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1

Mereology and Infinity

100%

Niebergall K.-G.

Logic and Logical Philosophy

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2016

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vol. 25

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issue 3 Mereology and Beyond (II)

309-350

EN

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.

2

Mereology then and now

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Gruszczyński R., Varzi A. C.

Logic and Logical Philosophy

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2015

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vol. 24

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issue 4 Mereology and Beyond

409-427

EN

This paper offers a critical reconstruction of the motivations that led to the development of mereology as we know it today, along with a brief description of some questions that define current research in the field.

3

Set-theoretic mereology

100%

Hamkins J. D., Kikuchi M.

Logic and Logical Philosophy

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2016

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vol. 25

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issue 3 Mereology and Beyond (II)

285-308

EN

We consider a set-theoretic version of mereology based on the inclusion relation ⊆ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of ∈ from ⊆, we identify the natural axioms for ⊆-based mereology, which constitute a finitely axiomatizable, complete, decidable theory. Ultimately, for these reasons, we conclude that this form of set-theoretic mereology cannot by itself serve as a foundation of mathematics. Meanwhile, augmented forms of set-theoretic mereology, such as that obtained by adding the singleton operator, are foundationally robust.

4

Sequents for non-wellfounded mereology

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Maffezioli P.

Logic and Logical Philosophy

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2016

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vol. 25

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issue 3 Mereology and Beyond (II)

351-369

EN

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.

5

Notes on models of first-order mereological theories

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Tsai H.-c.

Logic and Logical Philosophy

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2015

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vol. 24

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issue 4 Mereology and Beyond

469–482

EN

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.

6

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

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Tsai H.-c.

Logic and Logical Philosophy

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2015

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vol. 24

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issue 4 Mereology and Beyond

483

EN

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.

7

Composition, identity, and emergence

100%

Calosi C.

Logic and Logical Philosophy

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2016

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vol. 25

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issue 3 Mereology and Beyond (II)

429-443

EN

Composition as Identity (CAI) is the thesis that a whole is, strictly and literally, identical to its parts, considered collectively. McDaniel [2008] argues against CAI in that it prohibits emergent properties. Recently Sider [2014] exploited the resources of plural logic and extensional mereology to undermine McDaniel’s argument. He shows that CAI identifies extensionally equivalent pluralities – he calls it the Collapse Principle (CP) – and then shows how this identification rescues CAI from the emergentist argument. In this paper I first give a new generalized version of both the arguments. It is more general in that it does not presuppose an atomistic mereology. I then go on to argue that the consequences of CP are rather radical. It entails mereological nihilism, the view that there are only mereological atoms. I finally show that, given a mild assumption about property instantiation, namely that there are no un-instantiated properties, this argument entails that CAI and emergent properties are incompatible after all.

8

Notes on models of first-order mereological theories

100%

Tsai H.-c.

Logic and Logical Philosophy

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2015

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vol. 24

|

issue 4 Mereology and Beyond

469-482

EN

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.

9

Mereology and truth-making

100%

Simons P.

Logic and Logical Philosophy

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2016

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vol. 25

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issue 3 Mereology and Beyond (II)

245-258

EN

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.

10

Tractarian Ontology: Mereology or Set Theory?

88%

Lando G.

Forum Philosophicum

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2007

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vol. 12

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issue 2

247-266

EN

I analyze the relations of constituency or “being in” that connect different ontological items in Wittgenstein's Tractatus Logico-Philosophicus. 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.

11

Czy monady mają części? Witkiewicz i jego krytyka mereologii jako ontologii

88%

Szachniewicz A.

Analiza i Egzystencja. Czasopismo Filozoficzne

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2017

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vol. 37

79-99

EN

This paper reconstructs Stanisław Ignacy Witkiewicz’s understanding of logic, accentuating the differences in his evaluation of logic and systems of ‘logistics’. Leśniewski’s theory of collective sets (mereology) exemplifies logistics as understood by Witkiewicz. I present an outline of Leśniewski’s nominalism, which entails a belief in a non-abstract nature of sets. I focus on these features of mereology that could have led Witkiewicz to interpreting it as an ontological system. Witkacy (Witkiewicz’s penname) was skeptical of the usefulness of formal systems (or logistics), and of mereology in particular, for the purposes of designing a unified ontological system describing essential properties of objects (the world). According to Witkiewicz, such formal systems assumed the role of ontology but severely lacked in philosophical justification. I argue that regardless of his nominalism and corporeal conception of individuals, mereology cannot be considered a formal theory of Witkiewicz’s monads.

12

Composition as identity and plural Cantor's theorem

88%

Bohn E. D.

Logic and Logical Philosophy

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2016

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vol. 25

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issue 3 Mereology and Beyond (II)

411-428

EN

In this paper, I argue that the thesis of Composition as Identity blocks the plural version of Cantor’s Theorem, and that this in turn has implications for our use of Cantor’s theorem in metaphysics. As an example, I show how this result blocks a recent argument by Hawthorne and Uzquiano, and might be turned around to become an abductive argument for Composition as Identity

13

The Notion of the Diameter of Mereological Ball in Tarski's Geometry of Solids

88%

Sitek G.

Logic and Logical Philosophy

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2017

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vol. 26

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issue 4

531–562

EN

In the paper "Full development of Tarski's geometry of solids" Gruszczyński and Pietruszczak have obtained the full development of Tarski’s geometry of solids that was sketched in [14, 15]. In this paper 1 we introduce in Tarski’s theory the notion of congruence of mereological balls and then the notion of diameter of mereological ball. We prove many facts about these new concepts, e.g., we give a characterization of mereological balls in terms of its center and diameter and we prove that the set of all diameters together with the relation of inequality of diameters is the dense linearly ordered set without the least and the greatest element.

14

More on The Decidability of Mereological Theories

88%

Tsai H.-c.

Logic and Logical Philosophy

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2011

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vol. 20

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issue 3

251-265

EN

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.

15

Regions-based two dimensional continua: The Euclidean case

75%

Hellman G., Shapiro S.

Logic and Logical Philosophy

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2015

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vol. 24

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issue 4 Mereology and Beyond

499-534

EN

We extend the work presented in [7, 8] to a regions-based, two-dimensional, Euclidean theory. The goal is to recover the classical continuum on a point-free basis. We first derive the Archimedean property for a class of readily postulated orientations of certain special regions, “generalized quadrilaterals” (intended as parallelograms), by which we cover the entire space. Then we generalize this to arbitrary orientations, and then establishing an isomorphism between the space and the usual point-based R × R. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause” (to the effect that “these are the only ways of generating regions”), and we have no axiom of induction other than ordinary numerical (mathematical) induction. Finally, having explicitly defined ‘point’ and ‘line’, we will derive the characteristic Parallel’s Postulate (Playfair axiom) from regions-based axioms, and point the way toward deriving key Euclidean metrical properties.

16

The mereology of structural universals

75%

Forrest P.

Logic and Logical Philosophy

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2016

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vol. 25

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issue 3 Mereology and Beyond (II)

259-283

EN

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.

17

Dynamic relational mereotopology: Logics for stable and unstable relations

75%

Nenchev V.

Logic and Logical Philosophy

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2013

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vol. 22

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issue 3

295-325

EN

In this paper we present stable and unstable versions of sev- eral well-known relations from mereotopology: part-of, overlap, underlap and contact. An intuitive semantics is given for the stable and unstable relations, describing them as dynamic counterparts of the base mereotopo- logical relations. Stable relations are described as ones that always hold, while unstable relations hold sometimes. A set of first-order sentences is provided to serve as axioms for the stable and unstable relations, and representation theory is developed in similar fashion to Stone’s representation theory for Boolean algebras and distributive lattices. Then we present some results about the first-order predicate logic of these relations and about its quantifier-free fragment. Completeness theorems for these logics are proved, the full first-order theory is proved to be hereditary undecidable and the satisfiability problem of the quantifier-free fragment is proved to be NP-complete.

18

Spheres, Cubes and Simplexes in Mereogeometry

75%

Borgo S.

Logic and Logical Philosophy

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2013

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vol. 22

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issue 3

255-293

EN

In 1929 Tarski showed how to construct points in a region-based first-order logic for space representation. The resulting system, called the geometry of solids, is a cornerstone for region-based geometry and for the comparison of point-based and region-based geometries. We expand this study of the construction of points in region-based systems using different primitives, namely hyper-cubes and regular simplexes, and show that these primitives lead to equivalent systems in dimension n ≥ 2. The result is achieved by adopting a single set of definitions that works for both these classes of figures. The analysis of our logics shows that Tarski’s choice to take sphere as the geometrical primitive might be intuitively justified but is not optimal from a technical viewpoint.

19

Franz Brentano’s Mereology and the Principles of Descriptive Psychology

75%

Curvello F. V.

Dialogue and Universalism

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2016

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issue 3

109-123

EN

I analyse Brentano’s argumentative strategy from his lectures in the Deskriptive Psychologie and how he introduces and reframes his fundamental psychological theses. His approach provides us with the reasons why psychology can be distinguished into different domains of investigation and how the tasks of one of these domains—the de-scriptive-psychological one—imply a specific understanding about the structure of consciousness. Thereby a mereology of consciousness is developed, which offers the theoretical background to the aforementioned reframing of the Brentanian theses.

20

Abelian mereology

75%

Cotnoir A. J.

Logic and Logical Philosophy

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2015

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vol. 24

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issue 4 Mereology and Beyond

429-447

EN

In classical extensional mereology, composition is idempotent: if x is part of y, then the sum of x and y is identical to y. In this paper, I provide a systematic and coherent formal mereology for which idempotence fails. I first discuss a number of purported counterexamples to idempotence that have been put forward in the literature. I then discuss two recent attempts at sketching non-idempotent formal mereology due to Karen Bennett and Kit Fine. I argue that these attempts are incomplete, however, and there are many open issues left unresolved. I then construct a class of models of a non-idempotent mereology using multiset theory, consider their algebraic structure, and show how these models can shed light on the open issues left from the previous approaches.

Refine search results

Mereology and Infinity

Mereology then and now

Set-theoretic mereology

Sequents for non-wellfounded mereology

Notes on models of first-order mereological theories

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

Composition, identity, and emergence

Notes on models of first-order mereological theories

Mereology and truth-making

Tractarian Ontology: Mereology or Set Theory?

Czy monady mają części? Witkiewicz i jego krytyka mereologii jako ontologii

Composition as identity and plural Cantor's theorem

The Notion of the Diameter of Mereological Ball in Tarski's Geometry of Solids

More on The Decidability of Mereological Theories

Regions-based two dimensional continua: The Euclidean case

The mereology of structural universals

Dynamic relational mereotopology: Logics for stable and unstable relations

Spheres, Cubes and Simplexes in Mereogeometry

Franz Brentano’s Mereology and the Principles of Descriptive Psychology

Abelian mereology