Trope Theory Meets Bradley's Regress

This is the paper I am scheduled to present at the Bradley Conference at the University of Geneva in early December, warts and all. No doubt it needs more work.  So comments and criticism are welcome.

Trope Theory Meets Bradley's Regress

William F. Vallicella

1. Introduction

One of the perennial tasks of ontology is that of analyzing a thing’s having of properties. That things have properties is a datum consistent with different theories as to what properties are, what the things are that have the properties, and how best the having is to be understood. Any theory will have to provide a three-fold answer to this three-fold question. In so doing, it must show how the elements it distinguishes fit together to form the unified phenomenon of a thing having properties. Analysis is not enough for understanding; synthesis is also needed to show how the elements separated out by analysis form a unity. One of the criteria of adequacy for any theory is whether or not it can avoid the threat to unity known as Bradley’s regress.

This paper argues that trope theory may have trouble passing the Bradley test. In particular, what it argues is that (i) trope theory requires a compresence relation to account for the difference between a unified thing and its disparate property-constituents; (ii) the compresence relation is external and therefore open to Bradleyan challenge; (iii) the various attempts to defuse Bradley’s regress are unsuccessful; hence, (iv) Bradley’s vicious infinite regress is unavoidable and trope theory in its current versions may be untenable.

2. An Outline of Trope Theory

In terms of our opening question about properties, the things that have them, and the having, we can say that trope theory is a one-category ontology that assays properties as tropes, the things that have properties as bundles of tropes, and a thing’s having of properties as compresence of tropes where the relation of compresence is itself a trope, albeit a rather special one. Commitment to a one-category ontology is part of what is here meant by ‘trope theory’; so any theory that countenances irreducible nontropes, whether universals, or bare substrata, or whatever, in addition to tropes will not count as trope theory for present purposes. Thus a theory that has tropes inhering in substrata is not a trope theory in the strict sense here in play. The same holds for a theory that construes compresence as a universal relation. Favoring this strict reading of ‘trope theory’ is the fact that this reading is in the spirit of such seminal contributors to the theory as D. C. Williams (1952, 1986) and Keith Campbell (1990).

Trope theory includes a realistic theory of properties, one that eschews both conceptualism and nominalism. Its theory of properties is a theory according to which properties are independent of language and mind. Now it is clear that a theory could be a realist in this sense without implying that properties are universals. A universal is a repeatable entity, one that is repeated in each of the things that exemplifies it. By contrast, a particular is an unrepeatable entity, one about which it would make no sense to say that it is repeated.

Consider a red tomato. Is the redness of a tomato a repeatable (multiply exemplifiable) entity, or is it as unrepeatable as the tomato itself? This is a question that cannot be answered phenomenologically, but only dialectically. That is, the phenomenology of the situation is consistent both with the theory that what one sees when one sees an expanse of red in a particular location is an (immanent) universal and with the theory that what one sees is a particular. The answer to the question is therefore a matter of theory and argument. In this sense, it is a matter of ‘dialectic.’

For the trope theorist, properties are particulars; hence the redness of a tomato is as particular as the tomato itself. Yet there is an obvious distinction between this redness and the tomato: there is more to the tomato than redness, there is ripeness, spheroidness, etc. This distinction is captured in trope theory by saying that, whereas the redness is an abstract particular, the tomato is a concrete particular. This is an etymologically correct use of ‘abstract’: when we focus on the redness of our tomato, we ‘abstract from’ the rest of its features, features without which it cannot be a concrete tomato. Talk of tropes being abstract, however, is not to be taken to imply that they are products of mental acts of abstraction. Rather, as ultimate ontological building blocks, as the alphabet of being (D. C. Williams, 1953), their existence is ontologically prior to such acts. Note also that talk of the abstractness of tropes is not to be taken to imply that they are located in a realm apart from the spatiotemporal. A redness trope is quite obviously a spatially and temporally locatable item. It is this particular redness of this particular tomato right here in front of me.

A trope, then, is an abstract particular. It is a particular in that it is unrepeatable. It is abstract in that it is only a (proper) ontological part of a concrete thing. But there are abstract particulars that are not tropes. A state of affairs such as Tom’s being red is abstract because Tom has many other properties besides redness; it is also a particular because it is unrepeatable. The same is true of the Fregean proposition Tom is red. This proposition is particular because unrepeatable; it is abstract because it is only part of the truth about Tom. (It is also abstract in the currently irrelevant sense of being nonspatiotemporal.) To distinguish tropes from states of affairs and Fregean propositions, we must say that tropes, unlike the former, are ontologically simple. A state of affairs is a complex: at a bare minimum, it consists of a thin particular and an immanent universal, and perhaps also a nexus of exemplification to tie the particular to the universal. Something analogous holds for Fregean propositions. A trope, however, exhibits no such internal ontological structure. It follows that one cannot ‘factor’ a trope into its particularity and its quality, or its thisness and its suchness. To accept tropes is to accept entities in which there is an indissoluble unity of thisness and suchness. This is perhaps a problematic idea. The focus of this article, however, is on problems surrounding compresence, not on problems having to do with tropes as such.

A trope, then, is an ontologically simple abstract particular. But we should also add that tropes, as befits their building block status, are ontologically independent: they do not depend for their existence on other entities. In particular, they are ontologically independent of the concrete wholes of which they are the abstract parts. If the trope constituents of a concrete particular C depended for their existence on the existence of C, that would be as absurd as the supposition that the stones out of which a wall is composed depend for their existence on the wall.

It follows that a trope should not be confused with a property-instance, where the latter is the result of an ontologically irreducible thick particular’s exemplifying of an ontologically irreducible transcendent universal. So defined, a property-instance obviously depends for its existence on a thick particular, a transcendent universal, and perhaps also an exemplification relation. Nor should a trope be confused with an Aristotelian accident that inheres in a substance. Tropes are in no need of a substratum for their existence. To think otherwise is to needlessly extend the scope of ‘trope,’ thereby misusing the term. Bear in mind that trope theory is a one-category ontology: it aims to explain everything in terms of tropes and nothing else. If you like, you can take that as a stipulation as to the meaning of ‘trope theory.’

It should also be observed that a trope’s simplicity is consistent with spatial and temporal extension. (Maurin 2002, p. 15) The redness of a tomato is extended in space; the cacophany of a piece of discordant music is extended in time. Some consider this problematic, and it may well be; but my focus here is not on the very idea of a trope so much as it is on the logically subsequent question of compresence. Note also that the ontological simplicity of a trope does not entail that it cannot have properties, but it does entail that any properties it has cannot be (proper) ontological constituents of it. How a trope, a simple entity, can have properties is not a question that need concern us here.

3. The Compresence Relation

If properties are tropes, then how are we to translate our ordinary talk of a property’s belonging to a thing? What is this belonging? We cannot say of a trope that it is instantiated, since tropes are particulars (unrepeatables) and no particular can be instantiated. Only universals can be instantiated. Nor can we say that tropes inhere in substrata: there are no substrata on trope theory strictly construed. Frege’s talk of objects falling under concepts is also out of place, as is any talk of subsumption. On trope theory, properties of a thing are assimilated to its parts -- albeit ontological parts. What we have to say is that trope T is a property of something if and only if T is co-occurrent with, or compresent with, sufficiently many other tropes to form a concrete particular. Whereas instantiation (exemplification) and inherence are both asymmetrical, compresence is symmetrical. If a property is instantiated by a particular, then the particular is not instantiated by the property. But if one trope is compresent with another, then the other is compresent with the one. Within trope theory, then, ordinary language talk of things having properties is analyzed in terms of the symmetrical relation of compresence. It is easy to see that compresence is also transitive and partially reflexive. If T1 is compresent with T2, and T2 with T3, then T1 is compresent with T3. And if T1 is compresent with T2, then T1 is compresent with itself.

It is clear that one cannot get by without a relation of compresence. There is obviously a difference between a bundle B of tropes and those same tropes unbundled. This difference cannot be a brute fact; it requires an ontological ground. This ground is the relation of compresence. Compresence is the ontic glue that holds B’s constituents together, thereby distinguishing B from a mere collection of disconnected elements. Compresence is what makes of a sum of tropes a unified thing.

There is also this to consider. Even if no trope is unbundled, it doesn’t follow that each trope is bundled to every other one. If that were the case, there would be only one maximal bundle, only one concrete particular. Since there is a plurality of concrete particulars, there is need of an equivalence relation to partition the class of bundled tropes into equivalence classes. This relation is compresence. Accordingly, compresence serves both a unifying and a diversifying function. It unifies tropes into bundles, but into diverse bundles so that not every bundled trope is bundled with every other one.

Now given that trope theory is a one-category ontology, an ontology according to which everything is either a trope or a construction from tropes, it follows that the indispensable relation of compresence cannot be a universal. The fundamental role that compresence plays would also seem to dictate that it cannot be a construction from tropes. Compresence must therefore itself be a trope. To be exact, compresence as it occurs in reality must be parcelled out among many tropes, many compresence-relations, each of which is an instance or case of compresence. Clearly, the compresence-relation C1 in bundle B1 that connects T1 and T2 is numerically distinct from the compresence-relation C2 in B2 that connects T3 and T4. By ‘bundle’ here, I mean a maximally consistent bundle (a maxi-bundle) which is identical to a concrete particular such as a tomato. There are at least as many compresence-relations as there are maxi-bundles.

Of course, there may be many more compresence-relations than there are concrete particulars. Suppose compresence is dyadic, or two-termed. Then if T1, T2, and T3 are all compresent in the same maxi-bundle B, then there will be at least two C-relations in B, one that connects T1 to T2, and one to connect T2 to T3. Presumably, given the transitivity of the C-relation, it will logically follow that T1 is compresent with T3 thus obviating the need in reality for a third C-relation to connect T1 to T3. That is, the compresence of T1 with T3 will supervene upon T1's compresence with T2 and T2's compresence with T3. Similarly, given that T1 is compresent with T2, T1 is compresent with itself. But there is obviously no need for a separate C-relation to tie T1 to itself: T1's compresence with itself supervenes upon T1's compresence with T2. But these special considerations are not germane to the main thrust of this article. Nothing hinges on how many compresence-relations there are. And nothing hinges on the ‘adicity’ of the compresence relation.

4. Compresence Must Be an External Relation

Whatever the ‘adicity’ of C, it seems clear that it cannot be an internal relation. But there are at least two construals of ‘internal relation.’ One sort of internality is what we may call A-internality in honor of David Armstrong:

Two or more particulars are internally related if and only if there exist properties of the particulars which logically necessitate that the relation holds. (Armstrong, vol. 2, 85)

To put it another way, an A-internal relation is one that supervenes upon, or is founded in, monadic properties of its relata. To illustrate A-internality, let R1 and R2 be two red objects, two red balls say, or two distinct redness tropes. R1 and R2 stand in the same color as relation. This relation, however, is internal in that the relatedness in question is logically guaranteed by each item’s being what it is: there is no need for a tertium quid, a third item, to relate them, whether this be a universal relation or a relational trope. In a case like this there is a relatedness without a relation as ground of the relatedness; there is connectedness without a connector. There are just the two items with their monadic properties and nothing ‘between’ them. Because there is no connector, no question can arise as to what connects the connector to what it connects: Bradley’s regress can get no purchase.

Now, could compresence be an internal relation in this sense? Suppose redness trope R1 is compresent with sweetness trope S1. If compresence is an A- internal relation, then R1's being what it is and S1's being what it is would logically suffice for their being compresent. But surely R1's being a case of redness, and S1's being a case of sweetness furnish no ontological ground of their being present with each other in the same trope-bundle any more than a cat’s being furry, and a mat’s being flat, logically suffice for the cat’s being on the mat. Therefore, compresence cannot be an A-internal relation. It cannot be founded in, or supervene upon, the monadic properties of its relata. Could it be a B-internal relation?

A B-internal relation, so named in honor of Bradley and Blanshard, is one whose relata could not have existed apart from each other and the relation that relates them. This is clearly a much stronger sense of ‘internal relation.’ Consider again the red balls. In every possible world in which both exist, and both are red, they stand in the same color as relation. But this does not entail that the balls require each other to exist. There are possible worlds in which one ball exists but the other doesn’t, and vice versa.

Now if compresence cannot be an A-internal relation, then a fortiori it cannot be a B-internal relation. Compresence is a relation that is external to its terms. As such, compresence is an entity in its own right, an addition to being and not an "ontological free lunch" to borrow a phrase from Armstrong. Given the existence of distinct tropes R and S, it does not follow that their compresence is given. There is a difference between a trope-bundle and its constituent tropes taken collectively. The whole is more than the sum of its parts. Presence + presence does not equal compresence. Something more is required, a dab of ontological glue. Compare the case of the two redness tropes. The relational fact of R1's being the same color as R2 is identical to a conjunction of monadic facts, namely, the conjunction R1 is red & R2 is red. This conjunction ‘automatically’ exists when its conjuncts exist. A trope bundle, however, does not ‘automatically’ exist when its constituent tropes exist.

The difference between a trope-bundle and its constituent tropes taken collectively requires an ontological ground. But adding a special compresence-trope C to connect ordinary tropes R and S will not do the trick: there is still the difference between the bundle C(R, S) and the mere sum, C + R + S, or the set {C, R, S}.

5. Bradley’s Regress

At this point Bradley’s regress rears its churlish head. For if C connects R and S, what connects C to R and to S? This question is unavoidable once it is appreciated that C is an external relation, a relation external to its terms, and therefore an entity in its own right, an addition to being rather than an ontological free lunch. If an additional triadic connector C* is posited to connect C to R and to S, then a regress ensues that is clearly vicious. For the same question can be asked about C*. A similar vicious regress arises if we stick with dyadic connectors throughout. Then C* connects R1 to C, C** connects C to S1, C*** connects R1 to C*, ad infinitum.

There is nothing to the idea that this regress is benign. For if C connects R and S, but can do so only if a further entity connects C to R and to S, then we are in the presence of a vicious regress.  But is it quite obvious that the infinite regress, assuming it arises at all, is vicious?   Suppose there is an actual infinity of compresence-tropes that connect R and S.  One might think that precisely because there is an actual infinity of them, no compresence-trope remains unconnected.  But I fail to see how this solves the problem.  Consider any trope-bundle.  There is clearly a difference between any such bundle and the set or sum of its constituents, whether their cardinality is finite or infinite.  An infinity of tropes including an infinity of compresence-tropes does not add up to a bundle of tropes.  The actuality of the infinity of compresence-tropes does nothing to solve the problem of unity.

The natural response to the Bradleyan threat to unity is to deny that C needs any intermediaries to connect it to what it connects. The natural move is to accuse Bradley of failing to appreciate that the business of a relation is to relate. Relations, we will be told, are not inert ontological ingredients, but perfom a relating function. Relations as they occur in reality are participial rather than substantival: they are relating relations, connecting connectors, and it is only by an illicit process of hypostatization that we take them to be inert entities that need to be connected to what they connect. Since relations in concrete reality are relatings, relations relate directly. As Reinhard Grossman puts it, "Relations are the glue of the world." Exploiting the metaphor, one could add that relations are a glue that does not require superglue (superduperglue. . .) to glue together what they glue together. This holds whether we take relations to be universals, as Grossmann does, or particulars, as the trope theorist does.

Anna-Sofia Maurin (If Tropes, Kluwer, 2002) adopts this natural response to the Bradleyan threat. But since she has already (rightly) convinced herself that compresence is external to its terms, and thus an entity in its own right, she tries to accommodate the relating character of compresence by importing into the compresence-relation its relata. As she says, "the relation of compresence is external to the tropes it relates, but, simultaneously, the related tropes are internal to the relation of compresence." (Maurin 2002, 164) This means that a compresence-trope, "given that it exists, relate[s] exactly the entities it does in fact relate."

This is to say that a compresence-trope (C-trope) cannot exist without relating tropes, and indeed, without relating the very tropes it in fact relates. The existing of a C-trope just is its relating. This implies that C-tropes cannot exist unbundled, and indeed, cannot exist apart from the very bundles into which they enter. Of course, it is this very feature that blocks the Bradleyan regress. If C is exhausted by its connecting of R and S, then C just is the connecting of R and S rather than something that needs to be connected to them. So if Maurin’s conception is tenable, then Bradley’s regress cannot get started.

I am not convinced that Maurin’s conception is tenable. If I understand her position, it is committed to four propositions. (1) Compresence-tropes (C-tropes) are proper constituents of trope-bundles, constituents that are not identical to the bundles into which they enter as the ‘ontological glue.’ (2) C-tropes are unifiers: they unify collections of ordinary tropes so as to form unitary things. (3) C-tropes, like ordinary tropes, are particulars, unrepeatable entities. (4) C-tropes are relational trope that are individuated by their relata: their relata are internal to them. My difficulty is in understanding how these four propositions are consistent.

Propositions (2) , (3) and (4) together entail that a C-trope exhausts itself in the unifying or connecting of the tropes it unifies. As noted, this is precisely what blocks the regress. The regress cannot arise if there is no real distinction between the unifier and its unifying. And there cannot be such a distinction if the unifier is a relation-trope whose terms are, as Marin says, internal to it. But given (2) , (3),and (4), how can (1) be true? (1) states that C-tropes are proper constituents of trope-bundles. Consider again our redness and sweetness tropes R and S. Suppose C connects R and S. This gives rise to a mini-bundle C(R, S). By proposition (1), C is a proper part of this mini-bundle, but by the other propositions C is identical to this mini-bundle. But C cannot be both on pain of contradiction. C cannot be both a proper part of the whole into which it enters as the ontological glue and the whole itself.

One might try to avoid this contradiction by saying that R and S are internal to C only in the sense that if C relates R and S, then C does so directly, without the aid of intermediaries. On this weaker claim, C can exist without relating R and S, but cannot be compresent without relating R and S. Thus C is not exhausted by its actual relating of R and S, but is merely such that, if it relates R and S, then it relates them and only them directly. This would allow C to be external to R and S, while also assuring that no Bradley-type regress could arise.

But this is not the line Maurin takes, and if she did take it, then a different problem would arise, namely, we would be left with no explanation of the difference between the mere collection R + C + S and the same tropes actually bundled. If C is compresent, it is compresent with R and S and connects them directly, without intermediaries, and thus without igniting a Bradley-style regress. This conditional assertion merely assures us that no regress arises if we can get past the first hurdle, which is C’s connecting of R and S. But how can we render intelligible to ourselves C’s actual connecting of R and S? For if C is external to R and S, and thus an entity in its own right, then the mere sum R + C + S will fall short of the bundle C(R, S). Obviously, C cannot be the ground of the difference between these two complexes since C occurs in both of them. If you say that C in the bundle is a relating relation, as opposed to an inert ingredient, then you will have to explain how C gets related to R and S.

But this is not the problem Maurin faces given her actual views, at least as expressed in her book If Tropes. Her actual view stops Bradley before he can get started. The problem her view faces is different: it obliterates the difference between a compresence-trope and a trope-bundle. This is the price she must pay for saying that the relata of a compresence-trope are internal to it, which is what she must say to stop Mr. Bradley.

Could Maurin deny that compresence is a trope? Possibly. This would remove the contradiction I am taxing her with. She might say that it is just a brute fact that certain collections of ordinary tropes form trope-bundles, and that certain other collections do not, and that there is nothing in extralinguistic reality that corresponds to the word ‘compresent.’ In other words, she might take a nominalist line when it comes to compresence, where a nominalist is not one who holds that everything that exists is a particular, but one who holds that properties and relations are linguistic in nature. A nominalist of the latter sort would deny that a predicate ‘F’ applies to a particular a in virtue of anything extralinguistic such as a’s exemplifying of universal F-ness. Similarly, Maurin might deny that the dyadic predicate ‘compresent’ applies to tropes R and S in virtue of anything such as R and S’s being connected by a compresence-trope. She might just say that the predicate ‘compresent’ applies to R and S, full stop, in virtue of nothing extralinguistic. I rather doubt that theis way of avoiding contradiction is viable however: if compresence is not assayed as a trope, then why assay any ordinary property such as redness as a trope? Why not just say that it is a brute, ontologically unanalyzable, fact that things have properties and stand in relations, and leave it at that?

6. Coda: The Problem of the Unity of a Thing

Trope theory is a species of constituent ontology: it analyzes ordinary particulars into ontological constituents, tropes. But an ordinary particular is a unity as much as it is a plurality of constituents. This give rise to the ancient problem of the unity of a complex which first surfaces in the West in Plato’s Theaetetus at 202e-205e.  What Bradley’s regress shows, however, is that unity cannot be isolated by analysis: it cannot be grounded in a special constituent such as a compresence-trope. Something must function as unifier, but it cannot be anything internal to an ordinary particular. And of course, the unifier cannot be the ordinary particular itself. So unless sense can be made of the idea that the unifier of an ordinary partiuclar’s ontological consituents is something external to it, constituent ontology here suffer shipwreck. You could say it suffers shipwreck on the reef of unity.

Comments

[Vlastimil Vohánka]

Bill,

"... tropes, as befits their building block status, are ontologically independent: they do not depend for their existence on other entities. In particular, they are ontologically independent of the concrete wholes of which they are the abstract parts. If the trope constituents of a concrete particular C depended for their existence on the existence of C, that would be as absurd as the supposition that the stones out of which a wall is composed depend for their existence on the wall."

Here's the issue whether constituents of a complex (a unity) which is constituted by them are always independent on the complex.

Bricks constitute a wall, and are independent on it.

Barry Miller, however, claims that Fido and Fido's existence are constituents of Fido's existing which are dependent on this complex they constitute. See http://maverickphilosopher.powerblogs.com/posts/1180037630.shtml

Similarly, some philosophers say that ontological principles of, say, me -- my essence and my existence -- constitute me, but cannot occur out of (without) me.

Thus, it seems, some constituent is independent on its complex, and some constituent is independent on its complex.

Given such a division between kinds of constituents, one asks for its explanation (why some constituents are dependent, but other are independent?), and, more basically, also for the general explication of the nature of constituents as such.

[Bill Vallicella]

Vlastimil,

You raise an important question: is it true that for every type of complex the constituents (parts, components) are ontologically basic and the complex ontologically derivative? In the case of a brick wall it seems clear that the wall depends for its existence on the bricks, but the bricks do not depend for their existence of the wall. The same is true for all sorts of artifacts including TinkerToy and Lego constructions.

I plan to discuss Barry Miller's work later on especially since a philosopher at U of Toronto wrote me and expressed an interest in my take on Miller. So I won't say anything about Miller now. (In coming weeks and months I will be discussing Bradley, regress arguments, the meaning of life (need to finish a paper for the Monist, the potentiality argument (need to respond to an argument by Lupu), and then perhaps Miller's final statement of his position in The Fullness of Being.)

A lot depends on how we define 'complex.' We might define a complex as a whole of parts whose existence is not entailed by the existence of its parts. Plato gives the example of the syllable SO which is composed of the letters S and O. (See Theaetetus 202e-205e and compare Aristotle, Metaph 1041b10-30) This is a complex by the above definition since the existence of a token of SO is not entailed by the existence of tokens of S and O. (If S is on the left side of the page and O on the right, then the letters exist but the syllable does not.)

What about the fact of Tom's being tired? Is this a complex as above defined? Surely Tom can exist without the fact's obtaining. And if being tired is a universal, then it can exist without the fact's obtaining. This is so even if one maintains that all universals are exemplified. For it may be that being tired is exemplified by some other individual but not by Tom. It would then be the case that Tom exists, being tired exists, but Tom's being tired does not exist. So the fact would be a complex as above defined.

You mention the composition of essence and existence in an individual such as Fido. There is a sense in which these are ontological parts or constituents of Fido. But what could Fido's existence be apart from Fido? It seems that Fido's existence needs Fido to be individuated as precisely Fido's existence. And what would Fido's essence be apart from Fido? This Thomistic approach makes sense but I find it problematic. In any case, Fido so construed would not be a complex as above defined.

But suppose you think of an ordinary particular as a bundle of universals and the existence of the particular as the togetherness or compresence of those universals. Then that particular would be a complex as above defined: the existence of the universals would not entail the existence of the bundle.

[Vlastimil Vohánka]

Bill,

Thanks for these reasonable hints.

"You mention the composition of essence and existence in an individual such as Fido. There is a sense in which these are ontological parts or constituents of Fido. But what could Fido's existence be apart from Fido? It seems that Fido's existence needs Fido to be individuated as precisely Fido's existence. And what would Fido's essence be apart from Fido? This Thomistic approach makes sense but I find it problematic."

I am not a Thomist, still, I'd like to know what, according to you, is the problem.

[Bill Vallicella]

V,

Let's return to this discussion later, perhaps in January, in connection with Barry Miller's work. Right now I have other fish to fry.