In a previous post, I explored the question of whether reductionism about causation is a barrier to the project of natural theology. The answer to that question, we discovered, depends on what it means to reduce causation to natural laws, to regularities, or indeed to anything at all! In this post, I will propose my own definition of reductionism to set the stage for future posts on this topic.
By reductionism, I mean the claim that one domain of facts (call the “F2s”) is nothing over and above another domain of facts (call them “F1s”). In other words, F2s are nothing but F1s. This claim can be cashed out in terms of a complex relation known as supervenience.
Supervenience is a relation of covariance between two levels of facts. When facts at the bottom level are fixed, so are facts at the top level. Likewise, changes that occur at the top level must coincide with changes at the bottom level. This relation can be summarized in four propositions, with F1s at the bottom and F2s at the top:
(1) There can be no F2s without some F1s.
(2) There can be some F1s even without any F2s.
(3) F2s do not logically determine (or fix) which F1s there are, because different sets of F1s can fix the same set of F2s.
(4) F1s logically determine (or fix) which F2s there are, because only one set of F2s can be fixed by a given set of F1s.
For example, it is widely regarded that facts at the chemical level of description supervene on facts at the level of physics, which include physical laws and initial conditions. That is to say:
(1’) there couldn’t be any chemical facts without some physical facts – like the velocity or mass of quantum particles.
(2’) there could have been some physical facts without any chemical facts – like if our universe underwent gravitational collapse shortly after its initial expansion, but before any chemical elements could be formed.
(3’) different sets of physical facts can fix the same set of chemical facts – as when the same structure of H20 forms under different physical conditions.
(4’) if the set of all physical facts about the universe is fixed, then the set of all chemical facts about the universe is fixed too.
Similar to reductionism about chemistry, the reductionist about causation argues that all causal facts supervene on a more fundamental level. Reductionists disagree about which non-causal facts are fundamental, but they typically argue that natural laws, regularities, counterfactuals, probabilistic relations, or some hybrid of these will do the trick. Thus, the reductionist argues for the following:
(1”) there can be no causal facts without some non-causal ones.
(2”) there can be some non-causal facts without any causal ones.
(3”) different sets of non-causal facts can fix the same set of causal facts.
(4”) if the set of all non-causal facts is fixed, then the set of all causal facts is fixed too.
As it stands, the relation of supervenience helps us to understand the covariance between the causal and the non-causal levels. Unfortunately, propositions (1”) to (4”) are insufficient for a full-blooded reduction. Recall that reductionism requires that one domain of facts (F2s) be nothing over and above another domain of facts (F1s) – that F2s are “nothing but” F1s. Does our discussion of supervenience yield this conclusion? It seems not. First, it does not exclude the possibility that causal facts are emergent phenomena that co-vary with non-causal phenomena (such as laws, counterfactuals, etc) but nevertheless exist distinctly, over and above them. Second, supervenience does not explain why the covariance between these levels holds. Perhaps the supervenience is due to some mysterious “third factor” which explains why the top level of facts is fixed by the bottom level! To remedy this deficiency, something more that supervenience is needed.
We find clues about what more is needed by looking at the way reductionists deal with causal relations. Some reductionists argue that ‘causing’ is a probabilistic relation whereby one event increases the probability of another event – and in the right sort of way. To say that the ignition of gunpowder causes an explosion just is to say that the ignition increases the probability of the explosion. If this sort of claim is true, it will not be enough to say that instances of ‘causing’ co-vary with instances of probability-increase in every possible world. Rather, the reductionist is arguing that causal relations just are probabilistic relations, and nothing more. In other words, they are identical. 
The notion of identity can be incorporated into our previous analysis of reductionism by looking, once again, at distinct levels of facts. Identity requires that facts at one level (F2s) be identical to facts at another level (F1s). That is to say, there are no possible worlds in which F2s fail to be identical to some F1s. This notion also secures a full reduction of F2s to F1s because it explains why the supervenience between F2s on F1s holds true, and it eliminates the possibility that F2s are emergent phenomena which exist over and above lower level facts. Including the notion of identity in this way requires that a fifth proposition be added to our previous analysis, namely:
(5”) Causal facts are identical to non-causal facts in all possible worlds.
So far we’ve been working with a rather strong version of reduction vis-à-vis (1”) – (5”) that purports to hold necessarily, i.e. across all logically possible worlds – a view that I shall call “Necessary Reduction.”
However, there are some exceptions to this view in the literature on causation. Some reductionists prefer to limit the scope of the supervenience and identity relations to the actual world, instead of extending them to all logically possible worlds. This more modest approach says that causal facts happen to be reducible to non-causal facts in the actual world, but things might have been otherwise in other worlds. In future posts, I will refer to this position as “Contingent Reduction.”
 I define “facts” in a broad sense to include properties, events, relations, etc. It is of little consequence to my argument if there are those who prefer that “states of affairs” (or some other cognate) be used instead of “facts.”
 Allow me to make three qualifications about the place of identity here: (1) my claim is not that reductionism in every field of study (whether in psychology, biology, or philosophy of mind) should adopt the identity theory – only that this theory seems to capture how reductionists deal with causal facts; (2) for the sake of expediency, I will not discuss the relevance that token-type distinctions have for articulating specific species of reductionism, though these subtle distinctions are important; (3) I take it to be true that the relation of identity between two levels of facts entails supervenience, however broadly or narrowly the latter is construed.