Does Bradley’s Regress Support Nominalism?
Does Bradley’s Regress Support Nominalism?
Wolfgang Freitag, Konstanz, Germany

Abstract

Nominalists often derive indirect support for their position from an argument to the effect that realism about universals is vulnerable to Bradley’s regress. I argue independently (a) that realism is no more committed to the regress than nominalism is and, moreover, (b) that neither view is committed to Bradley’s regress. An analysis of the sources for the regress reveals that there are two ways of avoiding the regress. I argue that both ways are open to nominalist and realist positions alike.

Table of contents

    One of the standard arguments against realism about universals is based on Bradley’s regress. According to this argument, realism about universals is committed to a vicious regress of instantiation relations. If realism is false and nominalism the only alternative, then, so the argument concludes, nominalism is correct. The strength of this argumentation depends on three things: (1) that commitment to Bradley’s regress makes a position untenable; (2) that nominalism as the only alternative to realism is not committed to the regress; and, most importantly, (3) that realism is committed to the regress.

    I have three aims in this paper. My proximate aim is to show that if (3) is correct then (2) is incorrect: if the realist is committed to Bradley’s regress then so is at least one version of nominalism, namely, trope theory. The demonstration that neither theory is committed to the regress (and hence that (3) is false) is my second aim, attained by the proof that these positions have no commitment to a condition which is generally (and rightly!) held to be necessary for Bradley’s regress. As I move along, I shall also claim that there is a widely ignored second condition necessary for the regress, to which – again – neither nominalism nor realism has any commitment. The upshot is this: Bradley’s regress problem is independent of the problem of universals. I conclude with an attempt to explain why many philosophers have been misled into thinking otherwise.

    1. The regress argument, realism and nominalism

    Here, I shall discuss solely nominalism and realism concerning universals, which are understood to be nonrelational or relational properties.1 For the sake of simplicity, I will focus on nonrelational properties.

    Following the tradition, I take realism about universals to be the view that different objects may have the very same, repeatable property. If both the bike and the car are black, then the realist says there is one and the same property, blackness, instantiated by both the bike and the car. Thus, according to realism about universals, a single property may be multiply instantiated in a given world. Nominalism denies this. If the bike and the car are black, then they do not literally speaking have the same property in common. The class nominalist, for example, considers being black as no more than being an element of a certain class of particulars. Instantiation of a property then reduces to membership in a certain class. The trope theorist assumes properties to be much as the realist thinks them to be, except that they are not repeatable: in a given world, no two particulars have literally the same property.

    I have encountered the Bradley argument, employed against realism about universals, frequently in personal discussions, and sometimes in print. A very recent formulation of the argument by Gonzalo Rodriguez-Pereyra, a proponent of nominalism, gives me an opportunity to voice my own view on the matter:2

    [One argument against universals is this:] Suppose there are universals, both monadic and relational, and that when an entity instantiates a universal, or a group of entities instantiate a relational universal, they are linked by an instantiation relation. Suppose now that a instantiates the universal F. Since there are many things that instantiate many universals, it is plausible to suppose that instantiation is a relational universal. But if instantiation is a relational universal, when a instantiates F, a, F and the instantiation relation are linked by an instantiation relation. Call this instantiation relation i2 (and suppose it, as is plausible, to be distinct from the instantiation relation (i1) that links a and F). Then since i2 is also a universal, it looks as if a, F, i1 and i2 will have to be linked by another instantiation relation i3, and so on ad infinitum. (Rodriguez-Pereyra 2008)

    The argument asserts that instantiation of universals inevitably leads to a regress of ever more instantiation relations, i.e., to what is usually referred to as Bradley’s regress.3 The claim that a regress ensues seems to be based on the following two conditions:

    (Pu1)
    Wherever a universal is instantiated, there is an instantiation relation (not identical to one of the relata).
    (Pu2)
    The instantiation relation is a universal.

    Therefore it seems plausible to attribute to Rodriguez-Pereyra the following line of thought: According to (Pu1), instantiation of a universal demands an instantiation relation. Classifying this instantiation relation as a universal, as done in (Pu2), we are taken back to (Pu1), which then generates another instantiation relation, which together with (Pu2) again takes us back to (Pu1), which generates a further instantiation relation, and so on ad infinitum. Rodriguez-Pereyra concludes that realism about universals is in serious trouble. My first aim is to show that if the realist is in trouble, then so is at least one form of nominalism.

    One form of nominalism is trope theory. Trope theory distinguishes itself from realism not with respect to the reality of properties, but with respect to the view that properties can be multiply instantiated. Tropes can be instantiated – but only by the sole object having that particular trope. Tropes are “particularised” properties. Now, consider the following pair of conditions:

    (Pt1)
    Wherever a trope is instantiated, there is an instantiation relation (not identical to one of the relata).
    (Pt2)
    The instantiation relation is a trope.

    These two conditions differ from (Pu1) and (Pu2) in a single respect only: they contain the term ‘trope’ where (Pu1) and (Pu2) contain the term ‘universal’. If (Pu1) and (Pu2) lead to a regress, then (Pt1) and (Pt2) equally lead to a regress. Instead of speaking of universals or tropes, we can also formulate the matter in general terms, yielding the following pair of conditions:

    (P1)
    Wherever an entity is instantiated, there is an instantiation relation (not identical to one of the relata).
    (P2)
    The instantiation relation is an entity.

    The regress argument poses a threat only to those who are committed to these two conditions. The trope theorist may deny (P2) as little as the realist. He will understand ‘entity’ as referring to tropes because he is committed to the view that all relations are particularised relations, hence tropes. A difference between trope theory and realism concerning these conditions can thus at most be given by a difference in commitment to (P1). It will now be shown that there is no such difference.

    To see this, we must locate the motivation for (P1), the condition that instantiation demands an instantiation relation. In my view, the motivation lies in the lack of a strict supervenience relation between the existence of the relata of instantiation and instantiation itself: given a and F, it is not determined that a instantiates F. To illustrate this point, consider the situation in which there are exactly four entities, particulars a and b and properties F and G. If we assume that both a and b individually and contingently instantiate exactly one of the properties F and G, and nothing else, and if we assume that both F and G individually are (contingently) instantiated by exactly one of the objects a and b, and by nothing else, then two situations are possible:

    W1:
    a instantiates F; b instantiates G.
    W2:
    a instantiates G; b instantiates F.

    Both situations comprise exactly the same particulars and the same properties. Still, the situations differ; they comprise different facts, different instantiations. This means that the mere existence of particulars and properties does not necessitate a specific instantiation. The mere existence of the car and blackness does not necessitate that the car is black. It may still be that the car is green, and what is black is the bike. The existence of particulars and properties may determine that facts and instantiations obtain, as some authors (in particular Wittgenstein 19224 and Armstrong 1997) maintain. But it does not determine which facts, which instantiations obtain. As a recent author sums up this point:

    Even if a and F-ness cannot exist except in some state of affairs or other, there is nothing in the nature of a and nothing in the nature of F-ness to require that they combine with each other to form a’s being F. (Valicella 2000, p. 238)

    Instantiation between two entities does not strictly supervene on the existence of the entities alone, if these entities are considered to be contingently related. We need more than the relata of instantiation. This need is expressed by condition (P1). (P1) is the reaction to contingent instantiation. The properties F and G in my example can be understood both as tropes and as universals.5 It follows that, given contingent instantiation, the trope theorist is as much committed to (P1) as the realist is. David Armstrong has seen this very clearly:

    Suppose that the link between a particular and its tropes is not necessary. Then it is contingent. But if it’s contingent, then it seems that we have a clear case of a relation between a particular and its trope, and an external relation at that. But then a Bradleian regress ensues […]. (Armstrong 2006, p. 242)

    This concludes the argument for my first claim: realism is no more committed to Bradley’s regress than at least one form of nominalism, namely trope theory. I now proceed to the argument for my second claim: neither position is committed to the regress.

    2. How to avoid Bradley’s regress

    2.1 Avoiding commitment to (P1)

    Contingent instantiation leads to (P1) and starts the regress. In order to avoid (P1), avoid contingent instantiation. Make instantiation necessary. There is a variety of different positions, both nominalist and realist, which conceive of instantiation as being necessary and hence avoid – intentionally or not – commitment to (P1):

    (1) One position that makes instantiation necessary is class nominalism. This position, proposed inter alia by Anthony Quinton (1957), understands having a property as being a member of a certain class of particulars. The object a instantiates F iff a is a member of the F-class. Because classes are identified by their members and class-membership is a necessary relation, instantiation between a and F strictly supervenes on the existence of the F-class alone. In this way, class nominalism can avoid (P1) and thereby the regress. Class nominalism naturally escapes (P1).

    (2) Trope theory also has its means of avoiding (P1). In fact, a trope theorist has two options: (2a) Trope theory in combination with a bundle theory of particulars, as defended by, e.g., John Locke and, in more modern times, by D. C. Williams (1953), holds that particulars are sets or bundles of tropes. Consequently, a instantiates F iff the F-trope is in the a-bundle. Since the identity of the a-bundle is, I take it, defined by the constituting tropes, a’s instantiating F strictly supervenes on the existence of the a-bundle. (2b) The second type of trope theory combines a subject–attribute view with the doctrine of nontransferable tropes. A recent proponent of this view is John Heil (2003, chs. 12 and 13), although he prefers the term ‘mode’ to the term ‘trope’. According to this position, a trope is instantiated by the very same object in all possible worlds. Given the nontransferable trope F and the particular a, the instantiation between a and F follows by necessity.6 Again (P1) can be avoided.

    (3) Mutatis mutandis, realism has the same two options as trope theory: (3a) According to a bundle theory based on universals, of which Bertrand Russell (1948, part 4, ch. 8) is a proponent, particulars are understood as bundles of universals. In this view, a instantiates F iff F is a member of the a-bundle. Since F is a member of the a-bundle necessarily,7 the instantiation relation between a and F strictly supervenes on the existence of thea-bundle. (3b) The second type combines a substance–attribute view with a theory of nontransferable universals. According to this position – maintained by, e.g., David Armstrong (2004a, 2004b and 2006)8 – that a instantiates F supervenes on the existence of a and F alone.9

    Thus, neither nominalism nor realism is committed to the regress. Three of these positions, namely, (1), (2a) and (3a), agree in understanding instantiation to be constituted by class (or bundle) membership. For them necessity of instantiation – and hence the possible denial of (P1) – is built into the ontological conception of instantiation. For the substance–attribute views (2b) and (3b), necessity of instantiation is a feature additional to the basic conception of instantiation and devised, I presume, specifically to avoid (P1).

    All of these five options come with heavy ontological burdens. Ignoring their specific difficulties, I shall mention only the problem which they share: necessity of instantiation makes contingency impossible. Whether the substitutes on offer10 are satisfactory is at least doubtful. So it is worthwhile to investigate whether there might not be another way out of the regress.

    2.2 What is necessary for the regress? – A further condition

    So far I have acted as if (P1) and (P2) were sufficient for the regress, with the purpose of showing that realism is no more committed to (P1) than trope theory is, and that in fact neither of the two views is committed to (P1). Thus, I hitherto relied on the analysis of Bradley’s regress which seems commonly accepted. Now it is time to show that this analysis is flawed. (P1) and (P2) by themselves do not yet yield Bradley’s regress. It is quite obvious but frequently ignored: in order for the regress to obtain, it must be given that the instantiation relation is itself instantiated (by the entities it relates). Otherwise, given an instantiation relation, (P1) does not generate a further instantiation relation. To arrive at a regress, we therefore need the further premise

    (P3)
    The instantiation relation is itself instantiated (by the entities it relates).11

    Conditions (P1), (P2) and (P3) are jointly sufficient for the regress. Are they also individually necessary? I consider (P2) to be superfluous, since any instantiation relation is ipso facto also an entity. (P1) and (P3) are hence jointly sufficient for Bradley’s regress. I consider them also individually necessary: (P1) states the demand for an instantiation relation given any instantiation, while (P3) makes certain that this instantiation relation demands further instantiation. Thus (P1) and (P3) constitute, I think, the proper analysis of the basis of Bradley’s regress.

    Given this analysis, there is a second way of avoiding Bradley’s regress: accept (P1) and deny (P3); accept instantiation relations and therefore take the first step of the regress, but block the regress by denying that the instantiation relation is itself instantiated. This option should be the natural path to take for substance–attribute views operating with contingent instantiation, theories of types (2b) or (3b) albeit without the unnatural condition that instantiation is necessary. There is no space to develop this option here,12 yet the fact that (P3) is necessary for the regress should eliminate any remaining doubts: Bradley’s regress has nothing to do with the problem of universals.

    Conclusion

    To show that Bradley’s regress is neither specific to nor insurmountable for a realist about universals is one thing. To explain why the opposite view has been so compelling to many, is another. So let me end with a suggestion on this point.

    The source is the confusion of two different and logically independent senses of the problem of One over Many. There is the intraworld version of the problem, which concerns the question whether different particularsin a single world can have the very same property F. And there is the transworld version of the problem, which concerns the question whether different particulars in different worlds can have the very same property F.

    The traditional problem of universals is the intraworld version of the problem of One over Many. Universals can and tropes can’t be multiply instantiated within a single world. Bradley’s regress, on the other hand, concerns the transworld problem of One over Many. Transferable entities can and nontransferable entities can’t be multiply instantiated across different worlds. Keeping these two versions of the problem of One over Many apart, we get a clearer grip on the demands that a satisfying metaphysical theory must fulfil.

    Acknowledgments

    This paper draws on results from an ongoing research project commissioned by the Landesstiftung Baden-Württemberg (Germany). I am very much indebted for the support provided.

    Literature

    1. Armstrong, David 1997 A World of States of Affairs, Cambridge: Cambridge University Press.
    2. Armstrong, David 2004a Truth and Truthmakers, Cambridge: Cambridge University Press.
    3. Armstrong, David 2004b “How do Particulars stand to Universals?”, in: D. W. Zimmerman (ed.), Oxford Studies in Metaphysics, Vol. 1, Oxford: Clarendon Press, 139–154.
    4. Armstrong, David 2006 “Particulars have their Properties of Necessity”, in: P. F. Strawson and A. Chakrabarti (eds.), Universals, Concepts and Qualities: New Essays on the Meaning of Predicates, Aldershot etc.: Ashgate, 239–248.
    5. Bradley, F. H. 1893 Appearance and Reality, Oxford: Oxford University Press.
    6. Devitt, Michael 1980 “ ‘Ostrich Nominalism’ or ‘Mirage Realism’?”, Pacific Philosophical Quarterly 61, 433–439.
    7. Frege, Gottlob 1994/1892 “Über Begriff und Gegenstand”, in: G. Patzig (ed.), Gottlob Frege: Funktion, Begriff, Bedeutung, Göttingen: Vandenhoeck & Ruprecht, 66–80.
    8. Freitag, Wolfgang 2008 “Truthmakers (are Indexed Combinations)”, Studia Philosophica Estonica, forthcoming.
    9. Heil, John 2003 From an Ontological Point of View, New York etc.: Oxford University Press.
    10. Lewis, David 1968 “Counterpart Theory and Quantified Modal Logic”, in: David Lewis,Philosophical Papers, Vol. 1, Oxford: Oxford University Press, 26–39.
    11. Lewis, David 1986 On the Plurality of Worlds, Oxford: Basil Blackwell.
    12. Loux, Michael J. 1998 Metaphysics: A Contemporary Introduction, London: Routledge.
    13. Moreland, J. P. 2001 Universals, Chesham: Acumen.
    14. Quinton, Anthony 1957 “Properties and Classes”, Proceedings of the Aristotelian Society 48, 33–58.
    15. Rodriguez-Pereyra, Gonzalo 2008 “Nominalism in Metaphysics”, entry in the Stanford Encyclopedia of Philosophy.
    16. Russell, Bertrand 1948Human Knowledge: Its Scope and Limits, London: Allen and Unwin.
    17. Valicella, W. F. 2000 “Three Conceptions of States of Affairs”, Noûs 34, 237–259.
    18. Williams, D. C. 1953 “On the Elements of Being I”, Review of Metaphysics 7, 3–18.
    19. Wittgenstein, Ludwig 1922 Tractatus Logico-Philosophicus, London and New York: Routledge.
    Notes
    1.
    Sometimes the dispute is taken to concern not the question of universals in the above sense, but that of the existence of abstract entities. Quine even uses the term ‘universal’ as synonymous with ‘abstract entity’. I shall not enter this different dispute.
    ↵
    2.
    For other versions of the argument in print, see Devitt 1980, p. 437, Loux 1998, pp. 38–40, and Moreland 2001, pp. 114–116.
    ↵
    3.
    The attribution of such arguments to F. H. Bradley is historically problematic in at least two respects. Firstly, Bradley was concerned with relational properties specifically and not with properties in general. Secondly, he was not the originator of this line of thought. The general type of argument has been known at least since Plato’s dialogues. See in particular Parmenides, 127e– 130a.
    ↵
    4.
    Wittgenstein makes this claim with the help of the notion of incompleteness, which he borrows from Frege (1994/1892) but which he applies to all ‘objects’, properties and particulars alike. Together with the idea that incomplete objects cannot exist on their own, Wittgenstein arrives at his famous view that “[t]he world is the totality of facts, not of objects” (Wittgenstein 1922, 1.1).
    ↵
    5.
    That F and G, understood as universals, are, in the case discussed, instantiated only by a single entity, is not of relevance here. To see this, simply change the example accordingly.
    ↵
    6.
    This is simplified. There are at least three conceptions of the nontransferability of tropes: (i) F is instantiated in all possible worlds, and it is instantiated in all possible worlds by a. This presumably implies that a must exist in all possible worlds. (ii) F is not instantiated in all possible worlds, but where it is, it is instantiated by a. Option (ii) comes in two varieties: (a) in those worlds in which F is not instantiated, a does not exist; (b) in some worlds in which F is not instantiated, a does exist. The supervenience claim in the main text holds only for (i) and (ii.a).
    ↵
    7.
    Again, I assume that the identity of a bundle depends on the elements constituting it.
    ↵
    8.
    For Armstrong, not only properties but also particulars are nontransferable; particulars have their properties of necessity. Therefore, Armstrong has two independent means to secure the intended supervenience relation. 9 As in the case of tropes, there are at least three possible conceptions of the nontransferability of universals. The supervenience claim would have to be restricted to the analogues of (i) and (ii.a).
    ↵
    9.
    As in the case of tropes, there are at least three possible conceptions of the nontransferability of universals. The supervenience claim would have to be restricted to the analogues of (i) and (ii.a).
    ↵
    10.
    The best, and perhaps only, known way to achieve this is by replacing transworld identity with a counterpart relation for particulars (as David Lewis (1968 and 1986) and Armstrong (2004b) suggest) or for properties, depending on the demands of the theory. Given a suitable semantics, sentences may turn out to be contingent, although instantiations are necessary.
    ↵
    11.
    One of the few to recognize the need for this condition is Loux (1998, p. 38).
    ↵
    12.
    In (Freitag 2008) I have further explored this possibility.
    ↵
    Wolfgang Freitag. Date: XML TEI markup by WAB (Rune J. Falch, Heinz W. Krüger, Alois Pichler, Deirdre C.P. Smith) 2011-13. Last change 18.12.2013.
    This page is made available under the Creative Commons General Public License "Attribution, Non-Commercial, Share-Alike", version 3.0 (CCPL BY-NC-SA)

    Refbacks

    • There are currently no refbacks.