Three jobs for logical structure:

You might think the "logical structure" of a sentence is a way of cutting it up into parts [eg. "John is happy" becomes "is happy(john)"] that does three things:

1. gets used by the logical theory that best captures all the valid inferences.
2. matches the metaphysical structure of the world.
3. explains how we are able to understand that sentence, by breaking it down into these parts, and understanding them.

However, it's not obvious that the method of segmentation which does any one of these things best should also do the others. I don't mean that this idea is crazy, just that it is a bold and substantive claim that logic unites cognitive science with metaphysics in this way.

It's also not obvious that *any* method of segmentation can do 1 or 2.

Task 1 might be impossible to perform because there might not be a unique best logical theory. If we think that the job of logic is to capture necessarily truth-preserving inferences, then second-order logic is logic. Any recursive axiomatization of second order logic will be supplementable to produce a stronger one - since the truths of second order logic aren't recursively axiomatizable. (One might hope though, that all sufficiently strong logics that don't say anything wrong will segment sentences the same way)

Task 2 might be impossible because the world might not have a logical structure to reflect. What do I mean by the world "having a logical structure"? I think there are two versions of the claim:
a. The basic constituents of the world are divided between the various categories produced by the correct segmentation e.g. concepts and objects in Frege's case.
This is weird because "constituents of the world" sound like they should be all be objects. But presumably objects don't join together to produce a sentence, so the kind of expressions used in your chunking up can't all be objects.
Its also weird because it just seems immediately weird to think of the world as having this kind of propositional structure, rather than our just using different propositions with structure to describe the world.
b. The objects that really exist (as opposed to those that are merely a facon de parler), are exactly those which are quantified over by true statements when these are formalized in accordance with the best method of segmentation. To misquote Quine: "the correct logical theory is the one such that, to be, is to be the value of a bound variable in the formalization of some true sentence in accordance with that theory."
So, for example, if mathematical objects can't be paraphrased away in first order logic, but they can using modal logic, the question of whether mathematical objects exist will come down to which (if either) of these of logics has the correct segmentation.

Finally, Task 3 is ambiguous between something (imo) silly and something about neuroscience.
The silly thing is that a correct segmentation should reflect what components *you* break up the sentence "John is happy" into, when you hear and understand it (presumably none).
The neuroscience is `what components does *your brain* break up this sentence, when processing it to produce correct future behavior, give rise to suitable patterns of qualititative experience for you etc.?' This is obviously metaphorical, but I think it makes sense. It seems very likely that there will be some informative algorithm which we can use to describe what your brain does when processing sentences (it might or might not be the same algorithm for different people's brains). And, if so, it's likely that there will be some natural units which this algorithm uses.