TTP, 12. 'The formal mode of assertion'

Weir himself distinguishes three model cases where a claim's content is not transparently representational — to use my jargon for his idea — and I added a fourth case. (We are assuming, for the sake of argument, that the general idea of having NTR content is in good order.) The question left hanging at the end of the last post was this: Which of the models on the table, if any, is the appropriate one when it comes to elucidating Weir's idea that arithmetical claims have NTR content?

Well, we have some idea from the opening chapters what Weir wants — see the preceding blog posts in this series! For a start, (1) he wants to treat arithmetical claims 'at face value' in the sense that he doesn't construe them as being misleading in their suface form and as requiring unmasking as really representing some subject-matter not obviously revealed at the surface level. But it isn't that he thinks that arithmetical claims do actually represent numbers and their properties; rather (2) he wants them to be treated as belonging to a fundamentally non-representational mode of discourse.

So the sort of NTR content which is illustrated by cases with demonstratives — cases which are still fundamentally representational — can't provide us with a model of what's going on in arithmetic. Nor will my example of talk of colours be helpful: for Weir, it isn't that arithmetical talk deploys "confused ideas" and represents but in a way that calls for an unmasking story to lift the fog. Rather, "The mode of assertion of [arithmetical claims] … is formal, not representational".

The models to look at, then, for illuminating the story about the NTR content of arithmetic must indeed be the non-representational ones that Weir himself emphasizes, i.e. the cases of projectivist discourse and of fictional discourse.

What will make it plausible (I'm not saying right!) to tell a projectivist, non-representational, story about (say) moral discourse? It must look reasonably natural to tell a story about the mental states of speakers according to which moral assertions are keyed not to kosher beliefs representing the world but to evaluative attitudes. What will make it plausible to tell a projectivist story about probabilities? Again, we need to tell a natural story about how assignments of probability are keyed not to having a belief with a certain content but to the strength of another belief. Such projectivism about claims that p gets off the ground, then, when such claims can be seen as suitably keyed to mental states other than believing-that-p — and for this to be plausible, we'd need already have reason to discern such states. For example, we already have reason to think of agents as having attitudes pro and con various actions, and as having a desire that such attitudes be shared: it's not so surprising, therefore, if we should have acquired ways of talking whose purposes is to express such attitudes and facilitate their coordination. Similarly, we already have reason to think of beliefs as coming in degrees: no surprise, either, that we should have ready ways of expressing degrees of belief.

It's similar to the case of talking within a fiction, at least in the key respect that the claim that p (e.g. that Sherlock lived within five miles of Westminister) is not keyed to a common-or-garden belief but to something else, a pretence to be representing.

But now compare the "just so" stories in my post TTP 10. Take the extended abacus game, for example (the same applies to the other games, mutatis mutandis). We imagined children playing with an abacus, and then learning to "comment", first by learning to write '74 + 46 = 120′ when they have just achieved a certain configuration in a correct play of the abacus game. Now ask: at this stage, what mental states are those tokenings keyed to? Surely beliefs, common-or-garden beliefs about what has just happened in a correct play of the abacus game. There's no call for any story yet about a special non-belief state of mind behind such tokens as '74 + 46 = 120′. To be sure, the children may well lack the resources to frame a transparent representation of the correctness condition for their tokens and may not yet fully conceptualize the business of getting to an arrangement of beads by correct play. Fine. But that in itself would only make their claim a "confused" representation, in Leibniz's sense, not make it non-representational. It is surely beliefs, representational states, that are being expressed. (What alternative, non-belief, state that we already have reason to discern would their tokenings be keyed to?)

What about the end stage of the language game where the rules are relaxed to allow the children to write '74 + 46 = 120′ even when they haven't in fact just executed a (correct) play of the abacus game, so long as they could in (correct) practice achieve the configuration?

Well, there's a modality here, and if you are a projectivist, or other non-representationalist, about modality you could now take a non-representationalist line about the arithmetical tokenings in the developed language game. But that's seemingly not Weir's line. For a start he only fleetingly mentions the possibility of being a projectivist about modalities, which would be very puzzling if he intended to lean on such a view. But more tellingly, his own neo-formalist account of the correctness conditions for arithmetical claims comes to this: such a claim is true just if a proof of it (a concrete proof-token) actually exists or is practically possible. So Weir seemingly likes facts about practical possibility, and takes them to be available as inputs to his explanatory metaphysical-cum-semantic story about arithmetic. So again we might ask: why isn't it confused representations of such facts that are being expressed by the children's tokenings of '74 + 46 = 120′ at the final stage of the developed abacus game?

Here's the worry. Non-representational claims that p in other cases are non-representational because keyed to non-beliefs. In the abacus game, the children's tokenings by contrast do seem to be keyed to beliefs, albeit ones that may only foggily represent the structure of the facts that make them true when they are true. So it needs argument to show that, despite such appearances, the children's tokenings in fact aren't expressions of belief but belong to a different kind of non-representational mode of utterance, Weir's so-called 'formal' one? And then what type of non-belief state are such 'formal utterances keyed to? I'm not seeing that Weir offers us the necessary account here.

In sum: maybe arithmetical claims e.g. in the abacus game are not transparently representational. But it doesn't follow that they are outright non-representational, involving a different mode of assertion keyed to some new class of non-beliefs. What's the argument that they are so?