Given that GE-harmony is explicitly informed by Prawitz’s inversion principle, we should expect there to be a connection with normalisation. After all, we have seen that well-behaved conversion steps (together with auxiliary simplifications and permutations) lead to the normalisation theorem in a range of cases. Is GE- harmony a guide to normalisation?
Let us first observe how the two schema for intro- and elim-rules interact. Letting ϕ be the maximum formula in an application of the intro-rule followed by an application of the elim-rule, the Francez & Dyckhoff template gives the following reduction:16 [Σi]j1,...,jmi ˆ Πi ∆i ϕ (δIj1,...,jm)i Π1 Σ1 ... Πn Σn [∆1]1 Π01 ψ ... [∆n]n Π0n ψ ψ (δGE)(1,...,n)
which can be reduced to
Πi Σi ˆ Πi ∆i Π0i ψ
However, the schematic reduction glosses over crucial details. Given an instantia- tion of the schema, there is no guarantee that the resulting reduction will enable 16Importantly, Francez & Dyckhoff (2009) are not trying to establish a connection with nor- malisation. Rather, they are interested in the connection with what they call Local Intrinsic Harmony(see ibid., p. 7-8). Local Intrinsic Harmony consists oflocal soundenessandlocal com- pleteness, two notions adopted from Pfennig & Davies (2001). The upshot of their investigation is that GE-harmony entails Local Intrinsic Harmony.
an inductive step in a proof of normalisation. Whether or not this is the case de- pends, inter alia, on the distribution of connectives over the schema’s paramaters. Recall, for example, the discussion of E⊥C in Section 3.2.1. We return to the
question about harmonious rules for classical negation in Section 4.5.1, but for now we will look at another instructive example.
Read (2000) gives an illuminating example that shows that normalisation is not entailed by GE-harmony. In fact, the example shows more!
Example 4.6. Bullet: [•]u .. .. ⊥ • (I•)(u) • • [⊥]u .. .. C C (E•)(u)
It is straightforward to check that the connective•, calledbullet, is GE-harmonious.17
Yet, it displays a behaviour that led Read to call it a ‘proof-theoretic Liar’. It turns out that, in some respect, bullet is worse than tonk:18 In the presence of
EFQ (and contraction) it provesΓ `A, for any <Γ, A>(includingΓ =∅). First, we see that by letting C =⊥ we get the simplified derived rule (where ⊥ ⇒ ⊥ is considered trivial by reflexivity):19
• •
⊥ (SE•)
Assuming further that contraction allows us to simplify this to a rule with one instance of•, we can give the following derivation:20
17The connective was originally called ‘blob’ but I prefer ‘bullet’ following the LaTeX code for the symbol. In retrospect, perhaps it ought to have been called ‘the Scottish Constant’.
18Comparesuper-tonkin Section4.4.3.
19By the standard move we can permute the subderivation ofC from ⊥to show that we can retrieve the initial rule from the simplified one.
20This involves some cheating. For the more detailed look at the bullet-derivation of inconsis- tency, see Section4.4.4.
[•]1
⊥ (SE•)
• (I•)(1)
⊥ (SE•)
C (EF Q)
Allowing for negation in the language, and the rulesI¬andE¬, we can also derive bothA and ¬A for any formula A.
This is alarming. GE-harmony does not entail consistency, and, as a consequence, since normalisation entails that ⊥ is nonderivable (see Prawitz 1965, p. 44), GE- harmony cannot entail normalisation either.21 Obviously, neither the Subformula nor the Separation Property will hold, and•yields a non-conservative extension of any consistent system. In fact, Read reaches the same conclusion by observing that the conversion step for•cannot contribute to the inductive proof of normalisation:
[•]1 Π1 ⊥ • (1) Π•2 [⊥]2 Π3 C C (2) Π2 • Π1 ⊥ Π3 C
Roughly put, the problem is that a copy of•is transfered from the original deriva- tion to the converted derivation, and there is no guarantee that it does not form a maximum formula (again, this will depend on the structure of Π1 and Π1).22 We
return to the details of this conversion in a more general setting later (see Section 4.4.4 for details). For now, we are content to notice that there is noticeable gap between GE-harmony and other conceptions of harmony discussed in Chapter 2. Since revision due to failure of GE-harmony is separate from conservativeness, normalisation, and the Separation Property, it is worthwhile to return the revision of classical negation discussed in Chapter 3.
21Given some plausible assumptions about the logic for⊥. 22Compare the case ofE⊥
C as intro-rule where¬A appears in the converted derivation, but
Yet, the most puzzling result is that if GE-harmony is the correct formalisation of harmony, then harmony does not even entail consistency. Remember that Dum- mett called inconsistency ‘the grossest form of malfunction’ an inferential practice could have. Harmony, then, appears to fail as an all-purpose protection against semantic dysfunctionality. On closer inspection, however, the result might be less surprising. Belnap’s and Dummett’s original discussion of harmony and con- servativeness ran two thoughts together: First, that the inferentialist ought not to allow rules that lead to inconsistent (alternatively, that trivialise); second, that the intro- and elim-rules must somehow have corresponding deductive strength (or, specifically, must obey the inversion principle). The confusion has led Dummett astray—the right notion of proof-theoretic harmony is neither total nor intrinsic harmony.
Nevertheless, that does not mean that Dummett was not right in claiming that inconsistency is a gross malfunction, and that it should be ruled out. It just turns out that the task of ruling out inconsistency is independent from the task of ‘balancing’ intro- and elim-rules. Read’s achievement is to have separated the two issues clearly with a precise formalisation of proof-theoretic harmony. Our task is to elaborate on this observation by investigating the conditions under which GE-harmony also entails consistency.