Meaningless identities?

I run across the idea from time to time that some identity statements are “category mistakes” or “meaningless”—even when the terms flanking the identity have referents. In the philosophy of mathematics, it’s pretty common to see it suggested that “Julius Caesar=7” and “the natural number 2=the third von Neumann ordinal” are meaningless. Agustin Rayo’s recent book “The construction of logical space”, for example, advertises it as an advantage of his favoured approach to mathematical language that it allows for this phenomenon. Huw Price’s collection of papers on Naturalism and Representationalism suggests something similar for terms taken from different “discourses”. It’s an intriguing suggestion, but I can’t see how to make it work—for reasons independent of the “big picture” theories of ontology and meaning that it is supposed to articulate/challenge.

Here’s an argument against meaningless identities between referential singular terms. (The cogniscienti will recognize it as a variant of Evans’ famous argument against indeterminate identity; and in the light of that, I wouldn’t be surprised to learn that the point is already in the literature. References please!)

To start with, suppose that “A=B” is our candidate for a meaningless identity, and “A” and “B” have referents (thus, I’ll feel free to use those terms, rather than simply mention them). I offer two premises:

(1) A is an x such that there’s some term N referring to x, such that “N=A” is meaningful and true.
(2) B is not an x such that there’s some term N referring to x, such that “N=A” is meaningful and true.

By Leibniz’s law, since A possesses a property that B lacks, it follows:

(3) A is not identical to B.

I say that anyone who takes “A=B” to be meaningless is committed to (1) and (2), but must not commit herself to (3). Since (1) and (2) entail (3), this leaves her in an incoherent position.

The commitment to (1) is the easiest to defend. Just let the term N be “A”. I wouldn’t recommend resisting at this point. 

The need to avoid commitment to (3) is also pretty obvious, so long as nothing meaningless can be true.

The most involved case to make is that our target theorist is committed to (2).  Here’s the argument for that: Suppose that there was an N such that B possessed the property mentioned. Then, ex hypothesi “N=A” would be meaningful and true, where “A” refers to A and “N” refers to B. Hence B=A. So possessing that property would entail the truth of something that’s meaningless. Just as in the previous paragraph, that’s impossible. So B must lack the property, just as (2) reports. 

As mentioned, this is a variant of Evans’ argument. And various—but not all—the responses to Evans might have traction here. For example: a nonclassical logic might help resist certain of the moves; or endorse certain versions of Leibniz’s law but resist the one relied on above. On the other hand, what I think of as the most widely accepted caveat/objection to Evans’ argument—-inapplicability in cases of referential indeterminacy in the terms involved—doesn’t seem to me have any interesting analogue here.

There’s certainly new material in here that could be pinpointed as the problem. For example, the argument shoots up and down the semantic staircase, between the metalinguistic claim that “A=B” is meaningless, and the object language claims: A=B/~A=B. These are involved in (3) and (in more than one place) in the defence of (2). So questioning those moves, once they’re formalized in a fully spelled out form of the argument sketched, is one way forward our advocate of meaningless identities might favour.

However, though I see some options in the table, I’ve no idea exactly which way the friends of these “meaningless identity” view would jump in response. Ideas welcome!