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Figure 9.1. Upper, a string of bits encoding the prime numbers from 2 to 89, as described by Dembski. The last 73 1's are included to make the total number of bits supposedly 1000. Lower, the string of 1000 bits actually presented by Dembski. The encoding of the prime number 59 is missing from his string. After Dembski (2002b, pp. 143-44).

it is so vague and because he does not assess a cost based on the length of the specification, it is susceptible to a problem known as the heap paradox (Sainsbury 1995). The heap paradox asks, "When does a collection of grains of sand become a heap?" Clearly one grain of sand does not form a heap. And if n grains of sand do not form a heap, then adding one grain to get n+1 grains is not going to suddenly change a non-heap to a heap. We conclude that no heaps of sand exist. Yet they do.

This paradox can be resolved by realizing that the term heap is not a rigorous, black-and-white classification. There are degrees of heapness. A pile of ten grains of sand is only very slightly a heap, while a pile of a million grains is very strongly a heap. Once we measure heapness numerically, the paradox disappears.

In the same way, Dembski insists that specification is a black-and-white classification: an event is either specified or it isn't. But it doesn't make any sense to say, for example, that the text of Shakespeare's Hamlet is specified, but exactly the same text with an extra comma at the end is not. If x, a string of 0's and 1's, has a specification T, then the string x0 containing an extra 0 at the end can be specified simply by amending T to say "and add another 0 at the end." We can continue this process ad nauseum; without assessing a cost, every event is specified.

Thus, Dembski's notion of specification discards the crucial ingredient: a cost assessed on the length of the pattern's description. If we charge a cost based on the length of the pattern's description, then we get essentially the well-known Kolmogorov solution to the probability paradox we have already discussed. In this case, the specified complexity of a binary string x turns out to be essentially I x I- C(x), where I x I denotes the length of the string x and C(x) is the Kolmogorov complexity. We might call this quantity "specified anti-information" because it is close to the negation of what mathematicians and computer scientists usually mean by informationâ€”namely, C(x).

Finally, we point out one more inconsistency in Dembski's treatment of specification. According to its formal definition, a specification is supposed to consist of a lot of mathematical apparatus: a space of events, a rejection region, a rejection function, and certain real numbers. But when it comes to applying the definition of specification, Dembski doesn't use his own framework. Consider his discussion of the specification of the flagellum of Escherichia coli: "in the case of the bacterial flagellum, humans developed outboard rotary motors well before they figured out that the flagellum was such a machine" (Dembski 2002b, 289). What, precisely, is the space of events here? What are the rejection function and the rejection region? Dembski does not supply them. Instead he says, "At any rate, no biologist I know questions whether the functional systems that arise in biology are specified" (289). That may be, but the question is not "Are such systems specified?" but "Are the systems specified in the precise technical sense that Dembski requires?" This is equivocation at its finest (or worst). The bottom line is that Dembski's notion of specification is so muddled that we cannot say with certainty whether a given outcome is specified or not. Like Blondlot's N-rays, the existence of CSI seems clear only to its discoverer.

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