Sometimes, we do use truth-talk in such a way as makes it seem like we might be attributing truth to things that aren't assertions/descriptions/propositions/w/e. There is, for example, the phrase "your true self" (and "being true to yourself"). Technically, if etymologically, the word "betrothed" has such obscure significance, too. Or Aquinas/some scholastic would talk about such things (I have some memory of reading that somewhere, but I'm not sure where exactly, right now).
That kind of talk does seem flatly ruled out by the disquotational scheme as a constituent of the full, general truth scheme. For the disquotational scheme does pick out a technical property of sentences/propositions being isomorphic sans quotation marks to the facts that make them true. Or, that is, there is this isomorphism anyway, and we might as well use the word "truth" to refer to it. But so at any rate, the song of that scheme is easily played on a more complex instrument, by the correspondence theorist: for it is easy to say, and it does seem correct, that, "S is A," corresponds to a fact if and only if S is A.
On the other hand, one might want to balance the correspondence theory with a coherence theory, for the sake of balancing foundationalism and coherentism on the side of the theory of knowledge, or for some other reason. Or maybe we just want to specify the coherence theory of truth by reflecting on the plenitude of coherence relations in sophisticated coherentism? But that is my question: if the truth relation in alethic (not epistemic) coherentism mediates sentences/propositions without depending on the symmetry of the disquotational scheme, then is it possible for non-assertoric functions to enter into that relation? For example, we would not usually ever say that an imperative or a question were true or false. And if we are assuming disquotational symmetry as essential to the truth relation, it is obvious why. However, when that is exactly what we are no longer assuming, or are even actively negating in our system, where then does the denial of truth-aptitude to non-assertions come from, if anywhere????
???Though, to be fair, the only kind of general example(?) I can think of is something like the scheme of the question-answer relation. The way in which the concept of a question depends on the concept of possible answers, but also the reciprocal way in which the concept of something being an answer depends on the concept of a question, seems to illuminate an abstract erotetic-logical object whose erotetic functioning is "logically" accompanied by an assertion function. There is a "metaphorical" sense in which the Form of Answers is predicated of the Form of Questions so that the Form of Questions, though not an assertion/proposition, is yet true (and expressly not false), without this meaning that it "corresponds to a fact." Rather, it enters into an erotetic coherence relation with propositional space and its contents, which has to be sufficient (if anything is) in order for me to be justified in claiming that there are non-assertions that are truth-apt when truth is as in alethic coherentism.
Or would there be an option similar to (but not the same as) "truth-value realism," but in metaethics, whereby there is at least one pure imperative that nevertheless "happens" to sustain the truth predicate (or map to the Truth Value), despite not being an assertion and therefore not "corresponding to a fact" in the usual way? One does get that sense, sometimes, from reading certain theological statements, like, "God is the Truth," or, "God is the source of all truth," and so on. And it is unclear how the law of noncontradiction would be strictly formulated in a world where non-assertions could be true and paired with false assertions, or assertions could be true and paired with false non-assertions, and so on and on; so if it was possible for God to command the property of truth to be predicated of various non-assertions (though it was always necessary, regardless, that assertions would "also" sustain the truth predicate), then maybe that is how the law of noncontradiction could in some major way depend on the will of God?