Examples of M-facts might be things like:
1. "Schnee" refers-in-German to snow.In each case, the language relativity has been made explicit. I think that ignoring language relativity is a major fallacy in much writing about the foundations of linguistics and philosophy of language. Tarskian T-sentences are examples of M-facts. The bearers of the semantic (and syntactic) properties are types, not tokens. Again, I think that it's major mistake to be confused about this.
2. "sensible" is true-in-Spanish of x iff x is sensitive.
3. "Schnee ist weiss" is true-in-German iff snow is white.
4. "I" refers-in-English, relative to context $C$, to the agent of the speech act in $C$.
5. "kai" means-in-Greek logical conjunction.
Canonical examples of U-facts might be things like:
6. Speaker X uses the string "Schnee" to refer to snow.
7. Speaker Y has a disposition to utter the string "gavagai" when there are rabbits nearby.
8. Speaker Z tends to assert the string $\sigma_1 \ast$ "kai" $\ast \sigma_2$ just when Z is prepared to assert both $\sigma_1$ and $\sigma_2$.
One might initially think that U-facts explain M-facts. Or that U-facts provide evidence for M-facts. Roughly:
U-facts: evidence/data for linguistic theories.This is, I think, roughly right, but with a very important caveat, which is that M-facts cannot be explained by U-facts. The argument is this:
M-facts: theoretical content of linguistic theories.
(i) M-Facts are Necessities.An M-fact, such as the fact that "Schnee" refers-in-German to snow couldn't have been otherwise. The argument for this is a counterfactual thought experiment. Suppose $L$ is an interpreted language such that "Schnee" refers-in-L to sugar. Then it seems clear to me that $L$ isn't German. If one changes the meanings of a language, the language is simply a different one. Languages are very finely individuated.
(ii) U-Facts are ContigenciesA U-fact, such as the fact that Y has a disposition to utter "gavagai" when there are rabbits nearby, is contingent. It Y needn't have had that disposition. A U-fact is connected to properties of the speaker's cognitive system.
If the previous two claims, (i) and (ii), are correct, then
(iii) U-facts cannot explain, or provide evidence for, M-facts.This conclusion follows because contingencies cannot explain necessities.
I've given this argument is several talks since 2008 (originally in a talk "Meaning, Use and Modality" in at Universidad Complutense, Madrid). The audience frequently responds with considerable surprise!
If U-facts do not explain M-facts, then what explains M-facts? I say, "Nothing". Nothing explains why "Schnee" refers in L to sugar. It is simply an intrinsic property of the language L. There is some sense in which M-facts, along with syntactic and phonological and pragmatic facts, about a language $L$ are mathematical facts. Languages are complicated (mixed) mathematical objects. For example, suppose that
The string $\phi$ is a logical consequence in $L$ of the set $\Delta$ of $L$-stringsThen this fact, about $L$, is a necessity.
If U-facts do not explain M-facts, then what do they explain or provide evidence for? I think the answer to this is,
U-facts explain, or provide evidence for, what language the speaker/agent cognizes.So, let me call these C-facts, and these have the form:
(C) Speaker X speaks/cognizes LSo, for example,
The (contingent) U-fact that X has a disposition to utter "gavagai" when there are rabbits nearby is evidence for the (contingent) C-fact that X speaks/cognizes a language L for which the following M-fact holds of necessity: "gavagai" denotes-in-L rabbits.