Design Arguments and Probability - Reply to Ocham

I certainly don't mind being challenged, and I can usually count on my regular commenter, Ocham, to do just that. He seems to take issue with nearly everything I say. My last post on Intelligent Design was no exception. Here's his latest:

Ocham: I don't see how the designist argument counts as an explanation. As Dawkins says, it's a cop out. How do you explain how the designer came to exist, in all its complexity and so forth.

Another problem: as Einstein said, all arguments from probability disappear in the face of the improbable event actually happening. In the case that any highly improbable event occurs, namely the event that there exist conscious beings capable of asking how probable their existence is, naturally they will come to the conclusion that their existence cannot be chance at all. But in that case they would be wrong.

My reply: I'm surprised you're bringing in Dawkins here, Ocham. His favorite "Who designed the designer?" line, which he clearly takes to be a knock-down refutation of the design argument, is hardly that. When the ID theorist proposes that a mind or designer might serve as a basic explanation, she does not automatically commit herself, as Dawkins supposes, to such claims as

1. Basic designers must be complex beings.
2. Basic designers must be contingent beings.

What Dawkins' question does show is that the ID theorist is implicitly committed to a denial of both (1) and (2) - a truly basic designer must be an undesigned, non-emergent designer. But a being like God fits that bill perfectly, since God is a metaphysically simple, noncontingent being. So, far from refuting design arguments for theism, Dawkins' line actually helps the theist connect the dots from premise to conclusion. That Dawkins fails to see this shows only that he doesn't understand theism and refuses to take it seriously as a worldview.

Secondly, the probabilistic considerations you raise ignore the important distinction between prior and posterior probabilities. A priori, for example, any specified string of 10 poker hands is equally likely to be dealt. Hence, getting a string of 10 spade suite royal flushes in a row, say, is just as likely as getting any other string of 10 poker hands. In short, where X and Y are two arbitrary sequences of 10 poker hands, Prob(X)=Prob(Y). But when we are evaluating explanations we're not interested in prior probabilities but posterior probabalities. According to Bayes' theorem, the probability of a given explanatory hypothesis H given evidence E is

Prob(H/E) = [Prob(H)*Prob(E/H)]/Prob(E)

The prior probability of an arbitrary sequence of 10 poker hands gives us only Prob(E). The other quantities on the RHS also need to be evaluated before we can draw any interesting conclusions regarding the respective merits of chance-type hypotheses and design-type hypotheses. Intuitively, if a person gets one royal flush, we'd say he probably just got lucky. If he gets two in a single game, that's awfully suspicious. If he gets 3 or more, someone's likely to get shot for cheating. The reason is that a royal flush is not just any hand in poker. It's a very special hand, one that intelligent poker agents take (or are likely to take, or should take) particular interest in. So as the number of royal flushes in a game goes up, the design hypothesis (it's rigged; someone's cheating) rapidly becomes more plausible than the chance hypothesis (it's sheer luck or coincidence).

From the Mailbag: On Mind and Intention

Reader C Grace (Celinda) asked me to comment on the following argument:

p1 Intrinsically, the mind has no form or existence only intention
p2 Intention is a potential relation
p3 Representation occurs when the mind takes on the form of the object presented
p4 Willing is the act of actuating an intention thus creating an actual relation
therefore
c1 the mind gains form and existence from willing a representation of an actually existing object
c2 any misrepresentation that we will has no existence
c3 any actually existing thing that we do not will, that we disbelieve, reject etc, is not represented in the mind.

For background on (p2) see comments on this blog post, and for more details on the overall argument see Celinda's blog here. She's coming from a broadly Thomistic angle.

OK, now for comments. I hope, Celinda, that these will help you refine your thoughts on these issues.

First, I'm skeptical of (p1). It seems to say that that the mind is, in itself, just a tabula rasa, a blank slate waiting for some experience of being to etch something on it and thereby in-form it. The reason I'm skeptical is that I think the mind as blank slate idea has been conclusively refuted from a variety of different angles. I don't have time to go into details, but there are powerful arguments from thinkers like Plato, Reid, Peirce, Chomsky, et al. to the effect that we come into the world partially "hard-wired" with certain a priori conceptions. So I wouldn't say that the mind has absolutely no form. Some Thomists may object and say that allowing any a priori form into the mind will give rise to a Kantian-style skepticism of things-in-themselves. But that doesn't follow if the a priori structure of the mind is isomorphic with the intelligible structure of reality, as thinkers like Reid, Peirce, and Lonergan would argue.

That said, I would agree with Aquinas that our minds start out as largely indeterminate, having a primordial orientation toward being and a receptivity for the forms (or intelligible characters) of experienced objects.

In (p2) I wouldn't talk of a 'potential relation' as opposed to an 'actual relation'. There is either a relation or there isn't. If there is a relation then it is ipso facto actual. What I gather is meant by (p2) is that an intention in itself, considered in abstraction from the existence or nonexistence of its object, does not actually constitute a relation to its object, but if its object does exist and that intention is "fulfilled", then the intention serves to bring the mind into relation with that object.

Regarding (p3), it's not clear what it is for the mind to "take on the form" of the object presented. Is this form a type of qualia, concept, perceptual judgment, or something else? Qualia and/or concepts in themselves will not suffice to bring the mind into relation to any particular object, so I'm guessing that a perceptual judgment (e.g., "That's a red chair") is what's needed here.

Talk of willing "actuating an intention" (p4) sounds odd to me. And why would it create an actual relation? Better, I think, is to say that to will is to form an intention (or simply intend) to do X for the sake of Y. But that by itself won't suffice to bring about a relation between oneself and the ultimate object of one's willing. Perhaps the intention has a built-in time delay (I intend to get up in ten minutes). Or perhaps, unknown to me, I've become paralyzed, such that when I try to carry out my intention I find that I can't.

As for the conclusions (c1)-(c3), I find these rather implausible as they stand. Nor it is terribly clear how they are supposed to follow from the premises. I'll mention just a few issues I have concerning each.

Regarding (c1), I don't think it's right to say that the "mind" wills - persons will, minds don't. Nor does the mind "gain existence" in this way. You need to have an idea of X and thus a mind before you can will X.

Regarding (c2), what is the sense of "misrepresentation" - a false judgment? If so, how does one will a false judgment? Finally, if someone's wills a misrepresentation then doesn't the misrepresentation have to exist in the willing of it?

Regarding (c3), if I reject or disbelieve something then it has to be represented in my mind. I can't have a propositional attitude (like disbelief) without having the corresponding proposition in my mind.

Again, Celinda, I hope these comments help. As it stands, I don't think your argument is very effective. Perhaps you can fix it up for us, or clue me in if I've misunderstood you.

Which is More Like Alchemy - Darwinism or ID?

In my previous post I linked to a recent essay by William Dembski, one of the leading figures in the Intelligent Design (ID) movement. In the essay, he argues that Darwinism is significantly analogous to alchemy such that, like alchemy, its scientific merits should be suspect.

In should mention that Dembski doesn't use the term 'Darwinism' in this essay, but it is his preferred label for the view he wants to criticize. He defines 'Darwinism' elsewhere as follows:

Darwinism is really two claims. The less crucial claim is that all organisms trace their lineage back to a universal common ancestor. . . . This claim is referred to as “common descent” or “universal common ancestry.” Although evolutionary biology is committed to common descent, that is not its central claim.The central claim of evolutionary biology, rather, is that an unintelligent physical process can account for the emergence of all biological complexity and diversity. Filling in the details of that process remains a matter for debate among evolutionary biologists. Yet it is an in-house debate, and one essentially about details. (Link)

Dembski's comparision between Darwinism and alchemy runs something like this:

• Alchemy: Despite a paucity of positive evidence for the claim that lead can be transmuted into gold, and despite the lack of any "causal specificity" on how this transmutation could be effected, the alchemists were convinced by a prior commitment to neoplatonic metaphysics that transmutation had to be possible.
• Darwinism: Despite a paucity of positive evidence for the claim that life can arise from nonliving matter by purely physical processes, and despite the lack of any causal specificity on how this transformation can occur, Darwinists are convinced by a prior commitment to a materialistic metaphysic that such a transition must be possible.

Setting the merits of Dembski's comparison aside for the moment, a commenter on my previous post, William, suggests that the same charge that Dembski levels against Darwinism can, with equal if not more justification, be leveled against ID:

Demski [sic] risks being speared by his own argument. His objection to absence of "causal specificity" in evolution is extremely well-addressed in the literature of the science. His alternative hypothesis, "intelligent design" is justly criticised for failing to propose any kind of mechanism for the design it purports to detect. It has no more causal specificity than alchemy, in fact, somewhat less.

Now, I'm not entirely sure what William means by saying that the lack of causal specificity in evolution is "extremely well-addressed" in the literature. I make no pretentions to being well-versed in the origin-of-life literature, though I have read enough to know that the problem is widely regarded as being very far from a solution. But let's set that aside. What I want to focus on is William's claim that ID is "justly criticized for failing to propose any kind of mechanism for the design it purports to detect." I think this charge reflects a common and important misconception about ID, which I'd like to explain.

At bottom, the debate between Darwinism and ID is a methodological debate over whether mind (design) can ever be a basic (non-derivative) and scientific explanation of worldly phenomena.

According to Darwinism there must be a purely materialistic causal pathway from molecules to man. One possible ground for this confidence is a prior commitment to metaphysical materialism. Dembski seems to suggest that this is the only possible ground, but perhaps Jeffersonian deism could generate the same result. In any case, exclusive reliance on material causes as explanations of worldly phenomena requires a worldview that grounds and justifies that reliance, and materialism fits the bill nicely.

According to materialism, before the first mind all there was was matter, space-time, and laws of physics governing how matter gets rearranged over time. Eventually, some lump of matter got organized in just the right sort of way and, voila, life emerged. Gradually these living lumps of matter got organized in more and more complex ways until, viola, a mind emerged. Thus, on the materialist view, mind is nothing but an emergent product of matter. Since mind is wholly explained by the arrangements of matter that sustain it, mind cannot be a basic explanation of anything.

Of course, this is not to imply that minds and designers can never be derivative explanations - most materialists would, for example, explain Shakespeare's plays as products of the Bard's mind. But they would then go on to insist that if we could trace the causal chain back through Shakespeare's ancestry we would eventually arrive at a point at which no minds existed. Moreover, Shakespeare's mind, insofar as it exists and is capable of doing anything at all, derives all of its causal powers from the material substratum upon which it supervenes.

The foregoing explains why Darwinists have no problems with the scientific credentials of SETI (the Search for Extraterrestrial Intelligence) even though the SETI researchers are looking for signals that exhibit design and thus point to the existence of ETI's. The reason Darwinists have no problem with SETI is because the researchers are committed to the idea that whatever minds they might detect would themselves have to be reductively explained both diachronically, as evolutionary products from nonliving matter, and synchronically, as supervenient effects of purely material causes. So Intelligent Design (ID) proponents who point to SETI as an example of the scientific legitimacy of design-type explanations are to some extent missing the point. Darwinists are happy to admit design-type explanations into science, provided that the posited designers can then be given a reductive, materialistic explanation.

Now what about ID? I think the best way to understand ID is, primarily, as a methodological position in the philosophy of science; secondarily, as a research program; and tertiarilly as a set of theories about X, Y, and Z. As a methodological position, ID is defined by its rejection of the Darwinian methodological constraint that mind cannot be a basic explanation. As a denial of this, ID should not be taken to imply a commitment to the positive thesis that mind is in fact a basic explanation of some worldly phenomena. Rather, it is a commitment to the thesis that mind could be a basic explanation and that there would be nothing scientifically untoward should that turn out to be the case. As a research program, ID is defined by the claims that design is reliably detectable and that, with respect to the origins and development of life on earth, it is highly probable that design, even basic design, has played a significant role. To date, this research program has encompassed a wide array of specific design theories, for example, that theory that the bacterial flagellum is 'irreducibly complex' and thus an instance of (possibly basic) design. Critics who charge ID for its lack of specifics are very often guilty of conflating ID-as-methodology and ID-as-research-program with ID-as-theory-of-X. On the other hand, I think ID proponents could be more clear in keeping these levels distinct.

With that background in place, I think we can begin see the problem with William's charge that ID is "justly criticized for failing to propose any kind of mechanism for the design it purports to detect" (emphasis added). In the first place, the criticism expects from ID the kind of specificity appropriate to a theory-of-X, overlooking the more fundamental methodological and programmatic aspects of ID. In the second place, the criticism expects ID to become something it is not. As I've pointed out, ID is committed to a wider philosophy of science that allows (though doesn't require) design explanations to be basic explanations. But if design is a basic explanation for some worldly phenomenon, then there cannot be a "mechanism for the design". To ask for a mechanism at that point is to ask the ID theorist to give a reductive explanation of design or mind in terms of something else. This is tantamount to asking the ID theorist to give up the committment to the possibility of basic design explanations, and therefore tantamount to asking him to give up ID. Obviously, that is not something any self-respecting ID theorist ought to concede.

I conclude, then, that ID, properly understood, does not slip into alchemy. Whether Dembski's is right that Darwinism does so, is something I'll leave for the reader to decide.

Evolution as Alchemy

An interesting comparison by leading ID expert William Dembski. (Link)

Further Thoughts on the Probabilities of Conditionals

In an earlier post, I proposed the following hypothesis concerning the probabilities of conditionals:

(RH) (a) If p entails q then P(If p then q)=1. (b) If p entails ~q, then P(If p then q)=0. (c) If p entails neither q nor ~q, then P(If p then q) = P(X), where X is whatever information is contained in q that is not already contained in p.

And I suggested that the term X in last part (c) is to be understood as the conjunction of all propositions entailed by q that are not entailed by p.

Now, it'd be nice to be able to reduce conditions (a) and (b) to (c) by treating them as limit cases of the latter. I'm just not sure how to formulate a convincing argument for that reduction.

In case (a), X drops out entirely, since if p entails q then it also entails every proposition entailed by q. Perhaps we could think of X in this case as a proposition with no content, i.e., a tautology. As a tautology, P(X) = 1. So if we are entitled to construe X in this way, then (a) can be reduced to (c).

In case (b), we could take a different tack. If p entails ~q, then by (a) P(If p then ~q) = 1. Now, if we accept (as I do) conditional excluded middle (CEM), then "If p then either q or ~q" is a necessary truth. Its probability therefore equals 1. Since q and ~q are mutually exclusive and jointly exhaustive, however, it seems to follow that P(If p then either q or ~q) = P(If p then q) + P(If p then ~q), which means that P(If p then q) = 0.

So if that's right, then case (b) reduces to (a), which reduces to (c). But that's assuming that X is a tautology in case (a) and that CEM is true, which is controversial.

Here's another issue: What sort(s) of conditional could (RH) apply to? Clearly, (RH) will not work for material conditionals, for then if we let p = (r & ~r), we generate a contradiction since r & ~r materially implies both q and ~q. Hence by (RH), P(If p then q) would come out to both 1 and 0, which is absurd. The same result obtains for strict conditionals, since necessary falsehoods strictly imply everything.

I'm not bothered by this, however, since I don't think ordinary language conditionals are equivalent to either material or strict conditionals. I reject the ex falso quodlibet (a false proposition implies anything) assumption of material implication and the ex contradictione quodlibet (a necessarily false proposition implies anything) assumption of strict implication. Instead, it seems to me that ordinary language conditionals assume that the antecedent has got to be relevant to the consequent in terms of its semantic content (and not just in terms of its truth value in the actual world or its distribution of truth values across all possible worlds). Unfortunately, I don't have a developed theory of semantic relevance to offer.

Finally, reader Phil ("Oudeis Oudamou") asks a couple questions. I'll take the easier first.

Phil: First of all, do you accept the equivalence : prob[if a, then b]= [if a, then prob b]?

Alan: No. The equation doesn't make sense as it stands. The LHS is equivalent to a number between 0 and 1. The RHS is a proposition. Whatever else they are, numbers aren't propositions. I suspect when you wrote that you were thinking of the LHS as the proposition "Probably, if a then b" and wondering if I'd agree that that proposition is equivalent to the proposition "If a then probably b". Again, then answer is no. You aren't trying to catch me in the probabilistic equivalent of the necessitas consequentiae / necessitas consequentis fallacy, are you? That a conditional is true in most possible worlds (assuming the notion of "most" possible worlds is well-defined) does not imply that if the antecedent is true in the actual world then the consequent is true in most possible worlds.

Phil: Could I get you to illustrate how RH (c) will work in some concrete examples? Suppose, for example, I make the prediction “very probably if in Phoenix the overnight min temp > 83 F, the next day's high > 110 F.” This on the basis of my long-time personal experience with summer weather here, but without consulting any official weather data. And my prediction turns out to be correct except for a very cases, after we check the weather records for the last 50 years. Or suppose I'm doing stylometrics and I argue “ very probably if Plato wrote the VII Epistles, then he also wrote the II Epistle.” I'm curious to see how you'd construct X in these and assign the prob X.

Alan: Those are fairly complicated examples, and I'm not sure I'm up to the task of tackling them just yet. Let me try to work with something simpler. Let's take the conditional "If I were to roll this die repeatedly, then each number (1-6) would occur with a frequency of approximately 1/6." Here p="I roll this die repeatedly" and q="Each number (1-6) occurs with a frequency of approximately 1/6." Now we want an X that, together with p, will entail q without packing anything more than necessary into X. Matters are complicated by the vague word "approximately", but let's set that aside. It seems to me that X has got to be something like "this is a fair die and will remain so for the duration of the experiment". In normal cases of dice rolling, that's a reasonably safe assumption, so I would say that P(X) is fairly high and that, therefore, P(If p then q) is fairly high. Matters are not quite that simple, however, for any statistician will point out that even p+X doesn't entail q, unless we project the experiment into the infinite long run. But then X becomes implausible - any normal die will break down and start becoming 'unfair' way short of infinity. So we've got to idealize the situation to a high degree to make the formula work. This may show that (RH) needs to be modified, and perhaps complicated considerably, to handle more realistic cases. It make still be useful as a rough estimation tool, however.

My Dissertation

I've decided to make my dissertation available for download. Entitled "The Problem of Induction: An Epistemological and Methodological Response", I examine and reject most of the extant proposals for "solving" the problem and develop a novel approach to the issue that, I think, works, at least in broad outline.

If I were to rewrite it now, there are some things I would revise. For example, I wouldn't rely as much on Bayes' Theorem in Chapters 5 and 6. I still stand by my overall conclusions, though.

For those who aren't interested in reading a whole dissertation, here's a 15-page summary.

Conditional Probabilities and the Probabilities of Conditionals

A conditional probability is represented P(A | B), read "the probability of A given B", and is (by definition) equivalent to P(A & B) / P(B). To see how this works, consider a simple example. Let's take a fair die and roll it. What is the probability that we will role and even number? Clearly, it's 1/2. And what is the probability that we will role a 6? Clearly, it's 1/6. Now, what is the probability that we'll roll a 6 given that we roll an even number? In other words, what is P(roll a 6 | roll an even number)? By definition of conditional probability this is equivalent to P(roll a 6 and roll an even number) / P(roll an even number). This equals (1/6)/(1/2), which equals 1/3.

Some philosophers (including myself a few years back) have thought that conditional probabilities could be used to represent the probabilities of conditionals. This view is commonly referred to as Stalnaker's hypothesis (SH), after Robert Stalnaker, a prominent philosopher who explicitly proposed the idea. In other words,

(SH) P(If p then q) = P(q | p).

Unfortunately, while Stalnaker's hypothesis is prima facie plausible, it is demonstrably false, as famously shown by David Lewis (an achievement that went a long way toward establishing Lewis's reputation as one of the smartest philosophers on the planet until his death in 2001). I won't try to run through Lewis's proof here, but suffice to say that it is now universally agreed upon by scholars working on conditionals that Stalnaker's hypothesis fails.

But if the probabilities of conditionals cannot be equated with conditional probabilities, what can they be equated with? Intuitively, it seems like the notion of the probability of a conditional ought to make sense and that there should, in principle at least, be some way of estimating a value for large classes of conditionals, if not all. I think this is right, though I should mention that not everyone agrees. Philosopher Ernest Adams has famously defended the view that there are no probabilities of conditionals. In other words, he holds that there is nothing that can be called the value of P(If p then q). He goes on to propose that the conditional probability P(q | p) measures the assertibility (but not the probability) of the conditional if p then q.

I think Adams is wrong and that there's a very straightforward way of thinking about the probabilities of conditionals. Consider if p then q. If p entails q, then it seems obvious that P(If p then q) ought to equal one. Similarly, if p and q are mutually incompatible (p entails ~q), then it seems obvious that P(If p then q) ought to equal zero. But if those two cases clearly have probabilities, then it's hard to see why cases in which p is compatible with but does not entail q should not have probabilities. Suppose we think of this like an argument with p as a premise and q as the conclusion (a natural model because every argument can be written as a conditional). Since p does not entail q, we have an enthymeme:

p
(unstated premise)
∴ q

Now, what proposition do we need for an unstated premise to make this argument valid? Let's call this enthymematic premise the deductive complement of p in relation to q and represent it by X. X's job is to supply any information in q that is not already in p, such that (p+X) entails q. My proposal, then, is this (I'll call it Rhoda's hypothesis, RH):

(RH) (a) If p entails q then P(If p then q)=1. (b) If p entails ~q, then P(If p then q)=0. (c) If p entails neither q nor ~q, then P(If p then q) = P(X), where X is whatever information is contained in q that is not already contained in p.

The notion of information 'contained in' a proposition may be explicated via the notion of logical entailment. Propositions p and q contain the same information if and only if they have exactly the same entailments. If p entails q, then anything entailed by q is also entailed by p. X is equivalent to the conjunction of all propositions entailed by q that are not entailed by p.

In some cases, X will be logically equivalent to 'If p then q', but it will never be logically stronger than that, and often it will be logically weaker. I'll leave it as an exercise for the reader to explain why.

Toward a Probabilistic Model of Divine Providence

One of my long-term goals is to explore the consequences for divine providence on the assumption that open theism (OT) is correct. As I shall understand it here, OT is defined by the following commitments:

1. Monotheism: There exists one and only one God who is personal; necessarily existent; and essentially omnipotent, omniscient, and morally perfect. Every concrete reality metaphysically distinct from God is freely created by God ex nihilo and owes its continued existence to God's sustaining activity.
2. Future Contingency: There are some possible future states of affairs of which it is true that they both might and might not obtain.
3. Incompatibility of Future Contingency and Exhaustively Settled Divine Foreknowledge: Whatever both might and might not happen cannot be truly known as something that either will or will not happen.
4. Divine Epistemic Openness Regarding Future Contingents: Insofar as the future is contingent, God knows it as something that might and might not happen and does not know it as something that either will or will not happen. This follows from (2) and (3).
   
5. Divine Temporality: At least since creation, God stands in real temporal relations with his creation. This follows from (2) and (4).

Now, suppose that it is now T0, that God would like to bring about a certain state of affairs S at T2, and that S's obtaining at T2 is a future contingent. What can God do to affect whether S obtains at T2? It depends on S and on God's willingness to tolerate S's not obtaining at T2.

If S is an intrinsically indeterministic state of affairs, one that by its very nature precludes antecedent necessitation like Peter's freely (in a libertarian sense) denying Christ, then given (3) there is nothing God can do to guarantee that that state of affairs comes about. At most, God can act in ways that would influence whether that state of affairs comes about by directly altering the propensities. For example, if God at T1 were unilaterally to cause several people near Peter to accuse him of being a disciple and also to plant strongly in Peter's mind the idea that if he is found out as a disciple then he will likely meet with a painful death, then perhaps God could ratchet up the propensity of Peter's freely denying Christ somewhere close to 1. He would have to careful, though, not to overdo it. If God's influencing activities overwhelm Peter, then he ceases to be a libertarian free agent in the circumstances. And even if they don't quite overwhelm, but come very close to doing so, it becomes doubtful whether Peter could justly be held morally responsible for his actions.

If S is not an intrisically indeterministic state of affairs, however, then matters are much simpler, providentially speaking, since there is nothing preventing God from unilaterally bringing it about. He may not always want to do so, however. If God, say, were to unilaterally intervene at T1 so as to bring about Peter's verbally denying Christ at T2, then he compromises Peter's integrity as a moral agent and thereby compromises the quality of his relationship with Peter. Insofar as God values genuine, free relationships with his creatures he will refrain from unilaterally intervening to compel them to action. But perhaps on occasion the benefits outweigh the costs?

Anyway, for God of OT to make the right providential decisions, he needs to be able to assess the propensities of various possible states of affairs obtaining and to update these assessments as conditions change. Suppose, for example, that God wants to know at T0 what the propensity is for S to obtain at T2. In other words, suppose he wants to evaluate Prob(S at T2 | T0), i.e., the probability that S obtains at T2 given the state of the world at T0. How can he do this?

Well, presumably it would have to work something like this: At T0 God calculates the propensities for each possible states of the world at T1. To keep it simple, let's suppose that there are three such states (A, B, C) that are mutually independent and jointly exhaustive. Thus, God evaluates

Prob(A at T1 | T0) = a
Prob(B at T1 | T0) = b
Prob(C at T1 | T0) = c

Then, he can evaluate the conditional propensity of S on each of these possibilities. Thus,

Prob(S at T2 | A at T1) = d
Prob(S at T2 | B at T1) = e
Prob(S at T2 | C at T1) = f

Prob(S at T2 | T0) is then equal to

Prob(A at T1 | T0)*Prob(S at T2 | A at T1) +
Prob(B at T1 | T0)*Prob(S at T2 | B at T1) +
Prob(C at T1 | T0)*Prob(S at T2 | C at T1) = ad + be + cf

Once T1 arrives, and either A, B, or C occurs, then the other probabilities become irrelevant. If A occurs at T1, then Prob(S at T2 | T0) no longer matters. What's relevant now is Prob(S at T2 | A at T1).

Realistic cases, however, could be much more complicated. There might be no neat partition into a small finite number of mutually exclusive and jointly independent possibilities. Presumably the calculations wouldn't bother an infinitely intelligent being like God, but it would increase the number of providential variables, thereby requiring much more extensive intervention on God's part if he wants to achieve a specific outcome.

Quantum Indeterminacy and Miracles

In a comment on an earlier thread, C Grace posed me the question:

Do you think QI [quantum indeterminacy] is needed for God to manipulate the material universe without breaking His natural laws? It seems to me that without QI, God's freedom to affect the physical universe would be limited to what he could do through us.

No, I don't think QI is needed for God to interact with creation (say, by performing miracles) w/o breaking natural laws (e.g., conservation laws). Unlike Hume, I don't think miracles are properly thought of as "violations" of natural laws but rather as "exceptions" to those laws. Any natural law that we are able to formulate has a structure like "If conditions C obtain then, all other things being equal, with probability P result R will obtain". The "all other things being equal" condition is also called a 'ceteris paribus' clause and functions as a built-in exception clause. Thus, natural laws reflect the ordinary operations of created things. In the case of a miracle, all other things are not equal, so we have an exception to the law, not a violation of it.

Philosopher Richard Purtill makes the same point by using the example of a presidential pardon. If the President of the U.S., say, should decide to pardon a criminal, this in no way abrogates the law or changes the criminal code. It's an exception that preserves the rule. Just as the President of the U.S. has the executive authority to override the normal law of the land in special circumstances, so also God has the sovereign authority to override the normal operations of things in the natural order if He sees fit to do so. And if He does so, this in no way nullifies the normal operations of the natural order. It's still normal for things to fall in a gravitational field even if God decides for some reason to supernaturally levitate my couch.

William Lane Craig vs. Bart Ehrman

Here's a link to a transcript of a recent debate between William Lane Craig and Bart Ehrman entitled "Is There Historical Evidence for the Resurrection of Jesus?"

The Intelligent Designer Speaks!

Here's a recorded interview with the Intelligent Designer. I like his shades.