Another argument about simplicity

In an earlier post, I defended the idea (which Trent Dougherty also came up with independently and earlier) that only theory-unexplained entities, or kinds of entities, count against the simplicity of a theory. Here is another argument for this. Start with these two principles:

1. If theories T₁ and T₂ are otherwise equally evidenced and explanatorily powerful, but T₁ is simpler, then T₁ is more epistemically likely to be true than T₂.
2. The Principal Principle: Epistemic probabilities should (except in exceptional cases) be set to equal objective chances when the latter are available.

Now imagine that there is a powerful physical theory, T₀, according to which there is a special type of particle, U, that can only be produced through an exceedingly unlikely combination of events, so unlikely that it is unlikely that in the lifetime of the world the particle would ever be produced outside the lab. Scientists build the extremely expensive piece of apparatus to produce the particle. The apparatus is so expensive that it is unlikely it would ever be built again. But a rogue scientist gets hold of the apparatus at night and hooks up a bomb that will destroy the apparatus, and all results of any experiment, in five minutes. She also hooks an indeterministic fair coin flipper to the apparatus, so that if the coin comes up heads, U particle production is triggered, and if comes up tails, U particle production is not triggered. Consider now two theories:

• T_H: T₀ is true, no U particles ever get produced except perhaps in a moment by this apparatus, heads will come up, and a U particle will be produced by the apparatus.
• T_T: T₀ is true, no U particles ever get produced except perhaps in a moment by this apparatus, tails will come up, and no a U particle will be produced by the apparatus.

By a very plausible application of the Principal Principle, since the chances of heads and tails are equal as the coin is fair:

3. P(T_T)=P(T_H).

But if the number of explained kinds of objects counts against simplicity, then T_T is simpler than T_H, since according to T_T reality includes an extra kind of particle, the U particle. (If one doesn't think reality includes the future, run this thought experiment retrospectively after the explosion.) So by (2), then, P(T_T)>P(T_H). But this contradicts (3). Thus, by modus tollens, the number of explained kinds of objects does not count against simplicity.

Parental duty

Another excerpt from my forthcoming One Body book, this time from the discussion of gamete donation (challenge to the reader: find the relevance of this to gamete donation):

Now, it is not merely the duty of the parents to bring it about that the children are cared for and appropriately educated morally, religiously and academically. Rather, it is the duty of the parents to care for and educate the child—i.e., to do it themselves. In caring for and educating the child, parents will make use of the help of others, including that of family members, friends, and professionals. How much the parents can rely on the help of others before they have failed in their duty of caring for and educating the child will depend on the circumstances.
There are thus two aspects of the parental duty: (a) caring for and educating, and (b) ensuring that the child is cared for and educated. In other words, there is the aspect of parental activity and the aspect of results. These two aspects need to be balanced prudently, and, moreover, balanced with other duties the parents may have; how they are balanced will depend on particular circumstances. In no cases will it be desirable and rarely will it be possible for the parents directly to care for and educate the child in all respects with the help of no one else. Moral education, for instance, requires contact with virtuous people of a significant variety of different characters, not just the parents. Academic education should typically include education in subjects in which the parents lack competency. The need to work to earn money to provide for the child can force the parents to delegate a significant degree care to a third party.
Here is an observation worth making. In most couples, there will be specialization. Thus, the mother might be working long hours to earn the money needed to diaper, feed, clothe, and house the child, while the father might be changing the diapers, feeding, clothing, and otherwise taking care of the child for most of the day. It might seem that in such cases, each parent will be neglecting an aspect of the parental responsibility to himself or herself care for and educate the child. But we can respond to this by noting that parents should be friends of each other, and bringing in an idea from Aristotle’s Nicomachean Ethics. Aristotle considers what value there in having good friends. He observes that friends share a life, a friend is “another self,” and one can be active through one’s friend’s activity: what the friend does virtuously is something that accrues to oneself.

Conditional probability and nonmeasurable sets

Let P be Lebesgue measure on the three-dimensional cube [0,1]³. Assume the Axiom of Choice. Then there will be P-nonmeasurable sets (e.g, as there is no finitely additive (much less countably additive, rigid-motion-invariant probability measure on all subsets of [0,1]³ by the Banach-Tarski paradox. Now let F be the Lebesgue measurable subsets of [0,1]³. One might then hope that one can define P(X|Y) for all X and Y in F, as long as Y is non-empty.

Turns out that one can't, at least if one expects finite additivity and rigid-motion invariance. The reason for that is that if we let Y be (the surface of) a sphere in [0,1]³, then all subsets of Y will be in F, but there is no finitely additive rotation invariant probability measure on all subsets of a sphere, by the Hausdorff paradox. This problem disappears if we restrict ourselves to the Borel subsets of [0,1]³, but the extension to the Lebesgue measurable ones is epistemologically very plausible—obviously any subset of a null set should be a null set.

But here's a funny thing. While I haven't checked all the details—it's grading time so I can only give this so much thought—it turns out that one can sensibly define P(X|Y) for all X in F and all non-measurable Y. The easiest case is where Y is maximally non-measurable, i.e., all its measurable subsets have null measure and all its measurable supersets have full measure. In that case, one can simply define P(X|Y)=P(X). Moreover, one can naturally extend this measure to all X's in F_Y, where F_Y is the smallest σ-field generated by F and Y (basically by setting P(X∩Y|Y)=P(X) and P(X∩Y^c|Y)=0 for X in F).

This means that the Popper function approach to conditional probability on which P(X|Y) is defined for all X and Y in a single field or σ-field is not general enough, at least if we want P(X|[0,1]³) to be defined for all Lebesgue measurable X's. For in fact it seems we have more freedom as to what Y's we get to plug in and less as to what X's.

Limiting God to solve the problem of evil

Long ago, I remember reading with great curiosity Rabbi Kushner's Why Bad Things Happen to Good People? How disappointing that Kushner's intellectual answer seemed to be that God isn't omnipotent. (His practical answer not to worry about the question but just to do good is much better.) The idea of limiting divine attributes in part to answer the problem of evil has recently had some defense (e.g., here and in the work of open theists), so I guess it's time to blog the objection to Kushner—which applies to the others as well—that I had when I read him, with some elaboration.

Basically, the objection is that as long as God remains pretty good, pretty smart (he was smart enough to create us!) and powerful enough to communicate with us (Kushner at least accepts this), then serious cases of the Problem of Evil remain. Moreover, these cases do not seem significantly easier to solve than the cases of the Problem of Evil that were removed. Consequently, the intellectual benefit with regard to the Problem of Evil is small. And the intellectual loss with regard to the simplicity of the theory is great—the theory that God has all perfections is far simpler.

Start by considering a deity whose goodness is unlimited but whose knowledge and power are fairly limited.

Consider, first, the problem of polio. This is certainly a horrendous evil. And the limited deity could have alleviated a significant portion of the problem hundreds of years earlier simply by whispering into some people's ears how to make a vaccine—surely any deity smart enough to create this world would be smart enough to figure out how to make vaccines. Maybe the limited deity couldn't have prevented all cases, in the way that an unlimited God could. But given that neither did the wholesale prevention happen nor did the partial prevention by vaccines happen as early as it could have.

Consider, second, the many cases where innocent people suffered horrendously at the hands of attackers, where the attack could have been prevented if the people had been warned. Even a deity of limited power and knowledge should be able to see, for instance, that the Gestapo are talking about heading for such-and-such a house, and could then warn the occupants. (I am not saying that such warnings were never given—for all I know, they were in a number of cases. But I am saying that there are many cases where apparently they were not.)

Moreover, even if one limits the goodness of the deity, and only claims that he is pretty good, the problem remains. For unless the deity had a very serious reason not to tell people about vaccines and not to warn the innocent victims of horrendous attacks, it seems plausible that the deity did something quite bad in refraining from helping, so bad as to be incompatible with being pretty good. (If the deity had a reason that fell a little short of justifying the refraining, then that might be compatible with being pretty good; but a reason would have to be pretty serious for it to fall only a little short of justifying the refraining when the evils are so horrendous.) So even if one thinks that the deity has limited power and knowledge and is only pretty good, the problem of finding very serious reasons for the deity's non-interference remains.

Granted, the problem is diminished, especially if one has decreased the belief in divine goodness. But notice that the decrease in belief in divine goodness is the most religiously troubling aspect of a limited God doctrine. And even that does not make the problem go away.

Moreover, the sorts of things one can then plausibly say about the remaining problems of evil are things that, I suspect, the traditional theist can say as well about this and many other cases. Perhaps God does not prevent all attacks on innocent people (for all we know, he prevents many) because he wants humans to have effective freedom of will. Perhaps he wants to give victims opportunities for forgiveness of their aggressors in an afterlife. Perhaps God does not prevent disease because he wants us to help our neighbor and to develop medical science to this purpose. Or to give us an opportunity to join him on the cross in redeeming humankind. Or perhaps God prevents many evils, but his purposes do not allow him to prevent all, and some arbitrary line-drawing is needed. I am not saying that these answers are sufficient (though I think some contain a kernel of something right), but only that they can be equally used in the case of a limited and unlimited God, and in the case of an unlimited God such answers may well have rather general applicability.

Reducing sets

I find sets to be very mysterious candidates for abstract entities. I think it's their extensionality that seems strange to me. And anyway, if one can reduce entities to entities that we anyway want to have in our ontology, ceteris paribus we should. I want to describe a three-step procedure—with some choices at each step—for generating sets. I will use plural quantification quite a lot in this. I am assuming that one can make sense of plural quantification apart from sets.

Step 1: The non-empty candidates. The non-empty candidates, some of which will end up counting as sets in the next step, will be entities that stand in "packaging" relation to a plurality of objects, such that for any plurality, or at least for enough pluralities, there is a candidate that packages that plurality. There are many options for the non-empty candidates and the packaging relation.

Option A: Plural existential propositions, of the form <The Xs exist>, where a plural existential proposition p packages a plurality, the Xs, provided that it attributes existence to the Xs and only to the Xs.

Option B: Plural existential states of affairs (either Armstrong or Plantinga style), i.e., states of affairs of the Xs existing, where a plural existential state of affairs e packages a plurality, the Xs, provided that it is a state of affairs of the Xs existing. I got this option from Rob Koons.

Option C: This family of options generates the candidates in two sub-steps. The first is to have candidates that stand in a packaging relation to individuals, such that each candidates packages precisely one individual. Call these "singleton candidates". For brevity if x is a singleton candidate that packages y, I will say x is a singleton of y. The second step is to take our non-empty candidates to be mereological sums of singleton candidates, and to say that a mereological sum m packages the Xs if and only if m is a mereological sum of Ys such that each of the Ys is a singleton of one of the Xs and each of the Xs is packaged by exactly one of the Ys. We need the singleton packaging relation to satisfy the condition (*) that a mereological sum of singletons of the Xs has no singletons as parts other than the singletons of the Xs. (In particular, no singleton of y can be a part of any singleton of x if x and y are distinct.)

We get different instances of Option C by considering different singleton candidates. For instance, we could have the singleton candidates be individual essences, and a singleton candidate then packages precisely the entity that it is an individual essence of. I got this from Josh Rasmussen. Or we might use variants of Options A and B here: maybe a proposition attributing existence to x or a state of affairs of x existing will be our candidate singleton. (Whether the state of affairs option here differs from Option B depends on whether the state of affairs of a plurality existing is something different from the mereological sum of the states of affairs of the individuals in the plurality existing.)

There are many other ways of packaging pluralities.

Step 2: The empty candidate. We also need an empty candidate, which will be some entity that differs from the non-empty candidates of Step 1. Ideally, this will be an entity of the same sort as the non-empty candidates. For instance, if our non-empty candidates are propositions, we will want our empty candidate to be a proposition, say some contradictory proposition.

Step 3: Pruning the candidates. The basic idea will be that x is a member of a candidate y if and only if y is one of the non-empty candidates and y packages a plurality that has x in it. But the above is apt to give us too many candidates for them all to be sets. There are at least two reasons for this. First, on some of the options, there won't be a unique candidate packaging any given plurality. For instance, there might be more than one proposition attributing existence to the same plurality. Thus, the propositions <The Stagirite and Tully exist> and <Aristotle and Cicero exist> will be different propositions if Millianism is false, but both attribute existence to the same plurality. Second, some of the candidates will be better suited as candidates for proper classes than for sets and some candidates may be unsuitable either as sets or as proper classes. For instance, there might be a proposition that says that the plural existential propositions exist. Such a proposition packages all the candidates, including itself, and will not be a good set or proper class on many axiomatizations.

Feeling cold? Apply to our graduate program!

Are you thinking of grad school and feeling cold? As your December days get colder and darker, you may want to reflect on the warm weather in Waco. It was over 80F today.

Of course, the really great thing is the warmth of the graduate student community.

We've extended our deadline for the Baylor Philosophy PhD program this year until January 2.


The anole photo is from today, from the path by the river on campus (or just off campus?).  The butterflies are from Thanksgiving, though I saw a number today, too.

Scepticism and causeless events

Suppose that there is no First Cause. Then there can be uncaused events—the coming into existence of the universe is an example, for instance. Now consider the Ultimate Sceptical Hypothesis (USH): you are a nonmaterial being that is the only thing that ever exists; you came into existence the previous moment for no cause at all; and there is no cause of your having the presnet occurrent mental states you now have; and you have just these occurrent mental states and no other states.

Compare, now, USH to what its main nonsceptical alternative is if there is no First Cause. That main alternative will be SN: scientific naturalism, with the initial state of the universe being a brute, unexplained fact. USH is simpler than SN. It is simpler on the crude criterion of entity counting: USH has only one entity, you, while SN has many atoms, galaxies, houses, geckos, etc. However, if we count only unexplained entities, as I suggested in a previous post, USH has only you and your present occurrent mental states (which there aren't many of!), while PN has the universe and its initial state, so maybe we have a tie. But PN is much more descriptively complex: it includes a number of laws of nature with various constants, for instance, as well as a high-energy extremely low entropy initial state. While you just have whatever occurrent mental states you now have—which is not much at all (how much of a thought can you think in an instant). So USH seems to be preferable to PN on grounds of parsimony.

Thus, rejecting a First Cause leads to scepticism.

This is, of course, a variant of a Rob Koons argument.

Music, religion and appreciation

I typically do not appreciate music at all. While there are rare exceptions, music typically leaves me aesthetically cold or annoys me (though there may be a non-aesthetic emotional impact, say of creepy music during a scary part of a movie). This inability to appreciate music is a kind of disability, one that I hope will be gone in heaven (plus the music there will be better), since music seems an important part of the human good of aesthetic appreciation.

I suspect that how I typically feel about music is how many (though not all) non-religious people feel about religion: while it may be good for others, it's just not something one finds oneself getting anything out of. But I think there is a crucial disanalogy. For it is uncontroversial that to be properly benefited by receptive aesthetic goods, like those proper to listening to music or contemplating a painting, one needs to experience them with appreciation. One gets nothing from musical goods without listening to the music, and mere listening gets one nothing of the aesthetic good if one doesn't appreciate. (Though experiencing the art without appreciation can lead to later development of appreciation, and an analogous claim can be true of religious practice.) But according to many of the great religions, many of the goods of participation—say, innate transformations of the soul, the intrinsic value of praising God, etc.—can occur in the absence of experiential appreciation.

There is also another disanalogy. Participating in religious goods isn't exactly analogous to experiencing works of art. Rather, it is analogous both to experiencing and to creating them. And creating works of art is an aesthetic good that perhaps does not require appreciation of the works of art that one is creating. One could have a sculptor who manages to express her artistic vision in incredible ways, but who incorrectly experiences herself as producing junk. The artist need not understand her work.

"Love does not seek its own"

Here is another excerpt from Section 2.2. of my forthcoming One Body book.

One way love is humble is that the actions of love are not focused on agapê itself [...]. There would be something odd about a parent explaining why he stayed up the night with a sick child by saying: “I love my son.” Surely the better justification would be the simpler: “He is my son.” The latter justification puts the parent in a less grammatically prominent spot (“my” instead of “I”), and shows that the focus is on the son. Most importantly, however, the use of “I love my son” as a justification would suggest that if one did not love him, the main reason to stay up the night would be missing. But the main reason to stay up the night is that he is one’s son. That he is one’s son is also a reason to love him as one’s son, and that one loves him may provide one with a further reason to stay up with him. However, the main reason for staying up is not that one loves him; rather, the love, expressed in the staying up, is a response to a reason that one would have independently of the love. Thus, in an important sense, the parent acts lovingly—acts in a way that is at least partly constitutive of love—without acting on account of love. Love’s actions are not focused on love but on the beloved as seen in the context of a particular relationship.

However, to explain why we made some sacrifice for someone to whom we had no blood ties, we might well say, “I love him.” Nonetheless, I suggest, this may be an imperfection—it may be a case of seeking one’s own. Why not instead act on account of the value of the other person in the context of the relationship? It is true that love may be a central part of that relationship, but I want to suggest that love is not the part of the relationship that actually does the work of justifying the sacrifice. For suppose that I stopped loving my friend. Would that in itself take away my obligation to stand by him in his time of need? Certainly not. The commitment I had implicitly or explicitly undertaken while loving him, a commitment that made it appropriate for him to expect help from me, is sufficient for the justification. If I need to advert to my own love, then something has gone wrong.

Besides, there would a circularity in appealing to one’s own present love to justify one’s basic willingness to engage in loving actions for the beloved. For if one were not willing to do loving actions for the other, then one would not be loving the other, and hence a total failure to will to do loving actions for the other would not be a violation of love, for there would be no love there to be violated. Of course, such a failure might well be a violation of one’s duty to love the person (whether arising out of personal commitment, or a general duty to love everyone or some specific duty like those we have to our relatives), but that is a different issue. It is not love, then, that justifies the general willingness to act lovingly, but the value of the other and the kind of relationship that one stands in to the other apart from the fact of love.

Simplicity and multiple universes

In yesterday's post, I offered a criterion for when multiplicity of objects or kinds posited by a theory counts against the simplicity of a theory: namely, when it is a multiplicity of objects or kinds not explained by the theory.

Let's apply this to multiple universe theories, which people do tend to see as offending against simplicity.

Lewis's Modal Realism: Lewis's universes have no explanation of their existence. Their existence is simply a brute fact. Thus, the infinitude of Lewis's universes counts against the simplicity of a theory. Moreover, along with unexplained universes there will be unexplained kinds. Lewis's theory, for instance, implies that there exists a universe where an uncaused griffin exists from the beginning. Likewise, unicorns, Pegasuses, and so on. So Lewis's theory implies the existence of a great diversity of unexplained kinds of things. And that should count against the theory.

Theistic Multiverses: Theistic multiverse theories have an infinity of universes, but these universes are explained by God's goodness in creating all universes that it is worth creating. Thus there is only one unexplained entity in the theory--God--and there is no offense against simplicity.

Physicists' Multiverses: I don't know enough about string-theoretic multiverses to say anything about those here. But inflationary universes that bubble up out of other universes will be exempt from the worry if there is one root universe from which the others come, since then the cost in terms of unexplained entities is the same as that of single universe theories, and there is no offense against simplicity.

Simplicity

Consider two theories.

Theory 1: There is a single asexually reproducing ancestor of all life on earth, Ag, who came into existence at time A and split into two almost genetically identical descendants, Bof and Bok, at time B, and there was no biological life before Ag.

Theory 2: There is a pair of almost genetically identical ancestors of all life on earth, Bof and Bok, who came into existence at time B, and there was no biological life before Bof and Bok.

Both theories fit our observed data equally well, and will always do so, since we have basically no chance of identifying a fossil of Ag. The question I want to ask is which theory is simpler.

An easy thing to say is that Theory 2 is simpler, as it posits one fewer entity, namely Ag, but Theory 1 compensates for its greater complexity through having an additional explanatory merit--it explains the genetic similarity between Bof and Bok, which Theory 2 leaves unexplained. Moreover, the difference in complexity is small, because while Theory 1 posits one more entity, it is an entity of the same kind as Bof and Bok.

But I want to consider a different evaluation: Theory 1 is simpler, because when we consider the simplicity of a theory in terms of entity or kind counting, we only count the entities or kinds not explained by this theory (or, better, entities or kinds weighted by the degree to which they are unexplained by the theory, if explanation comes in degrees). Thus, Theory 1 posits one theory-unexplained entity while Theory 2 posits two theory-unexplained entities. So Theory 1 wins not only on explanatory grounds, but also on simplicity grounds.

Why go for this method of evaluation? First, in cases where the explanation is deterministic, it coheres with information-theoretic compression-based measures of complexity, which is a plus. Second, I think this fits with our intuitions about other examples. Consider two theories about the origin of life on earth.

Theory 3: A meteorite deposited some organic chemicals 4.1 billion years ago that combined to produce life.

Theory 4: A meteorite deposited some organic chemicals 3.9 billion years ago that combined to produce life.

As far as the details I gave of the two theories, there is no difference in complexity. But Theory 3 commits us to way more organisms in the history of the earth--200 million years' worth of organisms. And when we conjoin with evolutionary theory, Theory 3 will commit us to significantly more kinds of organisms as well--over those 200 million years, surely there would be a lot more species. But these added entities (or kinds) should not by themselves count as increasing the complexity of Theory 3 over Theory 4 (or Theory 3 + evolution over Theory 4 + evolution). Why not? The best explanation of why not seems to me to be that these added entities and kinds are easily explained by the theory in question.

The entities that are unexplained by one theory may, of course, be explained by another. It is only entities posited but unexplained by the theory in question that I am considering here. Nor am I saying that this is the only contribution to complexity--there are, no doubt, many others.

All this casts helpful light on the question whether theism or naturalism is simpler. At least it undercuts simple arguments that naturalism is simpler as it posits one fewer entity or even kind of entity.

Kant's argument from gratitude

Kant famously argued that we had a duty to believe in God, as this was necessary for us to fulfill the duty to give thanks for the universe. My understanding is that Kant thought that this was not an argument for the existence of God, but only an argument for the duty to believe in God. But surely it becomes an argument for the existence of God (or at least an agent who caused the universe) when one adds the very plausible premise:

• One only has the duty to give thanks for a product of agency.

Perhaps, though, Kant would want to subjectivize this premise into:

• One only has the duty to give thanks for what one believes to be a product of agency.

But if one says this, then the atheist doesn't have a duty to give thanks for the existence of the universe, and Kant's argument fails. So it seems that either Kant's argument fails (at least in the case of atheists--maybe you could argue that if you believe in God then you have a duty to believe in God, which is an interesting result, but I don't think it's what Kant was trying to argue for) or Kant is wrong that we can't argue for the existence of God or both.

But I am no Kant scholar.

Prayer and Thomistic accounts of chance and design

On Thomistic accounts of chance and design, God micromanages the outcomes of chancy processes by means of primary causation, ensuring that the processes secondarily cause precisely the results that God wants. (Thomists often say a similar thing about free will, too.) On such accounts we can distinguish between two different ways that God can achieve a result, which I will call the miracle and natural methods. In the miracle method, God suspends the causal powers of the chancy process and directly cause the specific outcome he wants. If he does this on a die toss (I'll assume that die tosses are indeterministic), then the hand tosses the die, but somewhere there will be a break in the natural chain of causes. In the natural method, God causes the the causal powers of the chancy process to cause, in the way proper to them, the specific outcome he wants. Presumably, given that the natural method preserves the value of finite causes' activity, much of the time God providentially acts using the natural rather than the miracle method.

Now suppose that I am about to toss a die. And suppose that I pray, for all the right reasons (say, a good to a friend will result from non-six, and nobody will be harmed by it) and in the right way, that the die should show a non-six, while no one else prays that it should show a six. Moreover, suppose that God in fact does not have any significant counterbalancing reasons in favor of the die showing six. Let C be a complete description of the state of the world--including all the facts about the universe on which God's reasons are based--just before the die toss result. This seems a paradigmatic case for God to be moderately likely to exercise providential control. Moreover, let us suppose with the typical Thomist that almost all the time, excepting cases of particularly spectacular demonstrations, God exercises providential control by the natural method. Suppose then:

• P(God wills non-six | C and no miracle) > 0.95.

And since God wills non-six if and only if non-six occurs on the Thomistic view:

• P(non-six occurs | C and no miracle) > 0.95.

Suppose that in fact three occurs. It is then obviously correct to explain the non-six by adverting to the above 0.95 probability. The question of interest to me is this: Can we also explain the non-six by the fact that natural causes described in C, in isolation from the facts about prayer and the like, had a probability of 5/6 of producing a non-six?

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.

Then:

2. 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:

3. P(the coin lands heads | no miracle)=1/2

and

4. 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.

Barr on chance and design

Stephen Barr has an article in First Things where he argues that there is no conflict at all between chance and divine design. The position seems to hinge on two claims:

1. A series of events is chancy if and only if the secondary (i.e., finite, non-divine) causes of the events are independent of one another.
2. God controls series of events by primary causation.

Given (1) and (2), there is no conflict between divine design, since (1) says nothing about dependencies among events induced by primary causation.

I wish Barr's account worked. I'd love for there to be a good account of the interplay of chance and design. But there are a number of serious problems with Barr's proposal.

I. The account of chanciness does not work in the case of a single event. Depending on how we read (1), a single event will either trivially count as chancy (since its cause is independent of the causes of all other events in the series, there being no others) or it will never count as chancy. But single events can be chancy (imagine a universe where there is only one quantum collapse happening) or non-chancy.

II. Chance is explanatory, both in gambling and in evolution. But independence of causes has no explanatory force--without probabilities or chances, it generates no useful statistic predictions or explanations (here is a very technical way to make the point). So Barr's account needs something more, something like objective tendencies of the secondary causes that give rise to probabilities. I will assume in some of the following criticisms that something like this has been added.

III. Suppose I go to the casino and I play the slot machine a thousand times, and each time win, due to independent secondary causes, because God so arranged it. It seems absurd to say that I won by chance, and yet on Barr's definition my winning is a matter of chance.

IV. The preceding case shows that it is difficult to see how one can make any probabilistic predictions about a chancy series of events. Consider an infinite sequence of coin tosses where the limiting frequency of heads is 1/2. God can just as easily make this infinite sequence of heads come out in the case of an ordinary fair coin as in the case of a coin heavily biased in favor of heads. When God controls series of events by primary causation—and as far as Barr's position goes, this could be always—it is not clear why we should expect frequencies to match the probabilities arising from the tendencies of secondary causes. The frequencies of events will be precisely what God needs them to be for his purposes. Why think his purposes match the probabilistic tendencies of secondary causes? Now it could be that God wills to ensure that the actual frequencies usually match the secondary causal tendencies, in order that the universe be simpler and more predictable. That is a reasonable hypothesis. But then the it seems that it isn't the secondary causal tendencies that are directly explanatorily of the observed frequencies, but rather the explanation of the observed tendencies is God's will. I.e., in the case of a fair coin, the reason the limiting frequency comes out as heads is because God willed to ensure that the limiting frequency match the secondary causal tendency of the coin. The secondary causal tendency of the coin is still explanatorily responsible for this outcome (because God willed to match the frequency to it), but it isn't causally responsible for this outcome (unless we take an occasionalist analysis of secondary causation).

V. In light of III and IV, no statistical prediction can be made from probabilistic facts about the causal tendencies of secondary causes without an implicit auxiliary hypothesis that God through primary causation willed a particular series of events whose statistical features match the stochastic features of the causes. This is not entirely special to Barr's account—-probably every theistic account requires an implicit auxiliary hypothesis that God works no miracle here. But in Barr's case there is a difference—it's not just a hypothesis that God works no miracle here, since in the case where God makes me win the slot machine a thousand times in a row, on Barr's view no miracle has occurred, just the ordinary chancy operation of secondary causes and God's primary causal oversight. So Barr's view needs two auxiliary hypotheses to generate empirical predictions from scientific data: a no-miracles hypothesis like in every theistic case and a hypothesis of stochastic-to-statistical match.

VI. Random processes need not involve independent causes. Take, for instance Markov chains or exchangeable sequences of random variables.

VII. The elliptical orbits of the planets in our solar system and of the planets in another solar system have independent causes—the gravitational influences of different bodies—and hence by Barr's criterion the two events are chancy. But it's not chance that the orbits are elliptical. Now maybe Barr will count these cases as not independent because they are governed by the same laws of nature. True, they are. But so are paradigmatically chancy events, like the results of successive quantum collapse experiments.