In Strong atheism, two of the classes of definitions of god were absolutely undetectable deities and presently undetectable deities. Commenter Ben Wallis argues that these two classes of definitions render strong atheism untenable because "we cannot speak to the probabilities of deities in general." Ben argues that the definition of essentially undetectable is not, strictly speaking, meaningless, because the existence of an absolutely undetectable deity matters to a deity itself. Wallis argues that in a similar sense to the Bertrand Paradox, we cannot rigorously and unambiguously define the probability of any presently undetectable deity existing. Since we cannot rigorously definite the probability of a presently undetectable deity existing, it is unwarranted to hold any kind of probabilistic belief; weak atheism or agnosticism is presumably the preferred position.
While I don't entirely agree with him, I don't think Wallis is really that far wrong. The undetectable deities are already in the grey area of philosophical hair-splitting; the distinction between strong and weak atheism with regard to undetectable deities is similarly a matter of very fine, hair-splitting distinctions. New Atheism is primarily a political and social movement, and the only definitions that have political and social implications are the detectable, paranormal definitions (which I would assert, contra Wallis, encompasses Yahweh, Jesus and Allah). No actual believer talks about a perfectly deistic god who passively observes the world, and no one actually believes in a scaredy-cat god who's hiding behind the couch. Since the real debate is just about detectable gods (and what, precisely, we mean by "detectable"), we're not giving up any important ground to simply declare weak atheism and agnosticism regarding undetectable gods while still maintaining strong atheism regarding detectable gods.
I do, however, enjoy splitting hairs as much as the next philosopher, so I want to address Wallis' arguments directly.
Wallis argues against strong atheism with regard to to presently undetectable gods by invoking the Bertrand Paradox, which argues that it is possible to have mutually exclusive definitions of "random" that definitely give different answers to questions of probability. But one outcome of a careful examination of the "paradox" is that we can add a qualifier to the definition of randomness — the "maximum ignorance" principle — that seems to categorically disambiguate competing definitions of random: we can consider only those definitions that satisfy the maximum ignorance principle to constitute "true" randomness. If we assume this qualifier, Wallis fails to rebut my original argument.
On another view, the Bertrand Paradox doesn't change our view. If there is some ambiguity in the determination of the probability of some hidden deity existing, the range is either large or small. if the range of probabilities is large, then the definition is too weak to actually name a concept about which anyone can have any sort of belief. If the range is relatively small (e.g. between 10-9 and 10-12) then the ambiguity is irrelevant: no matter what the actual probability is, all the probabilities are low enough to warrant disbelief. Just as science does not include absolute certainty in its definition of knowledge, neither does it include absolute precision.
One might form a definition of a deity for which there was sufficient precision to be coherent and encompass a range of probabilities sufficiently high to warrant at least agnosticism, but I have not yet seen such a definition. The best attempt I've seen so far is the Fine Tuning argument, which has been decisively rebutted in a number of ways.
Wallis' objection to the absolutely or essentially undetectable deity hinges on a particular metaphysical view of ontology and epistemology. The scientific metaphysical system is epistemically prior: scientific ontology is just the narrative of what the world must be like to account for our knowledge. All apparently differing narratives that account for the exact same body of knowledge are, by definition, exactly equivalent. For example, the ontological narrative of (parts of) General Relativity can be expressed in two seemingly different ways: on one view, objects themselves become distorted in a gravitational field; on another, objects retain all their properties, but space itself is distorted in a gravitational field. Although seemingly different, physicists have (I'm reliably informed) determined that these two narratives always have the exact same epistemic consequences, and are thus saying exactly the same thing.
When a pair of statements in conjunction equivalently describe our actual knowledge, it's notable that the alternatives are not inverses of each other. P and not-Q in General Relativity above is not the simple inverse of not-P and Q. (The inverse of P and not-Q is not-P or Q.) Holding them as mutually exclusive alternative formations does not entail any contradiction. We have a different situation, however, when a statement (even a compound statement) and its inverse are epistemically equivalent. In this case, admitting the meaning of the statement entails a contradiction: To say, for example, that God exists and God does not exist are epistemically equivalent statements is to say that P equals not-P. To avoid the contradiction, we have to deny meaning to P: it is a category error to call it truth-apt.
It is not the case that one must adopt an epistemically prior metaphysical system, but neither is it the case, I think, that one cannot reasonably adopt epistemic priority. If Wallis wants to adopt an ontologically prior metaphysical system, then he might find strong atheism untenable, but if he wants to argue that my adoption of strong atheism is unreasonable, then he must either argue that it is unreasonable under epistemically prior metaphysics or he must problematize epistemically prior metaphysics.
Strong atheism, while not necessarily a required position (although I think ontological priority is a much more problematic metaphysical concept than epistemic priority), is, I believe, a tenable position.