# Supervenience, Properties, and Relations

Supervenience-based arguments for moral naturalism have tended to apply only to moral properties, not to relations.  One might have thought that they could easily be generalised so as to apply to relations as well.  However, as I'll argue here, this may not be so easy.

In the case of properties, it can be shown that the following is a valid argument.

(1) The A-properties (strongly) supervene on the B-properties; i.e. for any possible objects x and y, if x and y instantiate the same B-properties, then x and y instantiate the same A-properties.

(2) The B-properties are closed under complementation and arbitrary intersections and unions.

Therefore:

(3) Every A-property is coextensive with a B-property.

(Of course, this doesn't yet establish that the A-properties are B-properties, but it's a step in that direction.)

Suppose, then, we try something similar with relations.  The natural way to revise (1), (2), and (3) so that they apply to relations instead of properties seems to be as follows.

(1*)  The A-relations (strongly) supervene on the B-relations; i.e. for any possible objects x and y, if x and y stand in the same B-relations to the same things, then x and y stand in the same A-relations to the same things.

(2*) The B-relations are closed under complementation and arbitrary intersections and unions.

Therefore:

(3*) Every A-relation is coextensive with a B-relation.

In this case, however, the argument is invalid.  The following is a counter-instance.  Let B contain only the four relations "is identical to", "is not identical to", "is either identical or not identical to", and "is both identical and not identical to".  And let A contain the relation "is taller than".  B is closed under Boolean operations, so (2*) is true.  And (1*) is also true: the antecedent is false whenever x ≠ y (because then, e.g., x is identical to x but y is not identical to x), and the consequent is true whenever x = y.  But (3*) is false: "taller than" is not coextensive with "is identical to", because nothing is taller than itself; nor is it coextensive with "is not identical to", because some things are not taller than some other things; and so on.

### 15 Responses to Supervenience, Properties, and Relations

1. Jamie Dreier says:

Interesting.
To me, the natural extension of monadic property supervenience to relations is:
For any possible x, y, if x and y stand in the same B relations to one another then they stand in the same A relations to one another.
Is there any particular reason you chose the extension you did choose?

2. Hi Jamie,
Here’s how I’m thinking of it. (Below, for generality, I’ll count properties as monadic relations.)
The concept of supervenience depends on the notion of indiscernibility. The A-relations (strongly) supervene on the B-relations iff for any possible objects x and y, if x and y are indiscernible with respect to B-relations, then x and y are indiscernible with respect to A-relations. Thus the question is: what is it for two objects to be indiscernible with respect to a relation R, or, as I’ll say, “R-indiscernible”?
In the case of monadic relations (i.e. properties), the answer is simple:
(I) If R is a monadic relation, then x and y are R-indiscernible iff: Rx iff Ry.
How might this be generalised to relations of arbitrary adicity? I suggest the following:
(I*) If R is a relation of adicity n, then x and y are R-indiscernible iff for any n-place sequence of objects s, and any number i with 1 ≤ i ≤ n: Rs[i/x] iff Rs[i/y] (where s[i/x] is the sequence that is identical to s except that the i-th element has been replaced by x).
Formally, I* seems the natural generalisation of I. (Notice, I* entails I.) Moreover, it seems to capture an intuitive sense of things being indiscernible with respect to a relation. Consider, for example, the dyadic relation “is taller than”. I* entails that x and y are indiscernible with respect to this relation iff (a) x is taller than all and only those things which y is taller than, and (b) all only those things which are taller than x are taller than y; or in other words, x and y are taller than the same things, and the same things are taller than x and y. That seems right to me.
Your suggestion, I take it, is something like this:
(J) If R is a dyadic relation, then x and y are R-indiscernible iff: Rxy iff Ryx.
I don’t think that’s right. Take the relation “is the capital of”, which some cities bear to some countries. According to J, Auckland and Wellington are indiscernible with respect to this relation, because Auckland is not the capital of Wellington, and Wellington is not the capital of Auckland. But that seems wrong. Wellington is the capital of New Zealand, but Auckland isn’t; so there is a way to tell them apart using this relation, so to speak.

3. Mark says:

I guess you should add that the B-properties must be closed under projection. (You project an n-place relation to a k-place relation, for k

4. Mark says:

&lt n by fixing n-k of its arguments to particular values.)

5. Hi Mark. That’s interesting, but I’m not seeing how it helps. In the counter-instance I suggested above, B contains only 2-place relations. So the effect of closing B under projection would be to add some 1-place relations. But none of these can be coextensive with “is taller than” either, because it’s 2-place.
I imagine something like the following might be true. Suppose (1) that the A-relations supervene on the B-relations, (2) that the B-relations are closed under arbitrary Boolean operations, and (3) that both the A-relations and the B-relations are closed under projection. Then every 1-place A-relation is coextensive with a B-relation. So in the example above, “is taller than Anne” would be coextensive with “is identical either to Bob or to Catherine or to … “. But that may still leave the proper relations irreducible.

6. Jamie Dreier says:

I see.
Well, if you want indiscernibility to be central, then I would have said that the extension should be this:
(J*) If R is a dyadic relation, then the pair <x, y> is R-indiscernible from the pair <v, w> iff (Rxy iff Rvw).
I might add: … and Ryx iff Rwv.
Suppose in a certain world all the married couples love one another and hate everyone else. (Marriage is always dyadic, and to get some necessity into the story we could add that the pattern is also a law of nature.) Then we might say that is married to is indiscernible (in this world) from loves; if John and Marsha are married-indiscernible from Al and Brunella, then the couples are also loving-indiscernible.
I can’t see that one way of extending the concept is better than the other. Here’s a kind of practical advantage to my way. Imagine that on a foggy day, a bunch of people are out strolling in a meadow. Visibility is very limited. Whether two of them can see one another depends only on how far apart they are. I want to say: the visibility relation supervenes on the distance relation, but the sibling relation does not. My way of characterizing supervenience gets this ‘right’ (that is, makes it true that visibility supervenes on distance and siblinghood does not). But your way makes it come out that siblinghood also supervenes on distance (in the trivial way that you make use of in your counterexample, since no two people are distance-indiscernible).

7. Jussi Suikkanen says:

Identity doesn’t seem to be a relation to me. Well, it might be that it’s a reflexive relation. But, some people think that when we say that x is identical with y, we are really saying either something about the constitution of x or something about the relation of between the terms ‘x’ and ‘y’.
But, you might think that relations proper to which the property supervenience applies are relations between two things. Of course some metaphysicists claims that not such things exist. Anyway, I have some difficulties in following the counterexample, but I would feel much more confident if the counterexample relied on A and B properties proper and not on identity which might be a special case.

8. Jamie,
Here’s a worry I have about your version of supervenience. I assume we can define it as follows (where indiscernibility is defined as in your J*):
The A-relations Dreier-supervene on the B-relations iff:

1. for all x and y, if x and y are indiscernible on all 1-place B-relations then x and y are indiscernible on all 1-place A-relations;
2. for all x, x’, y, and y’, if <x,x’> and <y,y’> are indiscernible on all 2-place B-relations then <x,x’> and <y,y’> are indiscernible on all 2-place A-relations;
3. for all x, x’, x”, y, y’, and y”, if <x,x’,x”> and <y,y’,y”> are indiscernible on all 3-place B-relations then <x,x’,x”> and <y,y’,y”> are indiscernible on all 3-place A-relations;
4. and so on …

Now suppose A contains just the 2-place relation “is taller than”, and B contains all the individual height properties (e.g.”is 5ft tall”, “is 6ft tall”). Intuitively, A supervenes on B. But A does not Dreier-supervene on B.
Jussi,
The example I just gave is a counterexample of the sort you asked for (i.e. one not involving identity): “is taller than” supervenes on the height properties, but is not coextensive with any height property.
I think, however, this may show how to fix the argument. We need to add a premise like this: if F and G are both B-properties then there exists a 2-place B-relation R such that (Rxy iff (Fx and Gy)).

9. Jussi Suikkanen says:

Are you sure? Height properties do not seem to be relational properties to me. I thought relations hold between 2 or more objects (barring reflexive relations). Heights don’t. I don’t even see how you could get height in (1*).

10. Jussi, I was counting properties as 1-place relations.
When I say that “taller than” supervenes on the height properties, I just mean this: if x and y instantiate the same height properties (i.e. are the same height) then, for any z, x is taller than z iff y is taller than z, and z is taller than x iff z is taller than y. Nonetheless, “taller than” is not coextensive with any of the height properties, because it’s 2-place and they, as you say, are 1-place.

11. Jussi Suikkanen says:

ok. That allows me to reformulate my question. Assume that someone made the following argument:
(1**) The A-Two-Or-More-Place-relations (strongly) supervene on the B-Two-Or-More-Place-relations; i.e. for any possible objects x and y, if x and y stand in the same B-Two-Or-More-Place-relations to the same things, then x and y stand in the same A-Or-More-Place-relations to the same things.
(2**) The B-Two-Or-More-Place-relations are closed under complementation and arbitrary intersections and unions.
Therefore:
(3**) Every A-Two-Or-More-Place-relation is coextensive with a B-Two-Or-More-Place-relation.
I take it that neither identity or height examples are relevant here. So, is there a counterexample against the reformulated argument?
I guess the reason that motivates me is that I think Bart Streumer might be right that you can run Jackson’s argument for moral properties also for reasons – it gets messy with disjunctions and whole world descriptions with relations added. But, I don’t see any principled reason that would block this.

12. Here’s a counterexample. Suppose there are only three possible objects, x, y, and z, such that x is taller than y, y is taller than z, x is wider than y, and z is wider than y. Let B be the Boolean closure of {“taller than”}, and let A contain “wider than”. Then A supervenes on B, because no two things are taller than the same things. But “wider than” is not coextensive with any B-property. It’s not coextensive with “taller than”, because y is taller than but not wider than z. Nor is it coextensive with “not taller than”, because y is not taller than but not wider than x. And so on.
I think Bart’s argument must depend, at least implicitly, on stronger premises than yours. In particular, it must include something the added premise I suggested in my previous reply to you (i.e. if F and G are both B-properties then there exists a 2-place B-relation R such that (Rxy iff (Fx and Gy))).

13. Jamie Dreier says:

Hm, that’s right, Campbell.
On the other hand, your formulation makes everything supervene on identity. That’s surely not the common philosophical notion of supervenience.
Oh, maybe your idea works better with transworld identity?

14. Jamie Dreier says:

It just occurred to me that there is something else surprising about the extension of supervenience to relations. It’s that global supervenience does not imply strong supervenience. (There are known counterexamples to the implication even for monadic properties, but I believe they all involve certain kinds of tricks that we don’t need for counterexamples to the implication for relations.)
Suppose two worlds are alike in their distribution of the “child of” relation (that is, whenever x is the child of y in w, x is the child of y in w’). Then they are alike in their distribution of the “sibling of” relation. The world could not have been different in respect of who is a sibling of whom without being different in respect of who is a child of whom.
However, Campbell-SS doesn’t hold. My brother and I have the same parents, but we do not have the same siblings.
Jamie-SS does hold, but plainly does not capture the same idea as global supervenience.
I wonder if maybe our common philosophical idea of supervenience for monadic properties might extend to a number of different ideas for relations, none of which is a great match for the original.

15. Jeffrey Ketland says:

Campbell, this sort of definition of indiscernibility goes back to Hilbert & Bernays and is discussed by Quine (see Philosophy of Logic and “Grades of Indiscriminability” (1976, J Phil), reprinted in Theories & Things 1981). There’s been a fair bit of discussion of this recently in papers about structuralism, with papers by Saunders, Ladyman and others (including yours truly, a couple of papers on it, and a forthcoming on in Review of Symbolic Logic). Quine formulates it in a language-dependent way and first-order. Campbell’s is a second-order equivalent (so long as the language has finitely many primitive symbols).
If a language L has a single unary symbol F and a single binary symbol G, then write “x ~_F y” to mean “x is F-indiscernible from y” and similarly for “x ~_G y”. They’re defined by:
1. x ~_F y iff [Fx < -> Fy].
2. x ~_G y iff [forall z [(Gxz < -> Gyz) & (Gzx < -> Gzy)]].
[The permutations here are what Campbell achieves using sequences and substitution.]
Then “x ~_L y” (“x is indiscernible in L from y”) is defined as:
3. x ~_L y iff [x ~_F y & x ~_G y].
If C is a collection of relations, write “x ~_C y” to mean “x and y are C-indiscernible”. If B is the collection of basic relations, and R is a non-basic one, then let WeakSup(R, B) mean “R weakly supervenes on the B-relations”. Campbell defines Sup(R, B) by saying:
4. WeakSup(R, B) iff [forall x, y (x ~_B y -> x ~_R y)].
But this is very weak, and imposes almost no conditions on the non-basic relation R.
As Jamie mentions, “On the other hand, your formulation makes everything supervene on identity. That’s surely not the common philosophical notion of supervenience”.
Yes, all relations supervene on identity if supervenience is defined as in (4). If = is a basic relation, then “x ~_B y” is equivalent to “x = y”. So the definiens in (4) reduces to “for all x, y [x = y -> x ~_R y]”, which is true, since “for all x, x ~_R x” is true (any indiscernibility notion is reflexive). In other words, we can prove:
5. For any relation R, WeakSup(R, =),
This shows one must be careful in formulating supervenience in this way. I noticed this odd consequence myself many years ago, when looking at a definition similar to Campbell’s (4).
Campbell, I’ve e-mailed you my old notes on this.
Sorry to ramble on …