UPDATE (1/14/11): I have added new information to this post, indicated below.
If you're not familiar with the subject, it's most famously touted by the "analytic philosopher" Bill Dembski. It's more commonly demonstrated as "the watchmaker argument." It basically states:
1. The complex inner workings of a watch necessitate an intelligent designer.
2. As with a watch, the complexity of X (a particular organ or organism, the structure of the solar system, life, the entire universe) necessitates a designer
I knew from the first time I heard this argument that there was something wrong with it. But I've always been hard-pressed to put it into simple language (as I have a tendency to drone on and on and go off on tangents). Today I will attempt to do just that --- I will explain why this argument is nonsense in as few points as possible:
1) The only reason we know that a watch must have been designed is because we know in advance that a watch was designed; as such, it is impossible to cite this argument without first referring to something which we know prior to be designed (such as a watch). Ergo, it is the prior knowledge of design which allows us to infer that it was designed, in the first place.
2) Therefore, it is not the function of the device which leads us to assume that it was designed, but rather the fact that we know it to have been designed.
3) Ergo, we do not infer specified complexity based solely on the functionality or complexity of the object; for "specified complexity" is defined not by the functionality or complexity of the object, but by the fact that it was designed. We infer specified complexity based on whether or not we know if something was designed.
In short, specified complexity is completely circular. There is simply no logical or mathematical proof which demonstrates that complexity (of any extreme) necessitates a designer. We require prior knowledge of design in order to infer whether it was designed; we cannot use such proof to conclusively infer that something which we do not know a prior to be designed was designed.
In summation: "Specified complexity" is not useful as a predictive tool; it is only useful in hindsight, which means that it can only be used to judge things which we already know to be designed or not designed. Which means that it's redundant, and therefore basically useless --- a "hypothesis" which requires the knowledge it supposedly "predicts" in order to predict it.
UPDATE: In Information Theory, it is said that if information being transferred is already known in advance of its transfer, then no information has actually been transferred. Wikipedia gives a solid example of this in strict numerical form:
Suppose one transmits 1000 bits (0s and 1s). If these bits are known ahead of transmission (to be a certain value with absolute probability), logic dictates that no information has been transmitted.
This is where "specified complexity" falls short. The question is, "is something which is complex, necessarily designed?" Yes, the watch is complex, and yes, it is designed; but is the watch complex because it was designed, or is it designed because it is complex? If it is designed because it is complex, then why? Why does complexity necessitate design? If we say, "because things which are complex are designed," then we are again begging the question and equivocating:
(1) Why does complexity necessitate design?
(a) because complexity cannot arise naturally; it's too improbable.
(2) Why is complexity too improbable to arise naturally?
(a) because the only things we know of that are complex were designed.
(3) How do we know that those complex things were designed?
(a) because we observed them being designed; ergo, we already know they are designed before making this inference.
The "information" contained in the conclusion, 3a, is already available before the inference is made. Therefore the "information" is redundant and Specified Complexity is a fallacy according to information theory.