*Tl;dr: there are drawbacks of what you propose, both algebraic and foundational, which I consider serious enough to indicate that the "valuation" in question really be a secondary rather than fundamental object. However, there are definitely more intricate approaches along the lines of what you suggest that are much more satisfactory, and even find application in classically-construed mathematics.*

The short version for me is that the map given by a "structure" (in whatever sense is meant) from sentences to truth values is **the** fundamental object in a given logic, and what you've described is really better considered a "derived" object.

First, let me push back against Joel's answer a bit. There *is* a perfectly meaningful semantics for your logic: a "structure" is just a set of sentences in the relevant language, and this fully determines a map from sentences into $\{P,D,I\}$ (= **p**rovable, **d**isprovable, **i**ndependent). While this is extremely far from the usual notion of first-order semantics, and countable structures (respectively *computable*, or etc., countable structures) aren't in my opinion any more objectinable than arbitrary sets of sentences (resp. *computable*, or etc., sets of sentences), and so this shift feels a bit unnatural to me, it's perfectly satisfactory. So let's go with it.

So what are the criticisms?

**First problem**: *Theory* and *semantics* are blurred. Self-explanatory. In my opinion, part of what makes logic interesting is the *differentiation* between semantics and syntax and their subsequent *unification*. Certainly some notions of semantics make philosophical commitments which are difficult to justify, but that doesn't mean that *all* semantices fall prey to this, and abandoning the whole semantic project seems inappropriate to me.

- That said, for completeness (hehe) I need to mention Jean-Yves Girard, whose "antirealist" stance is both highly interesting and on face value strongly opposed to at least some of what I've written here. I have my own thoughts about his work, and the extent to which it
*actually* pushes against what I've written here, but this isn't an appropriate forum for that; I mention him only because failing to mention a serious opposing viewpoint to what I'm saying would border on dishonesty.

**Second problem**: We lose *truth-functionality* of basic operations. The truth value of $p\wedge q$ is no longer determined by the truth values of $p$ and $q$: e.g. taking our "model" (in this sense) to be the theory ZFC, consider $(i)$ $p=CH, q=\neg CH$ versus $(ii)$ $p=CH, q=CH$.

- It's worth noting that truth functionality is by no means a
*requirement* of a logical system - non-truth-functional operations are extremely important, and in particular a modal interpretation of provability yields a rich and meaningful theory - but the degree to which we lose truth functionality here seems difficult to justify.

So while the map assigning $P,D$, or $I$ to a sentence depending on its status relative to a given theory is extremely important and interesting, in my opinion it has too big drawbacks to be considered the satisfactory **fundamental object** for a logical system. Rather, it's better thought of as a **derived object**, coming *after* the semantics has already been established. This semantics need not involve strong Platonist commitments, but does in my opinion need to be a bit richer.

However, all is not lost (in my opinion)!

In a comment to Joel's answer you write

perhaps there should be a theory where it's explicitly neither true nor false.

This is **not the same** as what you've proposed; it's a much less restrictive vision. And indeed, it's one which is realized by extremely interesting and useful approaches.

I'll mention in particular **Boolean-valued models**. First, a purely syntactic comment. In some sense the truth functionality issue I highlighted above was only due to *loss of information*: if I kept track of the *truth conditions* of the sentences involved, and not just their ultimate proof-theoretic status, I wouldn't have run into trouble. This can be made precise by considering the **Lindenbaum algebra** of a theory: given a theory $T$, the set of sentences in the language of $T$ modulo $T$-provable equivalence forms a Boolean algebra, and this Boolean algebra - rather than the set $\{P,D,I\}$ - is a reasonable set of truth values. Semantically speaking, a Boolean-valued structure should be one where the atomic formulas are allowed to take on truth values in an arbitrary Boolean algebra, rather than just $\{\top,\perp\}$; so a Boolean-valued structure is probably going to be a *pair* consisting of the structure side and the "target" Boolean algebra. This winds up being a very useful generalization - in particular, it has serious value in the Boolean approach to forcing in set theory.

And we could also consider other algebraic structures for our sets of truth values, like Heyting algebras or even wilder things. Thinking along these lines also motivates the general study of algebraic logic, and for a perspective both general and syntactic see Czelakowski's book.

The moral of this line of thought is: **mere provability status is not complete information**, *even if* we adopt a radically anti-realist stance. It's this loss of information, not the absence of a philosophical commitment, which is intension with your suggestion.

One final note: another important loss-of-information occurring in what you propose comes from *degrees of meaningfulness*. You write in a comment to Joel's answer that you distinguish between Goldbach's conjecture and CH, at least partly because there is a strong sense in which "independence implies truth" for the former but not the latter, but once you accept this as a possible phenomenon to any degree the strict approach you've proposed seems even more inappropriate. I've written a bit about my own thoughts on relative meaningfulness here, and some grown-ups' thoughts on the matter can be found here.

Let me end with a bit of clarification. My stance above may come off as "semantics-first," which is pretty hypocritical: if my major foundational interest is the *interaction between* semantics and syntax, then I should be treating them as equally interesting.

My response is that I **do** think that it is perfectly valid (and often more interesting) to go the other way: first determine a purely syntactic logic, and then look for a semantics for it. Similarly, it's often extremely valuable to identify multiple *equally good* semantics for a given syntactically-defined logical system. However, I also think that we cannot really be satisfied until we have a semantics which is both reasonably natural and with respect to which our proof system is sound and complete. Moreover, my definition of "semantics" is sufficiently broad in my opinion that nothing is lost by my previous claim. I would say that the nature of my stance is *aesthetic*.