Some careful philosophical discussion by Monti, Parsons, and Osherson (2009):
There may well be a “language of thought” (LOT) that underlies much of human cognition without LOT being structured like English or other natural languages. Even if tokens of LOT provide the semantic interpretations of English sentences, such tokens might also arise in the minds of aphasic individuals and even in other species and may not resemble the expressions found in natural language. Hence, qualifying logical deduction as an “extra-linguistic” mental capacity is not to deny that some sort of structured representation is engaged when humans perform such reasoning. On the other hand, it is possible that LOT (in humans) coincides with the ‘‘logical form’’ (LF) of natural language sentences, as studied by linguists. Indeed, LF (serving as the LOT) might be pervasive in the cortex, functioning well beyond the language circuit […].
Levels of analysis again. Just because something “is” not linguistic doesn’t mean it “is” not linguistic.
This calls for a bit of elaboration! (Thanks Martin for the necessary poke.) There could be languages—in a broad sense of the term—implemented all over the brain. Or, to put it another way, various neural processes, lifted up a level of abstraction or two, could be viewed linguistically. At the more formal end of cognitive science, I’m thinking here of the interesting work in the field of neuro-symbolic integration, where connectionist networks are related to various logics (which have a language).
I don’t think there is any language in the brain. It’s a bit too damp for that. There is evidence that bits of the brain support (at the personal-level of explanation) linguistic function: picking up people in bars and conferences, for instance. There must be linguistic-function-supporting bits in the brain somewhere; one question is how distributed they are. I would also argue that linguistic-like structures (the formal kind) can characterise (i.e., a theorist can use them to chacterise) many aspects of brain function, irrespective of whether that function is linguistic at the personal-level. If this is the case, and those cleverer than I think it is, then that suggests that the brain (at some level of abstraction) has properties related to those linguistic formalisms.
Monti, M. M.; Parsons, L. M. & Osherson, D. N. (2009). The boundaries of language and thought in deductive inference. Proceedings of the National Academy of Sciences of the United States of America.