The five abstraction levels above essentially represent two ways to structure the grammatical type inventory. Level 5, on the other hand, is somewhat different and strictly speaking does not belong to the same continuum of abstraction. So, I am basically just wrapping individual grammatical types with three extra layers of structure. That may sound like a mouthful, but the main point is simple enough to state in one sentence: From Level 1 to Level 4, the “output” at each level is the “input” at the next level. The level of all grammatical types, including both types inside the structures of Levels 1–4 and types outside them, such as logical connectives and syntactic “ roots,” both of which are essentially “ syncategorematic.”.The level of a structured set of all possible versions of the set from Level 3 (e.g., some languages may not have an adjectival extended projection, and some may have different verbal or nominal types).The level of a certain structured set of extended projections (e.g., a language’s grammatical type system may have a verbal extended projection, a nominal one, and an adjectival one, with some connections between them).The level of a certain structured set of types (e.g., an extended projection).The level of an individual type (e.g., V, T, or D).We can think about the human language grammatical type system at five levels of abstraction instead of three: In fact, the layered-abstraction line of thought can go further. As Fong & Spivak point out in their recent textbook on applied category theory- Seven Sketches in Compositionality-category theory is “unmatched in its ability to organize and layer abstractions.” The three concepts present us with a scenario of layered abstraction, which is also what has inspired my application of category theory in my research. In my discussion so far we have met three concepts about grammatical types: (i) the types themselves, (ii) the extended projections building on individual types, and (iii) the various abstract relations building on extended projections. And as in the previous post, I will still abstract away from technical details (which you can find either in my extended abstract or, in fuller forms, in my dissertation) and focus on my motivation and source of inspiration instead. In this post I continue to introduce my category-theoretic modeling of the human language grammatical type system. In the previous post I laid out the disciplinary background of my application of category theory in linguistics in more detail (than I had done in my explainer video). Level 4: a category of language varieties.
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