THE ASYMMETRIES OF PROPERTY SUPERVENIENCE

Ingvar Johansson

(Published in Sten Lindström and Pär Sundström, eds., Physicalism, Consciousness, and Modality, Umeå Preprints in Philosophy 2002:1, Department of Philosophy and Linguistics, Umeå University, pp. 95-124.)

‘Supervene’ means to come after, to be above, or to be in some sense higher than something else. In Anglo-American philosophy, a relation termed supervenience has been associated with a lot of overlapping characterisations. I will regard the ten ones below as desiderata of the original concept of a supervenience relation, i.e., the concept as, disregarding relata, it was used by Richard M. Hare and Donald Davidson. The main focus will be on cases where one kind of property supervenes on another kind of property, but in section five there are some remarks on relation supervenience, too. To be a supervenient property is to be a property that has a relation of supervenience to some other property, called subvenient property, or, as here, base property. Here are the desiderata:

1. Supervenient properties are determined by their base properties.
2. Supervenient properties are dependent on their base properties.
3. An entity has supervenient properties in virtue of its having base properties.
4. Base properties underlie supervenient properties.
5. Base properties realise supervenient properties.

6. Descriptions of base properties do not entail descriptions of supervenient
properties.

7. Supervenient properties cannot possibly exist without being connected to base
properties.

8. A supervenient property may have different base properties.

9. If two entities have the same base properties, then necessarily they have the
same supervenient property; or, base property indiscernibility entails
supervenient property indiscernibility.

10. If two entities have different supervenient properties, then necessarily they
have different base properties; or, supervenient property difference entails
base property difference.[i]

According to this list and the etymology of ‘supervenience’, a concept like ”symmetric supervenience” should not be used. It is as much a contradiction in terms as the concept ”being symmetrically above each other” is. In the second half of the nineties, however, ”symmetric supervenience” has got an established place in the philosophy of supervenience. Let me mention two books.

According to David M. Armstrong, in A World of States of Affairs (1997), an ”entity Q supervenes upon entity P if and only if it is impossible that P should exist and Q not exist, where P is possible”.[ii] This definition, as Armstrong himself explicitly says, leaves it open whether or not a specific relation of supervenience is asymmetric or symmetric.[iii] According to David Chalmers, in The Conscious Mind (1996), the ”template for the definition of supervenience is the following: B-properties supervene on A-properties if no two possible situations are identical with respect to their A-properties while differing in their B‑properties”.[iv] Here, as in Armstrong’s definition, the formulation allows the base properties to be supervening on its own supervening properties. Implicitly, Chalmers’ ”template” allows both symmetric and asymmetric supervenience.[v]

Jaegwon Kim has over more than two decades been a prolific and influential analyser of supervenience. At the beginning, and for a long time to come, he stressed the asymmetric or non-symmetric character of the supervenience relation,[vi] but nowadays he appears to be somewhat pessimistic about the possibility of making clear what the asymmetry of supervenience consists in. In Mind in a Physical World (1998), he distinguishes between covariance on the one hand and asymmetric dependence or determination on the other hand, and he says: ”What needs to be added to property covariance to get dependence or determination, or whether dependence/determination must be taken as an independent primitive, are difficult questions that probably have no clear answers”.[vii]

The aim of this paper is, firstly, to distinguish between different kinds of asymmetries that originally were connected to the relation of property supervenience; the concept of asymmetry is taken in a broad informal sense. I will take my departure in Chalmers’ ”template”. Secondly, the aim is to show what is involved in the move from asymmetric to symmetric supervenience. Of course, those who have proposed concepts of symmetric supervenience have not proposed self-contradictory concepts. Three kinds of asymmetries will be distinguished: asymmetry in covariance (section two), asymmetry between property sets (section three), and asymmetric internal relations (section four); the last asymmetry is by far the most important one. The fifth section is devoted to Armstrong’s concept of supervenience and D. Lewis’ concept ”Humean supervenience”. In the sixth and last section, I summarise my findings. Now, as a first section, I will make some introductory historical remarks.

1. The original concept of supervenience

The history of the concept of supervenience in the last fifty years of Anglo-American philosophy[viii] can be divided into three main ”starters” and corresponding problem areas: (i) the concept enters moral philosophy, (ii) it enters the mind-body problem, (iii) the analysis of the supervenience relation and its possible relata becomes in itself a philosophical problem. This paper belongs to the third area, but I will start with a few words about the other two.

Hare made the supervenience relation important in moral philosophy. In The Language of Morals (1952), he claimed that if there are two persons that are exactly alike and are acting in the same way in exactly the same kind of circumstances, then it is logically impossible to say that one of the persons is good but the other is not. In spite of this logical impossibility, Hare claimed, it is not the case that the persons’ characters and actions entail that they are good.[ix] Instead, he said, goodness supervenes on natural properties; the latter properties are good-making characteristics. Whatever the full meaning of Hare’s concept of supervenience is, it certainly contains both the view that moral goodness is not entailed by natural properties and the view that two persons in our world that are indiscernible with respect to natural properties are indiscernible with respect to moral goodness, too. In my opinion, the context makes it clear that he also was of the opinion that goodness is for its existence dependent on the existence of good-making natural properties.

The second problem area, that of psychophysical supervenience, can be regarded as initiated by Davidson’s two papers ”Mental Events” (1970) and ”The Material Mind” (1973).[x] Primarily, Davidson claimed that ”Anomalous monism shows an ontological bias only in that it allows the possibility that not all events are mental, while insisting that all events are material”, but he also said that this ”view is consistent with the view that mental characteristics are in some sense dependent, or supervenient, on physical characteristics. Such supervenience might be taken to mean that there cannot be two events alike in all physical respects but differing in some mental respect”.[xi] According to the latter idea, it is impossible for two persons to be in exactly the same kind of physical (neural) state but differ in mental state, i.e., two persons that are indiscernible with respect to physical states are indiscernible with respect to mental states, too. Like Hare, Davidson thinks that a description of the base properties in question does not entail a description of any supervenient property. He explicitly says that ”Dependence or supervenience of this kind does not entail reducibility through law or definition”.[xii]

As Hare and Davidson used the concept of supervenience, it contains at least the following three ideas, be they overlapping or not: (a) indiscernibility with respect to base properties implies indiscernibility with respect to supervenient properties (desideratum nine), (b) no description of base properties entail a description of a supervenient property (desideratum six), and (c) no supervenient property can (at least in this world) exist without being connected to a base property (desideratum seven). However, without argument, I claim that all the other desiderata fit their concept of supervenience rather nicely, too.

When a seemingly identical relation is found in two radically different contexts, two questions emerge quite naturally: (i) Is the relation really the same and only the relata different, or are there in fact two different relations?, and (ii) Can the relation(s) connect even more kind of relata?. In the late seventies and onwards, a lot of philosophers began to discuss both these problems. The third problem area was opened up, and a lot of different supervenience concepts (e.g. strong, weak, global) and proposed relata (e.g. properties, relations, predicates, theories) entered the philosophical scene. If one specific philosopher should be made the symbol of this explication problem, it ought to be Kim.[xiii]

In what follows, I will focus on supervenience as a relation whose relata are either properties or non-empty sets of properties; the properties are conceived of as simply inhering in things.[xiv] Also, I will disregard non-modal definitions of supervenience; such definitions do not at all fit the original conception.

According to Kim, ”Supervenience is standardly explained as a relation between two sets [italics inserted] of properties over a single domain of individuals”.[xv] I have already quoted Chalmers template for the definition of supervenience, according to which ”supervenience is a relation between two sets of properties”.[xvi] He calls it a template, because it is an attempt to sum up in a single formula most of the proposed different definitions of supervenience. It is good a starting point, but I will at once make a terminological change. I will exchange Chalmers’ terms ‘B-properties’ and ‘A-properties’ for ‘S(upervenience)-properties’ and ‘B(ase)-properties’, respectively. This terminology makes it easier to remember when a supervenient property is referred to. Another reason for the change is the fact that Chalmers is using the terms ‘A-properties’ and ‘B-properties’ in a way opposite to that of several other philosophers.[xvii] With my substitutions, Chalmers’ template reads as follows: S-properties supervene on B-properties if no two possible situations are identical with respect to their B-properties while differing in their S-properties. It is equivalent to what Kim and B.P. McLaughlin call the core idea of supervenience.[xviii]

In this template, the modal operator ‘possible’ is a variable that can take on different values such as ‘nomically possible’, ‘causally possible’, and ‘logically possible’ (the first two values give us natural supervenience, and the third one logical supervenience); the term ‘situations’ is a variable that can take on values such as individual things and persons (= local supervenience) as well as whole worlds (= global supervenience).[xix] Since the primary aim of this paper is to elucidate the original concept of supervenience, I will restrict myself to logical and local supervenience. Neither Hare nor Davidson meant that they were doing natural science. In fact, Hare were only indirectly concerned with properties. Directly, he was merely interested in supervenience as a relationship between evaluative and purely descriptive words.[xx] I will, to start with, use as my definition of supervenience a reformulation and specification of Chalmers’ template. Requirements with respect to spatial and temporal locations are left out of account, and it is taken for granted that the sets are distinct and non-empty. It looks as follows:

A set of properties S supervenes on a set of properties B, if and only if,
it is logically necessary that: any two individuals x₁ and x₂ that
have the same properties in B have the same property in S.

This is a definition of what I will call single-modal supervenience and single-necessity supervenience. One can easily relate it to a certain domain D, and require that x₁ and x₂ belong to D. The definition contains exactly one modal operator, a necessity operator. There is, however, also a weaker formulation that contains two necessity operators.[xxi] All sets that conform to the single-necessity definition conform to the double-necessity definition. Chalmers comments on the last definition only in a footnote.[xxii] The formulation that I will later use looks as follows:

A set of properties S supervenes on a set of properties B, if and only if,
    it is logically necessary that: (for any property in S, if x_n has S_j,
    then there exists a base property B_i such that x_n has B_i, and
    it is logically necessary that: (any x that has B_i has S_j)).

In sections two and three, I will only discuss single-necessity supervenience, but then double-necessity supervenience has to be brought in, too

2. Supervenience and asymmetry in covariance

As I have presented property supervenience, it is some kind of modal relation between sets whose members are properties (monadic universals, types) or property instances (tropes, tokens); and that is the way I want to discuss it. My conception of properties is what Kim has called a ”sparse” conception.[xxiii] This notwithstanding, some aspects of the supervenience relation can be read off more easily if one regards the relation as being representable by a set of ordered triples. Just as ordinary relations and functions can be represented by (and, some would say, even identified with) sets of ordered n-tuples, supervenience relations can be so represented. Any ordinary real valued function z = f(x, y) can be represented by a set of ordered triples <x, y, z>, where x, y, and z are variables for real numbers. A specific function formula can then be regarded as a requirement that generates the function-set from the set of all possible such ordered triples; e.g. the function x = y generates the set of all ordered pairs where the first and the second member is exactly the same real number. In the same way, the definition of single-necessity supervenience can be regarded as a general requirement that out of a given set may generate a specific (sub)set. Let me show.

Consider the set of all ordered triples <B, S, x>. S is a variable for one kind of supervenient properties (S_j = S₁, S₂, S₃, ...), but it will also be used as a name of the corresponding set of properties. If S is the set of all mental states, then both S and S_j will be used as variables for the specific mental states (S₁, S₂, S₃, ...) that are the elements of the set. I think this ambiguity will show itself to be innocent. Similarly, B is a variable for the corresponding base properties (B_i = B₁, B₂, B₃, ...) and a name of the corresponding set; note, though, that in this representation S_j and B_i cannot represent property instances (tokens). The third variable, x, is a variable for individuals (x₁, x₂, x₃, ...), but the name of the corresponding set will be D. When nothing else is explicitly said, D will be the universal domain, i.e., the set of all actual and possible individuals.[xxiv] All three variables are allowed to take the value zero. In this kind of representation, an ordered triple, e.g. <B₇, S₉, x₂>, represents two simple facts: the fact that x₂ has the property B₇, and the fact that x₂ has the property S₉. Note that if S is the set of mental states and if B is the set of human bodily states, then every specific base property, B_i, is a complex property that is constituted by several different kinds of physico-chemical properties.

Out of the set of all ordered triples <B, S, x> so defined, of course, immensely many subsets can be constructed. Some of these sets conform, and some do not conform, with the definition of supervenience in the sense that they do not contain any two triples like <B₁, S₁, x₁> and <B₁, S₂, x₂>, i.e., they do not contain two members that represent two individuals that instantiate the same property in B but different properties in S. Among the conforming sets, there is one set (or some equally large sets) that has more members than the other sets. Such a maximal subset, I will call a supervenience-set of <B, S, x>, and I think that for some purposes it is convenient to represent supervenience relations by such sets.

In this kind of representation, the desiderata nine and ten are equivalent. Both these desiderata (nine: ”base property indiscernibility entails supervenient property indiscernibility”, and ten: ”supervenient property difference entails base property difference”) generate the same the supervenience-set. Both of them forbid, and forbid only, that the supervenience-sets contain two triples like <B₁, S₁, x₁> and <B₁, S₂, x₂>. These desiderata are logically equivalent in spite of the fact that they need not in ordinary language be exchangeable. Utterances of desiderata nine and ten, respectively, may have different semantic presuppositions.[xxv] For instance, it would be odd to talk about desideratum ten in cases where it is presupposed that there is only one single property in the set S.

A supervenience-set can contain members like <B₁, S₁, x₁> and <B₂, S₁, x₂>, i.e., it can contain members that represent individuals that have different B-properties but the same S-property. Moreover, since D is the universal domain, it has to contain them; this is required by desideratum eight. Moral goodness can be realised by several different kinds of personal character traits, and it seems reasonable to assume that a certain mental event may supervene on some different kinds of neural events. Originally, supervenient properties were always regarded as being ”multiply realisable”. Of course, this requirement is not met by symmetric supervenience. In symmetric supervenience there is a one-one correlation between B‑properties and S-properties, but in a real supervenience-set there is a many-one correlation.

When supervenience is represented by supervenience-sets, symmetric supervenience can be regarded as a special case of supervenience in general. One gets a symmetric supervenience-set by adding a further requirement that creates a subset from the real supervenience-set, i.e., one gets a sub-subset of the set of all <B, S, x>. The requirement is of course that no two members like <B₁, S₁, x₁> and <B₂, S₁, x₂> are allowed into the set. Multiple realisability is forbidden, and one gets a one-one correspondence between specific B‑properties and specific S-properties. Symmetry has arisen.

Let us now leave this set-theoretic representation of supervenience and see what happens if, with the insight we have gained, we return to the ordinary modal formulation. What was called the real supervenience-set was not just any subset. It was the maximal set conforming to ”the supervenience restriction”. In order to get the corresponding modal formulation, a formulation that has multiple realisability built into itself, both a necessity operator and a possibility operator is needed, i.e., a double-modal formulation is needed:

A set of properties S supervenes on a set of properties B, if and only if,
(a) it is logically necessary that: any two individuals x₁ and x₂ that
have the same property in B have the same property in S, and
(b) it is logically possible that: two individuals x₁ and x₂ that
have different properties in B have the same property in S.

In order to capture the original conception of supervenience, the idea of multiple realisability has to be part of the definition. The necessity operator has to be complemented by a possibility operator.[xxvi] Chalmers’ template is too weak. It should have looked as follows: S-properties supervene on B-properties if (i) no two possible situations are identical with respect to their B-properties while differing in their S-properties, and (ii) there are possible situations with different B-properties but with the same S-property. Single-modal supervenience has to be replaced by double-modal supervenience. When this is done, one realises that symmetric supervenience cannot be regarded as merely a specification of an original and general definition of supervenience. In fact, in the move from supervenience to symmetric supervenience, a modal possibility operator is substituted by a modal necessity operator; clause (b) above is substituted by clause (b’) below:

A set of properties S supervenes symmetrically on a set of properties B,
if and only if,
(a) it is logically necessary that: any two individuals x₁ and x₂ that
have the same property in B have the same property in S, and
(b’) it is logically necessary that: any two individuals x₁ and x₂ that
have the same property in S have the same property in B.

To sum up. If supervenience relations between properties are regarded as being in all their aspects representable by ordered triples, then symmetric supervenience can be regarded as merely a special case of supervenience in general. However, if the modal formulations are, as they should be, taken into account, it becomes clear that symmetric supervenience is not a special case of supervenience in general. Those who take multiple realisability away from the supervenience relation do not create a species concept of the original genus concept of supervenience. They create, so to speak, another genus concept.

It might be said that I have spent many words on making a rather trivial point. In his overview paper ”Varieties of Supervenience”, B.P. McLaughlin rested content with saying more or less the same thing merely in a footnote. It reads:

A strictly terminological point: I speak of dependent-variation, rather than following Kim (1990) in speaking of ”co-variation.” The reason is that, to my ear at least, ’co-variation’ suggests both that A-respects cannot vary without variation in B-respects and that B-respects cannot vary without variation in A-respects. But the claim that A-respects supervene on B‑respects does not, of course, imply that there can be no difference in B-respects without a difference in A-respects. For example, even if there can be no difference in mental respects without a difference in physical respects, it may nevertheless be that physical respects can vary without variation in mental respects. We could distinguish one-way covariation from two-way covariation, and claim that supervenience entails only one-way covariation. But I think it is preferable to speak instead of ’dependent-variation’. My differences with Kim here are, of course, merely verbal.[xxvii]

This was published 1995, and I agree with everything that McLaughlin says, but I think that he was too optimistic. Sometimes, what starts as a mere terminological difference may in the end mark a small but substantial difference. In this case, the substantial difference has to do with how the content of the original supervenience concept is conceived. If the difference between one-way (= asymmetric) and two-way (= symmetric) covariation had been kept clearly in sight, then, I guess, no new construct of a dependence relation would have been dubbed symmetric supervenience. No doubt, the original supervenience relation contains, in its kind of covariance, an asymmetry.

3. The property set asymmetry in supervenience

When a supervenience relation is represented by a supervenience-set of ordered triples <B, S, x>, the existence of three non-empty sets (B,S, and D) is taken for granted. The domain of individuals, D, is, I have said, universal. It is the set of all actual and possible individuals. But, one may ask, are there in the original conception of supervenience any explicit or implicit requirements on the property sets B and S? The answer is ‘Yes’.

One requirement on B and S can be called the principle of coinstantiation. It says that a supervenient property and its base properties should be instantiated in the same individual at the same time.[xxviii] Moral goodness should be where its good-making characteristics are, and a mental state should be where its neural base is. In my opinion, this principle rules out, a priori, some property sets from being connected by a supervenience relation. Some properties are in their essence such that they cannot simultaneously be instantiated in one and the same individual. This state of affairs has been much discussed in the literature around the determinate-determinable distinction.[xxix] An ordinary thing cannot as a whole be both red and green, nor can it be both spherical and cubical, nor have two masses. Such an individual can have only one (overall) colour, one shape, and one mass. Therefore, a colour cannot supervene on another colour, a shape cannot supervene on another shape, and a certain mass cannot supervene on another mass.

When not regarded as naturally given, property sets can be constructed both by means of similarity relations and by picking properties completely at random. However, no one concerned with the supervenience relation has ever argued that sets of randomly collected properties can be either sets of supervenient properties or sets of base properties. Try to think, for instance, of the set consisting of the three members ’having a mass of 1.518 gram’, ’having a volume of 2.373 m³, and ‘being pink’ as being a supervenient property set. It seems impossible. All philosophers of supervenience have relied on some implicit distinction between natural sets of properties and artificial sets of properties. I will rely on it, too, but explicitly. This does not mean, though, that I will try to analyse the concept of a natural property set. Let it just be said, that in all such sets all the members are rather closely linked to each other by similarity relations.

Back again to the set of all possible triples <B, S, x>. We can now say that B and S must name natural property sets, and that each triple <B, S, x> represents a synchronic coinstantiation of B-properties and S-properties. But what about the relation between, on the one hand, the set of all possible <B, S, x> and, on the other hand, the natural sets B and S?

First, x ranges over all actual and possible individuals, and it is the normal thing to let S range over all the members of the set S. But what about B? In the set we start with, every value of S (apart from zero) is correlated with at least one value of B (apart from zero), but this correlation need not necessarily exhaust the members of the set B. With hindsight, it is always possible to construct the set B in such a way that, by definition, it contains no member that is not correlated with a member of S. However, this construction may very well turn the original and natural set B into an artificial set of properties. Let us take a quick look at the sets at work in Davidson’s and Hare’s writings, when these are realistically conceived.[xxx]

In the case of Davidson, S is the set of all mental events, and B is the set of all physical events; and he writes (as earlier quoted) that he ”allows the possibility that not all events are mental, while insisting that all events are material”. This means that there may be members in B that are not correlated with any S. In this sense, the set S is smaller than the set B; and if panpsychism is neglected, S has to be smaller. In relation to Hare, I think we have to say that S is a set with only one member, the property of being a morally good person. Then, because of multiple realisability, B has to contain more than one member. However, I very much doubt that there is a pre-given natural set B such that all its members will be connected with the member in S.

The point that I want to make is very simple. In the first applications of the original concept of property supervenience, there is a kind of asymmetry between the property sets that are the relata of the supervenience relation. The set of supervenient properties S cannot possibly contain more elements than the set of base properties B, whereas B probably is larger than S. In symmetric supervenience, there can be no such asymmetry.

4. Supervenience as an asymmetric internal relation

So far, my discussion has been related only to single-necessity supervenience, and I have focussed on the last three stated desiderata of supervenience, i.e., ”multiple realisability”, ”base indiscernibility entails supervenience indiscernibility”, and ”supervenience difference entails base difference”. Now I will try to take some of the other desiderata into account as well. For instance, one may ask whether the first two desiderata, the requirements that supervenient properties are determined by and dependent on their base properties, are really met by (double-modal) single-necessity supervenience conceived of as necessary asymmetric covariation. However, I think it is easier to start with a look at the seventh desideratum. It has, I claim, surely not been met from a strictly logical point of view. Let me explain.

According to desideratum seven, no single S-property in the set S can possibly be instantiated without a simultaneous coinstantiation of a base property B_n. This requirement is stronger than those of desiderata nine and ten are. The latter are only concerned with properties considered pair-wise. Therefore, they allow the supervenience-set to contain one, if only one, member like <0, S_j, x_m>, i.e., one individual that lacks a base property but has a supervenient property S_j. Desiderata nine and ten do not prohibit a supervenience-set from containing two members like <0, S_j, x_m> and <B_i, S_j, x_n> if there is no member <0, 0, x_m>. Of course, in every actual case where the property sets B and S are specified, it seems reasonable to assume that there are at least some (and even many) individuals that lack both B-properties and S-properties; this is so since the domain of individuals is the universal domain that allows possible individuals as well. When this is the case, the covariance component of supervenience requires that all individuals that lack a B-property lacks an S-property, too; i.e., the supervenience-set cannot contain a member like <0, S_j, x_m>. However, the covariance requirement is met by property sets B and S which are such that, for instance, (a) there is no actual or possible individual that lacks an S-property, but that nonetheless (b) there is one and only one individual that lacks a B-property. These sets meets the covariance requirement of supervenience (desiderata nine and ten), but they do not meet requirement number seven.

Let me now turn to double-necessity supervenience; see formulation at the end of section one. Here, explicitly, even the point <0, S_j, x_m> is prohibited from becoming part of a supervenience-set. The definition makes each and every S-property necessarily connected with a B-property.

The analysis made of single-modal and single-necessity supervenience has repercussions on the original formulation of double-necessity supervenience. Since a possibility operator had to be added to the original single-necessity formulation, it had to be turned from a single-modal into a double-modal formulation. For exactly the same reason, the double-necessity and double-modal formulation has to be replaced by a triple-modal but, still, double-necessity formulation. When the possibility operator that guarantees multiple realisability is added, the definition looks as follows (the numbering of the clauses reflects the order in which they are introduced in this paper):

A set of properties S supervenes on a set of properties B, if and only if,
(2a) it is logically necessary that: (for any property in S, if x_n has S_j, then
    there exists a base property B_i such that x_n has B_i, and
    (1a) it is logically necessary that: any x that has B_i has S_j, and
    (1b) it is logically possible that: two individuals x₁ and x₂ that
    have different B-properties have the same property S_j).

Now and then it has been claimed that the supervenience relation consists of two quite distinct relations. T.R. Grimes has argued that dependency and determination are distinct,[xxxi] whereas Kim keeps dependency and determination together but separates them from covariance.[xxxii] C. Macdonald has said that ”dependence is probably best seen as an independent component of psychophysical supervenience, to be justified independently of appeals to supervenience”.[xxxiii] In my opinion, Grimes conflates dependency and multiple realisability, Kim thinks mistakenly that the concept of dependence needed is inexplicable, and Macdonald (apart from wrongly dissociating dependence from supervenience) ties dependence too much to ordinary causality. Nonetheless, I think that all of them point in the right direction: the (original) relation of supervenience contains two relations. This direction lies implicit in the formulation of double-necessity supervenience. So, let us take a closer look at it.

In the last definition there are three modal clauses. Clauses number (1a) and (1b) have already been discussed. Together, (1a) and (1b) make up what I will call (1) the covariance component of supervenience. Clause number (2), I will call the dependence component of supervenience.

To start with, let us isolate clause number (2) from the rest. Separated, it says only that it is logically necessary for any property S_j, that if x_n has S_j, then x_n has to have some base property B_i, too. In the way explained in section two, the covariance component of supervenience can be represented by a supervenience-set; a specific set of ordered triples <B, S, x>. Looked at in this representation, the dependence component of supervenience can be regarded as just a further requirement that can be added to the (asymmetric) covariance requirement in order to create a new subset from the original supervenience-set; let me call this new set the supervenience-set proper. The new dependence-requirement is that such a supervenience-set must not contain any members like <0, S_j, x_n>.

At first, the adding of the dependence-requirement looks like a very small change. It seems merely to mean that no member like <0, S, x> can be part of a proper supervenience-set. If the sets B, S, and D are regarded as the three dimensions of an abstract three-dimensional space, a supervenience-set is a volume in this space. With respect to such a space, clause number two merely implies that a proper ”supervenience volume” cannot even in one single point touch the surface of the S‑D‑plane. Since I have already said that I do not think that there are any supervenience-sets that in fact have a single point in the S-D-plane, why bother about the possibility of this point?

The real reason to bother is that there might be a curious kind of relation hiding beneath the set-theoretic representation of the dependence component. First observation, the dependence component is at bottom not a relation between two sets. At most, it is a relation between one member of a set and a whole another set; whatever kind of relation that would be. The unique S_j cannot be instantiated if not some member of the property set B is instantiated. Second observation, the relation in question seems to be a close kin to the kind of internal relation that was very dear to nineteenth century British idealism, but that Russell, for one, tried to ban from philosophy. Bradley claimed that all relations are internal, Russell claimed, contrariwise, that all non-formal relations are external. The supervenience relation requires, I will argue, that some substantial relations are internal. What then is a Bradley-internal relation?

According to A.C. Ewing’s overview of British idealism, the best definition of internal relations is the following: A is internally related to B by the relation r when it is logically impossible for A to exist unless B existed and was related to it by r.[xxxiv] This definition should be compared with the following minor reformulation of the dependence component of supervenience: It is logically impossible for any S_j to exist unless some base property B_i exists. Obviously, there is a kinship, and that I will explore.

First a conceptual remark. What was originally called an internal relation is not exactly the same kind of relation that Armstrong and a lot of other contemporary philosophers call an internal relation. In order to keep these relations conceptually distinct, I will call them Bradley-internal relations and Armstrong-internal relations, respectively. In order to make this important difference more clear, I will return to Armstrong’s views in section five, but I will make one comment at once. In my opinion, the so to speak manifest intension of Armstrong’s concept of internal relation is the same as that of Bradley-internal relations, but there is a latent intension at work as well, an intension that makes Armstrong-internal relations identical with a kind of relations that I will call grounded relations. Both Bradley-internal and grounded relations can be contrasted with still another kind of relation, external relations. Let me give a very brief explanation of this tripartition.[xxxv]

Two individuals that stand in an external relation can change this relation but nonetheless remain exactly the same with respect to their monadic properties. Spatial relations between things are the prime examples of external relations. Obviously, a spatial distance between two things may change without any change in the monadic properties of the things considered. In case of grounded relations between individuals, this is impossible. ’Being longer than’ is such a relation. In order to change the fact that, for instance, a is longer than b, then either a or b has to be changed. Exact similarity (in a certain respect) is another grounded relation. If the fact that two individuals are exactly similar is to be changed, then necessarily at least one of the individuals has to be changed. Associated with this ontological difference between external relations and grounded relations, there is an epistemological difference, too. In principle, knowledge about all the monadic properties of two individuals allows one to derive and get knowledge of all their grounded relations. But the same knowledge about monadic properties allows no valid derivations that give rise to any knowledge of the external relations between the individuals in question. To be at a certain place in space is not a monadic property; it is a relation between an individual and space.

From an epistemological point of view, Bradley-internal relations are much closer to grounded relations than to external relations. Nonetheless, there is an epistemological difference even between Bradley-internal and grounded relations; a difference that reflects a deep ontological divide. When two relata of a grounded relation are known to be instantiated in two individuals, the instantiation of the grounded relation can be derived. Similarly, when the two relata of a Bradley-internal relation are known to be instantiated, the instantiation of the Bradley-internal relation can be derived. However, in the latter case, this derivation is possible even when one explicitly knows only that one of the relata is instantiated. This is so, because in a Bradley-internal relation it is logically impossible for one of the relata to exist without the other, and vice versa. In a grounded relation, e.g. ’being exactly similar’, such a derivation is quite impossible. Of course, a colour spot with a certain hue can exist independently of whether or not there is somewhere else another spot with exactly the same hue.

A Bradley-internal relation is a relation whose relata are necessarily coinstantiated, whereas the relata in both external and grounded relations can be instantiated independently of each other.

In the history of modern philosophy, the concept of Bradley-internal relations is not the only concept that seems to be close to the dependence component in the definition of double-necessity supervenience. What Bradley expressed by saying that between A and B there is an internal relation, Brentano could have expressed by saying that A and B are parts that are necessarily non-separable, and Husserl could have expressed it by saying that A and B are connected by a law of essence. In the phenomenological tradition, at least its realist part, these ideas of the founding fathers were later elaborated into a more general concept of existential dependence.[xxxvi] There are of course many differences between these concepts from Bradley, Brentano, and Husserl; differences that have to do with the rest of their very different philosophical outlooks. For the purposes of this paper, all these differences except one can be neglected.

When Brentano introduced his idea of complex entities that have non-separable parts, he claimed that such non-separability had two forms. It could be either mutual (symmetric) or one-sided (asymmetric). Similarly, Husserl and several later phenomenologists, Roman Ingarden in particular, distinguished between mutual and one-sided existential dependence. In Bradley, however, there is no corresponding distinction between symmetric and asymmetric internal relations. As far as I can see, nothing in the concept of internal relation itself prohibits it from being used to distinguish between two such forms of internal relations. But, of course, if Bradley had allowed it, it would have had serious repercussions on his very tight holism. Here, however, where we are interested neither in the metaphysics of British idealism or in the overarching aspects of phenomenological philosophy, but in the relation of supervenience, we can neglect such consequences, whatever they are.

According to the last formulation of the dependence component, such a dependence means that it is logically impossible for any S_j to exist unless some base property B_i exists. Consequently, a description of the existence of S_j entails a description of the existence of some B_i. Note that this is quite consistent with desideratum six, which goes in the other direction. It says that no description of base properties can entail a description of a supervenient property. I think that the dependence component had better be formulated in such a way that it not only is consistent with desideratum six, but that it has this desideratum as an explicit part of itself. One will then get a formulation that corresponds to the ideas of one-sided existential dependence and asymmetric internal relations. If one claims that it is logically possible for each B-property to be instantiated without a coinstantiation of an S-property, then one has also claimed that no description of the mere existence of B_i can entail the description of the existence of some S_j; and then desideratum six is met.

In section three, I claimed that, when fully spelled out, the covariance component of supervenience contains both a necessity operator and possibility operator. Now, I make a similar claim with respect to the dependence component. It should be regarded as containing both a necessity operator and possibility operator, too. In order to capture the whole of the original supervenience relation, at least a four-modal and double-necessity formulation is needed:

A set of properties S supervenes on a set of properties B, if and only if,
(2b) it is logically possible that: (VB_i)(VS_j)(Ex)(B_ix and ¬S_jx), and

(2a) it is logically necessary that: (for any property in S, if x_n has S_j, then
there exists a base property B_i such that x_n has B_i, and

    (1a) it is logically necessary that: any x that has B_i has S_j, and
    (1b) it is logically possible that: two individuals x₁ and x₂ that
    have different B-properties have the same property S_j).

Clause two, clauses (2a) and (2b) together, captures the idea of one-sided existential dependence or of asymmetric Bradley-internal relations. With respect to asymmetry in covariance and in property set asymmetry, there is no need to distinguish between a discovery of the asymmetry in itself and a discovery that the corresponding concepts are non-contradictory and coherent notions. With respect to asymmetric Bradley-internal relations things are not that easy. Empiricist-minded and atomist-minded philosophers may find the very idea of Bradley-internal relations crumbling. However, since the aims of this paper merely are to lay bare the structure of the original supervenience relation and to show that it contains asymmetries, I will here by and large rest content with pointing at the concepts of asymmetric internal relations and of one-sided existential dependence. But, of course, I regard the concepts as coherent, and will make one small attempt to show that they are. A non-coherent notion is not really applicable anywhere, but there is to my mind at least one application of the concepts of asymmetric internal relations and one-sided existential dependence.

In my opinion, phenomenal colour is one-sidedly existentially dependent on phenomenal (= perceived) extension. It seems (cf. 2a) to be logically necessary that when a phenomenal colour is instantiated, then it is instantiated in something that has a spatial extension.[xxxvii] This necessity seems to me just as certain as the necessity of getting three when I add the pure mathematical numbers one and two. But it seems (cf. 2b) quite possible to perceive a specific extension without perceiving any colour in it. In fact, we do it more or less continuously since, normally, we perceive an empty and colourless distance between us and the things we perceive. If moral goodness supervenes on natural properties, and if mental properties supervene on material properties, then, apart from covariance, moral goodness has to have an asymmetric internal relation to natural properties, and mental properties has to have such a relation to material properties.

5. Asymmetry in relation to Armstrong’s and Lewis’ concepts of supervenience

I will now bring in Armstrong’s definition of supervenience that I quoted in the introduction. Although it is a single-modal and single-necessity definition, it differs from the ordinary one that (in Chalmers’ formulation) I have discussed. According to Armstrong’s definition, an S‑entity S_j supervenes on a B‑entity B_i if and only if it is impossible that B_i should exist and S_j not exist; or, as he himself says: ”supervenience in my sense amounts to entity B entailing the existence of entity S”.[xxxviii] Since Armstrong does not speak of pair-wise instantiation but simply of instantiation, his definition is stronger than the ordinary single-necessity formulation. Everything that conforms to Armstrong’s definition conforms to Chalmers’. If it is impossible that B_i should exist and S_j not exist, then, of course, it is also impossible that two individuals should instantiate B_i but not instantiate the same S-property. However, the converse relation does not obtain. From the premise ‘it is logically necessary that: any two individuals x₁ and x₂ that have the same property in B have the same property in S’ one cannot draw the conclusion that ‘it is logically necessary that: if one individual x_n has a certain property in B then x_n has a certain specific property in S’. Chalmers’ template does not imply that B_i entails S_j, even though he himself sometimes lapses into talking about such entailments.[xxxix]

Armstrong’s definition is quite general, but when both the B‑entity and the S‑entity are properties, then, in fact, his explicit definition of supervenience would capture the intension of the concepts of one-sided existential dependencies and asymmetric internal relations if only the phrase ‘but not vice versa’ were added. However, in Armstrong’s single-necessity definition of supervenience, the one-sided existential dependence discussed in section four is turned around. In the double-necessity formulation discussed, the existence of a supervenient property entails the existence of a base property, but in Armstrong’s definition the supervenient property is entailed by the base-property. In fact, Armstrong’s definition of supervenience directly violates desideratum six, which, in conformity with Hare’s and Davidson’s concepts of property supervenience, says that no description of base properties should entail a description of a supervenient property. If someone doubts the importance of this desideratum, he should consult Hare’s second thoughts on supervenience.[xl] Armstrong’s concept of supervenience allows the existence of symmetric supervenience simply because his definition does not meet all the classical desiderata of supervenience.

From what has been said so far, one could think that Armstrong’s symmetric property supervenience is the same as a Bradley-internal relation. But this is wrong since Armstrong qualifies his explicit definition by adding that ”Symmetrical supervenience yields identity”.[xli] If we had been discussing predicate (not property) supervenience, this would have been an understandable position. Two different predicates can supervene on one another and be identical in the sense that they are intensionally and/or extensionally equivalent. But nothing similar can be said in relation to properties. To say that two different properties are identical is nonsense. This means that Armstrong, who is a realist, is forced to admit that two distinct properties cannot, in his sense of supervenience, be symmetrically supervenient on each other. In turn, this means that such symmetric supervenience relations should be kept distinct both from Bradley-internal relations and from mutual existential dependence relations.

Armstrong’s concept of supervenience, however, is both coherent and has applications; the kind of dependence relation it denotes will in what follows be called Armstrong-supervenience. With respect to properties, there is to my mind one good example of asymmetric Armstrong-supervenience. Determinables are Armstrong-supervenient on their determinates. If a certain determinate (e.g. ‘having a circular shape’) is instantiated in an individual, then, necessarily, its determinable (e.g. ‘having a shape’) is instantiated, too. Armstrong, I am sorry to say, cannot use this example since he is a realist only with respect to quantifiable determinables that are part of natural laws.[xlii]

Applied to external and grounded relations (i.e., to non-Bradley-internal relations), Armstrong’s definition says that a relation S_n supervenes on an ordered pair of properties <B₁,B₂> if and only if it is impossible that the ordered pair should exist and S_n should not exist. From the explications of the concepts of external and grounded relations made in section four, it follows that no external but all grounded relations are Armstrong-supervenient on the ordered pair made up of their relata. For instance, the relation ‘being longer than’ (= S_n) supervenes on the ordered pair <5 cm, 3 cm> since it is impossible that this ordered pair should exist and S_n not exist. If a 5‑cm-individual and a 3‑cm-individual exists, then necessarily the relation ‘being longer than’ exists, too. It is asymmetric supervenience since ‘being longer than’ can exist even if the ordered pair <5 cm, 3 cm> does not exist, a pair like <101 cm, 7 cm> can equally well be the base entity. If the relation had been more specific, e.g. ‘being 2 cm longer than’, the ordered pair <101 cm, 99 cm> could have been a base entity, too. Remember, though, as explained earlier, that all these relata can very well exist independently of each other. Grounded relations do not Armstrong-supervene on their relata taken distributively; they supervene on their relata only when these are taken collectively as in an ordered pair. This means that there is a special asymmetry between a grounded relation and each of the relata taken distributively. The relation is existentially dependent on each relatum, but no relatum is in itself for its existence dependent on the relation.

Now a quotation from Armstrong:

An important case for us that falls under these definitions, and which will also serve here to illustrate them, is that of internal relations. ’Internal relation’ itself is an ambiguous term. But in this work it will be said that a relation is internal to its terms if and only if it is impossible that the terms should exist and the relation not exist, where the joint existence of the terms is possible. Or again, the joint existence of the terms being possible, they entail the existence of the relation.[xliii]

When Armstrong’s general definition of supervenience is specified for relations, it looks a little different from the specification for properties; the base entities have to be differently described. As soon as this is noted, one realises that Armstrong’s definition of (Armstrong‑) internal relation is the same as his implicit definition of (Armstrong‑) supervenience for relations. It is no mystery that he finds internal relations (= grounded relations) to be good illustrations of supervenience.

Important for Armstrong is the view that ”What supervenes is no addition to being”, and that one gets supervenient properties as an ”ontological free lunch”.[xliv] He would like to have both supervenience and a reductive materialism. With respect to the examples of Armstrong-supervenience that I have afforded (i.e., determinables Armstrong-supervene on their determinates, and grounded relations Armstrong‑supervene on their relata), I find Armstrong’s two “sayings” adequate. However, I would like to keep them distinct, whereas Armstrong treats them as being almost synonymous. Let me explain why I do not agree.

If, at the beach, there is already a little sand castle when a kid arrives and makes a large castle, then the kid has to do nothing more in order to create the grounded relation ’being larger than’ between the two castles. The relation comes, so to speak, for free. Similarly, when the kid gives the castle a determinate shape, he gets for free the fact that the castle has the shape-determinable. However, even though these indisputable facts well fit the expression ”ontological free lunch”, they do not settle the issue whether the grounded relation and the determinable at hand are ”additions to being” or not. In my opinion, as soon as grounded relations and determinables are regarded as distinct universals or tropes, they must be regarded as something distinct from the relata and the determinates, respectively; and every distinct being adds something to being.

Let me summarise. Armstrong’s concept of supervenience allows symmetric supervenience, it is both coherent and can be exemplified, but it is a single-necessity formulation of supervenience that is much stronger than the ordinary formulation. In particular, it violates at least the non-entailment requirement (desideratum six) of the original concept of supervenience. In my opinion, Armstrong should have given the kind of dependence relation he defines another name. Now, a superficial look may give the false impression that when Armstrongian reductive materialists and nonreductive materialists say that the mental supervenes on the material, they mean the same thing. But that is not the case. They are using different concepts of supervenience.

My last claim may seem to be, but is not, contradicted by David Lewis’ philosophy. He is, just like Armstrong, a reductive materialist, but he is always using an ordinary single-necessity formulation of supervenience.[xlv]

In the introduction to the second volume of his Philosophical Papers, Lewis says (1984) that many of the papers seem to him ”in hindsight to fall into place within a prolonged campaign of behalf of the thesis I call ‘Humean supervenience’”.[xlvi] The same perspective is stressed by the editors of a recent book on Lewis’ philosophy, Reality and Humean Supervenience (2001).[xlvii]

In a world or part of a world[xlviii] where there is Humean supervenience, there are by definition no necessities in and between the individuals (particulars) as such. All the individuals of such an aggregate whole are in their spatiotemporal existence both logically and nomologically independent of each other. However, according to Lewis, even in such a contingent ”spatiotemporal arrangement of local qualities”[xlix] other qualities (properties) and relations can supervene. His concept of Humean supervenience represents, among other things, a restriction on what kind of relata that are allowed to enter the supervenience relation.

According to Lewis, ‘To say that so-and-so supervenes on such-and-such is to say that there can be no difference in respect of so-and-so without difference in respect of such-and-such’[l], and ‘Supervenience means that there could be no difference of the one sort without difference of the other sort. Clearly, this “could” indicates modality’.[li] Whereas most writers on supervenience are using formulations like desideratum nine (‘base entity indiscernibility entails supervenient entity indiscernibility’), Lewis is mostly using formulations like desideratum ten (‘supervenient entity difference entails base entity difference’). He does not take desideratum six (the non-entailment requirement) into account at all in his definition of supervenience. The indiscernibility requirement, which for Hare is merely one of at least two requirements on supervenience, becomes for Lewis the whole definition.

In Lewis’ opinion, laws of nature, counterfactuals, causation, persistence in time, mind, language, and chance can all be regarded as being Hume-supervenient properties on material particles and their spatiotemporal relations.[lii] In other words, they are all supervenient on independent individuals and the external relations that obtain between these individuals. My intention here is not to discuss any of these seven specific supervenience theses, [liii] but to explain why Lewis, mistakenly, can regard his supervenience theses as reductive thesis.

According to Lewis, there is a wholly uncontroversial example of Humean supervenience, a dot-matrix picture:

A dot-matrix picture has global properties – it is symmetrical, it is cluttered, and whatnot – and yet all there is to the pictures is dots and non-dots at each point of the matrix. The global properties are nothing but patterns in the dots. They supervene: no two pictures could differ in their global properties without differing, somewhere, in whether there is or isn’t a dot.[liv]

A dot-matrix picture is a pattern, and there is a little more to be said about patterns than Lewis does. For instance, I think that, necessarily, a dot is a union of two properties (a shape and a colour) and that, necessarily, the limit of a pattern is a fiat delimitation. Furthermore, I think it involves grounded (or Armstrong-internal) relations.[lv] However, I will neglect these facts, and then try to see what is involved in Lewis’ account. First, we should ask what is not supervening in a dot-matrix picture. What do the base entities look like? They are of two main kinds, dots (= individuals with monadic properties) and spatial relations (= external relations). Patterns are supervenient on individuals and external relations. In this they are similar to grounded relations; the main difference is of course that a two-term grounded relation supervenes on two individuals whereas a pattern supervenes on a larger plurality.

Having recourse, as now, both to single-necessity ordinary supervenience and to single-necessity Armstrong-supervenience, an interesting observation can be made. A dot-matrix picture supervenes on a plurality of dots not only in the ordinary sense of single-necessity supervenience; it supervenes in the sense of Armstrong-supervenience, too. If the dots are there, it is entailed that the pattern is there. When one paints dots, a pattern comes into existence simultaneously. I think this is the reason why Lewis says that a dot-matrix picture is ”nothing but” patterns in the dots. Even patterns come, like determinables and grounded relations, for free. Not because they are supervenient in the ordinary sense but because they are Armstrong-supervenient.

Armstrong-supervenience yields at least something like reduction, but Hare-Davidsonian supervenience does not. Even if Lewis would have managed to show convincingly that laws of nature, counterfactuals, causation, persistence in time, mind, language, and chance are, according to his definition, supervenient on independent individuals and external relations, he would not thereby have shown that they, like patterns, are ”ontological free lunch”. In order to show this, he has to show that they are Armstrong-supervenient as well.

6. Conclusion

At the beginning of the paper, I quoted a statement by Kim: ”What needs to be added to property covariance to get dependence or determination, or whether dependence/determination must be taken as an independent primitive, are difficult questions that probably have no clear answers”. I hope now to have shown that in making such a statement, one is implicitly denying the coherence and/or applicability of the concept of asymmetric Bradley-internal relations and the corresponding concept of one-sided existential dependence.

If there is something coherent in the old ideas of internal relations and existential dependence relations, then there is also a new way in which one can try to find in the original supervenience relation two components, one being an asymmetric internal relation and the other being an asymmetric covariation relation.

On the other hand, if there is some fundamental flaw in the concepts of internal relations and existential dependence relations, there are nonetheless asymmetries left in the remnant of the original concept of supervenience. There is both asymmetry in covariation and property set asymmetry. The term ‘symmetric property supervenience’ is a misnomer.

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THE ASYMMETRIES OF PROPERTY SUPERVENIENCE

Ingvar Johansson

References

NOTES