Wednesday, November 21, 2012

Probability and divine will

The Thomistic reconciliation of design with chance—of which Barr's reconciliation is a special case—holds that, necessarily, each particular chancy event occurs precisely because God causes it with primary causation. Now, if p and q are propositions that have the property that, necessarily, p is true if and only q is true, then P(p)=P(q) and P(p|r)=P(q|r) for any proposition r such that P(r)>0 and, in fact, intuitively, for any possibly true r for which the conditional probabilities are defined. Suppose:

  1. P(the coin lands heads)=1/2.
  1. P(God primarily causes the coin to land heads)=1/2.
But why should the probability of God's primarily causing the coin to land heads be exactly 1/2? Indeed, why should there be numerical probabilities of God's choices at all? (Note: It won't help to say that the probabilities are conditional on some background. For whatever background they are conditional on, as long as the background is possible, the conditional probabilities of the coin landing heads and of God primarily causing the coin to land heads will be 1/2.)

If the probabilities are epistemic, there may be less of a problem. For typically we have no reason to think God prefers the coin to land heads than to land tails or vice versa, and so the epistemic probability of his causing it to land heads may be 1/2. (Generalizing this to other cases may be problematic. How would this work for a Poisson or Gaussian distribution? To suppose a Poisson or Gaussian distribution on God's preferences would be weird.)

But if the probability in (1) is merely epistemic, then it isn't going to be useful for explaining why of a hundred tosses about fifty landed heads. Maybe one could still explain it by saying that God's preferences are likely to be randomly or quasi-randomly distributed, because of the great diversity of factors that affect God's choices about different coin flips. But then it is (2) rather than (1) that is the real explanation of why about fifty tosses landed heads: our explanation essentially involves a random distribution on the factors that God's decision is made on the basis of.

A non-Thomist (and by that I just mean someone who doesn't accept this reading of Thomas and the corresponding reconciliation—she might be a Thomist in all sorts of other ways) could say that God doesn't specifically choose which way each coin toss goes, but cooperates with the coin-tosses in a way that does not determine the specific outcome. Of course, God can still work a miracle and specifically choose a coin outcome, but then that outcome will be miraculous (in a weak sense of the word not implying God's self-revelation in the event) and not random. Such a non-Thomist will then say:

  1. P(the coin lands heads | no miracle)=1/2
  1. P(God cooperates in a way that results in the coin landing heads | no miracle)=1/2.
And there is no surprise about (4) since on this kind of a view God's non-miraculous cooperation with chancy processes (or free choices for that matter) does not micro-manage the outcome. But the non-Thomist will then have to work hard to reconcile design with chance.

I do not think this is available to the Molinist: I suspect it only works on simple foreknowledge (or open theist, for that matter) views.

Mike Almeida has told me that he has worried about the coincidence between (1) and (2) as well.


Mike Almeida said...

Here's a thought. We need not assume in chancy worlds w that there are any propositions that lack a truth value. God can know, I think we agree, that (i) the coin will fall tails if flipped in circumstances C and (ii) that chances that it did fall tails was .5. Chance does not crowd out bivalence. So God's choices need not be chancy in order to create a chancy world. He can create a chancy world in which he knows exactly what will happen (indeed, what would happen, were he to do something a bit different).

Alexander R Pruss said...

It seems to me that such choosing of circumstances destroys the stochastic explanation. If I know ahead of time which of the coins in the pile will land heads and which will land tails, and then I choose about 500 of the one and 500 of the other, precisely for that reason, then the standard probabilistic explanation of why about 500 landed heads no longer applies.

Heath White said...

What follows is held with low probability. :-)

I think what I want to say about this is that we have two different Ps going on. Start with the idea that the fundamental kind of probability is conditional probability, and there is a relation of conditional probability governing causal relations with the schema P(effect|cause). (For deterministic causes, P is 1.) Then I want to say there is a P governing the relation of secondary causation, and there is another P governing primary causation. The sense in which both are causal relations is only analogical, so it is not incoherent to split up the relevant notion of probability.

The Barr-Thomist might say that primary P for events that occur is always 1. But I can imagine a weaker view; perhaps God only jacks up primary P for special miraculous events; maybe that is what you think. An interesting possibility is that primary P is 1 for events which God intends but something less for side-effect events he doesn’t care about one way or the other. (He intends to create fish; he does not care if this fish eats that one.) In the latter case, some of these side-effect events might be deterministic given the laws of nature and prior (chancy) events. If so, primary P can be less than secondary P.

One might extend this distinction to other distinctions between types of causation. For example, early geneticists knew that the probability of a human child being male was about 0.5 without knowing anything about the mechanism for it. They might have framed this in terms of the causal indeterminacy of genes. Was this knowledge useful for explaining why a given child is male? (“Well, about half are, it’s not that surprising.”) I am inclined to think it was, in which case we need a P for biological explanation distinct from the presumably different P for explanations at the level of chemical interaction.

But there is always a mechanism behind the mechanism behind the mechanism, and I am a little worried that if we go down the route “The REAL explanation isn’t what you think, it’s [at the lowest level which explains all the higher levels]” it will turn out we have no real explanations.

Alexander R Pruss said...

That's an interesting solution indeed.

Here's one worry, though whether it's a problem may depend on how the details are worked out. If we allow too much freedom as to which causes we condition on, we will have too much freedom in explaining. For instance, suppose that Sally the casino owner is obsessed with the thought that perhaps one day her roulette wheel will make someone win so many times in a row that she'll go broke. So she makes a special crooked roulette wheel whose statistics for up to 100 turns are guaranteed to "look independent and random" to simple statistical tests (frequencies, frequencies of adjacent pairs, etc.) but in fact aren't. But because this is guaranteed (unlike on a real roulette wheel where you can win 100 times in a row), the probabilities on Sally's roulette wheel are different from those on a real roulette wheel.

Now, let E be the event that Sally makes money from her roulette wheel. We can give two kinds of statistical explanations:
P(E | Sally's special roulette wheel setup) = 1.
P(E | roulette wheel setup) > 0.9999.
The second only includes the generic information that Sally has a roulette wheel. Both conditional probability claims are true and applicable (i.e., Sally's mildly crooked wheel is still a roulette wheel, but it's sufficiently rare in the class of roulette wheels that it doesn't affect the statistics when conditioning on roulette wheel). But only the first one explains E.

This is related to your concern at the end that perhaps only the most fundamental stuff ends up explanatory. I certainly don't want that conclusion. I think sometimes the coarser-grained conditional probabilities are explanatory and sometimes they aren't. I have no necessary and sufficient conditions for when they are or aren't, but I do have a general heuristic that when the case at hand involves agential manipulation at a finer-grained level to produce such-and-such statistical results at the coarser-grained level, then these statistical results are no longer explained by the stuff at the coarser-grained level.

Heath White said...

After some thought, here is a potential diagnosis of the dispute. There does not seem to be a problem when we are dealing with epistemic probabilities; it is metaphysical probabilities that are the issue. These in turn depend on accounts of indeterministic causation (for metaphysical probabilities less than 1). The fundamental dispute is over what we say when there is more than one causal explanation for an event.

The Barr-Thomist is committed to saying that we can have both secondary and primary causal explanations. These do not conflict or compete, they are both correct, and as a result we can have different things to say about the probabilities of an event in the different contexts. A given event can be both secondarily chancy and primarily designed to the detail.

The Prussian :-) view seems to be that we cannot have overlapping secondary and primary causal explanations. We have to choose one or the other, or parcel out the explanatory work between them. (Maybe God primary-causes one to act virtuously and we secondary-cause just how we act virtuously.) So if an event is secondarily chancy it cannot be primarily designed to the detail and if it is designed to the detail this would undermine chancy secondary explanations of it.

It seems to me that relevant analogues of this dispute can be found when we have different kinds of causal explanation, e.g. biological vs. chemical explanations of animal traits, or psychological vs. sociological explanations of individual choices. Jaegwon Kim’s Exclusion Argument against mental causation is another interesting analogue (physical vs. mental explanations of action).

You say I do have a general heuristic that when the case at hand involves agential manipulation at a finer-grained level to produce such-and-such statistical results at the coarser-grained level, then these statistical results are no longer explained by the stuff at the coarser-grained level and it may be that this captures the main source of the disagreement. It is not overlapping causal explanations as such that are problematic, but agential manipulation in order to produce a certain sort of pattern, which undermines the explanatory force the pattern would otherwise have.

If that is the intuition, it strikes me as at least needing argument. An observation: it bears a strong resemblance to manipulation arguments against compatibilism about responsibility. One key premise of those arguments is that, if S is agentially manipulated into doing X, then S cannot be responsible for doing X, no matter what other conditions S meets. Obviously the Barr-Thomist is not going to subscribe to that premise either.

Alexander R Pruss said...


I don't have much of a problem with multiple causal explanations of one phenomenon. After all, causal overdetermination is entirely unproblematic.

But there is a distinctive mode of stochastic explanation, which is more than just causal explanation. This mode of stochastic explanation of some effect E, say that about half of the coins land heads, depends on citing some condition C such that P(E|C) is high. But not just any condition that makes E probable can be appropriately cited.

It has to be an explanatorily relevant condition C. A classic example is that P(x doesn't get pregnant | x is taking the Pill) is going to be high in general, but is not explanatory when x is male.

The question of which conditions C that probabilify an explanandum E are explanatorily relevant is a difficult one. One classic answer is that we should explain E in terms of the finest statistically relevant partitioning possible. (In the case at hand, a finer partitioning is: men on the Pill, and that's statistically relevant since they are less likely to get pregnant than women on the Pill.)

If the classic answer were right, I would have an easy argument here, since God's goals will provide a finer statistically relevant partitioning.

But I am inclined to think the classic answer is wrong, for instance because it would immediately prohibit stochastic explanation in deterministic cases (since in deterministic cases, the finest statistically relevant partitioning would get things down to necessitating conditions), which at least prima facie we should allow (we should be able to stochastically explain why about half of the coin tosses were heads, even if the coins are deterministic).

So I am left without a principle to make the distinction with. But I can still have case-by-case intuitions. I've been trying to pump intuitions by agential cases, because those seem clearest to me. But other cases work, too.

More in a second...

Alexander R Pruss said...

Suppose that the result of a coin flip by a mechanical device is determined by extremely fine detail of the mass of the coin. So fine, in fact, that one molecule changes the result--the coins with an even number of molecules land heads, and those with an odd number land tails. In this case, normally, we would want to say that we have a stochastic explanation of why half of the tosses are heads.

But suppose that the coins being successively fed into the launcher were produced by a new process at the mint that generates an alternating sequence of molecule count parities: even, odd, even, odd, even, odd, .... Then indeed about half of the coins land heads. But we should no longer explain that fact stochastically. The correct explanation involves the alternating molecule count parities.

On the other hand, if the coins were generated by a process insensitive to the molecule count parities, then the stochastic explanation might well work, even if there were a finer-grained deterministic explanation that explains how each coin got to have the number of molecules it did.

Suppose that I run the birthday paradox on a class of 40 people, and unsurprisingly find that some students share birthdays. Normally, there is a stochastic explanation. But suppose that it turns out that the class consists of 20 pairs of twins, because it is a practical relationship class on how to get along with your twin, and you're required to sign up along with your twin. Then the birthday paradox calculation is no longer a correct explanation of why some students share a birthday.

I think one (defeasible) mark of a good explanation is the manipulationist one. Suppose we have two facts, C1 and C2, each of which is proposed as an explanation. Then we look at what happens when we vary each while keeping the other constant. If the explanandum relevantly changes when we vary C1 keeping C2 constant, but not vice versa, then we have reason to think C1 but not C2 is the explanation.

Now in the theistic case, if we keep fixed the divine plan to produce a certain pattern but change the secondary causal tendencies, we keep the same effect; but if we keep the secondary causal tendencies fixed, and vary the divine plan to produce the pattern, we change the effect. This suggests, but does not prove, that it is the divine plan that is explanatory.

(The applicability of this criterion is complicated when the divine plan is based on the tendencies--when the content of the divine plan is simply to produce something that matches the tendencies.)

Brandon said...

While I don't think it's necessarily fatal, I think one difficulty with the formulation of the problem here is that on the sort of Banezian Thomism view you're discussing, there is actually no difference between p and q: q is just another way of glossing q, because no events at all occur without God's primary causation: whenever you are talking about secondary causes, you are already talking about primary causes, because the ability of secondary causes to act at all is due to primary causes actualizing their potential. Thus the Thomist would already deny that there is any difference between (1) and (2): with (2) you've just given (1) more verbosely.

In other words, the progression here is that (1) we suppose that a particular secondary cause is able to have a certain effect with probability 1/2; (2) we then question why a primary cause would actualize the secondary cause to have a certain effect with a specific numerical probability at all. This only makes sense if we assume that the probability in (1) depends on no primary causation at all; but since a coin is, in fact, a secondary cause, the whole point of the Thomistic view would be that this is an incoherent assumption.

If, on the other hand, we take (1) already to presuppose primary causation, then we cannot say that it is (2) rather than (1) that does the explanatory work, because the contrast is incorrect: (1) and (2) are just two different ways to say the same thing, not two distinguishable probabilistic events that can be given two different explanatory statuses. Thus if (2) is a real probabilistic explanation, so is (1), for reasons much stronger than truth-functional equivalence.

This is why I brought up the contrast with Molnism in the previous post: Molinists can easily distinguish (1) and (2) because (1) is about a particular order of nature and (2) becomes about which order of nature God happens to actualize. But Thomists don't think that separating these two questions is a coherent account of primary and secondary causation.

Alexander R Pruss said...


On reflection, one problem with multiple objective P's is that in the end we will want there to be some sort of a Principal Principle, I think, about how epistemic probability should, in appropriate cases, match our best estimate of the objective P. But if there is more than one objective P...


Maybe p and q are logically equivalent, but they aren't the same proposition. In any case, even if they were, the claim P(God primarily causes the coin to land heads)=1/2 is odd. After all, P(God does A), if it's defined at all (I am sceptical about numerical probabilities of divine actions--it doesn't fit with my vague picture of divine freedom) is determined by the weight of reasons that God has for and against doing A. But why should, of necessity, the weight of reasons come to the same number as the objective physical tendencies whenever an event results from the objective physical tendencies?

Brandon said...

I don't think you can get out of the problem by distinguishing propositions here. For (1) to be even coherent on the Thomistic view, the relevant proposition has to already be understood as meaning at minimum something like "The coin is caused by its primary cause to land heads". The Thomist will think that other descriptions are either shorthand for this or inaccurate. So the only thing that (2) adds is that the primary cause that is being considered in this particular kind of case happens to be God. But I don't see your argument as being that the problem goes away entirely if the first cause is not God; and if that's the case, generalize one level, and just talk about the first cause. Then p and q are, in fact, the same proposition, differently stated, unless you are begging the question by separating thing that the Thomist denies can really be separated. Denial that they are the same would already be a denial of the Thomistic account of secondary causes, according to which every secondary causal action is a secondary action to some primary causal action.

March Hare said...

Are you not arguing probability only from ignorance here? Granted, it's what we do for most things, but still...

Define God. Most definitions will include omniscience, so P=1.

Which doesn't mean probability is pointless, it's still pretty useful in a card game even if the cards are known to non-players, but (to continue the analogy) God might be playing with a stacked deck.

Alexander R Pruss said...

No, these aren't just epistemic probabilities.