More on Hartry Field

As I never quite got around to saying something about the interesting bit in this paper (that is, 'Metalogic and modality' (1991)), I give myself a second chance. Going Back to a famous paper by Georg Kreisel, 'Informal Rigor and Completeness Proofs' (1967), Hartry Field wants to investigate whether there are advantages with taking the notions of consistency and (logical) implication as primitives instead of treating them in the traditional Tarskian manner. (Kreisel's point was about how to understand completeness in light of such primitive notions.)

What does Field mean by primitive in this context? Here is a first helpful hint: "The claim that consistency should be regarded as a primitive notion does involve the claim that we can't clarify its meaning by giving a definition of it in more basic terms." (p. 5) In other words, anything like a reductive approach (e.g., defining consistency in terms of truth-in-structures) is excluded. In fact, Field points to what he perceives to be an analogy to other basic logical notions:

Similarly, logical notions like 'and', 'not', and 'there is' are primitive. We don't learn these notions by defining them in more basic terms. Rather, we learn them by learning to use them in accordance with certain rules; and we clarify their meaning by unearthing the rules that govern them. (p. 5)

Stuff like this is music in the ears of anyone inclined towards inferentialism. Yet, Field takes it a step further, promising to expand the inferentialistic approach to consistency and consequence as well.

The same holds for consistency and implication, I claim: there are "procedural rules" governing the use of these terms, and it is these rules that give the terms the meaning they have, not some alleged definitions, whether in terms of models or of proofs or of substitution instances." (ibid.)

This undeniably smacks of a more full-blown proof-theoretic semantics, where core meta-logical notions are associated with 'procedural rules'. Unfortunately, Field does not deliver anything resembling such a semantics. Rather, the "two intuitive principles" that he takes to govern our understanding of 'consistency' are not procedural in any interesting way. Let's take a closer look at them: The first is intuitively the sufficient condition for Gamma being consistent, the model-theoretic possibility principle:

(MTP) If there is a model in which Gamma is true, then Gamma is consistent.

Likewise, there is a necessary condition, called "modal soundness".

(MS) If Gamma is consistent, it is formally irrefutable in F (where F is some particular proof-system).

Note that 'consistency' here appears completely uninterpreted; our understanding of the notion derives from its occurrence in these intuitive principles. Field himself ensures us that under this explication, "consistency is neither a proof-theoretic notion nor a model-theoretic notion." (p. 6) True, Field is correct in saying that model- and proof-theory now is on par, but the analogy to the inferential approach to logical connectives is not as obvious as before. The principles are not inferential, so the rule-talk disappears in something more like an axiomatic approach.

Fortunately, it's no cost to the theory to restate the principles in rule-form. By doing this, Field would also emphasize the point that his 'procedural rules' are supposed to capture our use of the notion - a fact that is made clear by his insistence that the principles are connected to the revisability of logic. What Field seems to have in mind is something like harmony for the rules for 'consistency' and 'implication' respectively, mimicking the harmony of introduction- and elimination-rules for logical constants. In particular, he entertains the thought that the former rule-sets might "be found to conflict", thus opening for revision.