According to the author, explanations of events fall into three categories: regularity, chance, and design. An event is explainable by regularity if it is to be expected under the circumstances by virtue of a universal or high probability regularity. If an event is of 'intermediate probability,' then it is attributable to chance. A 'small probability' event (an idea dealt with officially in the final chapter) is attributable either to chance or to design. For design to be the proper mode of explanation, it is not enough for the event to be of small probability: it must also be 'specified'. A simple example here can convey the rough idea of specified small probability events. Any actual particular sequence of heads and tails on 100 fair flips of a fair coin has small probability, but if it is easily describable with a short description ('all heads' or 'alternating heads and tails', for instance) then it is also specified. Explanation by regularity is the 'default' kind of explanation. If this is not possible because the event is not of high probability, we should next try chance. If the event is of intermediate probability, we explain it by chance. If the event is neither of high nor of intermediate probability, we have to decide between chance and design: if the event is specified then the explanation should be by design, otherwise by chance. The design category of explanation is simply the negation of 'either by regularity or by chance'.
[[So far so good.]]
Chapters 1 and 2 introduce these ideas leisurely and with plenty of examples. Chapter 3 is on probability theory, 4 on complexity theory (where the idea of easily specifiable is treated), 5 on specification, and 6 on small probability. Thus, after the two introductory chapters, the remaining chapters clarify in a formal way the basic concepts involved (specification, small probability, etc.) and formulate in an explicit an symbolic way the design inference, i.e., the move through the 'explanatory filter' that is supposed to justify the design mode of explanation. Unfortunately, the bulk of the book (Chapters 3-6) is devoted mainly to the explication of the design inference; as to the justification of the inference (the rejection of explanation by regularity or chance), very little is offered. Basically, it seems that the justification part is supposed to rest in the reader's becoming convinced that the formal apparatus somehow matches or fits the pattern described in the intuitively compelling informal examples given.
[[I don't think this is accurate. The examples hammer home that design inferences are a very common form of rational inference. The formal exposition gives a rational reconstruction of what we do. Eells implies that I haven't adequately justified the inference because I don't adequately account for the rejection of chance, but as I develop them, design inferences are eliminative inferences, so explicating the design inference is to show what's needed to eliminate chance. As for regularity, it can be treated as a special case of chance in which the probabilities collapse to 0 and 1.]]
Setting aside for now the concept of 'small probability', the idea is that a small probability event cannot be attributed to chance if it is also specified. Basically, an event E is specified if E is known to have occurred and also (1) E is, given the relevant chance hypothesis H, probabilistically independent of certain 'side information' I we may know (independently of E) that might help us describe E; and (2) E is easily ('tractably' in the complexity theoretic sense) describable using the side information I. The best I can find in the way of a justification of the thesis that the specification of a small probability event should eliminate chance is found on p. 147, which I paraphrase to avoid having to introduce notation and concepts extraneous to the point here:
Because E is conditionally independent of I given H, any knowledge of I ought to give no knowledge about E so long as E occurred according to the chance hypothesis H. Hence any description of E formulated on the basis of I ought not to give any knowledge about E either. Yet the fact that it does so means that I is after all giving knowledge about E. The assumption that E occurred according to the chance hypothesis H, though not quite refuted, is therefore called into question.... To actually refute this assumption, and thereby eliminate chance, it must be shown that the probability of E given H is small enough, ... the subject of Chapter 6.
[[Eells seems to be on a "justification kick." Let me turn it around. We make design inferences all the time. I've given a rational reconstruction of them. It seems to me that this is sufficient justification so long as my rational reconstruction accurately mirrors our pre-theoretic practice of drawing design inferences and so long as no counterexamples are forthcoming. I have yet to see Eells produce a counterexample.]]
Perhaps Dembski has in mind this form of argument: (1) Pr(E/H&I) = Pr(E/H) (and these probabilities are small); (2) Pr(E/I) > Pr(E): therefore, (3) E did not happen according to the chance hypothesis H. However, the premises on this reading obviously do not refute the proposition that E happened according to H: the fact that E and I are correlated when not conditioned on H, and independent when conditioned on H, is of course no refutation of the proposition that the event happened in accordance with the probabilities proposed in H, even if the probabilities proposed are 'small' (recall that it is not the small probability of E that is in question, but whether E is to be explained by the 'small-chance' hypothesis H). In any case, the conceptual connection is unclear and not spelled out (except by saying that this models part of what is true in some examples described). In particular, it's not clear just what premise (2) is supposed to add in support of (3), and just how (2) is supposed to be justified.
[[Eells is misrepresenting me, and it seems that he's doing it because he's misunderstood me. He's talking about events E, but not about patterns which need to be reconstructed on the basis of side information I, where I is conditionally independent of E. Look at what I wrote in chapter 5 and look at what he has here. They're not even close.]]
On the other hand, Dembski might have this in mind: (1') If E happened according to H, then Pr(E/I) = Pr(E) (and these probabilities are small); (2) Pr(E/I > Pr(E); therefore, (3) E did not happen according to H. However, while this argument is clearly valid, it is also clear that (1') is not, and should not be, what Dembski has in mind: statements involving conditional probabilities are quite different from corresponding conditional statements; the conditional independence premise is clearly stated on p. 145 as about conditional probabilities (as in (1) above); and neither of (1) and (1') above follows from the other. Finally, maybe what Dembski has in mind is this: (1) Pr(E/H&I) = Pr(E/H) (and these probabilities are small): (2') Pr(E/H&I) > Pr(E/H); therefore, (3) E did not happen according to H. But of course here anything follows from the premises, so Dembski should not approve of this reading of his p. 147 argument.
[[Again, he's not accurately representing my treatment of specification--not even close. He's bringing in neither patterns nor complexity. What's there to say? He's critiquing something that I find unrecognizable.]]
Having explained specification in Chapter 5, Chapter 6 completes the explication of the design inference with an explication of the idea of 'small probability'. Though space is limited here, I think Dembski's idea can be fairly, though roughly, described as follows: an event E has small probability if and only if the probability is less than ½ that an event appropriately like E ever happens in the entire course of the actual world's actual history. The roughness has to do with the use here of 'appropriately like'. This in turn is explicated in terms of descriptions available in an assumed underlying language (though no justification of a choice of a particular language is given).
[[I offer a formal theory of what I call "probabilistic resources." It makes rigorous what Eells makes out to be impossibly vague--"appropriately like." I dislike this latter expression. The saturation of E by its relevant probabilistic resources is not appropriately like E--it is usually a much bigger event (in the set-theoretic sense) and it is the relevant event whose probability must be evaluated to determine whether the events actually are of small probability. As for the choice of underlying language, this is always relative to the subject/agent investigating for design. Eells is looking for justification where none is needed. I'm offering a rational reconstruction of a common human activity. Again, the proper question is whether I've done the reconstruction accurately and whether there are counterexamples--I'm still waiting for a counterexample from Eells.]]
The rationale for the value ½ is given in Section 6.3, titled 'The magic number ½'. The rationale seems weak and is based on a principle according to which we should reject F as a description of an unknown event in favour of its compliment Fc if the probability of F is less than ½ (p.195).
[[My argument is more subtle.]]
Conspicuously missing here is a discussion of the utilities of the consequences involved in accepting or rejecting (when these are our only choices) a description F (though utilities do enter into discussion later on in a different way), and there is no discussion of why any value less than ½ would be more or less acceptable than Dembski's choice of ½ (though ½ is sometimes called the highest value lower than which is 'small').
[[I'm well aware of utilities. I didn't employ them because they were not relevant. The question I was considering was whether we should think that an event did or did not happen by chance. This is different from basing a course of action on such an event, where the costs involved with that event may incline us to bet on it in certain ways which may seem inconsistent with its probability--insurance companies cash in on this difference all the time.]]
Moreover, given the way the value ½ enters into the characterisation of low probability events, it would seem that values greater than ½ would intuitively, do as well: if the probability is less than, say, 0.8 that an event of a kind appropriately like E would ever happen in the course of the world's history, then it would seem that any actual occurrence of E is of low probability. Thus, the number ½ seems, to me at least, arbitrary and not so 'magic' in this context.
[[He just doesn't seem to get it. The probability 1/2 is significant because an event of probability less than 1/2, in the absence of other information, is one we should not expect to occur--we should expect its complement to occur.]]
Because of the weaknesses described above in Dembski's analysis, and because of the fact that some design inferences seem perfectly reasonable, I conclude that neither the explication nor the defence of the design inference provided by Dembski is quite on the mark.
[[I'm unimpressed. I don't think Eells has tracked my argument. Nor has he given any counterexample where my analysis fails. This has all been too general, and he's not dealt with the nuts and bolts of my argument.]]
THE UNIVERSITY OF WISCONSIN-MADISON ELLERY EELLS
[[Interesting that this review comes out of UW-Madison. Elliott Sober, at that same school, has also just written a negative review of TDI for Philosophy of Science. This is a more intricate review, though equally misguided.]]