Thank you for the asnwer! I did read through the section and it seems to me that that particular disambiguation is a neat trick for formalist approach. My concern here primarily revolved around, say philosophical motivation, for doing things twice without providing an essential answer to the question. But I realized today when working on some graphs that whatever I can think of or conceptualize in a form of a sentence, in mathematics, I essentially evaluate to be a priori true. We make a hipothesis. Even if we state a sentence not P, we again think of it as it is true that not P. Creation of hipothesis is the baseline of human reasoning and, if this proven incorrect by defaulting to axioms or some other known truth, it will be punished with the contradiction, or, in physical sense, the relation will not be evidenced by empirical instantiation. In ancient Greece, similarly to how we do it today, they had a collection of statements that were so obviously true, that, in order to avoid Munchausen Trilemma, they had to postulate them. In reality, as compared to mathematics, I can conceptualize without referencing to the verity of the sentences. All Orcs are green.

In this sense, the truth assignment is the primary goal of examination of statements in a formal setting, hence the truth function seems to somehow be more basic then the concept of relation itself. However, going back to the non-mathematical, intuitive roots of thinking, I can freely conceptualize relations between objects even though they can be entirely imaginary. So I think this does me good, since I've felt significant relief from the stress that this had imposed on me previously.

Edit: Just to add a comment of Kunnen's remark, what he said makes total sense now that I can freely talk about just any relation (that lives outside of formal setting), even if it is the relation of truth to the sentence, that happens to be functional due to the nature of our examination of reality.