Epistemic truth and fuzzy logic

In the context of the existence of the external world, two objections spring to mind to the contention that it is not a yes/no answer.

My first objection would be Kantian. I do not believe that is it possible for the external world to not exist. Human rationality pre-supposes the existence of space and time, it is the structure of rational thought. We can argue about what we mean by space and time, but the very fact that we can ask the question seems to me to presuppose the external world. If I were to summarise this argument I might say that nothing would "make sense" if the external world didn't exist.

My second objection would be that the logic behind the question seems to be bivalent, i.e. two-valued. I accept that we can have different senses of the word "exist", for example things can exist in fiction, or in conceptual terms. However, when we ask the question "does the external world exist?" this question deserves a yes or no answer. How could it be that the answer is that the external world sort of exists and at the same time sort of doesn't exist? That response, at least to me, is non-sensical.
Fuzzy logic seems to apply to non-bivalent questions, such as "are there 30ml of water in this glass?" I can see how this question has a non-bivalent answer because, in part it depends on how carefully you measure the contents of the glass. I also accept that the contents of the glass change over time through evaporation.

But you will note that by debating this we are pre-supposing the existence of the external world, or at least of space and time. In so doing we have knowledge that the external world (in terms of space and time) does exist.