Is Universal Grammar a universal grammar? From Chomsky's hierarchy we deduce that for each grammar there is a Turing machine, and conversely. Following this equivalence, it is immediate to conclude that a universal Turing machine is equivalent to a universal grammar. Meanwhile, in linguistics, Universal Grammar is the human brain circuitry that implements the faculty of language. So the definitive answer is achieved only when we show that the human brain is Turing complete, and that language uses this capability. So yes: Universal Grammar is a universal grammar, because the human brain circuitry that implements the faculty of language is Turing complete.