M.A. Thesis

Jeffrey Grupp, 2003

Western Michigan University

SOME QUESTIONS ABOUT THE INTERCONNECTEDNESS AND INTERRELATEDNESS OF ENTITIES

Jeffrey Grupp, M.A.

Western Michigan University, 2003

Relations pervade the theories of analytic metaphysics: philosophy of mind, philosophy of region, philosophy of causation, philosophy of math, philosophy of space and time, philosophy of physics, and theories of objects (bundle and substance theories). Many of the sorts of relations that (are alleged to) exist, according to these theories, are relations between or among non-collocated spatial entities (entities that do not occupy the same spatial region or regions), and between or among non-identical basic units of space. I argue that relations between or among any non-collocated spatial entities, and between or among non-identical basic units of space, do not exist: if any entities in space are not at the same spatial location, they do not have any sort of interconnection or interrelation.

SOME QUESTIONS ABOUT THE INTERCONNECTEDNESS AND INTERRELATEDNESS OF ENTITIES

by

Jeffrey Grupp

A Thesis

Submitted to the

Faculty of The Graduate College

in partial fulfillment of the

requirements of the

Degree of Master of Arts

Department of Philosophy

Western Michigan University

Kalamazoo, Michigan

December 2003

© 2003  Jeffrey Grupp

ACKNOWLEDGEMENTS

I am extremely grateful to Quentin Smith, Bill Vallicella, and John Dilworth for helpful comments on the writing to this paper.

I would like to thank my wife. If it were not for her understanding that passion and bliss are important aspects of life, this project never would have been conceived.

                 Jeffrey Grupp

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TABLE OF CONTENTS

ACKNOWLEDGEMENTS   ii

INTRODUCTION   1

SPATIALLY LOCATED RELATIONS BETWEEN NON-COLLOCATED ENTITIES   3

Complex and Non-complex Relations, and Basic Units of Space   4

Spatially Located, Spatially Extended, Non-complex Relations   5

Spatially Located, Spatially Unextended, Non-complex Relations.   8

A Complex Relation as a Continuum of Non-complex Relations, Part 1   13

A Complex Relation as a Continuum of Non-complex Relations, Part 2   16

SPATIALLY UNLOCATED RELATIONS BETWEEN NON-COLLOCATED SPATIAL ENTITIES   20

Platonistic Interrelating Between Spatial Entities   20

Realm Crossing Exemplification: Relations are not Realm Crossers   28

Realm Crossing Relations and Unmediated Attachments   33

Objections to the Concept of Realm Crossing   36

OBJECTIONS   40

Objection 1: Statements, Such as “p₁ is taller than p₂”, Cannot Be Necessarily False   40

Objection 2: Absolute Diversity   43

CONCLUSION   45

NOTES   45

BIBLIOGRAPHY   53

1

INTRODUCTION

In this paper, I am concerned with any sort of relation, connection, or relatedness (alleged to exist) between or among any non-collocated spatial entities, and between or among any non-identical basic units of space.[1] (By “non-collocated”, I mean “not occupying the same region or regions of space, not occupying the same basic unit or units of space”.) I discuss hitherto unnoticed problems about all varieties of relations (platonistic, physicalistic, etc.) between or among any non-collocated spatial entities and between or among any non-identical basic units of space; and I discuss problems to do with monadic relatedness possessed by non-collocated spatial entities or non-identical basic units of space[2].

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between (physicalist, topological, platonistic, etc.) entities in a region larger than a basic unit of space. I do not discuss relations an entity may have with itself (loves oneself, etc.)[4]. Also, I do not discuss relations between or among collocated spatial entities, if there are any[5]. I only discuss that if spatially located entities do not occupy the very same basic units of space, or if basic units of space are non-identical, such objects or basic spatial units are unconnected.

I will argue that there is a specific problem to do with any sort of relation between or among non-collocated spatial entities and non-identical basic units of space: such relations are neither spatially located (S) nor spatially unlocated (~S). In relating non-collocated spatial entities and non-identical basic units of space, relations must be either located in space, or located outside of space. If relations between or among non-collocated spatial entities or non-identical spatial points were found to be neither of these, then relations between or among non-collocated spatial entities would apparently be contradictory, since they would be describable as being neither spatially located nor spatially unlocated, ~(S v ~S), which translates to ~S ^ S.

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must be spatially unlocated. To show this, I will consider the thesis (toward reductio) that there are spatially located relations between or among (or that there is monadic relatedness possessed by) non-collocated spatial entities or non-identical basic units of space. In section 3, I consider spatially unlocated relations (relations not in space) among non-collocated spatial entities, and among non-identical basic units of space. In that section, I also come to serious problems when considering them.

In section 4, I explore some of the objections readers may have with the argumentation in sections 2 and 3; and I explore some of the objections readers may have with the concept that reality is devoid of relations or relatedness of any sort. This paper is not about what reality is like if the reasoning I give in sections 2 and 3 is correct and relations do not exist; I do not offer a “replacement metaphysics”[7]. Rather, my goal in this work is specifically to discuss hitherto unnoticed problems to do with relations.

SPATIALLY LOCATED RELATIONS BETWEEN NON-COLLOCATED ENTITIES

4

non-complex relations, and I will discuss a few issues to do with the basic units of space.

Complex and Non-complex Relations, and Basic Units of Space

5

space, are not analyzed in detail greater than this, and there is very little analysis in the literature of the precise details of, and the specific nature of, relations, that goes further than this.

My arguments in this paper do not depend on whether spatially extended, or point-size, basic entities compose space[11]. If the basic units of space have non-zero size (such as the size of a “Planck length”)[12], then the statement, “a relation between non-identical basic units of space”, would denote a relation between or among two or more non-identical basic units of space. An extended continuum of spatial points (or an extended continuum of matter points), also consists of relations between non-collocated basic spatial units (between non-identical spatial points), since points in a continuum are not immediately next to other points. In either the case of spatially extended, or spatially unextended, basic units of space, although the interrelating between basic entities is very different, the relations between or among entities are relations between or among non-collocated basic entities.[13]

I will next turn to my arguments against spatially located relations between or among basic units of space, and between or among non-collocated spatial entities.

Spatially Located, Spatially Extended, Non-complex Relations

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of one of the lions (p₁=paw, p₂=lion), or p₁ and p₂ could be basic units of space. In such cases, the relations between p₁ and p₂ are the relations, brotherhood and parthood, and the last I will call is topological connectivity. These are the examples of spatially located relations I will use throughout this paper. (Any others could have been used.) In the case of the relation, parthood, the paw (p₁) is not collocated with all of the lion, such as where the lion’s brain and mane are. And for this reason, parthood is a connection of non-collocated spatial entities, since the whole and parts do not exactly collocate. In this paper, I am only concerned with part-whole relations in which the relation is a connection between or among non-collocated spatial entities.

In this subsection, I discuss non-complex relations that, in connecting p₁ and p₂, are in some way-or-another in-between p₁ and p₂.[14] It appears that there are two ways to conceptualize a relation, if the relation is located in space. (a) A spatially located relation resembles an ordinary material object, such as a rope that connects a boat and a dock. The second option is: (b) a spatially located relation does not resemble an ordinary material object and is not between entities. In this subsection, I discuss the first scenario, and in subsection 2.3, I discuss the commonly-held position, where spatially located relations are considered spatially unextended entities that do not resemble ordinary material objects.

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The issue of whether relations are spatially extended is an issue rarely discussed in the literature, if at all, since relations are typically considered to be spatially unextended: relations are considered to be either spatially unlocated (and for that reason, spatially unextended), or, when relations are considered spatially located, they are also considered spatially unextended in that scenario. But I am going to discuss spatially extended relations just to cover all the possibilities there might be.

I will now give an argument against spatially located non-complex relations between non-collocated spatial entities, such as p₁ and p₂.

Any spatially located non-complex relation between non-collocated spatial entities is a relation that, by being non-complex, is fundamental and irreducible: partless, primitive, and not analyzable in terms of simpler parts (sub-relations). Such non-complex spatially located relations, being partless, have a non-zero spatial size: they occupy an unbroken extent in space. Being of non-zero spatial size, such relations connect or occupy at least two non-identical spatial locations, for the remainder of this subsection, call them x and y. If non-complex, spatially located relations between non-collocated spatial entities, or between non-identical basic units of space, occupy at least two non-identical spatial locations, then non-complex spatially located relations between non-collocated spatial entities are apparently contradictory, for the following reasons.

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property (second-order property), then the whole relation has the property. For example, of the account just given, the entire relation would have the properties, located at x, and located at y. If the relation is located at y, and if y≠x, then by also being at x, the non-complex immanent relation has the property not located at y. This could be said of any non-y location that the immanent non-complex relation occupies, such as if the relation were larger than two basic units of space, and located at three spatial locations, x, y, and z. At locations x and z, the relation would have the (second-order) property, not located at y. These are, however, properties the relation cannot have: since the relation is one, partless entity, if it is located at y, and not located at y, each of these (second-order) properties must describe the entire partless and spatially located relation, and that implies the relation would be describable as having self-contradictory properties, located at y and not located at y

Spatially Located, Spatially Unextended, Non-complex Relations.

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complex interrelation of non-collocated spatial entities, or of non-identical basic units of space, is itself located in space, but is spatially unextended, since it is located where and only where p₁ and p₂ are. On this account, the spatially unextended, spatially located, non-complex interrelation of p₁ and p₂ can, it seems, be considered primitive; but I will discuss that my arguments in this section show serious problems to do with non-complex (or any other sort of) spatially located relation (and to do with any sorts of spatially located monadic relatedness) regardless if it is primitive or not. I will next argue that this position is also seriously problematic.

First I will consider the scenario where the relation, parthood, connects non-collocated spatial objects, p₁ and p₂, where p₁=paw, and p₂=lion. p₁ and p₂ involve connections among -non-collocated spatial entities, since pieces of p₂ are non-collocated with all of p₁. p₂ (lion) collocates with p₁ (paw) at p₁’s spatial locations, but p₁ does not collocate with many of p₂’s spatial locations, such as where the lion’s heart, brain, or mane are. For these reasons, the relation, parthood, connects non-collocated entities, and my argument below only focuses on the connections among the non-collocated aspect of a whole and its parts.

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immanent relation’s spatial locations of co-exemplification not at a_n, then the relation does not have anything to do with p₁ (p₁ is not one of its relata). p₂, being a spatially located entity, also cannot fail to be at a spatial location; call p₂’s location, the collection of basic units of space, b_n. This implies that p₂ only participates in the co-exemplification of n-adic properties at b_n and nowhere else, since spatially located object p₂ is nowhere else but at b_n. If one of immanent relation’s spatial locations of co-exemplification not at b_n, the relation does not have anything to do with p₂ (p₂ is not one of its relata).  

These restrictions imply that p₁ and p₂ could not be interrelated at the spatial locations that they are not collocated at. If p₁ is only at a_n, and if p₂ is only at b_n, and if many of p₂’s spatial locations are not identical to p₁’s spatial locations (they are not identical since if a_n Ì b_n, then a_n ≠ b_n)[15], and if and on this account the spatially located interrelation of p₁ and p₂ is not being considered as spatially between p₁ and p₂, then at those spatial locations where p₁ and p₂ do not collocate, p₁ and p₂ apparently cannot have any sort of dealings with one another (such as being interrelated by the relation, parthood). It appears that in order for p₁ to, for example, participate in the co-exemplification parthood with p₂, p₁, which is wholly at a_n, must also be at all of b_n’s spatial locations, and thus must apparently take on characteristics that are self-contradictory.

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it is, and nowhere else, since basic unit of space p₁ nowhere but p₁ If one of immanent relation’s spatial locations of co-exemplification not at p₁, then the relation does not have anything to do with p₁. If basic unit of space p₂ is a spatial location, then p₂ only participates in the co-exemplification properties where p₂ is and nowhere else, since basic unit of space p₂ is only where it is, and for that reason is not at p₁. If one of immanent relation’s spatial locations of co-exemplification not at p₂, then the relation does not have anything to do with p₂.

These restrictions imply that any non-identical basic units of space, such as p₁ and p₂, could not be interrelated, for the following reasons. Since p₁ ≠ p₂, and since on this account the spatially located interrelation of p₁ and p₂ is not being considered as spatially between p₁ and p₂, then p₁ and p₂ apparently cannot have any sort of dealings with one another (such as being interrelated by a spatially located relation, connectivity). It appears that in order for p₁ to, for example, share (co-exemplify) a spatially located relation with p₂, p₁ must be also be p₂, and thus must apparently take on characteristics that are self-contradictory. Similarly, in order for p₂ to share a spatially located relation with p₁, p₂ must also be p₁, and thus must apparently take on characteristics that are self-contradictory.

If my reasoning in this sub-section is correct, it is apparently the case that any non-platonistic, spatially located relations cannot account for any connection or relatedness among non-collocated spatial entities, or non-identical basic units of space, if my reasoning in this section is correct.

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 Some may argue that spatially located relations (or spatially located monadic relatedness) is primitive, and for that reason, this entire paper is unnecessary. But the arguments of this subsection apparently reveal that any spatially located relation or relatedness between non-collocated spatial entities, or between non-identical basic units of space, are contradictory, regardless of any alleged primitivism involved with spatially located relations.

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spatial locations, since, if they are non-collocational, p₂ cannot be involved in any way with p₁’s monadic property, connected to p₂ (which is a property that is only at p₁). If this reasoning is correct, monadic relatedness apparently cannot account for a relatedness possessed by p₁, connected to p₂, since such relatedness cannot involve both p₁ and p₂.

A Complex Relation as a Continuum of Non-complex Relations, Part 1

Since non-complex relations make up complex relations, the reasoning of the previous subsection, if correct, apparently shows that spatially located non-complex relations between non-collocated spatial objects are contradictory. But perhaps there are other sorts of non-complex spatially located relations that make up specific sorts of spatially located complex relations, that need to be considered. In the rest of the section, I will discuss a few remaining sorts of complex and non-complex spatially located relations that may not be susceptible to the problems discussed so far. I will come to serious problems with each.

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space) that the non-complex has on that scenario. In the later scenario (unextended immanent relations), the problems apparently draw from the relation not being between the connected non-collocated spatial entities or basic units of space. But perhaps another sort of extended, spatially located relation could be considered: a spatially located, complex, extended relation between non-collocated spatial objects, and between non-identical basic units of space, that is composed of an extended continuum of point-size, non-complex, spatially located sub-relations between p₁ and p₂. This is not a relation that I have seen discussed often in the literature, other than for specific scenarios[16]. I will next discuss reasons why such a relation apparently cannot constitute any immanent complex relation between non-collocated spatial entities between, or between non-identical basic spatial units.

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where the “completion” “at infinity”, some might claim, in fact never occurs, since an infinite set of items has no last item. Chisholm considers this sort of regress vicious; Moreland lucidly writes about Chisholm’s position: 

There are at least three forms of infinite regress arguments… [One form] involves claiming that a thesis generates a “vicious” infinite regress. How should “vicious” be characterized here?... Roderick Chisholm says that “One is confronted with a vicious infinite regress when one attempts a task of the following sort: Every step needed to begin the task requires a preliminary step”. [Chisholm, 1996, p. 53.] For example, if the only way to tie together any two things whatever is to connect them with a rope, then one would have to use two ropes to tie the two the two things to the initial connecting ropes, and use additional ropes to tie them to these subsequent ropes, and so on. According to Chisholm, this is a vicious infinite regress because the task cannot be accomplished.[17] (Emphasis added.)

Phillips also straightforwardly discusses the problem involved in this sort of regress:

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relation is a different sort of critter, it seems that if a third thing is required to relate two things, then the third thing requires equally a fourth and fifth to tie it up with the first two, ad infinitum. The regress is vicious: unlike an infinite series of causes that does not undermine the notion that a preset x has y as its cause, the relation regress does undermine the work proposed for the relator. The relator, the third thing, cannot relate the two items without help form the fourth and fifth things (ad infinitum) needed to tie it up with the first two.[18] (Emphasis added.)

A Complex Relation as a Continuum of Non-complex Relations, Part 2

Some philosophers consider infinities to involve paradoxes, and for that reason, they make a point to avoid infinities when describing physical collections. But others may object to such a position, and may object to the reasoning given in the last section, claiming that physical infinities can exist, and there is no problem in considering a physical collection to have a cardinality that is infinitely large. These philosophers might consider examples of such collections to be the collection of spatial locations, the collection of time-instants, or, perhaps, the collection of sub-relations constituting an extended complex immanent relation that connects p₁ and p₂, as described in the last subsection.

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An extended continuum of point-size sub-relations resembles an extended continuum of points. In comparing this sort of complex relation between p₁ and p₂ to topological space, both the complex relation composed of sub-relations, and topological space, each consist of À₁ spatially unextended, spatially located, but spatially non-collocated objects, that are considered to give rise to an extended entity. For these reasons, hereafter I will consider the complex relation that is composed of continuum many sub-relations, and that I am currently discussing, to be a complex relation that is a continuum of sub-relations.

Points in a continuum do not directly contact one another, since any point in a continuum is not immediately next to any other points. This reasoning would apply to an extended continuum of spatially located point-size sub-relations extending between p₁ and p₂: none of the sub-relations are immediately next to one another. For this reason, a complex relation merely composed of point-size sub-relations cannot give rise to a complex relational connection between non-collocated entities p₁ to p₂. If the complex relation between p₁ and p₂ is only composed of point-size sub-relations, the complex relations fails to give rise to a connection between p₁ and p₂.

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entities. If so, perhaps the reasoning of the previous paragraph, where sub-relations were considered to be the only constituents of a continuum, is misguided[20]. Instead of discussing the sub-relations as directly attached to one another (which is impossible), the sub-relations instead should be considered to as interconnected by a spatially located topological relation, call it connectedness (or connectivity), which is perhaps analogous topological accounts of connectedness of material or spatial points in the spatial manifold, or in a continuum of matter points, and which is an immanent relation between or among the continuum-many spatially located sub-relations.

If a continuum is extended and interconnected, since the point-size items of the continuum cannot account for the interconnectivity (or extension) of the continuum, there are two constituents of the complex relation between p₁ and p₂: (a) the collection of spatially unextended, spatially located sub-relations, and (b) a topological relation, connectedness, between or among the sub-relations. Considering points (point-sets) as connected (related) in neighborhoods or unions (some topologists might denote this interrelatedness with the words, “nearness”[21], “closeness”, or “connectivity”) is standard among topologists, since topology is concerned with structures that are composed of points and relations between points[22] (or what are often called point sets).

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I will next argue that a spatially located topological relation between or among the point-size sub-relations of the complex relation between p₁ and p₂ cannot connect the sub-relations.

Consider the following issues.

1. Since none of the spatially located point-size sub-relations are immediately next to one another, the topological relation, connectedness, between or among the points, is a relation between or among non-collocated spatial entities.
2. If the connectedness of the continuum of sub-relations were itself also a continuum of point-size sub-relations, it too would consist of continuum-many sub-relations that are disconnected (not directly attached, not immediately next to one another). If the connectedness between the point-sized sub-relations were also composed of point-sized sub-relations, the connectivity would itself provide no continuous connection between the non-collocated sub-relations of the complex relation between p₁ and p₂.
3. If connectivity is a connection between or among the non-collocated sub-relations (point 1 above), and if the connectivity is not a continuum of point-size sub-relations (point 2 above), in order to interrelate the sub-relations, the connectivity relation apparently must be a spatially located non-complex relation between non-collocated sub-relations. But this is exactly the sort of relation found to be contradictory in subsection 2.2.

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Given (3), a topological connectedness among continuum-many sub-relations of the complex relation connecting p₁ and p₂ is apparently contradictory.

If my reasoning is correct, any spatially located relation between or among non-collocated objects, or between or among basic units of space, or between or among point-size sub-relations of a complex relation, apparently cannot be a spatially located non-complex relation, nor a continuum of non-complex sub-relations. Given these problems, the only way (that I can think of) out of these dilemmas is to espouse an account of relations where relations are not in space (they are platonistic).

SPATIALLY UNLOCATED RELATIONS BETWEEN NON-COLLOCATED SPATIAL ENTITIES

Platonistic Interrelating Between Spatial Entities

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platonic realm). Considering platonistic relations as spatially unlocated is the standard position on platonia. Loux discusses this:

What are the issues separating the Aristotelian realists from Platonists? … Aristotelians typically tell us that to endorse Platonic realism is to deny that properties, kinds, and relations, need to be anchored in the spatiotemporal world. As they see it, the Platonist’s universals are ontological “free floaters” with the existence conditions that are independent of the concrete world of space and time. But to adopt this conception of universals, Aristotelians insist, is to embrace a two-worlds” ontology… On this view, we have a radical bifurcation of reality, with universals and concrete particulars occupying separate and unrelated realms… [T]here [is a] connection between spatiotemporal objects and being completely outside of space and time.[23]

Next I will argue that a platonistic account of relations is a problematic account of the interrelatedness of non-collocated spatial objects, and of non-identical basic units of space. I will not argue against the existence of spatially unlocated objects, nor will I argue for physicalism. Rather, I will argue that any sort of connection between physical objects and spatially unlocated platonic objects has serious problems.

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The exemplification of relations by p₁ and p₂, in the scenario where relations are spatially unlocated, is the platonistic exemplification tie, which is considered by some to be a “realm crossing tie”. The platonistic exemplification tie connects entities in the spatially unlocated platonistic realm (where the relations, connectivity or parthood, are) to entities in the spatial realm (where p₁ and p₂ are). I borrow the phrase “realm crossing” from one of D. M. Armstrong’s passages where he discusses platonistic exemplification (but where he refers to it as the instantiation relation) between or among spatially unlocated entities (platonic universals) and spatially located entities (platonistic thin particulars[24]):

Once you have uninstantiated spatially unlocated] universals you need somewhere to put them, a “Platonic heaven,” as philosophers often say. They are not to be found in the ordinary world of space and time. And since it seems that any instantiated universal might have been uninstantiated… then if uninstantiated universals are in a Platonic heaven, it will be natural to place all universals in that heaven. The result is that we get two realms: the realm of universals and the realm of particulars, the latter being ordinary things in space and time… Instantiation then becomes a very big deal: a relation between universals and particulars that crosses realms.[25] (Emphasis added.)

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I will discuss an objection to the concept of realm crossing in subsection 3.4.

In this section I will argue that there may be a problem involved with such “realm crossing” ties[26]. If I am correct, and if the problem is serious enough, a spatially unlocated platonistic relation cannot interrelate non-collocated spatial entities or non-identical basic units of space (p₁ and p₂ cannot be platonistically interrelated). 

Before discussing the realm crossing relation, I will discuss how I use the terms “exemplification tie” and “unmediated attachment”, which are terms relevant to the discussion of any platonistic interrelation of non-collocated spatial entities.

There are two types of realm crossing between spatially unlocated entities and spatially located platonistic thin particulars.

1. A realm crossing exemplification tie, which is an intermediary connecting a spatially located platonistic thin particular (the thin particularity of p₁ or p₂) and the spatially unlocated platonistic n-adic properties (properties such as, relatedness, or relations, such as, connectivity or parthood).

2. A realm crossing unmediated attachment, which the spatially located platonistic thin particular and the exemplification tie are involved in, and which a spatially unlocated platonic universal and the exemplification tie are involved in.

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Let “realm crossing exemplification tie” denote what is denoted by spatially located entities p₁ and p₂ “exemplify” R, or p₁ and p₂ “have” the polyadic property, R (R is a platonic universal). The realm crossing exemplification tie is the entity between the spatially located entities and the spatially unlocated platonistic entity.

The exemplification tie is not merely the unmediated attachment of a property with platonistic thin particular. Rather, the exemplification tie is an additional entity, in addition to the property and particular, which connects the platonistic thin particularity to the spatially unlocated universal. If exemplification were not a third entity involved, distinct from the property and thin particular, in the scenario where a particular having a property, a Bradley-esque regress would ensue. (I discuss this much more in paragraphs below.) Some might consider that exemplification is merely the very tying (unmediated attachment) of property directly to particular, but Bradley’s work showed that such tying is viciously regress, whereby a non-relational exemplification is needed in order to avoid the regress. When Loux mentions that exemplification is a “nexus”, his word choice is a good one since “nexus” clearly denotes how exemplification is a bridging intermediary between property and particular, distinct from property and particular, which keeps property and particular from being involved in an unmediated attachment, whereby a Bradley-esque regress would ensue.

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Exemplification is an intermediary between entities, and is the opposite scenario of unmediated attachment. Let “unmediated attachment” express the concept of an attachment which spatially located entities, and spatially unlocated platonistic entities, are involved in, and which does not involve an intermediary. An unmediated attachment is not a relation between entities, and it does not involve non-relational ties, or any sort of entity that is between the attached entities.

Unmediated attachment is normally how exemplification is conceived to attach to a property and to the platonistic thin particular. The concept of unmediated attachment comes from responses to F.H. Bradley’s work on the paradox of the relations regress. Loux lucidly explains:

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Unfortunately, exemplification₂ is itself a further relation, so that we need a still higher-level form of exemplification (exemplification₃) whose role it is to insure that a, F-ness, and exemplification are related by exemplifiaction₂; and obviously there will be no end to the ascending levels of exemplification that are required here. So it appears… that the only way we will ever secure the desired result that a is F is by denying that exemplification is a notion to which the realist’s theory applies.

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Exemplification is a non-relational tie or nexus[28] between or among properties and platonistic thin particulars, or between or among properties and other properties. Exemplification is not related to the relation (connectivity, parthoood) or the non-collocated spatial entities (p₁ and p₂); and exemplifi