I’m really not totally sure what to make of Etchemendy’s objections to Tarski’s account of consequence in *The Concept of Logical Consequence* – perhaps I shouldn’t admit that while I’m TAing a class that covers this book, and grading papers about it. In general, the particular points he makes seem largely right, but I’m not really sure how they add up to the supposed conclusion. I suppose this all means that at some point I should put in some more serious study of the book. But the middle of exam week may not be the time for that.

His objections basically seem to be that Tarski’s account is either extensionally inadequate (if the universe is finite) or adequate for purely coincidental reasons; and that it doesn’t seem to be able to guarantee the right modal and epistemic features.

The worry about finiteness runs as follows – Tarski says that a sentence is a logical truth iff there is no model in which it is false. If there are only finitely many objects (including sets and models and the like), then every model has a domain which is at most a subset of this finite set, so there is some finite size *n* that is not achieved. Thus, any sentence that says there are at most *n* objects must come out logically true. However, intuitively, this is just an empirical matter, and not one that is up to logic alone, so the account must be wrong. Even worse, sentences of this sort can be expressed in a language that doesn’t even involve quantifiers or identity, so we can’t blame this on some sort of error in identifying logical constants. (One might try to sneak out of this objection by pointing out that the sentences involved always have more than *n* symbols – so every sentence that would exist in such a case would get the “correct” logical truth-value. However, there are sentences that are true in every finite model, but not in all models, and these raise a similar problem.)

However, I think this isn’t really a terrible worry – Tarski’s account of consequence (like his account of truth) makes essential use of quantification over sets. Thus, anyone who’s even prepared to consider it as an account of consequence must be making use of some sort of set theory. But just about every set theory that has been proposed guarantees the existence of infinitely many objects (of some sort or another), so we don’t need to worry about finiteness. Etchemendy suggests that this is putting the cart before the horse, in making logical consequence depend on something distinctly non-logical. But perhaps this isn’t really bad – after all, Tarski didn’t claim that his definition was *logically* the same as the notion of consequence, but rather that it was *conceptually* the same. Just because the truth of set theory isn’t *logical* doesn’t mean that set theory isn’t a *conceptual* truth – and if some set of axioms guaranteeing the existence of infinitely many objects is conceptually necessary (as neo-logicists seem to claim), then Tarski’s account could be extensionally adequate as a matter of conceptual necessity, even if not of logical necessity.

As for the requisite epistemic and modal features, there might be a bit more worry here. After all, nothing about models seems to directly indicate anything modal or epistemic. However, it does seem eminently plausible that every way things (logically) could have been would be represented by some model. In fact, we can basically prove this result for finite possible worlds using an extremely weak set theory (only an empty set, pairing, and union are needed). It seems likely that the same would obtain among the actual sets for infinite possible worlds as well. However, ZFC doesn’t see to provide any natural way of extending this to all possible worlds – in fact, ZFC can prove that there is *no* model that correctly represents the actual world, because there are too many things to form a set! Fortunately, this problem doesn’t seem to arise for other set theories, like Quine’s NF, and certainly not for a Russellian type theory. And even in ZFC, the fact that Gödel could prove his completeness theorem provides some guide – any syntactically consistent set of sentences has a model, so that even if there is no model representing a particular logically possible world, there is at least a model satisfying exactly the same sentences, so that logical consequence judgements all come out right. But that’s a bit unsatisfying, seeing as how it makes the semantic notion of consequence depend on the syntactic one.

At any rate, it seems available to classical logicians to suggest that it is a matter of conceptual necessity that every way things could logically have been is adequately represented by the sets – and thus that Tarski’s account is correct and Etchemendy’s criticisms inconclusive. I’m pretty sure something like this has to be right.

Of course, the non-realist about mathematics has to give a different account of consequence (as Hartry Field tries to do starting with “On Conservativeness and Incompleteness”) – but this will just be part of paraphrasing away all the many uses we have for set theory. This one is remarkably central (especially given that linguists now suggest that something like it is at the root of all natural language semantics) and so it will be the important test case in a reduction. But the criticism will be substantially different from Etchemendy’s – before the paraphrase, the non-realist can still make the same arguments I’m suggesting above.

(Of course, if Etchemendy’s criticisms are right, they could themselves form the starting point for a useful dispensability argument for mathematical non-realism – if we don’t need sets for consequence, then the strongest indispensability consideration is gone, and we’re just left with the physical sciences, all of which seem to require something much weaker than full set theory.)

Ole Hjortland(10:37:08) :Interesting comments. However, I wonder if the set-theoretical considerations really address the epistemological worry that Etchemendy raises (at least in the way it’s formulated in ‘Reflections on conseqeunce’ (unpublished paper availbale on Etchemendy’s homepage)).

Of course, the advocate of a Tarskian consequence can respond that the epistemic features of consequence is not captured by model-theoretic consequence, but rather by the proof-systems. This doesn’t seem to get us anywhere, however, since the soundness of the systems are only relative to the corectness of the Tarskian account.

Kenny(10:58:23) :Well, the Tarskian could say (assuming that the model-theoretic account gets the modal stuff right) that the model-theoretic account is correct, and the soundness and completeness theorems show that particular proof-theoretic accounts are correct too – and they guarantee the special epistemic access. It doesn’t seem clear to me that the modal and epistemic features of consequence should have to come from the same way of thinking about it – we just need to establish that they’re both there.

Aidan(11:24:47) :Just in case you haven’t spotted it, there’s a paper by your very own Charles Chihara in M.Schirn’s ‘The Philosophy of Mathematics Today’ (OUP ’98). It gives a series of rebuttals to Etchemendy which I seem to remember finding pretty persuasive at the time I read it. And I thought it would be of particular interest because the second part of the paper contains a (crazy to my mind, but hey…) discussion of how a nominalist might approach an account of model-theoretic consequence (that might not be quite what he’s up to in that section – it’s been a while since I read this stuff – but close enough for present purposes).

Ole Hjortland(12:19:03) :Aidan:

I actually thought that the Shapiro paper in the same volume had a more interesting perspective. Btw, Shapiro has also got a paper on Fields nominalism and logical consequence (Noūs 37:1 (2003) 113-132), but I haven’t had time to read it yet.

Kenny:

(1) Although Etchemendy doesn’t seem to agree, I think the chief upshot of his argument against model-theoretic consequence is that the epistemic notions (proof, provability) are more fundamental to logical consequence than mere preservation of truth. Thus, these notion should be guidelines to a correct conception of logical consequence; not the other way around (i.e., soundness ). This would lead us to something like proof-theoretical semantics for logical consequence. However, as mentioned, Etchemendy seems to reject both a model- and a proof-theoretical approach.

(2) Have you read Gila Sher’s ‘Did Tarski committ ‘Tarski’s fallacy’?’? She seems to argue that model-theory does not adequately capture the modal features of logical consequence.

Aidan(15:55:17) :I’d need to reread Stewart’s paper in that volume. I didn’t quite get what the final position was meant to be. In general I’m somewhat sympathetic to the logic as modelling idea that he and Roy have been pushing (though it certainly seems to face problems that I haven’t really seen serious discussion of yet), but if my memory serves me correctly, there was more in that paper; doesn’t he end up with some final picture of logical consequence that’s model theoretic, but which appeals to explicitly modal notions too (trying to occupy middle ground between the Tarskian account and Etchemendy’s representationalist account)? I didn’t get how the pieces of the picture is meant to fit together, though I’d like to.

I wasn’t really all that thrilled by Chihara’s positive picture – he’s forced to say a lot of the things he does just because he won’t appeal to sets, and I’m not the type to have such scruples. I do remember thinking, though, that his critique of Etchemendy was spot on in a lot of places. So for example, I liked his response to the claim made by Etchemendy and Stephen Read that the model theoretic account undergenerates because ‘Steve is a bachelor: so Steve is unmarried’ and such arguments are not counted valid without further premises (whereas representational semantics given an easy verdict of validity as the argument stands).

Greg(07:44:42) :Hi Kenny —

A couple of little things caught my attention.

1. (Very small point) You say: “any sentence that says there are at most n objects must come out logically true. However, intuitively, this is just an empirical matter, and not one that is up to logic alone.” But you allow sets as ‘objects’ here, and I don’t think most people think of the number of sets in the world as an empirical matter. However, you immediately say afterwards that it is “not up to logic alone” how many objects there are — and that seems defensible… for it’s a matter of mathematics.

2. (Slightly less small point) You write: “Tarski’s account of consequence (like his account of truth) makes essential use of quantification over sets. Thus, anyone who’s even prepared to consider it as an account of consequence must be making use of some sort of set theory. But just about every set theory that has been proposed guarantees the existence of infinitely many objects (of some sort or another), so we don’t need to worry about finiteness.”

In the material I worked on for my dissertation, Tarski explores (during 1941) a hardcore notion of nominalism, that would yield a notion of consequence that would do without all of the higher reaches of set theory. In this system, we keep the old definition of consequence, but we change the definition of ‘model’: the domain of a model can only contain (actually existing) concrete objects — Tarski explicitly disallows sets, numbers, properties, etc. from the domain. This of course will not be exactly co-extensive with the old version of consequence, but oh well… Tarski apparently thinks his philosophical scruples might counterbalance/ outweigh that mismatch.

Kenny(17:48:25) :Greg –

re 1: definitely right – I just meant “a non-logical matter” rather than “an empirical matter”.

re 2: that’s an interesting position Tarski found himself in! However, I think it only makes sense in the sort of Russellian type theory they were using at that point. Once we’re talking about models (and thus implicitly quantifying over them and treating them as objects) there’s got to be infinitely many objects. I imagine Tarski is instead considering models to be in a higher type, and being a nominalist about what’s in the bottom type. If he then requires the domain of a model to be a class of type 1, then we really do get Etchemendy’s problem, about as clearly as possible!