It took awhile due to other activities, but here, at last, is the inaugural post for Scripta Scientificae pro Christiana Apologeticum.
What amazes me about atheist philosophers of mind is that they are material monists and yet seem to be pretty ignorant of physics and invariably fail to present a formal mathematical description of the "goings on" in their physical brain that yields anything they can show to be consciousness or any other so-called "mental" properties of that mass of stuff between their ears. Instead, what you find is a tangle of mere qualitative verbiage.
The closest I have found any come to describing their "emergentism" is that there must be "law-like connections" between supervenient mental properties and the underlying material brain. They never explain "law-like" in detail that I have found. I consider such loose language to be the equivalent of the equally silly phrase "pregnant-like."
If by "law-like" they mean probabilistic relations, those are still technically laws with an irrational residue of randomness. At any rate, if there were a quantitative relation between matter and emergent "mental" properties those could still be expressed mathematically. If probabilistic, the frequencies of random outcomes would be determined by the material properties. The materialists need to present their theory or admit defeat that their scientific ideal of encompassing all of reality within a web of physical laws is flawed and that reality doesn't fit their atheist notion of the uniformity of nature.
As we have said, the concomitant of their merely assumed assertion of the uniformity of nature is that the present is as the past, and that the future will be as the present. As a result, the properties/laws that assembled their brain in the past via physical evolution are the same laws that determine their brain and their thoughts (merely "mental" properties of the underlying matter) today. This is called bottom-up causation -- it is a theorem of material monism and their assumption of the uniformity of nature.
So then, we don't need to dot all the "i's" to give the specific form of any emergent property that the atheist philosopher of mind might propose.
As mentioned above, the philosophers of mind use a lot of qualitative verbiage. By essentially verbal ruses, atheist philosophers of mind have devised different theories of mental properties in a purely material universe. These theories are: (1) epiphenomenalism; (2) emergentism; (3) reductionism; and (4) functionalism. All of these become essentially equivalent once the atheist has bought into his assumptions of physicalism and the uniformity of nature. The products produced in the physical crucible of the big bang and its subsequent physical evolution (culminating in the purely physical man) all necessarily obey the laws of physics. To assert otherwise is to embrace a radical and irrational principle of discontinuity. Such a principle of discontinuity would contradict the so-called "rational" atheist's belief in the uniformity of nature. Such discontinuity should be anathema to the atheist who eschews any hint of the miraculous.1 To admit such would be to say the present is not as the past and that the future may not be as the present. With such an admission the atheist's "scientific program" would be rendered absurd and self-repudiated. It is to embrace metaphysical irrationalism.
In all of these theories of mind, regardless of philosophical terminology, a mind represented by the symbol "y" is a function "f" of base physical properties denoted by "x." Physical matter is the only ingredients available to the rational material monist. And those ingredients by the ideal of science evolve according to differential equations of physics. So then in equation form:
Hamiltonian mechanics. The end result, as we have said before in less technical terms, is that all of human action and experience are the necessitated effects of matter-in-motion. There is no "mind" in the driver's seat, no top-down causality, no autonomous rationality. To the materialist, what your "mind" thinks is the result of the random vibrations of the quantum fields in your constitution and what you say are just ultimately random utterances. This is in no way a rational worldview.
1 One atheist who implicitly denies the uniformity of nature is Richard Dawkins, as discussed here. Dawkins, by maintaining human autonomy, is apparently unaware that he has implicitly thrown out the uniformity of nature. Hence, human autonomy to Dawkins is an irrational miracle.↩