alright, this question is philosophical and somewhat fuzzy. i also admit to knowing little about logic. all in all, this question can possibly be easily resolved by either pointing to (perhaps even well-known) literature i haven’t found or by pointing out a fault in my reasoning.
joel david hamkins is a proponent of multiverse interpretation of set theory, where we should see ZFC and other formalisations of the concept of sets as theories of not one universe of sets, but of a multiverse of sets, in which there are many “universes” of sets, .. or let’s call them aliverses.
the reason he gives is that since we can perfectly model – within ZFC – different theories ZFC, so one in which CH is true and one in which it is false, we know how such “places” look like, so to say that .. we are in only “one true universe” in which CH is either true or it is false and we just don’t know .. would be to disregard either of these “places” as unreal – even though we can readily visit them. instead, he regards these “places” as just aliverses of a multiverse of sets.
that position is known as a pluralist view and i find his arguement compelling. but that’s not quite the way we think of set theory, is it? of course, prof. hamkins would propose we adapt our way of thinking here, but then, when doing any mathematics founded on set theory, we would never be in “one definite” of these aliverses, but rather always in “some arbitrary” aliverse, make an argument in them and arrive at statements, which by the arbitrariness of the aliverse, holds in any of them. just as when we're doing group theory, we are never in one definite group, but rather awalys in some arbitrary group. in fact, just as we prepend “let $G$ be a group” before arguing in some arbitrary group, we should prepend “let $U$ be a set-theoretic universe fulfilling ZFC” before doing any mathematics founded on set-theory at all.
however, i still think of sets as belonging to one universe of sets, just as i think of natural numbers as belonging to one structure of natural numbers. there may be, in some abstract sense, multipile competing definite concepts of sets or natural numbers, but i can’t ever quite get to know them. rather i accept that the one concept of sets and the one concept of natural numbers that i have in mind are perhaps amorphous and vague. taking that seriously, i would have to say that the independency of CH from ZFC and then from ZF, which i regard to be a fitting formalisation of my amorphous and vague concept of sets, tells me that, with respect to my concept of sets, CH just simply is neither true nor false, that is to say: classical logic is not appropriate.
ok, so i’m probably not the first to think of this. but i haven’t seen this argument so far. so i have to wonder: am i missing something? doesn’t the independency phenomenon make a case for non-classical logics?
remark. i realize that the axiom of choice implies LEM within ZF from say intuitionistic logic. so i’m ready give up full choice in favor of dependent choice in order to drop LEM.
As noted in the comments, you are - or at least are dangerously close to - mixing up syntax and semantics. That said, I think there is a way to make your feelings precise, and I at least sometimes share them; since I can't find an exact duplicate of this question (although similar things have been asked before, e.g. 1), I'll jot this approach down here.
I'll call this position ZFC-finalism. The idea is that, while we informally work in a universe of sets, our real stance is that the ZFC axioms and only the ZFC axioms are a priori justifiable.
I personally think that ZFC-finalism is perfectly coherent, even if I don't share it usually. I suspect that it captures your own stance at least sometimes. The point is that there is a well-understood gadget set up to handle "ZFC-finalist truth values," namely the Lindenbaum algebra of $\mathsf{ZFC}$. This is the Boolean algebra whose elements are equivalence classes of sentences modulo $\mathsf{ZFC}$-provable-equivalence.
If we accept ZFC-finalism, then - I claim - the right logical shift is not really towards nonclassical logic but rather towards a Boolean algebra of truth values. In particular:
By definition, the truth value of $\mathsf{CH}$ is $[\mathsf{CH}]$ (where "$[\cdot]$" denotes the equivalence class in the Lindenbaum algebra).
Highly nontrivially, we have $[\perp]\not=[\mathsf{CH}]\not=[\top]$.
Set theorists of wildly different stripes can all agree with the points above (well, assuming they grant $\mathit{Con}(\mathsf{ZFC})$ for the second one); the only point of disagreement is how we deploy the phrase "truth value." Getting back to your original question, since the laws of Boolean algebra are exactly those of classical logic in a precise sense (i.e. every Boolean algebra has the same equational theory as the two-element Boolean algebra) in my opinion the philosophical takeaway from the preceding is that incompleteness does not in any serious way push us towards nonclassical logic. But of course mileage will vary on this point, and much better mathematicians than me will disagree with me (and with each other!).