Formal Statement of the Argument
In evaluating any argument, it is always useful to formally write out the argument, explicitly listing the premises and making sure that the logical structure is valid. The main argument of this essay can be formally stated as follows, with premises indicated by "P" and inferences indicated by "I":
(P1) Extremely complex phenomena that cannot even in principle be explained as arising from simpler, more fundamental principles are extremely improbable.
(P2) God is by definition a being that is a) conscious, and b) fundamental in the sense that he is not evolved or derived from anything more fundamental.
(P3) Conscious beings are necessarily extremely complex.
(I1) From (P2a) and (P3), God is extremely complex.
(I2) God cannot even in principle be explained as arising from simpler, more fundamental principles since, from (P2b), God is defined as being fundamental.
Conclusion: The existence of God is extremely improbable [from (P1), (I1), and (I2)].
By writing the argument out like this, it is clear that the logic is valid and therefore that the conclusion will be correct if the three premises are correct. So now lets see if the premises can be justified:
I have already spent some time above in justifying (P1). To summarize, (P1) is certainly reasonable as it is always the case that when an explanation is found for seemingly complex phenomena, the explanation relies on relatively simple principles. It is also the case that (P1) is widely accepted (and it is not necessary to belabor a premise that everyone already agrees with). (P1) is widely accepted by scientists, for example, since a failure of (P1) would mean the non-improbable existence of complex phenomena that cannot even in principle be explained in simpler terms. The existence of this type of phenomena would mean that science could not proceed. (P1) is also widely accepted by theists who use this premise in an essential way, even if it is not always explicitly stated, in their Argument from Design.
(P2) gives two defining properties of God that most theists, I think, would accept. A person who claims that God does not satisfy (P2a) is abusing the English language by referring to whatever it is that he or she believes in as God. (P2b) is a necessary ingredient to the definition of God, because we would otherwise have to say that God could be a sufficiently advanced alien.
It is (P3) that I think theists would be most likely to attack. They could claim that human consciousness may be complex, but the supernatural, spiritual consciousness of God is not. This type of claim, however, cannot be correct. Consciousness by its very nature is complex; whether we are discussing the consciousness of biological organisms or the consciousness of a hypothetical supernatural being is irrelevant. To see that consciousness itself is complex, consider that consciousness requires the ability to store and access information that is linked together in many intricate ways as well as the ability to process that information and to reason. The web of intricately interconnected data that consciousness requires is extremely complex. One measure of the complexity of a system is the logarithm of the number of states of the system . Applied to a conscious system, this measure of complexity is proportional to the number of pieces of data that the conscious system knows times the degree of interconnectedness in the data. There are three interesting things to note here: 1) this measure of complexity is very large if a large amount of data is accessible; 2) the interconnectedness of data that consciousness requires greatly increases the complexity; and 3) for an omniscient being, this measure of the being's complexity diverges.
So (P2) is true by definition, and although (P1) and (P3) cannot be proven with absolute certainty, I think it is fair to say the I have shown that it is extremely likely that (P1) and (P3) are correct. It then follows that it is extremely probable that the conclusion is correct, i.e., the existence of God is extremely improbable.
The notion of "irreducible complexity" as propounded by Behe is insufficient for both theological and atheological arguments. I have shown that "fundamental complexity" is the requisite concept and that a being fitting a very general definition of God would have to be extremely and fundamentally complex. The existence of such a being is therefore extremely improbable.