Is There a Philosophical Problem of Reference?

Many philosophers worry about reference, and think we need to come up with a "theory of reference". This would explain how strings get "connected" to things: the string-thing relation. So, maybe this should be analysed in terms of causation. Causation glues cats to the word "cat". And so on. Or maybe one gets worried about how, e.g., "aleph_0" and $\aleph_0$ got glued together, when causation couldn't have done that---for $\aleph_0$ is a causally inert abstractum. In fact, the string "aleph_0" is also an abstractum. Worried by the unsurprising inability to reduce reference to causation, one might go deflationary and say that there is just a reference predicate, the binary predicate "$x$ refers to $y$", governed by disquotational axioms (e.g., ""red" refers to the set of red things") and one has this in one's language (which gives the illusion of genuine reference). Or perhaps one gives up, becomes a semantic nihilist and says outright: there is no reference relation at all.

Well, maybe there is no philosophical problem of reference at all: any relation of strings to things is a legitimate reference relation. So, the theory of reference is this:

$\mathcal{I}(\sigma) = x$

where $\mathcal{I}$ is some function: any function you like. Because $\mathcal{I}$ is a parameter here, reference is really a ternary relation,

$\sigma$ refers to $x$ relative to $\mathcal{I}$.

Many authors believe, however, that reference is a binary relation that strings bear to referents. That is,

$\sigma$ refers to $x$.

With no language or interpretation relativity! It seems to me that this violates the single most basic principle of language---that anything can mean anything in some language. The language relativity is essential. And yet the view that reference is a binary relation without language relativity seems fairly standard.

Chris Gauker gave a talk on a related topic at MCMP several months back. And Chris's view is an example of the conventional view of what is required from a theory of reference:

The problem
What is reference? Offhand, it appears to be a relation, just as being heavier than is a relation. Moreover, it is a relation that holds between words and other things. It holds between "chair" and the chair I'm sitting in, between "chair" and the chair next door, and between "chair" and the chairs of the past and future. It holds between "Socrates" and Socrates, between "meson" and mesons, and between "beautiful" and beautiful things. Of course, the reason it holds between these things is not just that it holds between every word and everything. For instance, it doesn't hold between "basketball" and daffodils. The fact that the relation being heavier than holds between an atom of oxygen and an atom of hydrogen, as well as between Pavarotti and Diana Ross, but not between everything and everything, ought to make us wonder what being heavier than amounts to, if we don't already know. Likewise, the combination of diversity and specificity exhibited by the reference relation ought to make us wonder what reference is. We mustn't just call it an "unanalyzable primitive".
Accounts of reference that might satisfy us fall into two classes. On the one hand, we might seek what I'll call an analysis of reference. An analysis takes the form:

$t$ refers to $a$ if and only if $\dots$.

(Gauker 1990, "Semantics without reference", Notre Dame Journal of Formal Logic, 438.)

This seems to me to be a bit like a grammatical mistake, because it omits the crucial language parameter,

$t$ refers to $a$ in $(\mathcal{L}, \mathcal{I})$ if and only if $\dots$.

And then the answer is:

$t$ refers to $a$ in $(\mathcal{L}, \mathcal{I})$ if and only if $\mathcal{I}(t) = a$.

There are no constraints at all on what the interpretation function $\mathcal{I}$ might be. It can be any old string-thing function. On this proposal, it isn't that the famous semantic indeterminacy puzzles disappear (e.g., Quinian inscrutability of reference, Kripkenstein, Putnam's "model-theoretic argument", ec.). Rather, the puzzles turn out, on reflection, to be puzzles about what language one speaks. The question isn't how languages get referentially glued to things---they already are, in every mathematically possible way (generating uncountably many different languages). The question is how the mind gets glued to some particular language: a problem of language cognition.