I thought Craig gave a perfectly good answer. But I'd go about it somewhat differently.
i) It's true that the atheist questioner overstated his case about universal negatives. There are situations in which it's possible to prove a universal negative. We don't have to search every square inch of Jupiter to know there are no giraffes on Jupiter. We only have to know the kind of environment an animal like a giraffe requires for survival, and know that Jupiter's environment is inimical to giraffes.
Likewise, we can know that no Tibetan monk has ever been president of the United States. And so on.
ii) That said, there are other situations in which it's not possible in principle to prove a universal negative. In addition, there are situations in which it's possible in principle but not in practice. Finally, there are situations in which it's technical possible, but very difficult.
iii) And this goes to the burden of proof, which is quite germane in some Christian/atheist debates. The questioner is right about that. Are we holding both sides to a reasonable standard? Is the burden of proof the same for each?
Ironically, the questioner's observation backfires. As I've often remarked, when it comes to the subject of miracles the burden of proof is radically asymmetrical. On the one hand an atheist must eliminate every reported miracle from consideration. On the other hand, a Christian only needs a few good examples to establish their reality. So the atheist must assume a self-imposed burden of proof that's completely unrealistic. He can't justifiably rule out every reported miracle in advance of examining the evidence.
iv) The questioner then asserts that invoking divine agency is an appeal to something inherently inexplicable, so it doesn't solve the problem but compounds the problem. Appealing to something even more incomprehensible to explain something we don't understand.
v) Unfortunately, he fails to just what he means or defend his claim. Does he mean the very concept of God is inexplicable? If so, there are detailed philosophical explications and models of the divine attributes. So on the face of it, God is not inexplicable in that sense.
Indeed, it's something of a dilemma for an atheist to say God is unintelligible. For what is the atheist denying?
vi) Then there's the God of biblical narratives. Does he mean that representation of God is inexplicable? But far from being inexplicable, biblical representations of God are often anthropomorphic. The Bible God acts like a recognizable agent. Intelligent. Having intentions. Analogous in some ways to human agents.
vii) Perhaps his objection is not to the concept of God but the concept of divine agency. If so, what's inherently inexplicable about the concept of divine agency? It's a variation on personal agency. We're certainly very familiar with that concept.
viii) Maybe his objection isn't to personal agency in general but mental causation in particular. Yet humans have direct experience with mental causation. And there's strong evidence that minds are immaterial, yet able to act on the physical.
ix) Maybe he means the concept of an agent who subsists outside of space and time is inexplicable. But we do have a concept of things that exist outside of space and time. Abstract objects are the paradigm example. Even if you're a physicalist or nominalist, abstract objects are not conceptually unintelligible.
x) But perhaps he means to say it's inexplicable how a timeless spaceless agent causes things to happen in space and time. If so, one problem with that objection is that the concept of causation is someway mysterious and resistant to final analysis.
For instance, we normally think of a medium that transmits the cause to the effect. But if we insist on that as a necessary condition for cause and effect, it leads to infinite regress or infinite divisibility. Must there always be some intervening medium to bridge the gap, or must it bottom out at direct causation to avoid an infinite regress or infinite divisibility?
xi) It can't be inexplicable because atheists lack a concept of the supernatural. Atheists reject certain reported phenomena because they regard them as naturally impossible. But that demands a point of contrast. Naturalism is typically defined in terms of the world as a closed system of matter and energy. That's all there is. So the concept of the supernatural involves concepts to the contrary.
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