(R) God is known only by causal inference from effects, in such a way as not to fall under a genus.
The basic implication of this is that we know God not by having a definition of His nature, which would require identifying a genus and specific difference for deity, but only by differences that are not specific differences, which themselves arise from specific causal inferences.
One thing that is fairly easy to see is that lots of arguments in analytic philosophy of religion appear to violate this principle; you can often recognize them by conditionals beginning, "If God exists..." (although conditionals of this form do not in themselves violate the principle of remotion). For instance, here is a standard simple version of an argument from gratuitous evils:
(P1) If God exists, then there is no gratuitous evil in the world.
(P2) There is gratuitous evil in the world.
(C) Therefore, God does not exist.
On its own, this is fine, but the obvious question is why one would accept (P1). And it is here that the violation often occurs. So, for instance, a commonly given reason given for (P1) is something along the lines that it follows from the nature of God, who as a morally perfect being would necessarily, given his power, wisdom, and goodness, not allow evil he has no morally adequate reason to allow. All three of the italicized phrases, or anything equivalent to them, raise immediate questions relevant to remotion: talking about what follows from the nature of God appears to require some kind of real definition of God's nature itself; the rationale appears to identify God as falling within the genus of morally perfect beings; and the necessity seems to be the kind of natural necessity that comes from knowing the real definition of a thing's nature. But what binds it all together is what we do not get, which is any establishment of the content of these terms from a causal inference that does not make any assumptions about the genus of the cause. All the terms can be used in a way consistent with (R); but they can only do so in combination with a causal inference of the kind noted in (R), which then fixes the content of the term.
Here is another one:
(P1) If a perfectly loving God exists, reasonable nonbelief does not occur.
(P2) Reasonable nonbelief occurs.
(C) Therefore a perfectly loving God does not exist.
All well and good, but what is the justification for (P1)? A common justification is that a perfectly loving God would necessarily be open to a personal relationship with every willing person. But, again, this 'necessarily' seems to require the kind of necessity resulting from its being implied by the real definition of a nature, and there is no hint of what kind of causal inference would give one the relevant content of the term 'perfectly loving'.
It is perhaps not especially surprising that atheistic arguments tend not to fix the content of the terms they apply to God by giving the causal inference to God's existence that would fix them; but there has to be at least some hypothetical or for-the-sake-of-argument inference, or the argument violates the principle of remotion.
Here is a theistic argument that also, as usually understood, violates the principle of remotion.
(P1) If God is perfectly loving, God saves all those who can possibly in any way be saved.
(P2) God is perfectly loving.
(P3) Therefore God saves all those who can possibly in any way be saved.
Again, all well and good, but the question arises as to how one knows (P1), and again the usual answers do not seem to appeal to a causal inference or avoid the assumption that we have a real definition of God's nature from which we draw a necessary conclusion.
There are many views on which violation of remotion is not a problem; my point is not that this is an issue arising all the way across the board. Rather, my point is that the principle of remotion is regularly violated, and the arguments thus created are inconsistent with, already ruled out by, any view that accepts the principle of remotion. It is in fact the case that there are a lot of historical theologians (Jewish, Christian, and Muslim) who accept the principle of remotion, either explicitly or by accepting things that effectively imply it, which is, of course, where I got it. And for anyone who accepts the principle of remotion, a great many arguments one finds in analytic philosophy of religion look as if they are, not tight arguments dealing with rigorous necessities, but sloppy reasoning involving pure speculation.