>However, I know that many people underestimate e.g. Propositional Logic because their brain's can't comprehend what an exponential semantics is like.

An intriguing claim. I am of the mind that language, being something that everyone uses every day, must be rooted in mechanisms that don't require extensive study. But I can appreciate the perspective that our brains are doing a lot more computational processing than our minds are aware of. It just seems most probable to me that if Propositional Logic or similar formal systems were relevant, the last two and a half thousand years of philosophy (and civilization) would have been quite different.

>I don't think there is anything special about natural language, or any special property, which can not be used as an interpretation of some formal language. Now, the problem might be that what you consider some kind of "intrinsic quality" of natural language is hard to make precise, since you only have natural language to appeal to (I guess?). What do you think?

I think the truth is that everything about natural language is "special", that language itself is a special property; in general it is identical to and coincident with consciousness, an apparently unique emergent property of human brains. It is this very speciality, the ability to "interpret" things, which enables us to invent formal systems of logic and call them (mistakenly, in my opinion) languages. Without the intrinsic quality of natural language, which is about accuracy rather than precision, how are we to create, develop, and communicate computational code systems, or appeal about anything with anything to anyone? Every beast with a brain performs logic as an inherent capacity of having a neural network, why would natural language even exist if formal logic could provide any useful information without the foundation of arbitrary reasoning to communicate emotions and experiences well enough to develop civilizations complex enough to allow mathematicians and analytic philosophers to survive.