An Issue With Metaphysical Reduction

Take a fact F. In general, what does it mean to say that fact F metaphysically reduces to fact F'? Note I am speaking of metaphysical reduction as opposed to conceptual reduction. First of all, the latter has to do with concepts and propositions rather than facts. For example, when we say that being a bachelor just means being an unmarried male, or when we say the proposition that Alfredo is a grandfather just means that Alfredo is the father of a parent, these count as examples of conceptual reduction. These explications of meanings are just the result of fully specifying the nature of our concepts as they stand. These are very simple examples, but the more complex instances of conceptual reduction in philosophy follow the same general idea as these ones.

Metaphysical reduction on the other hand has to do with facts in the world and how they stand in relation to each other. I take it that the following necessary condition imposes a restriction on the relation of metaphysical reduction:

• (R) If fact F metaphysically reduces to fact F' then (i) fact F holds in virtue of fact F' holding and (ii) the holding of fact F is nothing over and above the holding of fact F'.

As an example, physicalists often say that all mental facts are reducible to physical facts. I take it that this at least means that the mental facts hold in virtue of the physical facts and that they are nothing over and above the physical facts.

Now, (i) and (ii) seem to me to be in tension with each other. In fact, on the most straightforward reading of (ii) their simultaneously holding leads to a contradiction. Hence, we must find some other way to explain (ii), since it does not seem like a primitive relation. This is rather difficult. Let me explain.

By (i), reducibility must be an asymmetrical relation. This means that if F reduces to F' then F' does not reduce to F. For suppose F reduces to F'. Then F holds in virtue of F'. But the 'holding in virtue of' relation is asymmetrical, since otherwise there would be circular chains of ontological dependence. So if F holds in virtue of F', then F' does not hold in virtue of F, and thus by (R), F' is not reducible to F.

The problem is that the most straightforward reading of (ii) is that the holding of fact F is identical with the holding of fact F'. After all, suppose F and F' are not identical and we are dealing with a world of just F and F' (here I'm abbreviating, and I should really be saying the holding of F and the holding of F'). Then there is a perfectly clear sense in which F is something over and above F', viz. there are more things in the world than F! For if F =/= F', then for some x, x =/= F'. So there is something out there in the world which is extra-mentally distinct from F'. That seems to be a legitimate sense in which F is something over and above F'. So if F is not something over and above F' then F = F'.

But of course, if that were the case, then the 'in virtue of' relation here would not be asymmetrical, since if F = F' and F holds in virtue of the holding of F', then by substitution of equals F' holds in virtue of the holding of F. So reducibility would not, in fact, be asymmetrical. And that is a contradiction, since we earlier established it was.

One option is to say that the 'in virtue of' relation is not asymmetrical. But that seems deeply problematic insofar as it doesn't allow us to capture the reducibility we want to pick out. After all, every materialist will accept that all mental facts reduce to physical facts, but no materialist would ever dare say the physical facts reduce to the mental facts! (Personally I find the latter suggestion more plausible than the former, but regardless it is not something the materialist would ever claim.)

Instead, we have to find a sense in which one could say fact F is nothing over and above F' even though F is not identical to F'. And I'm not sure how to explain this. No idea if this works or not, or whether it is at all helpful, but here's a thought: Let us denote by 'a full truthmaker of P' a truthmaker of P which is not a constituent or part of some other truthmaker of P. Let Q be the proposition expressing the holding of F. Maybe we can say F is nothing over and above F' if the set of all full truthmakers of the proposition Q contains only F'. That would make (i) superfluous it seems. Or at least from pretty uncontentious premises (i) would follow as a consequence. This theory is a little weird though, since the question arises as to what, metaphysically speaking, explains why Q would be distinct from the proposition expressing the holding of F'.

With that said, I don't know if that's on the right track. And even if it gets the extension of the relation right it might not even produce a deeper understanding. The point being, I don't myself know how to explain (ii). Like I said though, it doesn't seem like this is a primitive or undefinable relation. I wonder then what we can say about it.

Basics of Quantifier Variance

When I say that there are tables is it unambiguous what I'm saying? Quantifier variantists say no. Or at least they would say that in certain contexts it is not. In particular, the sentence is ambiguous when we are engaging in metaphysical debate about the existence of the table, as in the following case.

Consider the debate between what I will call compositionalism and anti-compositionalism. Compositionalism is the thesis that there are composite material objects, while anti-compositionalism is the thesis that there are not. Take the case of a world with just a table and its parts, and suppose we are considering a form of compositionalism which says there are tables. Assume further that there are exactly n atoms which, according to this form of compositionalism, are proper parts of the table. Note that we are using a philosophical definition of 'atom', according to which an atom is a material object which has no proper parts. Anti-compositionalism says there is no table; there are just the n atoms. 

In essence, compositionalism says (A) there are n+1 distinct things (viz. the n atoms, plus the table), while anti-compositionalism says (B) there are n things and there are no more than n things. Note that (A) and (B) can be adequately translated into a quantified language which only contains variables, quantifiers, sentential connectives, and the identity sign with the usual interpretation. For example, (A) would be translated as follows:

∃x₁∃x₂...∃x_n((x₁≠x₂ ∧ ... ∧ x₁≠x_n+1) ∧ (x₂≠x₃ ∧ ... ∧ x₂≠x_n+1) ∧ ... ∧ (x_n≠x_n+1))

Naturalness in the World

I'm currently taking a course on a newly concocted discipline in analytic philosophy called metametaphysics. For Aristotle or Aquinas the topics grouped under this heading would probably just fall under the science of metaphysics. We are primarily reading from the recent Chalmers volume and Ted Sider's new book Writing the Book of the World. In the latter book quite a bit turns on a distinction which Sider calls variously 'naturalness', 'fundamentality', 'structure', and 'carving at the joints'. My professors claim to not understand what Sider is talking about. Admittedly, Sider could be quite a bit more clear. However, I think that an Aristotelian would want to agree at least in some respects with his general point insofar as we'd admit some boundaries in reality to be privileged over others.

Consider an example Sider gives which is represented in the figure above (I'll paraphrase these examples a little bit). Suppose there is a world filled with red and blue liquid. There are many true ways that we can divide this world up when we describe it. We can divide it into those parts satisfying the normal understanding of our predicates 'red' and 'blue' as in the first figure. We could also divide it into those parts satisfying different predicates, call them 'bled' and 'rue', which correspond respectively to the portions left and right of the diagonal in the second figure. Both of these ways of describing the world are true. There are indeed bled and rue portions of the world just as much as there are red and blue ones; or, to put it another way, there are divisions of the world along the lines of both the first and the second figure. Yet the first way of speaking seems in some way to be natural while the second seems bizarre. Sider wants to assert that the first way of dividing up the world, i.e. dividing it up along the lines of our 'red' and 'blue' predicates, is better because it describes those features of reality which are in some way privileged; to put it in his terms, it describes those features which are natural/ fundamental/carve at the joints/are part of the structure of reality.

It seems like Sider gets a lot of his account from David Lewis's work. It is at least similar to the natural vs. non-natural distinction Lewis makes in his paper "New Work for a Theory of Universals." Consider two properties: being green and being grue, where grue is defined here as the property of being green and observed before 3000 A.D. or blue and not examined before 3000 A.D. This property which we have called grue appears "gerrymandered" in a way the property green is not. In Lewis's term grue is a "less natural" property than green is. Sider agrees on this point with Lewis. It should be noted that what makes grue a less natural property is not the syntactic complexity of its definition--after all, we can just give it a simple name like grue--but rather that it is in some way privileged over these other properties as being a more fundamental feature of reality. (Lewis may actually disagree here but I think his view is highly implausible and Sider does not seem to endorse it.)

Take a final example. Consider two classes of things: the electrons, and the electron-or-cows (EoC's), the latter class consisting of everything which has the property of being an electron or a cow. The things in the first group seem to go together quite well in a way which the objects in the latter group do not. The electrons do not go together better simply because they share many properties. For one, the EoC's share many properties. In fact, the EoC's share infinitely many properties: they each have the property of being an EoC or four feet long, the property of being an EoC or five feet long, the property of being an EoC or six feet long...and so on.  So it's not simply the number of properties shared which distinguishes the two. It's the fact that a grouping of things into electrons and cows gets the way reality fundamentally is, whereas a grouping of things into electron-or-cows does not.

So these are just a few examples where we can contrast fundamental/natural features of reality in opposition to gerrymandered or non-natural ones. It is necessary to use such illustrations since fundamentality is taken to be a primitive distinction which is likely not definable in more basic terms. There are a lot of questions that can be asked here: is fundamentality/naturalness/carving at the joints/structure of reality/etc. itself fundamental? Is fundamentality supposed to be a property? Is fundamentality a feature of our thoughts and concepts, of entities, or both? What is fundamental? How can we know? Is the same notion at work in each case? These are all good questions, some of which are discussed in the book.

However my primary concern is this: Is Sider really getting at some objective distinction or not? What I mean by 'objective' is whether his distinction between the fundamental/non-fundamental really corresponds to something in the external world. Personally I think he is onto something and makes an interesting case regardless as to the connections with Aristotelianism. However, I think an Aristotelian would want to admit the distinction even if he might disagree about what the fundamental things are. I will try to explain why in later posts.

If anyone is reading, I'd especially like to hear your thoughts on (1) whether you think Sider's distinction makes any sense and (2) whether the examples illustrate the point.