Stochastic Supervenience
Carl F. Craver
Washington University, St. Louis
The thesis of physical supervenience (PS) is widely understood and endorsed as the weakest assertion that all facts are tethered to the physical facts. Here I entertain a weaker tethering relation, stochastic physical supervenience (SPS), the possibility of which is suggested by analogy with the apparent failure of causal determinism (CD) in certain areas of physical science. Puzzling over this possibility helps to clarify the commitments of and the motivations for accepting the PS thesis.
The CD thesis asserts that given a complete specification of the state of the world at time t0, there is one and only one physically possible state of the world for all tn> t0. More colloquially, there can be no difference in the state of the world at tn without a difference in an earlier state of the world. The past facts fix (or determine) all future facts. The CD thesis was once endorsed throughout science and philosophy.
The PS thesis asserts that given a complete specification of the physical state of the world at time t0, there is one and only one possible total state of the world at time t0. More colloquially, there can be no difference in any state of the world without there being a difference in the physical state of the world. That is, the physical facts fix (or determine) all the facts. This thesis is now widely accepted by many scientists and philosophers.
The CD thesis, as a general thesis about the causal structure of the world, is now widely believed to have exceptions. There are laws in quantum mechanics that predict, given a complete specification of the antecedent conditions, only probability distributions over possible outcomes. The posit that hidden variables turn these probabilities into certainties entails Bell’s inequalities, which are known to be false on the basis of robust experiments (Bell 1964). But one need not appeal to quantum mechanics. CD arguably fails for Newtonian mechanics as well (Alpers, et. al 2000). Newton’s laws allow objects decelerating from infinite speeds after t0 to interfere unpredictably with the course of events. And the same laws are consistent with stationary balls sliding spontaneously and unpredictably off the top of a dome in any direction without any motive force (Norton 2003). Regardless of whether one accepts these arguments, it is clear that close scrutiny of contemporary science has forced many to question CD, a thesis once held to be beyond empirical reproach.
Might the PS thesis someday suffer a similar fate? Might science discover that the same physical state of the world is consistent with a range of possible total states of the world? Consider in more detail what SPS would involve.
Simplifying matters, imagine a world W containing just three physical magnitudes, A, B, and C arranged in a causal sequence: AàBàC. Let f be a variable taking different values depending on (i) the values of A, B, and C and (ii) on the nature of the causal relations between A and B and between B and C. For simplicity, assume that A, B, and C are dichotomous such that A can have the values {a, ~a}, B can have the values {b, ~b}, and C can have the values {c, ~c}. And suppose for now that the arrows in the causal sequence stand for necessary and sufficient conditions such that p(b|a) = 1, p(b|~a) = 0, p(c|b) = 1, and p(c|~b) = 0. Let y be a dichotomous variable (with values {y1, y2}) that supervenes on f. If y supervenes on f, then there can be no difference in the value of y without a difference in the value of f. SPS occurs if a single value for f is consistent with multiple values for y. For example, let f1 = (aàbàc). One simple variety of SPS would allow that p(y1|f1) = 2/3 and p(y2|f1) = 1/3. In more complex cases of SPS, f1 could be consistent with a probability distribution over arbitrarily many y-values {y1, y2, …, yk}. For now, I confine my attention to the simpler, dichotomous case.
Cases of SPS should be contrasted with two other superficially similar kinds of case, each consistent with PS. The first case involves introducing indeterminacy into condition (i). Suppose that A is indeterminate such that p(a) = 2/3 and p(~a) = 1/3. And suppose as above that the arrows in the causal sequence are interpreted as necessary and sufficient conditions. In that case, the probability of y1 given all one knows about W, is 2/3. However, the probability attached to y1 arises from the indeterminate value of A, a component in the supervenient base, and not from any indeterminacy in the supervenience relation itself. That is, the probability of y1 reflects the facts that f has two possible values (f1, as specified above, and f2, representing the sequence ~aà~bà~c), that p(y1|f1) = 1 and p(y1|f2) = 0, and that p(f1) = 2/3. In the world we are imagining, once the f-facts are fixed, so are the y-facts, even if some of the f-facts are difficult or impossible to fix in advance, as it were.
The second case involves introducing indeterminacy into (ii). Suppose that A, B and C take determinate values but that the causal relationship between A and B is stochastic. For example, suppose p(b|a) = 2/3 and p(~b|a) = 1/3, and suppose that the other causal relation is necessary and sufficient as above. In this case, as in the first, the y properties would inherit the stochastic features of the supervenience base. But the probability distribution across possible values of y does not result from the stochastic nature of the supervenience relation itself. Rather, the probability distribution results from the stochastic nature of the causal relations in the supervenience base. When f takes on the value f1, as defined above, then p(y1|f1) = 1. And when f takes on the value f3 (where f3 is the sequence aà~bà~c), then p(y1|f3) = 0. Again, fixing the f-facts fixes the y-facts, even if the f-facts do not fix one another. In true cases of SPS as we are considering it, the stochasticity lies in the supervenience relation itself, not in the supervenience base. (It is surely useful to explore how PS might be framed to deal with relations in which the supervenient property, its base, or both are probabilistic in nature, as in population genetics or a casino, but that is not my topic).
The contrast between SPS and these two cases helps to emphasize that the possibility of SPS does not arise from considering how, for example, quantum effects scale up to other levels. Quantum indeterminacy provides merely an analogy and an (at least) apparent air of plausibility in the present context. True SPS would involve indeterminacy in the supervenience relationship itself rather than among the items in the supervenience base. To repeat, the thesis of stochastic supervenience would allow that even if the values of A, B, and C were determinate, and even if all of the causal relationships were necessary and sufficient as specified above, it might nonetheless be the case that p(y1|f1) = 2/3 and p(y2|f1) = 1/3.
The SPS thesis asserts a weaker tethering relationship between the non-physical and the physical than does the PS thesis. Specifically, it allows that some facts only stochastically supervene on physical facts. Is SPS a genuine metaphysical option?
Some will object to the SPS thesis on the grounds that acknowledging the possibility of SPS amounts to nothing more than the rejection of the PS thesis. Yet not all rejections of PS are equal. It is undeniable that if SPS occurs, the PS thesis as traditionally conceived is false. However, the motivation for the PS thesis has always been to express the idea that non-physical facts are tethered to the physical facts. The SPS thesis can satisfy this motivation. Consider again the analogy with apparent failures of causal determinism: Although a single value for a variable representing the antecedent state of the world at t0 might (if critics of CD are correct) be consistent with more than one effect, the antecedent value might nonetheless be consistent with a finite number of future outcomes or an otherwise manageable range of future outcomes. Indeterminism in causation does not entail that just anything can follow any antecedent condition. At least in many cases considered in the literature, the antecedent value entails a probability distribution over a (perhaps finite) set of future outcomes. By analogy, perhaps the same supervenience base could be consistent with only a finite number of supervenient properties (or a distribution of values of variables describing such properties), and the laws of nature might assign a determinate probability to each. In that case, accepting the possibility of SPS need not lead one to the conclusion that anything goes. Supervenient facts would still be tethered by physical facts; they would simply be more loosely tethered to the physical facts than PS allows. The possibility of SPS need be no more threatening to the search for tethering supervenience relations than the specter of causal indeterminism has been to the search for etiological, causal relations. In fact, as described above, the thesis of global causal determinism just is the supervenience of future facts on past facts. If CD is a species of supervenience thesis, then the (at least) apparent failure of CD in some areas of science is an existence proof for the (at least) apparent possibility of stochastic supervenience relations.
Some will find the idea of SPS logically incoherent on the grounds that they can find no slack that would allow for a supervenient property to differ once its base has been fixed. For those so disposed, SPS can serve as a reminder of how weak the PS thesis really is. The SPS thesis is no doubt counterintuitive as applied to standard examples of supervenience in aesthetics and ethics. It seems implausible, for example, that physically identical paintings in physically identical contexts viewed by physically identical observers could be sublime two thirds of the time and banal the rest. And it seems implausible that physically identical agents engaged in physically identical behaviors in physically identical social and political contexts should be praiseworthy two thirds of the time and evil the rest. Examples such as these seem to have none of the nomological slack required for SPS. But the SPS thesis is not committed to the conclusion that all instances of supervenience are stochastic. The thesis is committed only to the existence of stochastic supervenience relations, not to their universality.
So even if SPS is truly incoherent in such cases, this should not lead one to conclude that SPS is incoherent in all cases. Rather it should lead one to the conclusion that in such counterintuitive cases, something stronger than mere supervenience (e.g., exhaustive constitution or identity) tethers the supervenient property to its base. The incoherence follows from the underlying presumption that supervenience is underwritten in such cases by a stronger, metaphysical relation that precludes the slack required for stochastic supervenience.
The SPS thesis is also counter-intuitive in standard and straightforward cases of mechanistic explanation. Suppose we specify in all relevant detail the entities, activities and organizational features of a standard-model Victor Snaptrap for mice. When it fires (yf), the the trigger is depressed, the catch slides, the spring tension releases, and the impact bar arcs toward its target (fa). Suppose the sequence in fa is deterministic, that is, p(catch slides|trigger depressed) = 1, p(catch slides|trigger not depressed) = 0, and so on. Here, we find none of the slack required for SPS; if all of the steps in fa occur, the trap necessarily fires. The trigger was depressed, the catch slid, and the impact bar completed its arc. These steps fully constitute the trap’s firing. There’s nothing more to be done— no slack for chance. I suppose Chalmers (1996) has something like this in mind when he describes some problems in cognitive science as “easy problems” (relative to the “hard problem” of, e.g., understanding consciousness): in such cases, we cannot clearly and distinctly conceive of the “higher level” property (Craver 2007) differing once all of the lower-level steps in the supervenience base have been fixed. One is tempted to say that the higher-level capacity just is, or perhaps more perspicuously, is exhaustively constituted by the completion of each stage in the operation of the mechanism. Regardless of precisely how one wishes to express this relationship in the language of contemporary analytic metaphysics (token identity, exhaustive constitution, realization, or what have you), there is clearly a kind of intimacy our minds cannot render asunder. We cannot clearly and distinctly conceive of fa being instantiated without the firing coming along for free.
Indeed, something like this thought operates as a constitutive ideal in the search for mechanistic explanations (Haugeland 1996). If one knows all of the relevant entities, activities, and organizational features, and if one knows all the relevant features of the mechanism’s context of operation, then one knows how the mechanism will behave. If the facts about the parts, activities, organization, and context are determinate and deterministically related, then fixing the f facts fixes the y-facts. If, as in our contrast cases, facts about the parts are indeterminate, or if their interactions are indeterministic, then we expect this indeterminacy to be inherited (without amplification or remainder) in the probability distributions with which we express the y facts. So in attempting to discover how a mechanism works, we take it as an epistemic warning sign when there are significant and salient features of the behavior of the mechanism as a whole (or the probability distributions of its behaviors) that cannot be accounted for in terms of our understanding of its parts, activities, organization and context. This is because we seem to operate with a background assumption that the phenomenon is exhaustively explained (now in an ontic rather than epistemic or representational sense; see Craver 2014) by the organized activities of parts in context.