That is a very thoughtful post. Some responses:
There's a lot of "given" that goes into formulating something in predicate logic
The critique of logic you mention-- that we have to explicitly state relationships such as that the set "red" is a subset of "colored"-- seems to ignore that this is exactly the same way that we as humans also learn that red is a subset of colored. I grew up speaking english, I live in a 4 dimensional universe with cartesian geometry and photons, I have light and sound sensing organs, and so after a few years of existence I learned that when I hear "red" it means a certain wavelength of light, and when I hear "colored" it means the set of all visible wavelengths of light. The problem of the "given" you mention in logic is just because we don't have the same patience when formulating logical statements as we do when raising children.
Imagine if we assumed that the entities in question - red things like apples - were fully determined by how they were represented in predicate logic
I agree that the noumenal apple is very different from the logical representation of it. The logical representation is only supposed to abstract the phenomenon.
Now, if we say that our knowledge of how red things are always coloured things is represented by the implication that red things are always coloured by including that as an argument premise, there is the question of the means of that representation and how it is is enabled through the "given" practical competences we leverage.
Not sure I follow... The means of that representation would just be something like
Heidegger situates his ontology in the "given" revealed by that gap between representations (outputs of representational behaviour) and the means of representation's leverage of know how.
I don't see room for Heidegger between two things that shouldn't be separated. It seems like you're saying he focuses on phenomenology, which is certainly true, but where are the noumena? This is my difficulty. You can't have a sound ontology without addressing noumena. Discussions on the limits of logic seem to me unrelated to this issue.
Edit:
If you're familiar with Wittgenstein, it's a similar brand of error to using a word outside of its context without noticing the violence done to its meaning ("language running idle") - the error here being one of apprehension/theorising style, taking a present at hand mode of apprehension outside of its intended context and not noticing the violence done to its topics of concern (similar to hyper reflection in Merleau Ponty maybe).
Do you have any examples of this Heideggerian violence? I'm generally familiar with Wittgenstein but his criticisms of language, ironically for this discussion, are easily solved by using logic instead.