Arguments for a UG

It seems like there is strong evidence for the existence of a universal grammar, but I still struggle to follow some of the arguments for it. Many of the arguments for UG are based on the notion that language is infinite, including the logical problem of language acquisition and a counterargument to the statistical model of language acquisition. The logic uses the fact that even children, when presented with a sentence form they have never seen before, know immediately whether it is grammatical. It follows that their internal grammatical rules cannot have been built exclusively from everything they've heard, and this implies the existence of a UG. But I fail to understand why the child couldn't have learned the recursive rules governing the language–this knowledge would allow him or her to determine whether previously unheard sentences were grammatical. In fact, isn't this essentially what a UG is: a set of rules that guide the syntactical composition of language? If the set of syntax rules is too vast be learned because it covers an infinite amount of sentences, I don't understand how it can fit in the brain at all (innate or not). And if it is merely a finite set of rules that guides language formation, I would think it could be learned too.

Carnie uses similar logic to refute the statistical model for language acquisition. According to this model, children deduce which grammatical rules are most likely based upon speech they have heard. An absence of a certain kind of structure in common speech, for instance, might indicate that such a structure isn't grammatically correct. Here Carnie argues that the fact that individuals can determine the grammatical correctness of a sentence they have never heard before (e.g. a sentence with seven embedded layers such as, "John said that Mary said that Paul said...") suggests the statistical model isn't the case. To me this would seem to be a strong counterargument if children analyzed sentences on an individual basis, comparing each new sentence to every sentence they have ever heard. But Carnie's argument doesn't seem effective if children learn rules instead of sentences. Returning to the statistical model, a child could deduce that embedding (e.g. Matt claims that [complete sentence]) is a valid sentence structure. In this case, having not heard a seven-times-embedded sentence is not an impediment to determining the sentence's grammatical correctness, since the child can simply use the embedding rule recursively. This post isn't to say that language isn't innate, but rather to scrutinize the logic of arguments in favor of a UG.