DS-2012-10:
Meyers, Jeremy
(2012)
*Locations, Bodies, and Sets: A model theoretic investigation into nominalistic mereologies.*
Doctoral thesis, University of Amsterdam.

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## Abstract

Mereology, as a form of philosophical or applied research, begins with the

assumption that objects have parts. Cars, people, planets, and galaxies are

organized part-to-whole. Indeed the entire range of concrete entities will be

conceived as having a certain decompositional makeup. The study of mereology is

an attempt to discern this structure and to formalize these notions in a

regimented theory.

The theory of nominalism has always maintained close ties with formal mereology.

Nominalism is the view that abstract objects do not exist and that

spatiotemporal objects exhaust the domain of existing things. Formal mereology

evolved from attempts to provide a system sufficiently powerful to supplant set

theory as a foundation for mathematics without abstract references of any kind.

Ambitions such as these continue to the present in mereology and metaphysics

more generally. Thus two important questions emerge concerning the feasibility

of nominalistic mereology. Firstly, could it be, despite the assumptions of the

project, that any such system unavoidably will contain references to abstract

entities? And secondly, supposing we could erect such a system, how much of

reality's decompositional structure could any nominalistically acceptable formal

mereology capture? Ultimately, I claim that there can be no such thing as a

nominalistic formal mereology in the sense envisaged. Moreover any remotely

acceptable system will fail to capture the entire part-to-whole structure of

concrete objects.

In the dissertation, we encounter three major problems with the conception of a

nominalistic mereology. The first failure concerns the status of the parthood

relation itself. I argue that, although the parthood relation may be understood

in some sense as an in re universal or spatiotemporally located entity, it is

clearly not a particular concretum or trope of any kind. It must be multiply

located and repeated wholly amid its relata. The second failure concerns the

conceptions required to represent the cohesiveness of physical objects. A

universe is not merely a mereological whole. For its dimensional parts are

interconnected in complex ways. Either a formal topology or mereology with

connectedness predicates will be required to represent the topological

properties of concrete objects. And these will entangle us in commitments to

set-theoretic constructions. Finally, nominalistic systems will be far too

weak. To demonstrate this we take pains to select a language which is maximally

acceptable. But we find that the richness of infinite spatial structures exceeds

our ability to capture them in any first-order theory.

Ontology. We first identify what ontological distinctions a reasonably

expressive language must be able to make. An assumption of nominalism will imply

that sets be rejected in favor of extensional mereological fusions. Only

concrete entities must be assumed to exist. Among these are so-called locations

which I define as fusions of either material or material-free substances.

Locations are extensional in the mereological sense and are closed under

unrestricted fusions.

The existence of movements on the part of persons and motions of inanimate

objects imply that subparts of reality have less dimensions than that of the

entire system. Persons are observers figuring in a multitude of localized

mereological arrangements. And they are capable of enduring changes in their

proper parts. Although it might be thought that conceiving of reality in this

way supersedes a purely nominalistic account or falls outside the pales of

formal metaphysics, I claim, based on features of our relation to our bodies,

that some such account must be adopted.

Perhaps ironically, a view of the physical world as a comprised of situated

persons provides a way to obviate explicit commitment to the topological

properties and sets required to represent the interconnectedness of physical

universes. Persons have intrinsically interconnected locations within a single

spatiotemporally closed universe. Hence we arrive at a view that the objects

postulated by nominalism are those connected via locations to our bodies.

The status of the parthood relation as multiply located entails that our

nominalist accept some notion of mereological state of affairs. A maximally

nominalistic ontology will therefore consist of concrete individuals and

mereological arrangements involving them. Some states of affairs are localized

and obtain at various sub-locations of reality, but others will hold regardless

of one's immediate location. I suggest that the distinction between localized

and non-localized situations helps to explain issues related to time,

simultaneity at a distance, and tense.

Mereologic. Having provided a maximally nominalistic ontology, we can then turn

our attention to defining formal notions and modeling reasoning over the

selected domain. Our pilot system is a modal logic of mereology tailored

precisely to the ontology. We employ a hybrid modal language. Hybrid languages

are extensions of standard modal languages in which references can be made to

individual objects by so-called `nominals'. The latter are atomic formulae

functioning like constants in the first-order language. We adopt an extension

H_m of Arthur Prior's nominal tense language with additional operators for

various part and extension relations. Although expressively weak in comparison

to first-order mereologies, it is shown formally that H_m is capable of denoting

nominalistically acceptable states of affairs. Given the modal nature of hybrid

languages, both localized and non-localized types of situation are

representable. Formulas are evaluated relative to a particular location. But

there are, in addition, those which "lift" the interpretation to a global

perspective.

Each formula of H_m is shown to represent an acceptable state of affairs

relating individuals part-to-whole. In nominalistic spirit, arithmetical

features and principles are thereby eliminated. In contrast to first-order

systems, in H_m, counting expressions are undefinable and arithmetical facts

hold only over distinguished objects. As for logics, we provide axiom systems

and demonstrate the existence of various mereologics for classes of extensional

mereological structures. In a novel Henkin construction, I demonstrate an axiom

system analogous to Leonard and Goodman's "General Extensional Mereology" is

complete with respect to the traditional classes of partial orders up to

zero-deleted Boolean algebras. General completeness results for varieties of

infinite atomic and atomless Boolean algebras are also demonstrated.

Morphisms germane to modal logic are of equal importance in formal mereology.

They allow us to gauge precisely the structural details seized by our adopted

languages. Essentially, mereological reasoning is encapsulated in the notion of

what I call mereobisimulation---a morphism stronger than bisimulation but weaker

than a strong homomorphism. And I situate mereo-reasoning with its targeted

nominalistic restrictions in relation to first-order logic: it is exactly the

mereo-bisimilar fragment of the first-order logic.

Can H_m detect the subtle differences between distinct parts of reality? I argue

that the best way to answer this is first to identify suitable models

representing the structural features we wish to preserve - in particular those

that represent the decomposition of space. Then one proceeds to test how much

structure the language can ``see'' of them. Two mathematical models are

indistinguishable by H_m-formulas if there is a mereo-bisimulation amid them.

Thus if H_m detects no differences between two models - one which has the

structural features of locations and another which clearly does not - then, a

fortiori, H_m will fail to capture the corresponding structural details in

reality.

Well-known, adequately proved results in the theory of Boolean algebras indicate

that certain mathematical structures called complete Boolean algebras have the

requisite features of the structure of unrestrictedly fused locations. Taking

some results proven by Tarski and MacNeille in the thirties for granted, I show

that any infinite n-dimensional atomless or atomic Boolean algebra expanded with

a finite distinguished elements is mereo-bisimilar to its corresponding Boolean

completion. In particular, I show that there is a sound and complete proof

system for the class of regular open sets of R^n for finite n and the class of

infinite atomic complete Boolean algebras.

Our answer to the second question can then be summarized as follows. If reality

contains infinitely many locations, then we will lose the ability to

discriminate between uncountably many of them. Indeed if there are infinitely

many locations and these decompose to a floor of atoms, single H_m-formulas will

conflate reality with a finite structure. If, however, there are infinitely many

locations and some of these contain no atoms (or if all are completely

atomless), then up to mereo-bisimulation, portions of reality will be conflated

with "pixelated" or geometrically extended, unanalyzable objects. Crystallizing

on the structure of an infinite dimensional system will therefore be impossible.

In conclusion, I urge that a formal language for mereology should not be

restricted on nominalistic grounds. We should be inclined, despite any

reluctance, to incorporate terms for sets and set-quantifiers.

Keywords:

Item Type: | Thesis (Doctoral) |
---|---|

Report Nr: | DS-2012-10 |

Series Name: | ILLC Dissertation (DS) Series |

Year: | 2012 |

Subjects: | Language Logic |

Depositing User: | Dr Marco Vervoort |

Date Deposited: | 14 Jun 2022 15:16 |

Last Modified: | 14 Jun 2022 15:16 |

URI: | https://eprints.illc.uva.nl/id/eprint/2116 |

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