If the foregoing is right, then we have reason to think that universals satisfy Restricted Identity of Indiscernibles. Further, if particulars fail to do so (and we will see in the next chapter that this is likely), then Wise UP is successful as a universal-particular distinction.
7.2.1.2 Russell's Intensional Paradox
To get at the second problem facing Realism, recall the second bit of near-universal agreement noted in Section 7.1, namely, the connection between something's satisfying a predicate and its exemplifying a property. We there noted the plausibility of the following account: the sentence formed by combining a name for x with the predicate P in a grammatically appropriate way is true if and only if x exemplifies the property correlated with P. While this account seems like a natural, if not inevitable starting point, it is not plausibly universally applicable. Indeed, it cannot be.
To make our way to a defense of that claim, consider the true sentence ‘Elsie is not a cat’. Here, we have a name for Elsie, ‘Elsie’, combined in a grammatically appropriate way with the predicate ‘is not a cat’. If we apply the foregoing account, the result is that we are committed to there being a property of not being a cat. While this may be the right account of this truth, it doesn't seem so inevitable as the account of the truth of ‘Elsie is a dog.’ Why? Because the property of not being a cat is a surprising sort of property. Indeed, it seems like the property of being a cat ought to play a role in the truth of the sentence ‘Elsie is not a cat,’ which it would not were this account correct. In fact, it is initially very tempting to treat ‘Elsie is not a cat’ as true if and only if it is not the case that Elsie exemplifies the property of being a cat. This account also avoids commitment to the negative property of not being a cat. Other logically complex predicates admit of similar treatment. For example, disjunctive predicates like ‘is F or G’ seem to be satisfied by an object x if and only if (simplifying a bit) one gets a truth either when one combines a name for x with the predicate ‘is F’ or one gets a truth when one combines a name for x with ‘is G’; that is, if and only if either x exemplifies the property correlated with ‘is F’ or x exemplifies the property correlated with ‘is G’. No mention of a disjunctive property is needed! Conjunctive predicates like ‘is F and G’ can be treated analogously and without appeal to conjunctive properties. If we go these ways, though, then there are some predicates, among them logically complex negative, disjunctive, and conjunctive predicates, which do not correlate uniquely with individual properties.8
In fact, a version of Russell's Paradox (the so-called “intensional” version, in contrast to the “set-theoretic” version) can plausibly be used to show that it can't be the case that every predicate is uniquely correlated with a single property.9 Suppose that there is a unique property for every predicate. According to many views of properties, properties do not always exemplify themselves. Redness, for example, is not self-exemplifying. (The property of being red does not exemplify itself; the property of being red isn't red.) Here we have a predicate, ‘is not self-exemplifying’, that is true of redness, and thus according to our supposition, there must be a property of not being self-exemplifying. There should be a fact of the matter, for any property, whether that property exemplifies the property of not being self-exemplifying. But if we ask, about the property of not being self-exemplifying, whether it exemplifies the property of not being self-exemplifying, we face a dilemma. Either this property is self-exemplifying or it is not. Suppose that the property of not being self-exemplifying exemplifies itself. Then it is self-exemplifying. But it exemplifies itself if and only if it has the property of not being self-exemplifying. If that is right, then it is not self-exemplifying. Contradiction. It must be that the property of not being self-exemplifying does not exemplify itself. Then it is not self-exemplifying. Which means, by our account of the truth of that sentence, that it exemplifies the property of not being self-exemplifying. Which means it is self-exemplifying. Contradiction again! Whether this property is self-exemplifying or not, we get a contradiction. So either there is no fact of the matter about whether this property is self-exemplifying or our account of the truth of sentences involving the predicate ‘is not self-exemplifying’ is wrong. Most metaphysicians are unhappy with the first option, and so go for the second. A promising way to make good on the second option is to just deny that there is any property of not being self-exemplifying.
If we follow these philosophers, we face a very general question. Just how many properties are there? Answers to this question fall on a spectrum from super-abundant theories to super-sparse theories. Super-abundant theories maintain that there are a great many properties, while super-sparse theories maintain that there are very few properties. Abundant theories of properties characteristically maintain that there is at least one property for every possible predicate (or concept), while spare theories characteristically claim that there are a great many predicates—logically complex predicates, for example—that are not correlated with a unique property. At any rate, Realists will need to be careful not to draw arbitrary lines, so as not to run afoul of Ockham's Razor.
7.2.1.3 Bradley's regress
A third challenge for Realists arises because they takes the character of an object to be determined by its relationship to universals. This challenge was raised powerfully by F.H. Bradley (1897/1930). We will present one version of Bradley's objection in this section (a version somewhat loosely based on Bradley's text), but in Section 11.2.3, we will return to it as an objection to the existence of multiple entities.
First, we observe that Realists seem committed to what Michael Loux (2006) has called “Platonic Schema”:
Platonic Schema. Something o is F if and only if o exemplifies the universal F.
There is a worry that Platonic Schema results in a regress, and this worry can take a definite shape in at least two ways.
First, consider the following:
(3) Lyle is sweet.
Here we have a true claim about THP's son, Lyle, a claim that concerns Lyle's character. (3) is a simple subject-predicate sentence of the form ‘o is F’. Realists, then, will offer the following account of Lyle's having this character, given their commitment to Platonic Schema:
(4) Lyle exemplifies the property of being sweet.
One way to read (4) is as another subject-predicate sentence of the form ‘o Fs’, where ‘Lyle’ is ‘o’ and ‘exemplifies the property of being sweet’ is ‘Fs’. But then, given Platonic Schema, Realists will be forced to say that (4) is true if and only if (5) is:
(5) Lyle exemplifies the property of exemplifying the property of being sweet.
But (5) is yet another subject-predicate sentence, and will generate a still further commitment given Platonic Schema, and so on. We have generated a regress.10
There is a second way a regress might get going. We can point out that Loux's Platonic Schema does not fully capture the Realists' commitment to properties. Realists believe not only o exemplifies F-ness whenever o is F—they also believe that o is F because o exemplifies F-ness. Let's call this the Strengthened Platonic Schema:
Strengthened Platonic Schema. Whenever something o is F, o is F because o exemplifies the universal F.
What can Realists say about these regresses? There are two ways to blunt the trouble created by them. First, one can accept that these are infinite regresses but deny that they are problematic. Second, one can find a way to stop the regress by restricting or rejecting Platonic Schema. Let's look a bit more at each of these strategies.
Suppose that one accepted these regresses but wanted to say that they are not problematic. Isn't this tack doomed from the beginning? Aren't all infinite regresses problematic? Maybe not. First, notice that there are infinite sequences at which no one balks. For example, the natural numbers (0, 1, 2, 3,…) form an infinite sequence, but no one takes this as a reason to worry about them.11 Now, this sequence isn't really a regress in any important sense. Regresses occur when, in the process of explaining or exp
licating something, one discovers a further, similar thing that must be explained or explicated as well, and that when one goes to explain or explicate that further thing, one discovers a still further thing that must be explained or explicated, and so on. Vicious regresses require that one thing is explained by or somehow dependent on something else, which something else is itself explained by or dependent on another thing further down the regress. This is problematic because one can never fully ground or explain the initial thing in the regress, the thing one is trying to ground or explain. The further requirement for viciousness will become a bit clearer as we consider our two regresses.
Consider the first regress, which starts with (3) and moves to (4), (5), and so on. Given a commitment to Platonic Schema, a commitment to (3) entails a commitment to (4), (5), and so on. However, Platonic Schema does not require that Realists explain (3) by appealing to (4), nor must Realists say that (3) is dependent on (4) in any way. Realists can simply insist that (3), (4), (5), and so on are related facts, and that (3) can be clarified by invoking (4), but that this relationship shouldn't be thought of as any kind of explanation or as charting any sort of metaphysical dependence. This first regress, then, is plausibly not vicious.
What about the second regress, which corresponds to the Strengthened Platonic Schema? Here there are explanations: (3) is true in virtue of the truth of (4), and (4) is true in virtue of the truth of (5). Thus, this regress is vicious because we cannot fully ground (3) without fully grounding (4), which cannot be fully grounded without (5), and so on to infinity. This regress looks a great deal more troubling.
However, there are other ways to respond to this regress. Most of these ways reject, in one way or another, the Strengthened Platonic Schema. One seemingly easy route is to deny that there really is a fundamental, universal relation of INSTANTIATION or EXEMPLIFICATION. This view is a form of Nominalism (more specifically, Ostrich Nominalism—see the next subsection) about that relation. If it's correct, then truths like (4) are not to be treated using the Strengthened Platonic Schema. Similarities that correspond to natural properties (like sweetness) are grounded in the sharing of a universal, but the way in which two instantiation pairs—take the pair a and F and the pair b and G, whenever a exemplifies F and b exemplifies G—are similar is not grounded in their sharing the universal EXEMPLIFIES. One might go in more generally for a sparse theory of universals, either by denying that any relations correspond to universals (restricting universals to monadic properties) or by denying that second-order, logical relations like exemplification correspond to universals. If one goes on to restrict the Strengthened Platonic Schema in the same way, then one will have solved Bradley's regress.
There is still a third way of understanding Bradley's regress, a way that is in fact closer to Bradley's actual text. This approach to the regress looks at the problem of what explains the unity of the truths or facts involved. This version of the regress is not immediately relevant to the evaluation of Realism, so we will take it up again in Section 11.2.3.
7.2.1.4 The Challenge of Ostrich Nominalism
We have seen that appealing to a sparse theory of universals is imperative for eluding some of the most challenging troubles facing Realism. The sparsest theory of universals, and one that eludes all of the troubles highlighted above, is one that posits no universals at all. An easy way to think about Nominalism is as just the denial that there are universals, and we'll think of Nominalism that way in this section. (We'll take this back a bit in our discussion of Resemblance Nominalism in Section 8.1.3.) One virtue of Nominalism is that it avoids the need to deal with the Universal-Particular Distinction and with Russell's Paradox. But further, there is an obvious challenge to the positing of any universals, grounded in Ockham's Razor.
An Ostrich Nominalist is one who denies the need for any general explanation of putative facts involving attribute agreement. The term was coined by opponents of the view who compared such Nominalists with the proverbial ostrich that sticks its head in the sand when faced with the similarity facts. Recall the red and square RedSquare, the red and circular RedCircle, and the blue and circular BlueCircle. Consider the following:
(7) RedSquare and RedCircle are both red/are similar with respect to color.
(8) RedCircle and BlueCircle are both circular/are similar with respect to shape.
According to most theories of properties, (7) is true because RedSquare and RedCircle exemplify one and the same (shareable) property, the property of being red, while (8) is true because RedCircle and BlueCircle exemplify one and the same (shareable) property, the property of being circular. Ostrich Nominalists, however, deny these explanations. Indeed, they deny the need for positing shared properties at all. Instead, Ostrich Nominalists simply insist that the fact that RedSquare is red is metaphysically fundamental, the fact that RedCircle is red is metaphysically fundamental, and (7) is made true by these two facts together. Similarly, the fact that RedCircle is circular is metaphysically fundamental, and the fact that BlueCircle is circular is metaphysically fundamental, and (8) is made true by these two facts together. The pattern is to explain putative cases of attribute agreement by exploiting metaphysically fundamental facts of character-having without ever appealing to properties that get instantiated. Notice that the Ostrich Nominalists cannot do their explaining of facts of attribute agreement in any generalized way. That is, they cannot say that claims of attribute agreement are made true because the things whose attributes agree share a property. They are precisely trying to avoid talking about properties at all! So, while there are explanations for every particular fact of attribute agreement in terms of other facts of character-having, there is no general explanation of facts of attribute agreement.
Not all Nominalists are Ostriches, and so we get a division of the Nominalist landscape:
7.1A.1T Reductive Nominalism. Nominalism is true, and there is a general explanation of the fact that some particulars have properties in common.
7.1A.1A Ostrich Nominalism. Nominalism is true, and there is no general explanation of the fact that some particulars have properties in common.
We take up Reductive Nominalism in Section 8.1. For now, we focus on Ostrich Nominalism.
Ostrich Nominalists face one obvious challenge, namely, a potential conflict with Classical Truthmaker Theory. Consider again (3):
(3) Lyle is sweet.
Ostrich Nominalists take (3) to be fundamental, in no need of further metaphysical grounding. However, what is the truthmaker for (3)? It can't be Lyle himself, since Lyle could exist and not be sweet. If we suppose that the truthmaker of (3) is something like the fact that (3), we begin moving beyond the scope of Ostrich Nominalism, and we face the problem of explaining what this fact is and how it differs from Lyle, on the assumption that there is no such thing as sweetness. If there are no properties, it is hard to see what could differentiate one fact from another. In light of this, some prominent philosophers (including Frege, Alonzo Church, Quine, and Donald Davidson) have argued that there could be only One Big Fact.
However, most Ostrich Nominalists reject Classical Truthmaker Theory. Spectral Truthmaker Theory and Truth Supervenes on Being seem tailor-made for Ostrich Nominalism, since on these views the truth of (3) is grounded in Lyle's existing and being as he is, which requires no further entity.
In fact, it seems that Ockham's Razor demands that we prefer Ostrich Nominalism to Realism, unless there are some further facts that the Nominalist cannot explain. This is because Realism demands more things than Ostrich Nominalism. Realists about universals and particulars—UP-Realists—posit two classes of things, particulars and universals, and a fundamental relation of exemplification or instantiation between things and universals.
7.1T.1T UP-Realism. There are fundamental universals and fundamental particulars, and the latter instantiate the former.
(We will consider some varieties of UP-Realism in Chapter 9.) On the other hand, Ostrich Nominalists posit only one class of things, particulars.
Let's suppose that there are k fundamental particulars and n fundamental kinds of things. Ostrich Nominalism requires only k things, falling into n irreducible kinds. UP-Realism, on the other hand, requires k+n things, falling into two fundamental kinds (things that instantiate something and things that are instantiated).12 If n is very large, then the domain of Ostrich Nominalism is much smaller quantitatively.
However, this argument from Ockham's Razor is too hasty, as Bryan Pickel and Nicholas Mantegani have recently argued (Mantegani 2010, Pickel 2010, Pickel and Mantegani 2012). This is because Ostrich Nominalists must posit a very large number of metaphysically fundamental sorts of things: spheres, cubes, blue things, red things, etc. For UP-Realists, each of these kinds of things can be reduced to universals and instantiation pairs:
(9) x is a red thing if and only if x instantiates the universal REDNESS.
(10) x is a sphere if and only if x instantiates the universal SPHERICALITY.
Thus Ostrich Nominalism is much larger qualitatively, since UP-Realism has only two fundamental kinds, whereas Ostrich Nominalism has n fundamental kinds. This leads us to an important question relevant to evaluating Ostrich Nominalism: is quantitative economy more important than qualitative economy or vice versa?
There are good reasons to think that qualitative economy is much more important than quantitative economy. Both are important, but the kind of simplicity that comes with reducing the number of fundamental kinds or categories of things is much more valuable. Simply adding more entities to already existing fundamental kinds is a relatively trivial addition, compared to adding entirely new kinds to our theory. This preference can be seen to be at work throughout modern science. We are always willing to add new entities (such as atoms, subatomic particles and quarks) in order to reduce the number of fundamental kinds of things (for example, reducing the 100+ elements to the three kinds of subatomic particles—protons, neutrons, and electrons). Thus, UP-Realism represents exactly the kind of scientific advance over Ostrich Nominalism that modern atomic physics holds over Daltonian chemistry.
The Atlas of Reality Page 23
