2.3 Dummett on Harmony
2.3.1 Philosophical Backdrop
Belnap does little towards motivating conservativeness as the constraint on le- gitimate meaning-conferring rule-sets. True, the conservativeness test does rule out tonk, and a range of tonkish connectives, but even if we grant that conser- vativeness is necessary—which it might not be—why think that it is sufficient? In other words, are there illegitimate connectives that are not weeded out by the conservativeness test.
Before we assess the formal prowess of the conservativeness test, let us intro- duce a more detailed account of the relationship between conservativeness and meaning provided by Dummett. His The Logical Basis of Metaphysics (1991) is indisputably the most systematic attempt at deploying proof-theoretic resources to give a theory of meaning for logical constants. In fact, Dummett casts his net much wider. His aim is to characterise conditions for successful linguistic prac- tice in natural language tout court. It just happens that on Dummett’s view, the logical constants lend themselves to a rigid analysis because of their system- atic nature.15 In particular, through their participation in deductive argument, logical constants are “connected at start and finish with the ordinary assertoric use of language” (ibid., p. 193). With the hope that the semantically significant part of our linguistic practice for logical constants is exhausted by the inferential practice, Dummett proposes to study the constraints on successful practice in a proof-theoretic setting.
Why think that properties of inference rules are a good guide to the semantics of logical constants? Dummett offers the following meaning-theoretic backdrop: In general, linguistic practice is conditioned by two broad classes of principles:
• Verificational principles;
• Pragmatic principles.
The first class of principles control when we are entitled to make an assertion, and under what conditions we are required to acknowledge it. The second class of principles govern what sort of commitments an assertion engenders through its consequences.16 Earlier, in Dummett (1973a), he says:
15That is not to say that there is no inferential noise in natural language connectives like ‘and’, ‘or’, ‘not’, etc. Any attempt to formalise inferential behaviour must accept some major concessions in comprehensiveness. For a discussion of the pragmatics of logical connectives, see Edgington (2006).
16Brandom (1994,2000) has developed the idea of entitlement and commitment as core seman- tic notions, set in the framework of a Sellarsiangame of giving and asking for reasons. Brandom suggests that we ought to understand the notion of assertion derivatively from such a game. See also Brandom (1983).
Learning to use a statement of a given form involves, then, learning two things: the conditions under which one is justified in making the statement; and what constitutes acceptance of it, i.e. the consequences of accepting it. (ibid., p. 453)
Dummett rightly observes that, in general, identifying such principles in our lin- guistic practice is a tall order. However, he adds that, fortunately, logical constants are governed by principles that may plausibly be spelled out in a natural deduc- tion framework. More precisely, verificational principles for a logical constant λ are given by the set of λ-intro rules; the pragmatic principles by the set of λ-elim rules (see in particular Dummett 1991, p. 216). If this is correct, the natural ex- tension of the thought is to mimic the constraints on verificational and pragmatic principles with proof-theoretic constraints on natural deduction rules. For this purpose Dummett adopts conservativeness.
It is in the context of verificational and pragmatic principles that the term ‘har- mony’ is first coined by Dummett.17 He stresses the possibility of a malfunctioning
linguistic practice, that is, a practice where these principles are somehow at odds with each other. Inconsistency, Dummett suggests, is the ‘grossest’ type of such malfunction, but there are a number of ways in which the practice might seman- tically misfire. In other words, a linguistic practice might subtly fail to deliver a semantic content to an expression. It is in the presence of such malfunction that it is legitimate to criticise and, potentially, revise an established linguistic practice (and, thus, an inferential practice).18 This can happen for instance in cases where a discourse turns out to beparadoxical (e.g., for a naïve truth predicate), but also 17As far as we know, the term makes its first appearance in Dummett (1973a, p. 454): “Such change [in the linguistic practice] is motivated by the desire to attain or preserve a harmony between the two aspects of an expression’s meaning.”
less visibly if there is failure of harmony at the level of the principles governing the practice.19
The possibility of failure arises primarily because of the multiplicity of principles governing our linguistic practice. For the language to function as intended, these principles must be in harmony with one another; but the mere fact that certain principles are observed in no way guarantees that the necessary harmony will obtain. (ibid., p. 210)
Thus, harmony is a relation between verificational and pragmatic principles. For logical constants, in particular, it is a relation between a set of intro-rules and a set of elim-rules. Interestingly, even if harmony is a hopelessly many-faceted relation for semantic principles in general, Dummett thinks it can be given an elucidating formal characterisation when we restrict our attention to deductive arguments and the accompanying inferential practice.20
[A]lthough we have no right to assume it a priori, we may at least hope that, in their case [logical constants], the matter can be treated entirely in terms of logical laws. (ibid., p. 215)
An informal idea of harmony is now taking shape. Dummett is suggesting that intro-rules and elim-rules must somehow be balanced off against each other. It is undesirable—and, according to Dummett, semantically undermining—to have rules for a logical constantλsuch that the consequences that can be drawn with the elim-rules outstrip the grounds for introducing the formulae with the intro-rules; or,vice versa, elim-rules that will not allow you to wholly recapture those grounds. 19Dummett contrasts his view—which is comparable to Belnap’s view about tonk—with the late Wittgensteinean approach where a linguistic practice is immune to revision, even in the face of inconsistency (see ibid. p. 209).
20One of the few non-logical cases Dummett discusses is the slur ‘Boche’, a pejorative term for Germans. See Dummett (1973a, p. 454). This case also receives some interesting treatment in Williamson (2003).
In the first case, the intro-rules are too weak (or alternatively, the elim-rules too strong); in the second, the intro-rules too strong (alternatively, the elim-rule too weak). Either way, the equilibrium is upset—the verificational and pragmatic principles are in tension.
Obviously, tonk is a good candidate for a connective that is disharmonious in this informal sense. Indeed, tonk appears to allow consequences “not warranted by our methods of arriving at the premises” (ibid., p. 217). More, given the assumptions discussed above, tonk is the limit case where the disharmony is so overwhelming as to trivialise the practice. Yet, it should also be clear from the discussion that it is perfectly possible to have a disharmonious logical constant that does not trivialise. First, the elim-rule might be just slightly stronger than what is licensed by premises in the intro-rules.21 Second, the elim-rules might be too weak, and thus obviously not trivialising. Although Dummett says that this latter disharmony “will not produce so deleterious an effect” (ibid.), the two ways in which the equilibrium can be upset will be taken equally seriously in what follows. It is not evident why one would think that rules which ‘undergenerate’ conclusions are any less harmful than rules that ‘overgenerate’.