## Abstract

Michael Dummett introduced the notion of harmony in response to Arthur Prior’s tonkish attack on the idea of proof-theoretic justification of logical laws (or analytic validity). But Dummett vacillated between different conceptions of harmony, in an attempt to use the idea to underpin his anti-realism. Dag Prawitz had already articulated an idea of Gerhard Gentzen’s into a procedure whereby elimination-rules are in some sense functions of the corresponding introduction-rules. The resulting conception of general-elimination harmony ensures that the rules are transparent in the meaning they confer, in that the elimination-rules match the meaning the introduction-rules confer. The general-elimination rules which result may be of higher level, in that the assumptions discharged by the rule may be of (the existence of) derivations rather than just of formulae. In many cases, such higher-level rules may be “flattened” to rules discharging only formulae. However, such flattening is often only possible in the richer context of so-called “classical” or realist negation, or in a multiple-conclusion environment. In a constructivist context, the flattened rules are harmonious but not stable.

Invited paper for *Dag Prawitz on Proofs and Meaning*, edited by Heinrich Wansing, in the *Studia Logica* series *Trends in Logic*. I owe a substantial intellectual debt to Dag Prawitz, to whose writings on proof theory I was first introduced by John Mayberry at Bristol, and which have inspired and guided me repeatedly throughout my career. This work is supported by Research Grant AH/F018398/1 (Foundations of Logical Consequence) from the Arts and Humanities Research Council, UK.

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## Notes

- 1.
- 2.
- 3.
For an extended discussion of GE-harmony, see Read (2010).

- 4.
\(m\) may be zero, as Prawitz (1973, p. 243) notes is the case for the absurdity constant, \(\bot \), which has no grounds for its assertion.

- 5.
If \(m=0\), the empty product predicts one E-rule, to infer an arbitrary conclusion from \(\bot \). If \(n_i=0\) for some \(i\), the product is 0. E.g., if we introduce \(\top \) by an I-rule with no premises (even if we give alternative, more restrictive, grounds for its assertion), \(\top \) is a tautology, and nothing can be inferred from it which is not already provable.

- 6.
- 7.
To obtain harmonious rules, the right response is, of course, not to strengthen \(\lozenge \)-E to match \(\lozenge \)-I, but to find some way to weaken \(\lozenge \)I. For one possible solution, see Read (2008).

- 8.
- 9.
- 10.
See, e.g., Girard et al. (1989, p. 152).

- 11.
The form of representation here is inspired by Gentzen’s notation in the draft of his dissertation, (Gentzen 1932).

- 12.
Note that in \(\alpha \Rightarrow \beta \), the assumption \(\alpha \) is closed, either by a rule discharging the assumption (e.g., \(\rightarrow \)-I) or by a derivation of \(\alpha \) (e.g., in the proof of the minor premise of \(\rightarrow \)-E).

- 13.
- 14.
See, e.g., Proposition 3.5.4 (vi) in Troelstra and Schwichtenberg (2000, p. 79).

- 15.
But see Murzi and Hjortland (2009).

- 16.
If we require the succedent to be non-empty, we can capture “empty” succedent with an instance of \(\bot \).

- 17.
Recall from Sect. 13.2 Dummett’s minimal I-rule for ‘\(\lnot \)’:

GE-considerations justify as E-rule:

which flattens to

But this adds nothing to what was already derivable using \(\lnot \)-I.:

- 18.
See Read (2000, pp. 149–150).

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Read, S. (2015). General-Elimination Harmony and Higher-Level Rules. In: Wansing, H. (eds) Dag Prawitz on Proofs and Meaning. Outstanding Contributions to Logic, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-11041-7_13

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