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!

My paper “The Julius Caesar Objection” is more or less devoted to arguing that Frege’s Theorem doesn’t depend upon our assuming that “0 = Caesar” is in fact meaningful, though I don’t really take a stand on whether it is meaningful or not. I think Crispin and Bob may somewhere question whether it is really coherent to think it isn’t meaningful, or whether saying it isn’t helps all that much. Ultimately, I agree with them, in the sense that one still needs to say why exactly it is supposed to be meaningless.

Roy raises some worries about counting what we might call “heterogeneous collections” in his review of “Frege’s Theorem”, e.g.: How many things are there that are either Caesar or zero?

I seem to recall that Dick Cartwright has some things to say about this issue somewhere, but I don’t recall where, and don’t have my books here at home.

A quick thought about Price-like reasons for `meaningless’ identities:

On the antirepresentationalist view, our commitment to the entities quantified over is supposed to fall out of a prior explanation of the function of the discourse in question. So, for example, the property of being good is something we are indirectly committed to by virtue of our participation in a certain type of endorsement activity. (Insert favourite projectivist story here).

Judged by these criteria, one question liable to strike the antirepresentationalist is the following: Is there any point in making the identity claim? Well, consider `pleasure = the good’. It is tempting to think that the projectivist will either read it itself as a further endorsement — too crudely: Hurrah for pleasure! — or as pointless, insofar as it has no obvious relation to the practice of endorsement which makes the evaluative activity worth engaging in in the first place.

Does that make it meaningless? Probably not, but it isn’t clear that the Pricean needs to go that far. Rather, it is just of dubious function. On the antirepresentationalist model, that is a strong indictment.

Adam—yeah, that seems right and fine. And I guess this is compatible with taking them to be “category mistakes” in some sense that is compatible with them being true or false (some e.g. you might think they’re false, but also that the fact that they’re false has no interesting role to play). I do think that perspective does change the dialectical situation for Price though. You can’t then say that people who discuss or argue over whether this or that identity is true (e.g. whether reference=causation; or whether rightness=maximizing hedons) are literally asking meaningless questions. The most you can say is that they’re asking questions whose answers have no function in the discourse. And that is a big shift, and whether it actually makes arguing over such questions pointless has to be looked at really carefully (for example, it might be that learning the answer to such identities might be instrumentally valuable, in that it forms part of a theory that entails answers to questions that *do* have value).

How about restricting the relevant instances of Leibniz’s law to cases where the embedded identity claim is (univocally) meaningful?

Andy—yep, you could do that. You’d then be left with a situation where A and B (determinately) have different properties. But (somehow) they don’t manage to be distinct. And of course, we could define up a new term—distinctness*; which relates two terms when they’re either distinct or have different properties. I guess I wonder what the argument is for finding distinctness more interesting than distinctness*; or for thinking that metaphysicians and others were making claims about the former and not the latter.

Very cool argument.

But naturally, I’m writing because I thought of an objection. Suppose first your use of ‘=’ expresses the identity-function, defined only on numbers, Then, it’s uncontroversial that ‘0 = Caesar’ is undefined. Moreover, it does not follow that ‘~A = Caesar’ is true. After all, it is composed from an undefined formula and negation, so it too is undefined.

You might reply that ‘=’ is instead supposed to express whatever ‘is identical to’ expresses in English (give or take some regimenting). Yet it seems like this would not affect the point. If ‘0 = Caesar’ is undefined, then so too is ‘~0 = Caesar’, for the same reason, assuming the denial in English is compositional a parallel way.

However, perhaps ‘meaningless’ means something different than ‘undefined’. Is that how you’d respond?

Hi Ted—yup, that’s a natural model for understanding the picture these folks have. And of course, it’s formally very like 3-valued models for vagueness, which were in the sights of the original Evans argument against vague identity. So I’d expect some of the moves that were tried there—I’m thinking particular of Terence Parsons’ book on indeterminate identity, and its discussion of Leibniz’s law. That’s definitely a way forward.

The one thing I’d emphasize is that we can’t just endorse a 3-valued model (or something isomorphic, where the 3rd value gets replaced with an “undefined” status) and leave it there. We also have to pick a point to resist the argument whose conclusion is one we are now committed to regarding as meaningless/undefined. That might involve restricting Leibniz’s law, as Andy suggests (and Parsons’ work on the analogous stuff for indeterminate identity will certainly help). Or it might involve denying a premise. Or diagnozing some subtle invalidity in the account. And we would then have to see whether what’s being said is plausible.

Another way to put it—given that the undefined approach can’t endorse the conclusion, then in effect the Evansian argument is an argument *against* thinking *identity* is a partial function. So one can’t simply endorse the partial function approach; one needs to pinpoint what’s wrong with the argument in a way that wins over the neutral audience.

Thanks all! Sorry for the delay in replying (I think my comments-notification emails were being filtered out).

Does using sorted quantifiers help your opponent resist 1 and 2?

Maybe one thing this brings out is the tight connection between identity and predication. If we are going to think of semantic role of predicates as in effect functions from objects, then it looks like we already have to be thinking of the objects as distinct from each other, for the apparatus of functions to get a grip. That’s why it seems that you’ve already given the game away when you allow that ‘ is identical with A’ is true of A but not B. Maybe the friend of meaningless identities should hold that strictly speaking there is no single semantic relation of reference, but a special analogous such relation for each sort of object? The sorted quantifiers would then just be picking up on that distinction. I think the opponent needs to find a way to have at least one of 1 and 2 come out nonsense on each valid reading. (If it’s reference1 both times then 2 is meaningless and if it’s reference2 then 1 is meaningless. If different semantic relations are cited then it’s equivocation on the premises.). Not sure how stable/expressible that is though – sounds like some failed solutions of the concept horse type stuff.

Heya Andy. Yup, I do think in the end the sorted quantifier strategy is a strong option (various people on facebook were pressing me on this). In particular, it looks like either the x that appears in the property abstracts in (1) ot (2) will either be of the number sort or the person sort…. If we have a sort match between name and predicate in each of (1) and (2), then they will need to be of different sorts, and so we diagnose equivocation. That seems right to me. I do personally think that having to insist on sorted quantification (or better: the nonsensicality of cross-sort quantification) is a major cost (and indeed: just really implausible). For starters, there’s all sorts of limitations in quantifying over the things others have talked about…. Similar to worries about type restrictions on truth familiar from the liar, but even less recherche. But perhaps the folks who are tempted by meaningless identities already commit to this stuff so it’s not a new cost.

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