It is undisputed that Large Language Models (LLMs) produce meaningful sentences, for example in responding to questions asked by humans. The meaningfulness, however, is standardly taken to be of a derived kind: LLMs can refer to things, and mean things because they operate on linguistic symbols semantically imbued by people. Yet they are incapable of meaning in a non-derived sense. Since they lack a proper embedding in a non-linguistic environment they exhibit no genuine semantic grounding. As opposed to this, a growing number of philosophers argue that LLMs indeed do display genuine semantic grounding and meaning in a non-derived sense and are capable - at least potentially - of thought comparable to human thought in relevant regards. They point out that LLMs are properly embedded in a non-linguistic environment in virtue of applying words standing in causal relations to non-linguistic objects, much like humans do. This paper considers this discussion through the prism of the Medieval dispute between Ockham's nominalism and Aristotelian-Thomistic positions regarding semantic content. It suggests that arguments to the effect that LLMs can achieve genuine semantic grounding presuppose a nominalist metasemantics where the semantic relation between a representation and a represented object is a mere matter of causation and the subject's recognitional capacities. This is contrasted with a Thomistic account of mental content according to which the formal identity of thought and the object of thought provides an essential tie between the metaphysical structure of reality and the representational capacities of humans. This allows for an attractive and precise characterization of the exceptionality of human thought in distinction to the tacit nominalism presupposed by defenders of genuine semantic grounding for LLMs.