We have argued that classical logic has a place in PTS. However, offering this place comes with a price. The traditional boundaries of inferentialist proof-theory is shifted towards unfamiliar grounds. If one accepts that the Separation Property is an important tenet of PTS, the onus is on the classicist to provide formalism in which it can be combined with classical negation. Both Read and Rumfitt succeed in doing this, but not without introducing proof-theoretic complications. Alter- natively, the classical logician sympathetic to PTS might simply resist the claim that Separation plays a crucial role in successful meaning-fixing. The danger is, however, that such a classicist is waxing holistic: One cannot isolate the meaning of logical constants in PTS; only the system in full provides the meaning.
Multiple-conclusion is the better option. It leaves classical logic without a remain- der, and even if reproached by most intuitionist revisionist, there is no knock-down argument against including multiple-conclusion in PTS. In fact, in anticipation of Chapter 4and Chapter6, multiple-conclusion greatly enhances the chances of de- veloping PTS into a proper formal semantics, ridden of an unnecessary revisionist bias.
62Rumfitt is himself a proponent of bilateralism, but fails to appreciate the connection with multiple-conclusion. See also Rumfitt (2008b).
Generalised Elimination Harmony
4.1
Introduction
Independently of the revisionist debate between the intuitionist and the classicist, there are other aspects of harmony that remain. Recall that the original moti- vation for harmony and its formal counterparts was Prior’s tonk argument, and the consensus that PTS cannot allow for unconstrained introduction of inference rules. If we are to allow for defect rules, and, in extension of that, revision of defect rules, there has to be some restrictions on which inference rules are deemed acceptable.
Dummett, Prawitz and others argued that when the demarcation between per- missible and impermissible rules has been drawn, the classical logicians will find themselves outside the scope of PTS. Against this, we saw in Chapter 2that PTS as a revisionary strategy for the intuitionist is hampered by some flexibility in the classicist’s axiomatisations. By enhancing the expressive power of the structural language in the proof-theory (by multiple-conclusion and primitive denial, respec- tively), the proof-theoretic gap between classical and intuitionistic logic is closed. That is not to say that there are not other resources the classicists employ, but at least these two are both live options.
But, even if the intuitionist project is at an impasse, it is a separate issue whether PTS constraints have a broader semantic and revisionary significance. In the beginning of Chapter 2 we learned how proof-theoretic constraints were aimed at ruling outtonkand related connectives. The general problem of an embarrassment of riches for PTS—of which tonk is a symptom—does not stand or fall with the more specific revisionary ideology promoted by the intuitionist camp. In fact, one might argue, irrespective of which logical flag one is sailing under, tonkand its ilk are a problem.
Thus, even if the foregoing proof-theoretic constraints cannot motivate a divide between intuitionistic and classical logic, they might still hold the key to divorcing PTS from Prior’s objection. We know from Belnap that conservativeness is suf- ficient (given some background assumptions about the context of deducibility) to avert disaster. Yet, it was equally clear that harmony-as-conservativeness comes short of what we expect from our formalisation of harmony. Specifically, conser- vativeness does nothing to prevent elim-rules that are too weak. In the larger picture, then, conservativeness is a lopsided constraint—it only ensures that the consequences drawn from a λ statement do not outstrip the grounds for asserting it. More, since conservativeness is not necessary for harmony, one might suspect that a better proof-theoretic analysis of the concept is still forthcoming.
Is there any improvement to be had in turning to intrinsic harmony, i.e., ‘the lev- elling of local peaks’ ? We have already seen a host of (primarily classical) natural deduction systems which are normalisable although they do not in general support conservativeness for every order of introducing their connectives (e.g., Peirce’s Law again). As discussed above, the central issue is that the normalisation theorem is a global property of a proof system. There might be permutational or simpli- ficational moves available to institute normalisation even if there are intro- and elim-rules that do not display the sort of neat conversions that Prawitz identified. Classical logic is a case in point.
Another example is that of the modal logicS4. Prawitz (1965, ch. VI) shows that
S4 is normalisable with the following rules adopted from Curry (1950):
Γ.. .. A A (IS4) A A (ES4)
where for IS4 each B ∈ Γ is modal (i.e., B = C or B = ⊥). Read (2008b)
made the point that whereas normalisation does hold for these systems, one is hard pressed to say that the modal rules are in harmony. Actually, the details of Prawitz’s proof reveals that adopting an alternative—and restricted—definition of an S4-derivation (i.e., what is to count as a derivation in the proof-system—the rules remain the same) is critical for the success of the result. The rules themselves, on the other hand, appear to be disharmonious in so far asIS4is restricted by the
side-condition. In fact, S5 shares the same elim-rule as S4 but the corresponding intro-rule IS5 only requires that the members of Γ be modal or co-modal (i.e.,
¬⊥ or¬).1
The pertinent question is whether normalisation has any direct impact on the legitimacy of tonk. In general, the answer is unfortunately no. Even if tonk in standard environments (say, if added to classical logic) wreaks havoc in the form of irremovable maximum formulae, there is nothing preventing us from having surprising definitions of derivation that block the unholy coupling of tonk-intro and -elim. In particular, non-transitive systems where some derivations cannot be chained together might provide a normalisable habitat for tonk. In Section 6.4.4 we return to the details of this observation.
However, in the presence of an ordinary consequence relation, we might still profit from Dummett’s idea of intrinsic harmony and Prawitz’s inversion principle. The challenge is to give a local formulation of the harmony that encapsulates the spirit of the earlier proposals. In the process, we might find that there are systematic
connections with normalisation, but that normalisation entails that such a new harmony constraint will not be one of them.