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.