5.2.1
PTS Again
Before we can return to the issue of revisionism, let us first try to sharpen some of the semantic underpinnings of the meaning-change argument. In order to give some content to Field’s remark above, let us revisit some of the fundamental ideas from foregoing chapters: What is the connection between INF, i.e., that the meaning of a logical constant is fixed by the constant’s behaviour in the inferential rules which govern its use, and revision of logic? Even more specifically, what is the connection between revision and INF as captured by PTS?
In order to shed light on the significance of proof-theoretic harmony, and, ulti- mately, the limitations of proof-theoretic harmony, we will investigate the inter- play between meaning-constitutive rules and revision of logical beliefs in a PTS
setting. As before, we will understand talk of inference rules as talk of formal rules in a proof system. The result is that we carry out the discussion of logical beliefs—and revision of them—in the context of formal systems. Some authors, noticeably Resnik (2004), have made much out of the distinction between revision of formal systems and revision of logical beliefs (about reasoning in our vernacu- lar).7 Without hesitation we admit that there is a significant gap, but, firstly, the present debate is not merely about “revising" our formalism—whatever that might mean—it is about studying ordinary reasoning through a mathematical idealisa- tion, i.e., it is formalism together with an application; and, secondly, many of the fields for which revision of logical belief is most pertinent deal to a large extent with formal logic as a model of ordinary reasoning, e.g., the semantic paradoxes, vagueness.
Of course, discussing revision in a formal setting means that outcome is to some extent hostage to the limits of the formalism—even if we stay neutral about choice of logic. Because of this the choice of formal framework is far from innocuous.8
For the most part, we stay neutral about whether natural deduction or sequent calculus (or another form of axiomatization) is preferable. Yet, as we shall see, which choice is made has affected the debate about what meaning-constitutive rules are.
5.2.2
Gentzenianism and Hilbertianism
With PTS we can now revisit Field’s remarks about the distinction between meaning-constitutive and non-meaning-constitutive principles. There are at least two important issues that need to be elaborated before the connection to revi- sionism becomes clear. The first is a question about the structure of meaning- constitution: Which inferential rules ought we consider meaning-constitutive? The 7In general, another prominent critic of formal methods applied to ordinary reasoning is Harman (1986).
8Recall, for example, the difference in non-conservativeness between natural deduction and sequent calculus (Section2.3.2).
second is about the nature of meaning-constitution: In what sense can a set of inferential rules fix the meaning of an expression, and—given the fuzziness of the label ‘meaning’—whatexactly is being fixed in the process? We start by addressing the former question. In Chapter 6 we return to the second question.
In the chapters so far we have investigated the Gentzenian tradition in some de- tails, focusing especially on the notion of proof-theoretic harmony. As we have seen, a key feature of what we call Gentzenian PTS is that specific subsets of the primitive inference rules of a system are considered meaning-constitutive. If we assume a standard natural deduction framework, for example, Gentzen’s orig- inal proposal was to think of the intro-rules as an (implicit) ‘definition’. There is, however, no immediate reason why one ought not think of the elim-rules as meaning-constitutive instead (e.g., Schroeder-Heister 1985), or even a mixed ap- proach (e.g., Milne 1994 and Rumfitt 2000). If you think, as has been proposed for instance by Peacocke (1987), that theobviousness of a basic inferential rule is a symptom of its meaning-constitutive power, you might conclude that modus po- nens is meaning-constitutive for ‘→’ whilst for ‘∨’ it is the intro-rules.9 But, a rule appearing to be obvious seems to be too much of an unstable phenomenological criterion to have real semantic import.
Alternatively, one might suggest, with Dummett, that a choice of which subset of rules is semantically primitive has to do with assertion-conditions, commitments, and entitlements (see Section 2.3.1). If one subscribes to such a thought, and, ad- ditionally, that intro-rules encode the semantically central aspect of such notions, then there is a pro tanto reason for ascribing priority to them. Irrespective, we have seen, in Section 3.2.3, that such a division of inference rules is tendentious: There is no reason for assertion to dominate the theory, rather than, say, denial. In fact, in Chapter 6, we will take this thought further and explore the idea of introducing other speech acts as well.
9Formally, this might be becausehypothetical rules, i.e., using discharge of assumptions, are considered more involved thancategorical rules.
Whatever choice one makes with respect to subsets of inference rules, however, the key aspect of Gentzenianism might be considered independent. The idea, following Dummett, is that an inferential practice might be dysfunctional—and so susceptible for revision—because of internal conflict between basic principles. At the core of Gentzenianism is the idea that implementing proof-theoretic harmony as a functional constraint is a systematic fashion in which logic can (and should) be revised. GE-harmony, as explored in Section4.3, focuses on intro-rules, but this perspective is contingent: There is no reason why one should not devise functional constraints which operate in the opposite direction.
In contrast to the Gentzenian tradition there is a less discerning approach which refuses to divide the inferential rules into meaning-constitutive and non-meaning- constitutive. Such a tradition—clearly part of the Quinean heritage to which Field’s above remark tacitly subscribes—is typically labelled holism. But to avoid confusion with other positions also termed ‘holism’ we prefer calling this Hilber- tianism (as opposed to Gentzenianism). The main claim, that the system of rules contributes tout court towards the meaning of the involved logical expressions, is reminiscent of the pre-Programmatic Hilbert’s Foundations of Geometry where the axioms of the system are both foundational principles and (implicit) defini- tions.10 Some further content is lent to the label by the fact that Gentzen wrote his proof-theoretic work as a student of Hilbert and the formalistic school.11
The obvious advantage of the Hilbertian approach is that the problem of motivat- ing a distinction between meaning- and non-meaning-constituting disappears—and with it the technical problems posed by harmony and other Gentzenian tricks. The Hilbertian is happy to admit that the entire axiomatic system (and think now of axioms as degenerate cases of inferential rules) fixes the content of the involved logical constants. A bit anachronistically, then, we can include under the label 10For Hilbert’s own outline of the idea, see his famous correspondence with Frege in Frege (1980).
11There is an analogy here to the philosophy of mathematics debate about neo-Fregeanism. In Ebert & Shapiro (Forthcoming) the term ‘neo-Hilbertian’ was coined.
‘Hilbertianism’ content-fixing by other formalisms like natural deduction and se- quent calculus. The idea, however, remains the same: No proper part of the axiomatisation is semantically responsible. Dummett remarks critically:
The meanings of all the expressions of the language are, on this view, determined by our linguistic practiceas a whole. If we change any part of the practice, we may change the meanings of indefinitely many—in the limiting case, of all—the expressions in the language. (Dummett 1991, p. 228)
For example, the Hilbertian might take a more lenient view on non-conservative extensions of the system: True, introducing a new operator, λ, might lead to new consequences in the original λ-free language, but this just reflects the plasticity of the semantic content when the system at large is updated. This could be the attitude towards, say, the non-conservativeness of classical negation with respect to the implicational fragment of intuitionstic logic. Of course, such a concession is impossible for Dummett and his followers as it blunts their chief argument against the classicist. Treating the semantics of logical constants in isolation (or at least not en masse) is considered pivotal by the Gentzenian.
Why the insistence on segregation? Because with Hilbertianism comes the threat of a re-conjuration of the anti-revisionism that the Gentzenian sought to dis- pel. Dummett made the case that the holist comes in a broadly Wittgensteinian shape.12 There is no sense in which the practice can semantically malfunction; it bestows meaning on the expressions blindly. In other words, the Hilbertian is willing to admit that an inconsistent practice is bad for independent reasons, but that it is does not impinge on the semantics.
That is not to say that the Hilbertian cannot combine semantic holism with con- sistency as a requirement on proper meaning-forming. What the Hilbertian balks 12Recall the mention of inconsistency and disharmony in Section2.3.1. See Dummett (1991, pp. 209-10).
at is the possibility of local constraints on fragments of the language (such as GE- harmony). One manner in which to phrase the situation, then, is to say that the Hilbertian, in virtue of only accepting semantic malfunction in the whole system, has less possibility of locating—and compartmentalising—the semantic effects of revision. Perhaps the Hilbertian is willing to live with that since since the upshot of the position seems to be that revision of logic cannot be motivated by semantic malfunction.