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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
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).
References
Curry, H. (1950). A theory of formal deducibility. Notre Dame: Indiana: University of Notre Dame Press.
Davies, R., & Pfenning, F. (2001). A modal analysis of staged computation. Journal of the ACM, 48, 555–604.
Dummett, M. (1973). Frege: philosophy of language. London: Duckworth.
Dummett, M. (1991). Logical basis of metaphysics. London: Duckworth.
Dyckhoff, R. (1988). Implementing a simple proof assistant. In J. Derrick & H. Lewis (Eds.), Proceedings of the workshop on programming for logic teaching, leeds 1987, Centre for theoretical computer science and departments of pure mathematics and philosophy (pp. 49–59). Leeds: University of Leeds.
Dyckhoff, R. (2013). General[-ised] elimination rules, http://rd.host.cs.st-andrews.ac.uk/talks/2013/Tuebingen-PTS2-talk.pdf.
Fitch, F. (1952). Symbolic logic: an introduction. New York: The Ronald Press Co.
Francez, N., & Dyckhoff, R. (2012). A note on harmony. Journal of Philosophical Logic, 41, 613–28.
Gentzen G (1932) Untersuchungen über das logische Schliessen. Manuscript 974:271 in Bernays Archive, Eidgenössische Technische Hochschule Zürich.
Gentzen, G. (1969). Investigations concerning logical deduction. In M. Szabo (Ed.), The Collected Papers of Gerhard Gentzen (pp. 68–131). Amsterdam: North-Holland.
Girard, J. Y., Taylor, P., & Lafont, Y. (1989). Proofs and types. Cambridge: Cambridge University Press.
Hermes, H. (1959). Zum Inversionsprinzip der operativen Logik. In A. Heyting (Ed.), Constructivity in Mathematics (pp. 62–68). Amsterdam: North-Holland.
Lorenzen, P. (1955). Einführung in die operative Logik und Mathematik. Berlin: Springer.
Murzi, J., & Hjortland, O. (2009). Inferentialism and the categoricity problem: reply to Raatikainen. Analysis, 69, 480–88.
Negri, S., & von Plato, J. (2001). Structural proof theory. Cambridge: Cambridge UP.
Prawitz, D. (1965). Natural Deduction. Stockholm: Almqvist and Wiksell.
Prawitz, D. (1973). Towards the foundation of a general proof theory. In P. Suppes, L. Henkin, A. Joja, & G. Moisil (Eds.), Logic, methodology and philosophy of science IV: proceedings of the 1971 international congress (pp. 225–50). Amsterdam: North-Holland.
Prawitz, D. (1974). On the idea of a general proof theory. Synthese, 27, 63–77.
Prawitz, D. (1975). Ideas and results in proof theory. In J. Fenstad (Ed.), Proceedings of the second scandinavian logic symposium (pp. 235–50). Amsterdam: North-Holland.
Prawitz, D. (1985). Remarks on some approaches to the concept of logical consequence. Synthese, 62, 153–171.
Prawitz, D. (2006). Meaning approached via proofs. Synthese, 148, 507–524.
Prior, A. (1960). The runabout inference ticket. Analysis, 21, 38–39.
Read, S. (2000). Harmony and autonomy in classical logic. Journal of Philosophical Logic, 29, 123–154.
Read, S. (2008). Harmony and modality. In C. Dégremont, L. Kieff, & H. Rückert (Eds.), Dialogues, logics and other strange things: essays in honour of Shahid Rahman (pp. 285–303). London: College Publications.
Read, S. (2010). General-elimination harmony and the meaning of the logical constants. Journal of Philosophical Logic, 39, 557–76.
Read, S. (2014). Proof-theoretic validity. In O. Hjortland (Ed.), Caret C. Foundations of Logical Consequence: Oxford University Press.
Schroeder-Heister, P. (1984). A natural extension of natural deduction. Journal of Symbolic Logic, 49, 1284–1300.
Schroeder-Heister, P. (2006). Validity concepts in proof-theoretic semantics. Synthese, 148, 525–571.
Schroeder-Heister, P. (2007). Generalized definitional reflection and the inversion principle. Logica Universalis, 1, 355–76.
Schroeder-Heister, P. (2014). Generalized elimination inference, higher-level rules, and the implications-as-rules interpretation of the sequent calculus. In E. Haeusler, L. C. Pereira, & V. de Palva (Eds.), Advances in natural deduction (pp. 1–29). Berlin: Springer.
Tennant, N. (n.d.) Inferentialism, logicism, harmony, and a counterpoint. To appear in ed. Alex Miller, Essays for Crispin Wright: Logic, Language and Mathematics (in preparation for Oxford University Press: Volume 2 of a two-volume Festschrift for Crispin Wright, co-edited with Annalisa Coliva), http://people.cohums.ohio-state.edu/tennant9/crispin_rev.pdf.
Troelstra, A., & Schwichtenberg, H. (2000). Basic proof theory (2nd ed.). Cambridge: Cambridge University Press.
von Plato, J. (2001). Natural deduction with general elimination rules. Archive for Mathematical Logic, 40, 521–47.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-11041-7_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11040-0
Online ISBN: 978-3-319-11041-7
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)