4.2.1
Harmony-as-Reduction
In the diagnosis of where harmony-as-conservativeness and -normalisation go wrong, we have already hinted at a more appropriate take on harmony. The common problem with the two approaches explored so far—total and intrinsic harmony—is that they only constrain inference rules to the extent that they contribute toglobal
properties of the proof-system, i.e., conservativeness and normalisation. But, in- ference rules which appear blatantly to disregard our pre-formal conception of harmony might facilitate normalisation and other global properties as long as they are introduced into an appropriate proof-theoretic framework. This is one of the lessons learned by the above exploration of PTS for classical logic.
In order to keep the inference rules behind a normalisable system in check, we must impose a local constraint which tracks the relation between grounds for and consequences of its logical constant. Read suggests that “[w]hat is needed is a rethink of the notion of intrinsic harmony, to discern the true relationship between the introduction- and elimination-rules” (Read 2000, p. 129). Rather than concerning ourselves with normalisation directly, we should revisit Prawitz’s inversion principle and conversion steps. Motivated by the fact that these were constraints that operated locally on the inference rules, the hope is that a formal notion of harmony can be teased out. To separate it from its predecessors we call the new approach harmony-as-reduction.
4.2.2
Harmony and Functionality
To set the scene for harmony-as-reduction, let us revist Gentzen’s brief remarks on PTS. Following the famous passage on introductions represening the ‘definitions’ of logical connectives (Gentzen 1934, p. 80), he continues by saying:
By making these ideas more precise it should be possible to display the
E-inferences asunique functions of their correspondingI-inferences, on the basis of certain requirements. (ibid., emphasis added)
There is an important idea here. Proof-theoretic harmony, in whatever formal guise we prefer, ought to be a function yielding elim-rules as output for intro- rules. In other words, harmony is not simply a matter of a test telling us whether a pair of rule sets <I, E> obeys some formal constraint, there ought to be a set of elim-rules as functional value set of intro-rules as functional argument. Even more, it ought to give us a unique set of elim-rules for each well-formed input. This is non-trivial. There are examples of normalisable natural deduction systems where different intro-rules yield the same elim-rule. More formally, we have anon- injective function f: There might be two inputsx, y such that x6= y but f(x) = f(y).
Recall, for example, the rules IS4 and IS5 above (Section 4.1), and the fact
that they are both paired with the same elim-rule. Similarly, the modal systemK
can be given by requiring that ifA depends onΓ, then the conclusionA ofIK
depends on Γ, i.e., {B | B ∈ Γ}:2
Γ⇒A
Γ⇒A (IK)
Worse, if the system has ♦ as primitive the ensuing inference rules are not even functional. The two standard S4 rules are due to Fitch (1952):
2The rule is in sequent style in order to explicate the difference. See Read (2008b) for further details.
A ♦A (I♦) ♦A [A]u .. .. C C (E♦)(u)
where forE♦, every assumption on which the minor premiseC depends on (except A itself) is modal, and C is co-modal. Again, as with the -rules, by loosening the restriction, this time on the elim-rule, we get an S5 system. This time, how- ever, the result is even more puzzling: For someone (like Gentzen or Read) who takes intro-rules to be conferring meaning on connectives, and the elim-rules to be mere ‘corollaries’, it is presumably unacceptable that the same set of intro-rules could harmoniously be paired with two distinct sets of elim-rules. Of course, if your preference is for elim-rules as semantically prior, the problems have been dualised—the rules fail on account of functionality, the ♦-rules on account of injectivity. Regardless, the situation is uncomfortable.3
Dummett (1991, pp. 285-87) discusses another example. If we compare the stan- dard disjunction elimination rule with its counterpart in the quantum logicNqp,
EY, we see that they are only separated by the presence of contexts (Θ, ∆) in the subderivations:4
Example 4.1. Quantum logic, Nqp:
Γ.. .. AYB [A]u .. .. C [B]v .. .. C C (EY) Γ.. .. A∨B Θ,[A]u .. .. C ∆,[B]v .. .. C C (E∨)
The difference, although merely structural, underpins the nonderivability of the law of distributivity (for ∧ over Y) in Nqp. Again, it is a situation where the elim-rules are different, but the intro-rules for ∨ and Y are the same. This is the case despite the fact that EY equally well supports the conversion step with
3Read suggests a solution for modal operators that we revisit in Section5.3.6.
4Proof-theory for quantum logic is dealt with in, e.g., Nishimura (1980), Cutland & Gibbins (1982), and Faggian & Sambin (1998).
I∨. Dummett suggests that the pairing of EYwith I∨ isunstable since a system with the connectives {∨,Y ∧} will collapse as we can derive A ∨B from A Y B. Consequently, {∨,Y ∧} is a non-conservative extension of {∨, Y} since the law of distributivity, for Y, becomes provable.
Finally, an interesting example that is not usually mentioned is⊥in Prawitz’sNip
and Ncp. Recall that classical and intuitionistic logic here have different elim- rules (EFQ andE⊥C respectively), yet the same ‘introduction rule’. Granted, this
is a truth with modification since ⊥ apparently does not have an intro-rule, but Prawitz prefers saying that the nullary connective has a limit intro-rule, namely the null-rule. Prawitz’s response is a bit hard to grasp. Unlike for example Tennant, he insists that ⊥ does have determinate propositional content, so, as with other connectives, he needs a semantic story. Since Prawitz has all the meaning-theoretic eggs in the intro-rule basket, he prefers the somewhat artificial reply that there is a default null-rule which is harmony with the respective elim-rules for ⊥. However, unless there are two structurally distinct null-rules, it appears that both elim-rules (the classical and the intuitionistic) are harmonious with respect to the same rule.5
In general, then, expecting the elim-rule output to be a unique function appears to put pressure on a number of rule-sets. We might identify two different strategies handling the issue of functionality in light of this: First, the hard-liners might propose that we simply decree that non-functional operators are non-logical. Thus, modal operators, and perhaps a series of more or less exotic connectives will not meet the criterion of logical constanthood. I have little sympathy for this approach. The overarching principle for formulating PTS should—as earlier stated—be to include a wide variety of logics. The fact that modal operators are tricky to treat proof-theoretically (while usually easy to treat model-theoretically) is hardly an excuse.
Second, one might simply feel obliged to drop functionality as a property of har- mony, irrespective of what the other details of the relation turn out to be. This comes with a cost, however. Harmony, in the informal sense described by Dum- mett, appears to presuppose functionality. Assuming that two distinct elim-rules yield different classes of consequences, it appears inevitable that at least one of them will either be too weak or too strong. For, if harmony is a matter of the elim-rule providing exactly the consequences which are already warranted by the grounds for the intro-rule, one of the rules must be under- or overshooting. Simi- larly, if one prefers a semantics where the elim-rules are privileged, the same moral ought to apply to intro-rules.
Either way, we are left with the awkward situation that the harmony constraint does not determine which inference is the correct one. A initial thought might be to suppose that whenever harmony provides more than one rule set as output, we always go with the deductively stronger set. But this is a strategy that should give us considerable pause: Increased deductive strength often (but not always) comes with a corresponding weakening of discriminatory strength, i.e., the ability to to make logical distinctions.6 Think, for example, of the difference between
modal logics, where stronger systems tend to collapse strings of modalities that are deductively distinct in weaker systems. Model-theoretically, we observe that the weaker systems have more models, and thus more potential counter-models. Who is to say that the practice of refuting, as opposed to that of proving, is not to be preferred when we adopt one inferential practice over another?
Evidently, none of the harmony constraints investigated so far involves a func- tionality constraint in this sense considered here. Yet, there is the feeling that since it is directly inspired by Prawitz’s inversion principle, harmony-as-reduction is anticipating precisely this idea. Even if Prawitz simply displays the pairs and observes that they share a feature, namely loyalty to the inversion principle, the 6For details for and complications with the notion of discriminatory power, see the excellent paper Humberstone (2005).
suggestion is to investigate the principle further to uncover the method by which we can produce the correct elim-rules on the basis of intro-rules.