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.)