So far we have outlined the Dummett-Prawitz approach to an inferentialist crite- rion for inference rules. Their investigation is clearly geared towards intuitionstic logic and the thought that with proof-theoretic constraints informed by meaning- theoretic considerations, a problem with classical logic will become apparent. We will call that attitude revisionism. It needs to be made absolutely clear that revisionist tendencies, be it against classical logic or anything else, are an issue separate from PTS and logical inferentialism. Yet, given the prominence of revi- sionists in the PTS tradition, it is worth going through the arguments to convince ourselves that even if the revisionism must be abandoned, there is something to be said for PTS in general. Let us now turn to the details of their argument. How does classical logic fail to pass the inferentialists’ bar?
First of all, it is crucial that there is a certain fluidity involved in what precisely classical logic is. There is no deep problem here, it is simply that the expression ‘classical logic’ picks out a number of things, e.g., a class of theorems, a model- theoretic relation, a provability relation, classicalsemantics. No danger is involved except when properties you are interested in might apply to, say, one classical
proof-system but not another. The revisionist’s burden is to convince us that whatever formal reasons there are for thinking that classical logic is bad, are not just presentation-dependent properties, but something that gets at—so to say— the heart of classical logic. Alternatively, their strategy could be to defend one presentation as the correct one. Different proof-systems are good for different purposes, and different formalisms have their advocates and detractors. Typically, such debates are coloured by claims about which formalism best captures ordinary reasoning, a line of argument that I find both obscure and obscuring.1 To the best
of my knowledge, no serious data has been put forward to support one framework or another. However, we will revisit this debate briefly in Section 4.5, then with reference to the issue of multiple-conclusion. We focus on natural deduction in this chapter because of the trends in the inferentialist literature.
We saw above in Section 2.3.2 that standard natural deduction for classical logic lacks total harmony. Ncp fails on this account because classical negation (or ratherE⊥C, with classical negation as defined) is non-conservative over the system
restricted to the implicational rules, I→ and E→, i.e.,Ncp is a non-conservative extension over the positive intuitionistic fragment, Nip+. On the other hand,
intuitionistic negation (in the shape of E⊥I) yields a conservative extension for
the same system.
Similarly, Gentzen’s negation rules I¬ and E¬, together with E⊥I also produce
a conservative extension over minimal logic Nmp.2 These negation rules, how- ever, only give intuionistic logic. In order to get classical logic, Gentzen suggests adding LEM (law of excluded middle) or DNE (double negation elimination), both of which will results in non-conservativeness. Prawitz’s system takes Nip
and replaces E⊥I with E⊥C (often just called classical reductio and contrasted
with the intuitionistically validreductio ad absurdum rule), which also yields non- conservativeness.3
1Revisit, for example, Gentzen’s original motivation for developing natural deduction. 2See AppendixA.3.
All these results are connected with the fact that introducing classical negation in some guise or other typically impacts provability for the implicational fragment of the calculus. Some remarks on this are in order. First, note that when we investigate Gentzen’s negation rules, I¬ and E¬, it becomes apparent that if
p¬Aq is defined as pA → ⊥q these rules are instances of the →-rules already in the system: [A] .. .. ⊥ A→ ⊥ A→ ⊥ A ⊥
Now contrast the classical reductio rule,E⊥C. When we read negation as defined
we get: [A → ⊥] .. .. ⊥ A
The resulting inference is clearly not an instance of a valid implicational inference in the sense that ⊥ could be substituted for any formula. In other words, adding
E⊥C toNipis tantamount to helping oneself to another inference rule for ‘→’.4
Second, the above observation can be further entrenched by noticing that the →- fragment of Nip can be extended to the →-fragment of Ncp by adding another implicational rule,Peirce’s Rule, corresponding toPeirce’s Law above (see Section 2.2.3): [A→B]u .. .. A A (P R)(u)
4Whether or not this is an elim-rule is an interesting issue. We return to the question below in Section3.2.1.
This rule is suggested in Curry (1950). Of course, this is trivially a conservative extension since the language remains the same. It does not vindicate classical negation for Dummett and Prawitz, but it does give us a natural deduction system characterising the classical implicational fragment.5
What we have seen so far is that there is a range of standard natural deduction presentations of classical logic that all fail with respect to Dummett’s total har- mony. Further, since the Separation Property entails conservativeness (see Section 2.3.3), all of these systems have ‘inseparable’ connectives (in particular, ‘¬’ and ‘→’). This analysis, however, includes no mention of intrinsic harmony. For many revisionist inferentialists, the deeper analysis of the problem with classical negation involves the details of conversion steps for classical negation.6
We saw in Def. (2.6) that theI¬-E¬formulation of intuitionistic negation can be given a conversion step as follows:
Π1 A [A]u Π2 ⊥ ¬A (u) ⊥ Π1 [A] Π2 ⊥
Consequently, both Prawitz’s systemNip(without negation rules) and Gentzen’s
Nip¬ normalise. But, importantly, this result does not carry over to classical logic in any straightforward way.
Take for instance the classical systemNcp¬ with DNE in addition toI¬andE¬:
[¬A]u .. .. ⊥ ¬¬A (u) A 5Note that classical reductio in axiom form,
p((p→ ⊥)→p)→pq, is an instance of Peirce’s Law.
The derivation cannot be converted into a derivation without the maximum for- mula like its intuitionistic counterpart. Dummett, in his discussion of negation, considers the similar system NcpD with I¬D, E¬D, and DNE (where no nullary
connective ⊥ is needed): [A]u .. .. ¬A ¬A (I¬)(u) A ¬A B (E¬) ¬¬A A (DN E)
Dummett is quick to point out that there can be no leveling of local peaks for the same reasons as in Ncp¬: [¬A]u .. .. ¬¬A ¬¬A A (u)
Correspondingly, NcpLEM yields the same problem, but parred withE∨:
Π A∨ ¬A [A]u Π1 C [¬A]v Π2 C C (u,v)
Prawitz observes that for the systemNcp(andNipfor that matter) theE⊥C-rule
does not fit the intro- or elim-rule mold of the other connectives. He entertains the thought of replacing the rule with:
[A]u .. .. B [A]v .. .. ¬B ¬A (Dil) ¬¬A A (DN E)
and thus allowing negation as primitive. Prawitz quickly concludes that this will not remedy the situation. First, he dislikes the fact that the intro-rule isimproper, that is, it has the principal connective occurring in one of the subderivations.7
Second, it does not satisfy Prawitz’s inversion principle since there are derivations where an occurrence of a maximum formula cannot be reduced. His example is
LEM:8 [A]1 A∨ ¬A [¬(A∨ ¬A)]2 ¬A (1) A∨ ¬A [¬(A∨ ¬A)]2 ¬¬(A∨ ¬A) (2) A∨ ¬A
Here the formula¬¬(A∨¬A)occurs as both the major premise of an elim-rule and as the conclusion of an intro-rule, and there is no way in which we can preserve the provability of LEM while removing the maximum formula.
Does this mean that normalisation fails for classical logic? No, Prawitz (1965) proved the result for the ∨,∃-free fragment of Nc (or, more generally, del-rule free classical logic). Since these logical constants are definable in the remaining language (¬, ∧, →, ∀), the result can still be said to hold of classical logic. The trick is to deal with classical negation as a special case. Prawitz first showed that anything provable the ∨,∃-free fragment of Nc is provable using only atomic for- mulae as consequences inE⊥C. The proof works by transforming arbitrary proofs
using E⊥C into proofs where the application of the same rule involves a formula
of lower complexity.9 Actually, given the absence of ∨ and ∃, the normalisation
result for classical logic is more straightforward than the intuitionistic result: Nor- mal form can be defined over maximum formula without bothering with segments and permutation conversions.
Theorem 3.1 (Ncp Normalisation). An Ncp derivation (in the ∨,∃-free lan- guage) Π from Γ to A reduces in some number of conversion steps to a derivation
Π0 in normal form from Γ to A.10
8See Prawitz (1965, p. 35) 9See ibid., p. 39-40.
10Actually, there is an improved normalisation result due to Stålmarck (1991). By adding extra reduction rules covering the cases of∨and∃, normalisation is proved for the full language.
Unsurprisingly, however, the corollaries are not the same as for Thm. (2.1).11
E⊥C must be taken as an exception, as its negated discharge formula might not
be a subformula of any premise, nor of the conclusion. This happens for example in a derivation ofLEM, where the formula¬¬(A∨ ¬A) is not a subformula of the conclusion.
Corollary 3.2 (Subformula Property Ncp). Every formula occurring in a normal form Ncpderivation (in the ∨,∃-free language) of A from Γis a subformula of A or some B ∈ Γ, except for assumptions discharged by applications of E⊥C and
occurrences of ⊥ immediately below such assumptions.12
Furthermore, as we already had reason to anticipate, adding E⊥C also affects
the Separation Property. In particular, since the extension of the intuitionistic implicational fragment is non-conservative, there are normal derivations ofAfrom
Γ where we need to apply rules for connectives not occurring in either A or Γ. Peirce’s Law is an example:13
[A→ ⊥]2 [(A→B)→A]3 [A→ ⊥]2 [A]1 ⊥ (E→) B (E⊥C) A→B (I→)(1) A (E→) ⊥ (E→) A (E⊥C)(2) ((A→B)→A)→A (I→)(3)
This is a normal derivation of Peirce’s Law. There are applications of intro-rules immediately followed by applications of the corresponding elim-rule, but not where the conclusion of the former features as the major premise of the latter. The con- clusion is in the→-fragment, and yet there is no way to do without the applications of E⊥C.14 Consequently, the Separation Property fails (exactly as expected given
11One corollary of Thm. (3.1) that we do get is the consistency of classical logic. See Prawitz (1965, p. 44). Of course, this extends to intuitionistic logic since it is deductively weaker. The consistency corollary is inspired by Gentzen’s consistency proof using cut elimination.
12From this it follows that the qualification will be restricted to negations of subformula. 13Recall that in Prawitz’s system negation is defined,¬A =A→ ⊥. We treat the rules here accordingly.
the non-conservativeness). The upshot is important: The normalisation theorem is a global property of a proof system that is to some extent independent of conver- sion steps for the logical connectives. It is these conversion steps that guarantee the Separation Property in the intuitionistic case, so the move from normalisa- tion to Separation does not hold in general. One might then suspect that like conservativeness, normalisation is not the ideal test for the inferentialist.15
As it becomes apparent that several of the PTS desiderata that have been asso- ciated with harmony (or with meaning-determining inference rules in general) are independent of each other, the inferentialist must pronounce on precisely which of the aforementioned properties must feature in her semantic theory. This is crucial if the inferentialist revisionist is to have a convincing case against any logic, say, classical logic, as falling short of proof-theoretic demands. We return in Chapter 4 to the analysis of proof-theoretic harmony. In what follows we will evaluate different classicist responses with respect to the proof-theoretic demands explored so far. Two main insights are reached: First, PTS is not intrinsically revisionistic; in particular, classical logic is perfectly compatible with any reason- able proof-theoretic demand. Second, negotiating the proof-theoretic framework required for different logics to fulfill the demands will inform the development of an exact harmony criterion.
15Peter Milne (1994) raises an interesting point regarding the conversion steps for intuition- istic logic. What guarantee do we have that there are no further rules that can be added to intuitionstic logic which displays the same sort of conversion, but which still does not yield clas- sical logic? In other words, is there an intermediate logic which is harmonious in the sense of
harmony-as-normalisation?
In order to prove that no extension was possible, one would, I suppose, have to go through every possible rule-extension of intuitionist logic that yields either classical logic or one of the uncountable infinity of intermediate propositional logics. The suggestion that completeness has been proved proof-theoretically may therefore be premature. (ibid., p. 56)