (shrink)

Hybrid languages are introduced in order to evaluate the strength of “minimal” mereologies with relatively strong frame definability properties. Appealing to a robust form of nominalism, I claim that one investigated language \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}_{\textsf {m}}$\end{document} is maximally acceptable for nominalistic mereology. In an extension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}_{\textsf {gem}}$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}_{\textsf {m}}$\end{document}, a modal (...) analog for the classical systems of Leonard and Goodman and Leśniewski is introduced and shown to be complete with respect to 0-deleted Boolean algebras. We characterize the formulas of first-order logic invariant for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}_{\textsf {gem}}$\end{document}-bisimulations. (shrink)

We show that a standard axiomatization of mereology is equivalent to the condition that a topological space is discrete, and consequently, any model of general extensional mereology is indistinguishable from a model of set theory. We generalize these results to the Cartesian closed category of convergence spaces.

This paper provides a detailed examination of Kit Fine’s sizeable contribution to the development of a neo‐Aristotelian alternative to standard mereology; I focus especially on the theory of ‘rigid’ and ‘variable embodiments’, as defended in Fine 1999. Section 2 briefly describes the system I call ‘standard mereology’. Section 3 lays out some of the main principles and consequences of Aristotle’s own mereology, in order to be able to compare Fine’s system with its historical precursor. Section 4 gives (...) an exposition of Fine’s theory of embodiments and goes on to isolate a number of potential concerns to which this account gives rise. In particular, I argue that Fine’s theory threatens to proliferate primitive sui generis relations of parthood and composition, whose characteristics must be stipulatively imposed on them, relative to particular domains; given its ‘superabundance’ of objects, Fine’s system far outstrips the already inflated ontological commitments of standard mereology; and there is a legitimate question as to why we should consider Fine’s primitive and sui generis relations of parthood and composition to be genuinely mereological at all, given their formal profile. These three objections lead me to conclude that we ought to explore other avenues that preserve the highly desirable, hylomorphic, features of Fine’s mereology, while avoiding its methodological and ontological excesses. (shrink)

The signature of the formal language of mereology contains only one binary predicate P which stands for the relation “being a part of”. Traditionally, P must be a partial ordering, that is, ${\forall{x}Pxx, \forall{x}\forall{y}((Pxy\land Pyx)\to x=y)}$ and ${\forall{x}\forall{y}\forall{z}((Pxy\land Pyz)\to Pxz))}$ are three basic mereological axioms. The best-known mereological theory is “general extensional mereology”, which is axiomatized by the three basic axioms plus the following axiom and axiom schema: (Strong Supplementation) ${\forall{x}\forall{y}(\neg Pyx\to \exists z(Pzy\land \neg Ozx))}$ , where (...) Oxy means ${\exists z(Pzx\land Pzy)}$ , and (Fusion) ${\exists x\alpha \to \exists z\forall y(Oyz\leftrightarrow \exists x(\alpha \land Oyx))}$ , for any formula α where z and y do not occur free. In this paper, I will show that general extensional mereology is decidable, and will also point out that the decidability of the first-order approximation of the theory of complete Boolean algebras can be shown in the same way. (shrink)

The emerging field of quantum mereology considers part-whole relations in quantum systems. Entangled quantum systems pose a peculiar problem in the field, since their total states are not reducible to that of their parts. While there exist several established proposals for modelling entangled systems, like monistic holism or relational holism, there is considerable unclarity, which further positions are available. Using the lambda operator and plural logic as formal tools, we review and develop conceivable models and evaluate their consistency (...) and distinctness. The main result is an exhaustive taxonomy of six distinct and precise models that both provide information about the mereological features as well as about the entangled property. The taxonomy is well-suited to serve as the basis for future systematic investigations. (shrink)

As it is indicated in the title, this paper is devoted to the problem of defining mereological (collective) sets. Starting from basic properties of sets in mathematics and differences between them and so called conglomerates in Section 1, we go on to explicate informally in Section 2 what it means to join many objects into a single entity from point of view of mereology, the theory of part of (parthood) relation. In Section 3 we present and motivate basic axioms (...) for part of relation and we point to their most fundamental consequences. Next three sections are devoted to formal explication of the notion of mereological set (collective set) in terms of sums, fusions and aggregates. We do not give proofs of all theorems. Some of them are complicated and their presentation would divert the reader’s attention from the main topic of the paper. In such cases we indicate where the proofs can be found and analyzed by those who are interested. (shrink)

En aquest article pretenc fer una crítica al nihilisme mereològic, al·ludint que les expressions «simples agrupats en-tant-que-F» tenen unes conseqüències desastroses per als seus compromisos ontològics. Primer, explicaré què és el nihilisme mereològic -que és part de l’eliminativisme- el qual pretén negar l’existència dels objectes compostos (objectes amb parts pròpies) i l’estratègia de la paràfrasi: l’ús que fan de les expressions «simples agrupats en-tant-que-F» per referir-se als objectes ordinaris sense comprometre’s amb l’existència d’objectes compostos, però posaré l’èmfasi en aquells nihilistes (...) mereològics que pretenen explicar l’existència de les propietats emergents. Seguidament, presentaré una classe de crítiques, les de Bennett i Tallant, sobre l’ús d’aquesta estratègia per part dels nihilistes, i quina és la resposta crítica que ofereix Brenner sobre aquesta classe de crítica. Finalment, presentaré la meva crítica on mostraré que el nihilisme mereològic utilitzant l’estratègia de la paràfrasi s’està comprometent amb l’existència d’objectes compostos i que la crítica de Brenner no funciona amb la meva postura. -/- In this paper I intend to critique mereological nihilism, alluding that the expression «simples arranged F-wise» has disastrous consequences for its ontological commitments. First, I will explain what mereological nihilism is -which is part of Eliminativism- which tries to deny the existence of composite objects (objects with proper parts) and the Paraphrase Strategy: the use they make of the expression «simples arranged F-wise» to refer to ordinary objects without committing to the existence of composite objects, and I put the focus on the mereological nihilist that pretends to explain the existence of emergent properties. Second, I will present a kind of criticism, by Bennett and Tallant, of the use of this strategy by nihilism and which is the response of Brenner about this kind of criticism. Finally, I will present my criticism where I will show that nihilism by using the Paraphrase Strategy is committing itself to the existence of composite objects and that the Brenner’s critique does not work with my position. (shrink)

I analyze the relations of constituency or ``being in'' that connect different ontological items in the Tractatus logico-philosophicus by Wittgenstein. A state of affairs is constituted by atoms, atoms are in a state of affairs. Atoms are also in an atomic fact. Moreover, the world is the totality of facts, thus it is in some sense made of facts. Many other kinds of Tractarian notions -- such as molecular facts, logical space, reality -- seem to be involved in constituency relations. (...) How should these relations be conceived? And how is it possible to formalize them in a convincing way? I draw a comparison between two ways of conceiving and formalizing these relations: through sets and through mereological sums. The comparison shows that the conceptual machinery of set theory is apter to conceive and formalize Tractarian constituency notions than the mereological one. (shrink)

Within the framework of Meinongianism, nothingness turns out to have contradictory features—it seems to be an object and not. In this paper, we explore two different kinds of Meinongian accounts of nothingness. The first one is the consistent account, which rejects the contradiction of nothingness, while the second one is the inconsistent account, which accepts the contradiction of nothingness. First of all, after showing that the consistent account of nothingness defended by Jacquette fails, we express some concerns on the general (...) possibility of consistently characterizing nothingness. Secondly, starting from Priest’s inconsistent characterization of nothingness :146–158, 2014b), we will introduce our own inconsistent account. The key idea of our account is to take nothingness as the complement of the totality. Finally, we will make formal sense of it by constructing an inconsistent mereological system, which is the development of the paraconsistent mereology proposed by Weber and Cotnoir. (shrink)

Many bioethical arguments rely implicitly on the assumption that the concept of “human part” is one on which everyone must agree, because it is unambiguous. But various parties interpret this “unambiguous” term in incompatible ways, leading to contention. This article is an informal presentation of a topomereological system on whose preferred interpretation several distinct but related meanings of “human part” can be isolated: part of a human body, part of the completion of a human body, and part of a human (...) being. A case is analyzed (the first total artificial heart (TAH) implantation), demonstrating in the process much of the apparatus of the system. By means of a casuistic methodology, the analysis is translated into recommendations for the ethical conduct of future TAH research. The more general conclusion, however, is that formal methods may provide useful tools for clarifying thought processes and organizing arguments in debates over bioethical issues. (shrink)

Kathrin Koslicki argues that ordinary material objects like tables and motorcycles have formal proper parts that structure the material proper parts. Karen Bennett rejects a key premise in Koslicki's argument according to which the material ingredient out of which a complex material object is made is a proper part of that object. Koslicki defends this premise with a principle motivated by its power to explain three important phenomena of material composition. But these phenomena can be equally well explained by (...) a weaker principle that does not support the questioned premise in Koslicki's argument, Bennett argues. I show that Bennett's weaker principle, together with an appropriate strengthening of a different premise in Koslicki's original argument, still yields a sound argument for the existence of formal parts. (shrink)

We are concerned with formal models of reasoning under uncertainty. Many approaches to this problem are known in the literature e.g. Dempster-Shafer theory [29], [42], bayesian-based reasoning [21], [29], belief networks [29], many-valued logics and fuzzy logics [6], non-monotonic logics [29], neural network logics [14]. We propose rough mereology developed by the last two authors [22-25] as a foundation for approximate reasoning about complex objects. Our notion of a complex object includes, among others, proofs understood as schemes constructed (...) in order to support within our knowledge assertions/hypotheses about reality described by our knowledge incompletely. (shrink)

We are concerned with formal models of reasoning under uncertainty. Many approaches to this problem are known in the literature e.g. Dempster-Shafer theory [29], [42], bayesian-based reasoning [21], [29], belief networks [29], many-valued logics and fuzzy logics [6], non-monotonic logics [29], neural network logics [14]. We propose rough mereology developed by the last two authors [22-25] as a foundation for approximate reasoning about complex objects. Our notion of a complex object includes, among others, proofs understood as schemes constructed (...) in order to support within our knowledge assertions/hypotheses about reality described by our knowledge incompletely. (shrink)

It is accepted that certain mereological concepts and phenomenological conceptualisations presented in Carl Stumpf’s Über den psychologischen Ursprung der Raumvorstellung and Tonpsychologie played an important role in the development of the Husserlian formal ontology. In the third Logical Investigation, which displays the formal relations between part and whole and among parts that make out a whole, one of the main concepts of contemporary formal ontology and metaphysics is settled: ontological dependence or foundation (Fundierung). My main objective is (...) to display Stumpf’s concepts of partial content, independent content, spatial wholes, sound wholes, and the different kinds of connection among parts, in particular, fusion (Verschmelzung). Second, I will show how Husserl improved this background, in particular with regards to the exact nature of the theory of manifolds (Mannigfaltigkeitslehre), in discussion with Georg Cantor, the father of set theory. Third, I will focus on Ingarden’s use of formal ontology and on the different modes of being that can be justified by appealing to the concept of ontological dependence in its Ingardenian variations. If my interpretation is adequate, it should be inferred that formal ontology is the operative theory of phenomenological philosophy, and this must be acknowledged in its full significance with respect to the supposed independence of the phenomenological method since 1913. A further consequence, not developed in this essay, is that formal ontology can be mathematised. (shrink)

In his book, History as a Science and the System of the Sciences, Thomas Seebohm articulates the view that history can serve to mediate between the sciences of explanation and the sciences of interpretation, that is, between the natural sciences and the human sciences. Among other things, Seebohm analyzes history from a phenomenological perspective to reveal the material foundations of the historical human sciences in the lifeworld. As a preliminary to his analyses, Seebohm examines the formal and material presuppositions (...) of phenomenological epistemology, as well as the emergence of the human sciences and the traditional distinctions and divisions that are made between the natural and the human sciences. -/- As part of this examination, Seebohm devotes a section to discussing Husserl’s formal mereology because he understands that a reflective analysis of the foundations of the historical sciences requires a reflective analysis of the objects of the historical sciences, that is, of concrete organic wholes (i.e., social groups) and of their parts. Seebohm concludes that Husserl’s mereological ontology needs to be altered with regard to the historical sciences because the relations between organic wholes and their parts are not summative relations. Seebohm’s conclusion is relevant for the issue of the reducibility of organic wholes such as social groups to their parts and for the issue of the reducibility of the historical sciences to the lower-order sciences, that is, to the sciences concerned with lower-order ontologies. -/- In this paper, I propose to extend Seebohm’s conclusion to the ontology of chemical wholes as object of quantum chemistry and to argue that Husserl’s formal mereology is descriptively inadequate for this regional ontology as well. This may seem surprising at first, since the objects studied by quantum chemists are not organic wholes. However, my discussion of atoms and molecules as they are understood in quantum chemistry will show that Husserl’s classical summative and extensional mereology does not accurately capture the relations between chemical wholes and their parts. This conclusion is relevant for the question of the reducibility of chemical wholes to their parts and of the reducibility of chemistry to physics, issues that have been of central importance within the philosophy of chemistry for the past several decades. (shrink)

A compact overview of the main formal theories of parthood and of their mutual relationships, up to Classical Extensional Mereology. Written as an Appendix to the other essays included in the volume.

In fields such as medicine, geography, and mechanics, spatial reasoning involves reasoning about entities that may coincide without overlapping. Some examples are: cavities and invading particles, passageways and valves, geographic regions and tropical storms. The purpose of this paper is to develop a formal theory of spatial relations for domains that include coincident entities. The core of the theory is a clear distinction between mereotopological relations, such as parthood and connection, and relative location relations, such as coincidence. To guide (...) the development of the formal theory, I construct mathematical models in which nontrivial relative location relations are defined. (shrink)

I examine the link between extensionality principles of classical mereology and the anti‐symmetry of parthood. Varzi's most recent defence of extensionality depends crucially on assuming anti‐symmetry. I examine the notions of proper parthood, weak supplementation and non‐well‐foundedness. By rejecting anti‐symmetry, the anti‐extensionalist has a unified, independently grounded response to Varzi's arguments. I give a formal construction of a non‐extensional mereology in which anti‐symmetry fails. If the notion of ‘mereological equivalence’ is made explicit, this non‐anti‐symmetric mereology recaptures (...) all of the structure of classical mereology. (shrink)

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. (shrink)

LeSniewski?s systems of Ontology and Mereology, considered from a purely formal point of view, possessstriking algebraic parallels, ascan be seen in their respective relations to Boolean algebra. But there are alsoimportant divergences, above all that general Mereology is silent, where Ontology is not, on the existenceof ?atoms? (individuals). By employing plural terms, LeSniewski sought to accommodate talk of (distributive)classes, without according these an autonomous ontological status. His logic also ? like predicate logic? has no place for mass (...) predication in its raw state. It is argued that reference both to pluralities and tomasses is ineliminable, and that one must therefore separate the grammatical distinction singular/pluralfrom the ontological distinction individual/class/mass. A kind of language, modelled on kiniewski?s, issuggested which enables these distinctions to be held apart and at the same time, building on the work ofRichard Sharvy, allows us to express the most important relationships between individuals, classes andmasses. (shrink)

Axiomatic Formal Ontology is a fairly comprehensive systematic treatise on general metaphysics. The axiomatic method is applied throughout the book. Its main theme is the construction of a general non-set-theoretical theory of intensional entities. Other important matters discussed are the metaphysics of modality, the nature of actual existence, mereology and the taxonomy of entities.

The present article provides a mereological analysis of Euclid’s planar geometry as presented in the first two books of his Elements. As a standard of comparison, a brief survey of the basic concepts of planar geometry formulated in a set-theoretic framework is given in Section 2. Section 3.2, then, develops the theories of incidence and order using a blend of mereology and convex geometry. Section 3.3 explains Euclid’s “megethology”, i.e., his theory of magnitudes. In Euclid’s system of geometry, megethology (...) takes over the role played by the theory of congruence in modern accounts of geometry. Mereology and megethology are connected by Euclid’s Axiom 5: “The whole is greater than the part.” Section 4 compares Euclid’s theory of polygonal area, based on his “Whole-Greater-Than-Part” principle, to the account provided by Hilbert in his Grundlagen der Geometrie. An hypothesis is set forth why modern treatments of geometry abandon Euclid’s Axiom 5. Finally, in Section 5, the adequacy of atomistic mereology as a framework for a formal reconstruction of Euclid’s system of geometry is discussed. (shrink)

This paper surveys mereology, the theory of parts and wholes, focussing on its origins in Leśniewski, and noting its intended employment as a surrogate for set theory. We examine parallel and independent work by Whitehead, Leonard and Goodman, and outline the subsequent adventures of mereology, both in its formal guises and in its now intensive application within philosophical ontology.

It is usually stated that quantum mechanics presents problems with the identity of particles, the most radical position—supported by E. Schrödinger—asserting that elementary particles are not individuals. But the subject goes deeper, and it is even possible to obtain states with an undefined particle number. In this work we present a set theoretical framework for the description of undefined particle number states in quantum mechanics which provides a precise logical meaning for this notion. This construction goes in the line of (...) solving a problem posed by Y. Manin, namely, to incorporate quantum mechanical notions at the foundations of mathematics. We also show that our system is capable of representing quantum superpositions. (shrink)

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 . The attempt failed, but there is another, prima facie more promising attempt by Jerzy Słupecki , who employed his generalized mereology to build mereological foundations for type theory. In this paper I situate Leśniewski's attempt in the development of (...) set theory, describe and evaluate Leśniewski's approach, describe Słupecki's strategy without unnecessary technical details, and 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 sk.. (shrink)