The Atlas of Reality
Page 24
We can codify this into a methodological principle. First, we need to articulate a fourth corollary Ockham's Razor:
PMeth 1.4 Fourth Corollary of Ockham's Razor. Adopt the theory that jointly minimizes the following classes: (i) facts about the existence of fundamental things, (ii) facts about the holding of fundamental (natural) properties of and relations between fundamental things, and (iii) facts about brute metaphysical necessity. If more than one theory minimizes these classes (if some trade-off between them is inevitable), adopt that theory which makes the best trade-offs.
Assuming that qualitative simplicity is more important than quantitative, we can add the following addendum.
PMeth 1.4.1 Prioritizing Qualitative Economy (Addendum to Ockham's Razor). When trading off quantitative economy (minimizing the number of fundamental things) and qualitative economy (minimizing the number of fundamental properties of and relations between fundamental things), always prefer the latter.
One might worry that Ostrich Nominalism makes hash of the Fourth Corollary of Ockham's Razor and Prioritizing Qualitative Economy, eschewing as it does the very existence of properties and relations. Can Ostrich Nominalists even make sense of these principles? In a word: yes. They can and should do. Ostrich Nominalists can put the principle into what is known as the ‘formal’ mode, speaking about ‘predicates’ or ‘general names’, in place of ‘properties’ or ‘kinds’. Ostrich Nominalists should prefer theories that can be expressed in languages with the fewest number of basic, underived general terms or names.
But can Ostrich Nominalists make sense of names, predicates, or other linguistic types? Mustn't they refuse to accept anything but particular tokens, particular episodes of speaking, or writing? No, they need not take such an extreme position. It is true that, for Ostrich Nominalists, there is no one thing common to every occurrence of the word ‘red’, for example. Nonetheless, Ostrich Nominalists can make sense of the practice of counting-by-type, versus counting-by-token. For example, consider the following sentence token:
(11) The red door is red.
Ostrich Nominalists can make perfect sense of each of the following claims:
(12) Sentence (11) contains five words (counting by tokens).
(13) Sentence (11) contains four words (counting by types).
So long as the Ostrich Nominalists can make sense of counting by types, they can apply the principle of qualitative economy to the expression of scientific (and metaphysical) theories.
If that is right, then Ostrich Nominalism is in a surprising amount of trouble vis-à-vis Ockham's Razor. This is a surprising and serious problem for the Ostrich, since an important, if not fundamental driving force behind Ostrich Nominalism is the offer of a metaphysically simple theory that powerfully accommodates the data a theory of properties is meant to accommodate.
This debate over qualitative economy between Realists and Ostrich Nominalists connects with another issue, one that we discussed in Chapter 6: are universals individuated by their causal roles or by their causal roles together with their systemic categorial relations to other universals or does each universal have its own sicceity or thusness? If each universal is individuated from others by its own, unique sicceity, then these sicceities would greatly add to the qualitative complexity of Realism, neutralizing the advantage over the Ostrich Nominalism gained by positing universals and instantiation. To use this argument against Ostrich Nominalists, it seems that Realists should reject sicceities and embrace the individuation of universals by their causal and categorial properties.
7.3 Universals and the Problem of Intentionality
What connects the predicates of our language (including adjectives and common nouns like ‘white’ or ‘dog’) with properties in the world? What connects our corresponding mental concepts with those properties?
The Ostrich Nominalists have no general explanation of this fact. They must rely on something like the homonymic clauses of Tarski's truth theory:
(14) The predicate ‘_____ is white’ applies to x if and only if x is white.
(15) The predicate ‘_____ is a dog’ applies to x if and only if x is a dog.
In fact, they must claim (since they are doing metaphysics) that the applicability of the predicate ‘___ is white’ to x must be grounded by x's whiteness. This lack of general expanation of concept applicability looks like a serious defect, if we compare the Ostrich Nominalism with Realism. Realists can offer a single, general explanation:
(16) A predicate of the form ‘___ is F’ applies to x if and only there is a concept C that is conventionally associated with ‘___ is F’ (as its meaning), and C applies to x (and when it does, it applies because C applies to x).
(17) A concept C applies to x because there is a universal U such that U is the one and only universal that is a part of C, and x instantiates U (and when it does, it applies because x instantiates U).
Of course, this explanation will only work for those predicates that have a single meaning, and even then only for those predicates whose meaning involves a single universal. It is probable, given a sparse theory of universals, that most of our concepts involve some complicated logical structure that includes more than one universal. Nonetheless, (16) and (17) should work for an important set of basic cases of intentionality and so could form part of a general theory of the intentionality of predicates.
Ostrich Nominalists could respond by pointing out that there is one important case of predication that the Realists' account cannot handle without incurring an infinite regress: the predicate ‘__ instantiates ___’. We're assuming that Realists will avoid Bradley's regress by denying that there is such universal as INSTANTIATION. This means that they cannot make use of the relevant instance of schema (17):
(18) The concept instantiation applies to the ordered pair
We recommend that Realists reject (18) as part of their theory of intentionality. In this one case, Realists should employ exactly the kind of Tarskian, homonymic axiom that Ostrich Nominalists use in all cases:
(19) The concept instantiation applies to the ordered pair
Thus, the Realists' account of the intentionality of simple predicative concepts will have two axioms, (19) and (20).
(20) For all simple concepts C other than instantiation, C applies to x if and only if there is a universal U such that U is the one and only universal that is part of C, and x instantiates U.
A theory with two axioms is simpler, and therefore better, than a theory with many axioms doing the same job. If our thought includes many simple predicative concepts, the Ostrich Nominalists' theory will be that much more complicated than the Realists'.
It would also be possible to build a relatively simple theory of predicative intentionality with the resources of Resemblance Nominalism.
(21) For all concepts C, C applies to x if and only if, for every mental token c of C, there is one and only one part of c that belongs to a resemblance class, and x and c belong to the same resemblance class.
Resemblance Nominalists will have to make one concept the exception to rule (21): namely, the concept of resemblance in terms of which we define resemblance classes. Just as the Realists had to do, Resemblance Nominalists will have to explain the application of resemblance homonymically (in terms of resemblance itself):
(22) The concept of resemblance applies to x and y if and only if x resembles y.
Once again, a theory with two rules will be preferable to one with many independent rules.
Realists face a further problem: namely, they must explain how a concept can contain a universal without instantiating the universal. We don't want the concept of a horse to be itself a horse! I don't have a little horse running around in my mind every time I think of a horse.
Realists will need some way of distinguishing the way in which a property is part of an ordinary particular and the way in which it is part of a concept in the mind. Aristotle and many of the scholastic philosophers who followed him made this distinction by arguing that properties exist with matter in ordinary particulars and without matter in mental concepts. This seems to work for properties that are essential properties of material objects, like properties of mass, velocity, chemical composition, and size. However, what about simple properties that aren't essentially properties of a body, like thinking or feeling sad? Surely I can have the concept of feeling sad without actually feeling sad myself at that moment. One more potential problem: Aristotle's solution only works on the assumption that the human mind is immaterial (as Aristotle argues in De Anima III).
Realists have at least two further options, beyond the traditional Aristotelian solution. First, they might suppose that concepts are wholes that include, besides the universal that the concept expresses, a universalizing element, which marks the whole as a concept rather than as particular instance of the property. These universalizing or conceptualizing elements would function in a way similar to the functioning of a bare particular. They would make the concept as a whole be the truthmaker for the mind's ability to think of the contained property, rather than the truthmaker for a proposition asserting the existence of a particular that instantiates the property.
Alternatively, Realists could give up on the theory that the simple concept of F-ness contains the F-ness universal or an F-ness trope as a part. They could instead suppose that the intentional connection between the concept and the property it expresses involves some other relation. For example, one might think that a concept C expresses a property P just in case (i) there was some relevant exercise by some particular thing of a power conferred upon that particular by P that resulted in the formation of the concept C (in the minds of the members of the relevant community or other population), and (ii) the concept C belongs to a system of concepts S1, property P belongs to a system of properties S2, and there is a formal isomorphism between S1 and S2 (i.e., the two systems exhibit the same formal structure). This is still the sort of general account of intentionality that would be unavailable to Ostrich Nominalists. Indeed, it would also be unavailable to Neo-Humeists, who deny the fundamental existence of powers.
Clearly, the problem of intentionality is a difficult one. Until a fully satisfactory account of intentionality in terms of universals or tropes is developed, the issue cannot count decisively against Ostrich Nominalism. However, it does point to a potentially grave challenge to that theory.
7.4 Properties as the Ground of Causal Powers
In Chapters 4 through 6, we examined four theories of causal powers and dispositions: Neo-Humeism, Strong Hypotheticalism, Strong Nomism, and Strong Powerism. As far as we can tell, Neo-Humeism and Strong Hypotheticalism have no implications for our general theory of properties. They are fully compatible with all of the various forms of Nominalism and Realism.
Nomism, in contrast, favors the truth of Realism. Nomists need properties as constituents of the laws of nature. This rules out Ostrich Nominalism. A law of nature is by its very nature something timeless and universal, connecting properties to one another. These properties certainly cannot be individual tropes, since then each law would be a merely particular fact about those particular tropes. One could, perhaps, take the laws of nature to be relations among resemblance classes of tropes, so long as these classes included merely possible tropes along with all of the actual ones (since laws of nature are supposed to govern merely possible, counterfactual situations). This seems like an unnatural combination of views, however. It is not surprising that Nomists overwhelmingly favor Realism.
Since Powerists take causal powers and dispositions to be grounded in the essences or natures of properties, Powerism is inconsistent with Ostrich Nominalism, which includes the denial of the real existence of properties altogether. In fact, it is hard to see how Powerism could be combined with any form of Extreme Nominalism. Mere classes of ordinary particulars would not seem to be the sort of thing capable of having a power-conferring essence. Realism and Powerism make a good fit, since universals could well have essences that confer powers and dispositions on the things that instantiate them. It seems that Powerism could be combined with Moderate Resemblance Nominalism, with the understanding that it is essential to each trope that it bestow a certain set of powers on its bearer, with exactly similar tropes conferring exactly similar powers (John Heil defends such a view).
Notes
1. This isn't quite enough. We would need to generalize this definition to include predicates/properties with more than one place. A specifying description of an object is a description which only that object satisfies.
2. See Aristotle's On Interpretation 17a37, Hochberg (2004), Lowe (2004), Oliver (1996), and cf. MacBride (2004, 2005), Ramsey (1925).
3. See Russell (1911), and cf. Lewis (1983, 1986a), MacBride (1998).
4. This view is mentioned by Wisdom (1934). See also Ehring (2004), Williams (1986), and Pickavance (unpub.).
5. There is one account of spatial location, the Theory of Spatial Qualities (17.1T.1T), on which a thing's location is an intrinsic characteristic of it. See Chapter 17.
6. Our usage of ‘natural’ will diverge from the particularities Lewis attaches to the idea. But we believe the spirit is similar. Space prevents us detailing the (interesting!) literature on naturalness.
7. See Pickavance (forthcoming) for a more thorough defense of this strategy.
8. Relational predicates, like ‘is two meters east of Elsie’ and ‘is taller than Wilt’, are plausibly treated as not corresponding to a unique property. Rather than saying that the fundamental level involves a thing exemplifying the property of being two meters east of Elsie, one can say that the fundamental level involves a thing standing in the relation of being two meters east of to Elsie.
9. We ignore attempts, like that of Russell himself (cf. Russell and Whitehead 1962 or Russell 1959), to dissolve the paradox by “typing” predicates and properties.
10. The regress could also move forward by thinking of EXEMPLIFICATION as a relational universal that links Lyle and the universal SWEETNESS (see Chapter 10.1). In that version of the regress, we would move from 4 to 5*:(5*) Lyle and the universal SWEETNESS exemplify EXEMPLIFICATION.
11. Well, there are some people who worry about the natural numbers for this reason. But they are in a very, very small minority.
12. There is one complication to consider in applying Ockham's Razor to the evaluation of UP-Realism: how to count the primitive binary relation x instantiates y. Does this relation introduce a single primitive sort (a sort of ordered pair) or two primitive monadic sorts (instantiating and being instantiated)? Or does it introduce a potentially infinite number of sorts, two for every entity A: instantiating A and being instantiated by A? We can rule out the last of these suggestions, since counting sorts in this way would give priority to quantitative economy over qualitative economy, since every individual entity would give rise to two primitive sorts. The first suggestion seems to under-count the commitments of Classical Realism, especially since instantiation is a non-symmetric relation, with a difference between being an instantiator and being instantiated. So, we'll assume that fundamental binary relations like instantiation correspond to exactly two fundamental sorts, each with a variable parameter x: the sort of instantiating x, and the sort of being instantiated by x. These two sorts can be used to define ‘universal’ and ‘particular’: a universal is something that can be instantiated, while no particular can be instantiated.
8
Reductive Nominalism and Trope Theory
8.1 Reductive Nominalism
As we noted, Ostrich Nominalism is only one type of property Nominalism. The other form is Reductive Nominalism:
8.1T Reductive Nominalism. Nominalism is true, and there is a general explanation of the fact that some particulars have p
roperties in common. (8.1T is identical to 7.1A.1T.)
There are a number of different versions of Reductive Nominalism, versions distinguished by the way in which each accounts for facts about having and sharing properties. We will discuss three broad varieties of Reductive Nominalism: Predicate Nominalism (Section 8.1.1), Class Nominalism (Section 8.1.2), and Resemblance Nominalism (Section 8.1.3). We will spend most of the first section on Resemblance Nominalism, which is (by our lights) the best alternative to UP-Realism.
Resemblance Nominalists believe that the general explanation of the fact that particulars resemble each other appeals to the existence of a general relation of resemblance between particulars. This relation is a basic or fundamental one. The fact that one particular resembles another is not grounded in their instantiating any common universals. Resemblance Nominalists come in two sub-varieties, depending on whether they take the resemblance relation to hold between particular properties (called ‘tropes’) or particular things that have properties (which we will call ‘ordinary particulars’). Trope Nominalists ground all similarity between ordinary things in a resemblance relation between tropes, while Extreme Nominalists take the resemblance relations between ordinary particulars to be fundamental.