SUPERVENIENCE, REDUCTION, A N D

I N F I N I T E D I S J U N C T I O N *

NICK ZANGWILL

Can a certain sort of property supervene on another sort of property without reducing to it? Many philosophers find the supervenience/irreducibility combination attractive in the philosophy of mind and in moral philosophy (Davidson 1980 and Moore 1903). They think that mental properties supervene upon physical properties but do not reduce to them, or that moral properties supervene upon natural properties without reducing to them. Other philosophers have tried to show that the combination is ultimately untenable, however attractive it might initially appear. Thus Ted Honderich and Jaegwon Kim argue that the combination cannot explain the causal efficacy of the supervening properties (Honderich 1982, Kim 1984b). And Simon Blackburn argues that the combination creates a mystery for a realist about the supervening properties (Blackburn 1971 and 1985). Kim also has a formal argument for the conclusion that supervenience brings reduction in its train (see Kim 1978, 1984a and 1990), and it is this argument that I shall discuss here.

I Begin with the claim that A properties supervene on B properties.

(I use the letters A and B tbrfamilies of properties - such as mental or moral properties; and I use the italicized letters F and G for particular properties within families - such as being in pain or being despicable.) Now, in any interesting case, B properties are likely to be complex conjunctive properties. How complex, we shall consider in a moment. Let us give such B properties a '*' to signify their conjunctive

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complexity. A 'B*' property is the 'subvening basis' of a particular instantiation of an A property.

What 'strong' supervenience gives us is this: if something has an A property F, then it has a B* property G* such that anything which is G* at any time and in all possible worlds is also F (see Kim 1984a). If strong supervenience holds, there are necessities linking B properties to A properties: there are some B* properties such that necessarily if anything has a certain B* property G*, at any time, in any world, then it has a certain A property F. The fact that something has a particular B* property G* is sufficient for it having the particular A property F. Its being G* necessitates its being F. This means that there are necessarily true conditionals which describe such necessitation relations. This strong notion of supervenience entails 'intra world' versions: A differences between two things (in the same world) entail B differences between them; and A changes entail B changes. Strong supervenience entails these 'weak' notions of supervenience.

I shall proceed here on the assumption that supervenience is best read in the strong fashion. We should be dealing with a strong notion of supervenience, which makes demands on what goes on in all worlds, not some puny weak notion which merely lays down constraints on what happens within a world. Metaphysics worth doing is worth doing properly! Weak supervenience is not committed to B-to-A necessities. I shall assume that supervenience is strong supervenience in what follows. This is essential at a certain point in the argument, and I shall mention this when we get there.

Supervenience means that every instantiation of an A property involves one-way B-to-A necessities. By contrast, 'reduction', as it is usually characterized, involves a two-way necessitation relation tying A and B properties to each other, not merely in one direction. Reductionism is the thesis that if something has some A property F, then it has some B property G such that necessarily something is F if and only if it is G.' Such two way necessitation relations can be described by necessarily true biconditionals. Sometimes philosophers put this another way by saying that the predicates 'F' and 'G' are necessarily coextensive.

Now it looks, on the face of it, as if supervenience-- with its B-to- A necessities-- is compatible with the absence of necessities going the

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other way, from A to B. A 'variable realization' or 'multiple realization' doctrine says that A properties can be realized by divergent B properties, and thus there are no A-to-B necessities. Since there are no A-to-B necessities, there are no two-way necessities linking A and B properties; and thus there are no necessarily true biconditionals. Variable realization means that there is no reduction. However, even though there may be no necessity that instantiations of A properties will always instantiate the same B property, it might still be necessary that something which instantiates a certain B property always instantiates a particular A property (given that the B property is a B* property). There can be B-to-A necessities without A-to-B necessities; A instantiations may have B sufficient conditions without having B necessary conditions. That's the whole point of employing the notion of supervenience. The hope is that we can have a determination relation without reduction, which many view with suspicion for a number of reasons.-"

However, Kim has an argument to the effect that, contrary to this appearance, strong supervenience does bring two-way necessities and necessarily true biconditionals in its train. Kim argues as follows. He asks us to construct a disjunction of the B bases of all actual and possible realizations of some particular A property F. Call that constructed B property "G#". G# = {G*~ or G*2 or G*3.,.}, where any member of that set would suffice for the instantiation of the property F. What Kim then points out is that this disjunctive B property, G#, necessitates and is necessitated by A. So the predicates 'F and 'G#' are necessarily coextensive. Thus we have a necessarily true biconditional describing the relation between the properties" G# and F: necessarily (Vx)(G# ~-> Fx). Since we have two-way necessitation relations and necessarily true biconditionals, and these are the stuff reduction is made of, it seems that supervenience yields reduction. That, in brief form, is Kim's formal argument.

1I if Kim is right, supervenience collapses into reduction. I shall try

to preserve the gap. On the way, we will learn something about the traditional notion of reduction.

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My argument will not hinge on the ungenuineness of disjunctive properties. Almost everyone who has discussed Kim's argument has gone down this road [e.g. Teller 1983]. In my view, this road is a dead-end, since no one has found a principled way of deeming some properties genuine and some second-rate [Zangwill 1993]. ~ It tends to be pure assertion. Most importantly, it is difficult to show that disjunctive properties are not causally efficacious and that they are not explanatory; for they are all that with respect to other similarly disjunctive properties [Contrast Owens 1989].

It is of ten said that not all Boolean constructions of physical properties are themselves physical properties. For example, maybe we do not want to say that not having any physical properties is a physical property (Post 1983). But this does not show in particular that disjunctive physical properties are not physical properties.' The fact that propertyhood (or physical propertyhood) is not preserved under some Boolean operations does not show that it is not preserved under others. If we are not to be completely liberal about property construction, we need some way of spotting which logical operations preserve propeltyhood?

In my view, and by contrast with the usual responses to Kim, it is best to admit straight off at least for the sake of argument-- that Kim is right that we can logically construct a perfectly respectable B# property which stands in a two-way necessitation relation with every A property. Call that 'Kim-reduction'. The question is: is Kim-reduction really reduction?

There is no point fighting over a word. In some sense Kim has definitely shown that supervenience entails reducibility. But there is intuitively more to the traditional philosophical notion of reduction than is captured by Kirn's construction. We can reconstrue Kim's argument as showing that this is so. The traditional notion of reduction involves more than the mere existence of such a two-way necessitation relation and necessarily true biconditionals.

What is this "more'"? This 'more', I suggest, is a certain epistemological dimension. ~ Of course, we might not know that a certain A property reduces to a

certain B property. To that extent, the notion of reduction is not an

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epistemological one. But the notion of reduction is not entirely devoid of epistemological significance.

The epistemological fall-out I am interested in is this: it must be the case that if we know that A properties are reducible to B properties, then we can use that knowledge in order to bwrease our A knowledge on the basis of B knowledge, and vice versa. Knowledge of reduction enables us to make predictions about unexamined cases. For example, once we know that water is reducible to H20, we know~ that any samples of H20 ( that we might come across will be water, and we also know that any samples of water that we might come across will be H20.

This, I shall argue, is what we cannot do with Kim-reduction2

111 The argument which follows will swing on the infinitude of the B#

property in the cases that interest us. Let us assume that we are dealing with the mind/body case, so A properties are mental properties and B properties are physical properties. (The argument applies just as well to moral and natural properties.)

We need to distinguish maximal B* properties from total relevant B* properties.

Suppose first that B* properties are maximal properties. Such properties are described by a complete description of every B property of the thing. Then a very tiny B difference--say, one molecule being minutely to the r ight--would generate a new B* property, which would nevertheless determine the A property. Since there are an infinite number of spatial locations, there are an infinite number of B* properties which can do the business of determining any A property. Thus B# properties are infinite disjunctions of such B* properties.

On the other hand, suppose that B* properties are not maximal properties but total relevant properties. Such properties are the coniunction o f only those B properties which are relevant to the determination of some particular instantiation o f the A property. Not all B properties that a thing actually has are necessary to determine the A property. Only some are. So we cannot argue, as we did with maximal B* properties, that a B set-up with one molecule slightly to the right would generate a new B* property, since that difference may

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not be a relevant difference." However, variable realization arguments are supposed to deliver the verdict that there are an infinite number of different kinds of B things which could realize (and thus determine) the A property. Mental properties are allegedly individuated by their abstract causal role. But it is said that the number of possible realizing Substances which can realize a causal role has no limit. Therefore if variable realization arguments go though and mental properties are individuated by their abstract causal role, there are an infinite number of kinds of substance which could realize an A property. So a disjunction of the possible total relevant determining B bases of an A property would be an infinite disjunction.

That there are an infinite rather than a very large finite number of possible kinds of realization tends to be assumed in the literature, probably because no one has ever given a reason why not. However, I suppose it is conceivable that, given the fundamental laws of nature, there are only a finite number of possible kinds of realizations of some complex causal role. I shall here drift along with the consensus that there are an infinite number of possible realizing substances. But those who are inclined to reject this premise can replace 'infinite' with 'indefinite' or 'finite but unknowably large' in the argument that follows. This will not make any difference to the argument.

So I am going to assume from here on that whether B* properties are maximal B properties or total relevant B properties, B# properties are infinitely disjunctive, or if they are not infinitely disjunctive, they have an indefinite or unknowably large number of disjuncts. It is at this point that the argument depends on supervenience being strong and not weak supervenience, because there are only a finite number of actual realizations of any mental property.

IV Let us now consider the demand that knowledge of reduction is

predictive in the light of Kim's infinitely disjunctive reduction. I shall first look at whether knowing that there is a B#

Kim-reduction for every A property could help us to predict the B* nature of something, given that we know that it has some A property, and vice versa. Then I shall look at whether knowing that there is a B# Kim-reduction tbr every A property could help us to predict the B#

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nature of something, given that we know that it has some A property, and vice versa.

Begin with the predictability of B* properties. A B# property is the disjunction of all possible B* determining

bases of an A property. So a particular B* property G* determines a particular A property F if and only if B* is a disjunct of the B# property to which F reduces. Now, how could we know the B# basis of an A property in such a way that it will be of some use to help us to predict which A property something will instantiate given that we know that it instantiates some particular B* property? Consider some particular B* property G*. Given a Kim-reduction of some A property F to some B# property G#, how can we know whether an object which satisfies G* also satisfies G#? The answer, surely, is that we cannot - - just because the disjunction is infinite (or indefinitely large). Two considerations support this. Firstly, we cannot even entertain such a reduction. So we cannot consciously search through the infinite number of disjuncts of G#, each of which would individually be sufficient to determine that F is instantiated, in order to find out whether the property G* is one among that infinite number. This infinite scanning is beyond our finite capabilities. And secondly, we have no more systematic means of seeing whether G* is a disjunct of G#, since, given variable realization at the B level, the B* disjuncts of G# have nothing in common beyond determining F. So for all we would know, G* would determine that F is not instantiated.

The predictive impotence of Kim-reduction is much more obvious if we take matters the other way round. We cannot predict what B* property something has given that we know only that it has a certain A property, since A properties may be variably realized in B properties. So merely knowing that it has some A property F does not allow us to infer anything about which B properties it instantiates.

Thus, unlike the water/H20 case, it seems that knowing that A properties reduce to B# properties does not allow us to predict B* properties on the basis of A properties or A properties on the basis of B* properties. Kim-reduction contrasts with normal scientific reductions. Given the knowledge that some unexamined thing is H20, we can predict that it is water, and vice versa. Not so with Kim- reduction.

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So far, I have also argued that Kim-reduction is non-predictive because, given that something is F, there is no way to find out if G* belongs to the reducing B# property G#. But we might get lucky. We might be wondering whether G*, say, determines F, and thus whether G* is a disjunct of G#, and we might casually walk into a room, and there, lo and behold, slap in front of us, there is something which is both G* and F. Then we would know that G* is indeed one of the disjuncts of G#. But the question we were concerned with is the question of how we can know whether it is a disjunct if we do not

come across an object with the G*/F combination. The problem is that of predicting whether unexamined G* things are F. If we are outside the room which houses something which is G'43 , say, we will be clueless as to whether it is F. The only exception would be where we have examined some A/B* combination, say, of F with G'42 in a previous room (room 42, presumably), where the G'42 basis differs only slightly from G'43. Then we could imagine inductive grounds, if such tiny differences had been irrelevant in the past, for predicting that G'43 will share the A property of G'42. So in such tiny-difference inductive cases there can be some prediction given Kim-reduction. However, what we cannot do is predict what A properties are possessed by something with a very different B* basis.

V Someone might complain that the only predictive knowledge we

should expect Kim-reduction to yield concerns B# properties, not their B* disjuncts. So let us consider the predictability of B# properties.

All we know is that any A property has some B# property to which it reduces. We cannot know which B# property it is. We are never going to be able to say: "Ah. I see that this thing has the B# property G#, so I reckon it must be F". And we are never going to be able to say "Ali, I see that this thing is F, so I reckon it must be G#". This is a considerable disanalogy with common-garden reduction.

The problem is exacerbated by the potential confusion between B# and G#, For every A property we know that there is some B# property to which it reduces. But we can never know any particular A-B# reduction because there is no B# property G# such that we can know that it is the reducer of some A property F. We can only know that

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there is some reducing B# property. But what the particular B# property is cannot be known. So even if we know in principle that some such reduction holds, It is quite impossible for us to know precisely what the reduction is.

Why, exactly, is it impossible to acquire knowledge of a particular B# property? As before, the answer is two-fold. Firstly, we cannot acquire knowledge of a particular B# property by knowing each of its B* disjuncts, for there are an infinite number (or indefinite number) of these. Secondly, we can know about some infinite sets without enumerating all their members, but only if we have some principle for generating the members. Bu t - -as we have noted-- this is not feasible for A properties which are variably realized in B properties, since at the B level there is no unifying principle.

Suppose---contrary to what we accepted in section I l l - - tha t we know that G# is a finite disjunction of B* properties. Suppose- - to go to the extreme-- that it is just a modest binary disjunction, and we know this. So we know that G#={G*~ or G*_,}" Then if we know that something is F, we cannot know if it is G*~. We can only know that it is G*, or G*2. But if we know that something is G*~ we can know that it is F. This is because we know the terms of the disjunction and that one of the disjuncts is satisfied. And if we know that something is F then we can predict that it will have the disjunctive property of being G*~ or G* 2, and vice versa. So in a finite binary disjunctive reduction, some prediction is possible. But this is just what is never possible in the case of Kim's infinitely disjunctive reductions. For in the case of Kim-reduction, we do not even know the terms of the disjunctions, unlike in the modest binary disjunctive reductive property just imagined. All we know is that in principle there is some B# property which determines and is determined by F, but we cannot know what it is. The infinitely disjunctive case is more like a case where A reduces to some simple binary disjunction which for some mysterious reason is unknowable.

Compare the situation with water and H20. If we examine one case where we know that we have the combination of water and H20, then we know that all actual and possible samples of water are H20 and that all actual and possible samples of H20 are water. (See Salmon 1981.) So we can make the requisite predictions. Now consider our situation.

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Of course, given one case where we know that we have a combination of F with G*, we can infer that all G* things are F, but not that all F things are G* (because of variable realization). But what about B# properties? If we could know that we had one case of a particular F / G #

combination, then we could indeed infer that all F things are G# and that all G# things are F - - a s in the water/H20 case. But the trouble is that we cannot know that something has some particular B# property G#. We are doomed forever to remain in the position that we were a

f e w centuries ago with respect to water and H20. No prediction of unexarnined cases is possible.

My argument has been a two-stage one: first the predictive infirmity of Kim-reduction with respect to B* properties; and then with respect to B# properties.

VI Before we conclude, let us briefly contrast the above line of

argument with other responses that there have been to Kim-reduction. I already noted, by way of advertizing, that I do not proceed, as

others have done, by ruling out all disjunctive properties as illegitimate. I regard that as cheating. If we are to believe that certain logical operations on properties do not preserve propertyhood, that must be where we arrive at the end of an argument, not where we begin. So I do not reject disjunctive properties per se. What I do reject are infinitely disjunctive reductions. So far as I can see, I am not thereby committed to rejecting infinitely disjunctive properties and infinitely disjunctive laws in general. '~ And I am not committed to rejecting simple finite disjunctive properties, laws and reductions. I argued only from the predictive infirmity of infinitely disjunctive reductions. If the argument carries over more generally to other infinitely disjunctive properties and laws and to simple finite disjunctive properties and laws, then my thesis ought to be expanded. But as far as I can see, my argument does not generalize in this way.

My strategy was not to rule out disjunctive properties and laws in general but to make a distinction within properties and laws--between those with and those without a certain epistemic virtue. Those with the virtue attain the status of reduction. Alternatively, we could say that the epistemic virtue also marks a distinction among reductions. And

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we can equally well reconstrue the argument as showing that only the properties, laws and reductions with the epistemic virtue are the real and true and genuine ones, while the others are not. The obvious thing to say about this, surely, is that it does not matter much which we say.

As I have said, many responses to Kim's challenge were question- begging. However, honorable exceptions to the question-begging response were proposed by William Seager (Seager 1991), and also by Kim himself (Kim 1992). Both Seager and Kim argued that disjunctive reductions are not genuine laws because they cannot be confirmed in the way that genuine laws can be confirmed. They worried about how we might find evidential support for Kim-reductions. My response is different, but their argument bears some relation to mine. I argued that even if Kirn-reductions hold, they are predictively useless, and in this respect they tell us so little that they are not worthy of the name 'reduction'. Both worries are broadly epistemological. But Seager and Kirn focus on confirmation whereas I focus on prediction. Moreover, unlike the confirmation argument, my argument appeals to the infinitude (or perhaps indefinite largeness) of the disjunctive reduction whereas Seager and Kirn worried that all disjunctions---even binary disjunctions--are not confirmed by their instances. My argument only applies to infinitely disjunctive reductions whereas their confirmation argument applies to simple disjunctions. The confirmation argument is thus much stronger than mine, and hence that much less likely to succeed. My solution to Kim's problem is safer than the one that Kim himself proposed. However, I shall not pursue any further analysis and assessment of the confirmation response in this paper.

VII To sum up then: knowledge of reduction should be something of an

epistemic accomplishment. We should know more when we have uncovered a reduction, and we should be able to put that knowledge to work. But beyond the bare formal knowledge that there exist Kim-style infinitely disjunctive reductions for every A property, we are not able to use that knowledge in order to increase our A knowledge given B knowledge, or to increase our B knowledge given A knowledge. Kim-reduction is no help when it comes to predicting which A

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property something will instantiate given that we know that it instantiates some B property, or vice versa. Hence the reduction which Kirn succeeds in constructing from supervenience cannot play the useful epistemological role that the traditional philosophical notion of reduction plays. The kind of reduction that Kim-reduction involves is epistemologicaily sterile." In a purely binecessitarian and biconditional sense of reduction, Kim's formal argument does show that supervenience entails reduction. But I hope to have suggested that, so long as we are dealing with a conception of reduction which is of some epistemologicai consequence, the supervenience/irreducibility combination is quite comfortable. Kim ' s formal argument does not show that supervenience entails reduction in this philosophically important sense.'-"

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NOTES ' Together with reasonable assumptions, it can be argued that this

account of reduction entails the usual formulation of theoretical reduction according to which one theory is reducible to another if and only if the causal laws of one theory are derivable from the causal laws of the other. (See Zangwill 1995.)

-" But see Zangwill 1992a, for an argument to the effect that the arguments for the variable realization of mental properties in physical properties were never very impressive. No one has definitely shown that mental properties are not type-type reducible to biological properties.

' Kim is right that turning from predicates to properties makes it less obvious that conjunctions and disjunctions are dubious. (1984, pp. 72-73.)

4 That there cannot be a quite general ban on disjunctive properties is shown by Kim's example of being an African emerald or a non-African emerald, which is a perfectly respectably property (1992, p. 32 1).

' 1 have some sympathy with James Van Cleve in (1990, 228-29) where he sees no reason to deny that P properties are closed under conjunction and disjunction, at the same time as thinking that they are not closed under negation.

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So I sympathize with Kim when lie says that reduction is an epistemic activity (1984, p, 74). 1 aim to explain why Kim-reduction falls short on epistemic grounds. My explanation differs from Kim's. Since submitting this paper, I see that William Child takes a similar line (but without supporting argument) in his paper "Anornalism, Uncodifiability, and Psychophysical Relations," Philosophical Review, 1993. See Zangwill 1992b and 1994 on a problem that this might be thought to generate. The case is then the same as Kim's Jade example: jade reduces to Jadeite or Nephrite. See his (1992). Kim gives the excellent example of the property or being more than a meter long, which may be construed as a perfectly legitimate infinitely disjunctive property (1984, p. 73). But in this case, we do have a principled way of knowing about the members of the infinite set. This is no objection to Kim, of course. He works with current accounts of reduction. Maybe Kim's argument serves to demonstrate the inadequacy of those accounts. 1 am grateful for comments from Jarnes Klagge, Jerry Levinson, Roger Teichmann, and especially Jim Edwards. This paper was delivered at the Pacific Division meeting of the American Philosophical Association, in Los Angeles, April 1994. Ronald Endicott gave a very helpful and thought-provoking reply. 1 am grateful for support from Glasgow University's Art's faculty research funds.

REFERENCES Blackburn 1971, "Moral Realism", reprinted in S. Blackburn, Essays on

Quasi-realism, Oxford: Oxford University Press, 1993. Blackburn 1985, "Supervenience Revisited", reprinted in S. Blackburn,

Essays on Quasi-Realism, Oxford: Oxford University Press, 1993. Davidson 1980, "Mental Events", Actions awl Events, Oxford: Clarendon. Honderich 1982, "The Argument for Anomalous Monism", Analysis. Kim 1978, "Supervenience and Nomological Incommensurables",

American Philosophical Quarterly. Kim 1984a, "Concepts Of Supervenience", Philosophy and

Phenomenological Research. (Reprinted in Kim 1993.) Kim 1984b., "Epiphenomenal and Supervenient Causation", Midwest

Studies in Philosophy, VoL IX, Causation and Causal Theories, University of Minnesota Press, 1984. (Reprinted in Kim 1993.)

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Kim 1990, "Supervenience as a Philosophical Concept", Metaphilosophy. (Reprinted in Kim 1993.)

Kim 1992, "Multiple Realization and the Metaphysics of Reduction", Philosophy and Phenomenological Research. (Reprinted in Kim 1993.)

Kim 1993, Supervenience and Mind, Cambridge: Cambridge University Press, t 993.

Moore 1903, Principia Ethica, Cambridge: Cambridge University Press. Owens 1989, "Disjunctive Laws", Analysis. Post 1983, "Comments on Teller", Southern Journal of Philosophy

Supplement. Salmon 1981, Reference and Essence, Princeton: Princeton University

Press. Teller 1983, "Comments on Kim", Southern Journal of Philosophy

Supplement. Zangwill 1992a, "Variable Realization: Not Proved", Philosophical

Quarterly. Zangwill 1992b, "Long Live Supervenience", Journal of Aesthetics and Art

Criticism. Zangwill 1993, "Supervenience and Anomalous Monism", Philosophical

Studies. Zangwill 1994, "Supervenience Unthwarted", Journal of Aesthetics and Art

Criticism. Zangwill 1995, "Psychophysical Reduction and Psychophysical Causal

Laws", lyyun.

* [This version of the paper replaces an earlier one, mistakenly published in vol. 24, nos. 3 - 4 - Ed.]

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