Just a quickie. Lots of people are sympathetic to the Fregean thought that the identity statement ‘a = b’ is not a trivial truth, even when a is the same thing as b. One line of thought seems to be: being told that ‘a = b’ is at least to be told that ‘a’ and ‘b’ co-refer; since the fact that they co-refer can be informative, so too ‘a = b’ can be informative, even when a is b and hence the Millian proposition (by which I mean just a set of possible worlds) expressed by ‘a = b’ is the same as that expressed by ‘a = a’, namely, the trivial proposition.
But this seems an odd line of thought to me. Why would anyone ever think that ‘a = b’ entails the proposition that ‘a’ and ‘b’ co-refer? Well, apparently Frege thought it in the Begriffsschrift, where he claims that
⊢ (A ≡ B) is therefore to mean: the symbol A and the symbol B have the same conceptual content, so that A can always be replaced by B and vice versa. (Begriffsschrift §8)
But, as everyone has recognised, including the later Frege, an identity statement is about a and b, and since a is b, it turns out to be about a alone; the latter is about English, or formally enriched English, and its words, and the reference relation, none of which appear in the compositional semantics for identity statements. (That is, one need not make use of such notions in the content of ‘a = b’, though one will surely make use of them in stating the content, and in particular the dependence of that content on the contents of its parts.) So the metalinguistic conception of identity is incorrect.
Without this thought, I find it difficult to run an objection to Millianism from informative identities—for if one antecedently admits that the metalinguistic claim is different from the identity claim, one already provides room for an explanation for the perceived informativeness of ‘a = b’ that doesn’t appeal to that sentence expressing anything other than the necessary proposition if true. For example (and only for example; I don’t endorse this proposal): an utterance of ‘a = b’ will appear to violate Gricean maxims (of relevance?) unless it is assumed that the speaker intends thereby to convey the metalinguistic fact. So it may be plausible that an utterance of ‘a = b’ implicates the metalinguistic fact that ‘a’ refers to the same thing as ‘b’. Or another example: we might (like Stalnaker ) go two-dimensional to explain the informativeness of identity statements, letting the diagonal content of an identity claim be something metalinguistic, which can play a role in explaining the cognitive impact of necessary truths. These are pretty different suggestions, but even on Stalnaker’s theory, something like Gricean processing goes on: his claim is that
under certain conditions, the content of an assertion is not the proposition determined by the normal semantical rules, but instead the diagonal proposition of the propositional content determined. To make this hypothesis precise, we need only spell out the conditions under which the operation is to be performed.… The general strategy is a Gricean one. (‘Semantics for Belief’, in Context and Content, p. 124)
Here’s another way to think about it. Suppose that ‘a = b’ can be non-trivial even when a is b, as the Fregean requires. There seems no reason why this substantive truth should come to seem trivial—whatever feature, sense or otherwise, that generates the substance of the identity claim is still present when we come to know that ‘a’ denotes the same thing as ‘b’. As Frege says,
In spite of [the referent of ‘a’ being the referent of ‘b’], the sense of ‘b’ may differ from the sense of ‘a’, and thereby the thought expressed by ‘a = b’ will differ from that expressed by ‘a = a’. In that case the two sentences do not have the same cognitive value… we can also say that the judgements are different. (‘On Sense and Reference’)
But this identity statement is trivialised when we recognise the metalinguistic fact. And this is not simply because the metalinguistic fact entails it; once we are in possession of the metalinguistic fact, we can still recognise that it is possibly false had the history of English gone differently. But we can recognise no such sense in which ‘a = b’ could have turned out false depending on the vagaries of the history of English, or on anything else. Coming to know the metalinguistic fact endows us with the ability to recognise the necessity of ‘a = b’ without believing the metalinguistic fact to be necessary.
I think I must just be confused about this. Once one rejects a metalinguistic theory of identity, one does seem to have the resources to reconcile the triviality of identity statements with their felt informativeness without making reference to senses. So why would someone want to add Fregean senses—especially since we observe the phenomenon of the trivialisation of identity statements, where the Fregean theory seems to predict no such trivialisation should occur?