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WEAK DISHARMONY: SOME LESSONS FOR PROOF-THEORETIC SEMANTICS

Published online by Cambridge University Press:  08 August 2016

BOGDAN DICHER
BOGDAN DICHER*
Affiliation:
Department of Pedagogy, Psychology, Philosophy, University of Cagliari
*
*DEPARTMENT OF PEDAGOGY, PSYCHOLOGY, PHILOSOPHY UNIVERSITY OF CAGLIARI CAGLIARI, IS MIRRIONIS 1, 09123 SARDEGNA, ITALY E-mail: bdicher@me.com
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Abstract

A logical constant is weakly disharmonious if its elimination rules are weaker than its introduction rules. Substructural weak disharmony is the weak disharmony generated by structural restrictions on the eliminations. I argue that substructural weak disharmony is not a defect of the constants which exhibit it. To the extent that it is problematic, it calls into question the structural properties of the derivability relation. This prompts us to rethink the issue of controlling the structural properties of a logic by means of harmony. I argue that such a control is possible and desirable. Moreover, it is best achieved by global tests of harmony.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 9 , Issue 3 , September 2016 , pp. 583 - 602
Copyright
Copyright © Association for Symbolic Logic 2016 

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References

BIBLIOGRAPHY

Belnap, N. D. (1962). Tonk, plonk and plink. Analysis, 22, 130–134.CrossRefGoogle Scholar
Cook, R. T. (2005). What’s wrong with tonk(?). Journal of Philosophical Logic, 34(2), 217–226.CrossRefGoogle Scholar
Dicher, B. (2016). On a generality condition in proof-theoretic semantics. (In preparation).Google Scholar
Dicher, B. (forthcoming). A proof-theoretic defence of meaning-invariant logical pluralism. Mind, forthcoming.Google Scholar
Došen, K. (1989). Logical constants as punctuation marks. Notre Dame Journal of Formal Logic, 30(3), 362–381.CrossRefGoogle Scholar
Došen, K., & Schroeder-Heister, P. (1985). Conservativeness and uniqueness. Theoria 51(3), 159–173.Google Scholar
Dummett, M. (1977). Elements of Intuitionism. Oxford: Oxford University Press.Google Scholar
Dummett, M. (1991). The Logical Basis of Metaphysics. London: Duckworth.Google Scholar
Francez, N. (2015). Proof-theoretic semantics. London: College Publications.Google Scholar
Francez, N., & Dyckhoff, R. (2011). A note on harmony. Journal of Philosophical Logic, 41, 613–628.CrossRefGoogle Scholar
Gentzen, G. (1969). The Collected Papers of Gerhard Gentzen. Amsterdam: North Holland.Google Scholar
Hjortland, O. T. (2009). The Structure of Logical Consequence: Proof-Theoretic Conceptions. Ph. D. thesis, University of St. Andrews.Google Scholar
Hjortland, O. T. (2012). Harmony and the context of deducibility. In Dutilh Novaes, C. & Hjortland, O. T., editors, Insolubles and Consequences, Essays in Honour of Stephen Read, pp. 105–117. London: College Publications.Google Scholar
Humberstone, L. (2007). Investigations into a left-structural right-substructural sequent calculus. Journal of Logic, Language and Information, 16(2), 141–171.CrossRefGoogle Scholar
Humberstone, L. (2010). Sentence connectives in formal logic. In Zalta, E., editor, Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/archives/sum2013/entries/connectives-logic/.Google Scholar
Humberstone, L. (2011). The Connectives. Cambridge, MA: MIT Press.CrossRefGoogle Scholar
Kürbis, N. (2008). Stable harmony. In Peliš, M., editor, The Logica Yearbook 2007, pp. 87–96. Filosofia.Google Scholar
Kürbis, N. (2013). Proof-theoretic semantics, a problem with negation and prospects for modality. Journal of Philosophical Logic OnlineFirst, 1–15. DOI: 10.1007/s10992-013-9310-6.Google Scholar
Milne, P. (2012). Inferring, splicing, and the stoic analysis of argument. In Dutilh Novaes, C. & Hjortland, O. T., editors, Insolubles and consequences: Essays in Honour of Stephen Read, pp. 135–154. London: College Publications.Google Scholar
Milne, P. (2015). Inversion principles and introduction rules. In Wansing, H., editor, Dag Prawitz on meaning and proofs, pp. 293–312. Berlin: Springer.Google Scholar
Naibo, A., & Petrolo, M. (2015). Are uniqueness and deducibility of identicals the same? Theoria, 81(2), 143–181.CrossRefGoogle Scholar
Paoli, F. (2002). Substructural Logics: A Primer. Berlin, Heidelberg: Springer.CrossRefGoogle Scholar
Prawitz, D. (1965). Natural Deduction: A Proof Theoretical Study. Stockholm: Almqvist and Wiksell.Google Scholar
Prawitz, D. (1994). Review of Dummett: The Logical Basis of Metaphysics. Mind, 193, 373–376.CrossRefGoogle Scholar
Prawitz, D. (2007). Pragmatist and verificationist theories of meaning. In Auxier, R. E. & Hahn, L. E., editors, The Philosophy of Michael Dummett, pp. 455–481. Chicago, La Salle: Open Court.Google Scholar
Prior, A. N. (1960). The runabout inference-ticket. Analysis, 21(2), 38–39.CrossRefGoogle Scholar
Read, S. (2000). Harmony and autonomy in classical logic. Journal of Philosophical Logic, 29(2), 123–154.CrossRefGoogle Scholar
Read, S. (2010). General-elimination harmony and the meaning of the logical constants. Journal of Philosophical Logic, 39, 557–576.CrossRefGoogle Scholar
Read, S. (2015). General-elimination harmony and higher order rules. In Wansing, H., editor, Dag Prawitz on meaning and proofs, pp. 293–312. Berlin: Springer.CrossRefGoogle Scholar
Restall, G. Proof theory and meaning: on second order logic. In Pelis, M., editor, The Logica Yearbook 2007, pp. 157–170. Prague: Filosofia.Google Scholar
Restall, G. (2007). Proof theory and philosophy. In progress. Draft available at http://consequently.org/writing/ptp.Google Scholar
Restall, G. (2014). Pluralism and proofs. Erkenntnis, 79(2), 279–291.CrossRefGoogle Scholar
Restall, G., & Paoli, F. (2005). The geometry of non-distributive logics. Journal of Symbolic Logic, 70(4), 1108–1126.CrossRefGoogle Scholar
Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139–164.CrossRefGoogle Scholar
Ripley, D. (2015). Anything goes. Topoi, 34, 25–36.CrossRefGoogle Scholar
Sambin, G., Battilotti, G., & Faggian, C. (2000). Basic logic: Reflection, symmetry, visibility. Journal of Symbolic Logic, 65(3), 979–1013.CrossRefGoogle Scholar
Schroeder-Heister, P. (1984). A natural extension of natural deduction. The Journal of Symbolic Logic, 49(4), 1284–1300.CrossRefGoogle Scholar
Steinberger, F. (2009). Not so stable. Analysis, 69, 655–661.CrossRefGoogle Scholar
Steinberger, F. (2011a). Harmony in a sequent setting: A reply to Tennant. Analysis, 71(2), 273–280.CrossRefGoogle Scholar
Steinberger, F. (2011b). What harmony could and could not be. Australasian Journal of Philosophy, 89, 617–639.CrossRefGoogle Scholar
Steinberger, F. (2013). On the equivalence conjecture for proof-theoretic harmony. Notre Dame Journal of Formal Logic, 54, 78–86.CrossRefGoogle Scholar
Tennant, N. (1994). The transmission of truth and the transitivity of deduction. In Gabbay, D., editor, What is a Logical System?, pp. 161–177. Oxford: Oxford University Press.CrossRefGoogle Scholar
Tennant, N. (1997). The Taming of the True. Oxford: Clarendon Press.Google Scholar
Tennant, N. (2005). Rule-circularity and the justification of deduction. The Philosophical Quarterly, 55(221), 625–648.CrossRefGoogle Scholar
Tennant, N. (forthcoming). Inferentialism, logicism, harmony, and a counterpoint. In Miller, A., editor, Essays for Crispin Wright: Logic, Language and Mathematics, pp. n/a. Oxford: Oxford University Press.Google Scholar
Weir, A. (2015). A robust non-transitive logic. Topoi, 34, 99–107.CrossRefGoogle Scholar