If a physical event p has a cause c that occurs at t, then p has a physical cause p' that occurs at t such that p is not overdetermined by c and p'.
Kim does not express Closure-Overdetermination anywhere in the text, but we attribute it to him because 1) it very neatly fills a gap in one of his arguments, and 2) it is highly plausible given Closure, which he does accept. For suppose that a physical event p has a cause c that is not physical. Then by Closure p has a physical cause p' synchronous with c. Is p overdetermined by c and p' ? If it is, then every event that has a nonphysical cause is overdetermined, and this seems absurd. It is of course logically possible that the physical causes posited by Closure sometimes but not always overdetermine their effects jointly with their concurrent nonphysical causes, but this seems bizarre and arbitrary. The most natural conclusion is that Closure-Overdetermination is true.12
4. The arguments
Before giving us his arguments, Kim announces (p. 39):
Properties as such don’t enter into causal relations; when we say “M causes M*”, that is short for “An instance of M causes an instance of M*” or “An instantiation of M causes M* to instantiate on that occasion.” Also for brevity we suppress reference to times.
We will follow Kim in adopting these abbreviating conventions whereby “an instance of property M” and the like will be replaced by simply “M” and the like, as if properties were identical to their instances. In reading the argument to follow, it will be useful to pretend that the world is such that there are just two times, call them “t” and “t”, and that anything that is a cause of anything else occurs at t, and anything caused occurs at t—references to times thus becomes unnecessary.
Kim offers two versions of his argument, and we will discuss them in order. Our exposition differs from Kim’s in two respects: First, we present his “versions” of the argument as two distinct arguments, rather than as a “Stage 1” followed by two alternative “completions”, as he does. Secondly, we will eliminate certain unnecessary steps, which we will comment on after we give each reconstruction. We will use sorted variables, so that “P”s, supplemented with primes as needed, are variables for physical properties and “M”s, likewise supplemented, are variables for mental ones.
4.1. Kim’s first argument
In the first argument we introduce for a reductio the assumption that a case of mental-to-mental causation occurs:
(1) M causes M'.
By Supervenience, we conclude that:
(2) There is a physical property P' such that P'is a supervenience base of M'.
By SC I we conclude from (1) and (2) that:
(3) M causes P'.13 Again, by Supervenience:
(4) There is a physical property P such that P is a supervenience base of M.
By SC II, it follows from (3) and (4) that:
(5) P causes P'.
But Irreducibility tells us that:
(6) M P.
From No Overdetermination and (4) we get:
(7) P' is not causally overdetermined by M and P.
Applying Exclusion to (5), (6), and (7), we conclude that:
(8) M does not cause P'.
(8) contradicts (3), and we have a refutation of (1). In other words, we have derived (9) from our assumptions.
(9) It is not the case that M causes M'.
Since M and M' were arbitrary mental properties, we must conclude the universal closure of (9):
(10) For all mental properties M and M', it is not the case that M causes M'.
In other words, there is no mental-to-mental causation. To show that there is no mental-to-physical causation, we need a separate argument, which can be obtained from the argument just given by deleting lines (1) and (2) and taking (3) (“M causes P' ”) to be the reductio hypothesis. These arguments together commit the nonreductive physicalist to a pervasive epiphenomenalism: mental properties do not cause any properties, mental or physical, to be instantiated.14
The argument just given fits into a larger reductio: one against the “substantive premises” of section 1 and the “general metaphysical constraints” of section 2. The conclusion (10) directly contradicts Causal Efficacy. We seem to have a demonstration that the “substantive premises” and the “general metaphysical constraints” cannot all be true. Supposing the metaphysical constraints to be off the table, we seem to be forced to reject either Causal Efficacy or one of the other “substantive premises”, which characterized nonreductive physicalism—in other words we must either accept epiphenomenalism or reject nonreductive physicalism.
The above reconstruction differs from Kim’s presentation of the argument in one important respect: in Kim’s presentation, Exclusion tells us that “we must eliminate either M or P as [P']’s cause”, whereafter Closure is called upon to tell us that it must be M that gets “eliminated” (pp. 42-43). But as can be seen above, Exclusion together with the preceding three lines yields the conclusion that M is not a cause of P, contradicting (3) [in Kim’s numbering, (8) (p. 41) contradicts (5) (p. 43)]. Citing Closure after a contradiction has been derived cannot contribute anything to the argument. Closure, then, is not actually needed as a premise in the first version of Kim’s argument.15