Hartry Field's 'Metalogic and Modality'

Reading Hartry Field's paper 'Metalogic and Modality' (Philosophical Studies 62: 1-22, (1991)), I came over some comments reminding me of an argument against a proof-theoretic account of logical consequence that has always bothered me. In fact, I first read it in Etchemendy's The Concept of Logical Consequence, but I recall seeing it elsewhere as well. And now - in Field's article. But first, here is the Etchemendy quote:

It has long been acknowledged that the purely syntactic approach does not yield a general analysis of the ordinary notion of consequence, and in principle cannot. The reason for this is simple. It is obvious, for starters, that the intuitive notion of consequence cannot be captured by any single deductive system. For one thing, such a system will be tied to a specific set of rules and a specific language, while the ordinary notion is not so restricted. Thus, by “consequence” we clearly do not mean derivability in this or that deductive system. But neither do we mean derivability in some deductive system or other, for any sentence is derivable from any other in some such system. So at best we might mean by “consequence” derivability in some sound deductive system. But the notion of soundness brings us straight back to the intuitive notion of consequence. (Etchemendy [1990, p. 2-3])

Similarly, Hartry Field provides the following line of thoughts:

These model-theoretic definitions [Tarskian] are much more natural than proof-theoretic definitions of implication and consistency - for instance, a proof-theoretic definition proceeds in terms of some definite proof procedure, but it seems prette arbitrary which proof procedure one picks, and it isn't very satisfying to rest ones definitions of fundamental metalogical concepts on such highly arbitrary choices. (p. 2)

In my master thesis, I argued that although this type of criticism - that a proof-theoretic account turns on a specific proof-system, and thus a specific language - applies to some out-of-date approaches, in particular, Hilbert's attempt at giving an axiomatic definition of logical consequence (whether or not Hilbert actually thought of himself is not all that relevant; Tarski certainly read something like this into the axiomatic project), the criticism is not applicable to semantic accounts framed in proof-theory, i.e., proof-theoretic semantics. The problem here is that many authors appear to be associating proof-theory merely with syntax, refusing it any semantic capacities. However, the tradition allowing 'proof' as a semantic notion (e.g., Prawitz, Dummett, Schröder-Heister, Wansing, Sundholm, Restall) typically appreciates proof-theory as something more than simply a collection of particular systems/languages.

Central notions in proof-theoretic semantics, such as proof-condition, harmony, normalizability, cut-elimination, recursiveness, etc., are all applicable to a wide range of systems, just as the rival notions, model, truth-in-a-model, truth-condition, satisfaction, etc. To take an example, Prawitz's definition of logical consequence in terms of the former notions fare no worse than Tarski's model-theoretic definition in terms of langauge-dependence. Another issue althogether is that the model-theoretic definition is more frequently applied, and thus has a wider range of actual appliactions. This, however, does not reflect on the potential.

If you're interested in the details of Prawitz's definition, here are some of his papers on the topic:

• "Ideas and results in proof theory", in: Proceedings of the 2. Scandinavian Logic Symposium, pp 237-309, J. Fenstad (ed), North-Holland, 1971.
• "On the idea of a general proof theory", Synthese 27, 1974, pp 63-77.
• "Remarks on some approaches to the concept of logical consequence", Synthese 62, 1985, pp 153-71.
• "Meaning approached via proofs", Synthese 148:507–524, 2006.