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On the equivalence conjecture for proof-theoretic harmony. (English) Zbl 1301.03014

The author discusses and analyzes the logical relations between three prominent accounts of harmony: harmony-as-conservative extension (see [N. Belnap, Analysis 22, 130–134 (1962); M. Dummett, The logical basis of metaphysics. Cambridge, MA: Harvard University Press (1991)]), harmony-as-levelling procedure (see [D. Prawitz, Theoria 43, 2–40 (1977; Zbl 0361.02008); Dummett, loc. cit.]), and harmony-as-deductive equilibrium (see [N. Tennant, Anti-realism and logic. Oxford: Oxford University Press (1987); The taming of the true. Oxford: Clarendon Press (1997; Zbl 0929.03001)], and concludes that there is no core notion of harmony toward which all three accounts of harmony converge.

MSC:

03A05 Philosophical and critical aspects of logic and foundations
03F03 Proof theory in general (including proof-theoretic semantics)
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References:

[1] Belnap, N., “Tonk, plonk and plink.” Analysis , vol. 22 (1962), pp. 130-34.
[2] Dummett, M., The Logical Basis of Metaphysics , The William James Lectures, 1976, Harvard University Press, Cambridge, Mass., 1991.
[3] Gentzen, G., “Investigations into logical deduction,” pp. 68-128 in The Collected Papers of Gerhard Gentzen , edited by M. Szabo, Studies in Logic and the Foundations of Mathematics , North-Holland, Amsterdam, 1969.
[4] Hodes, H., “On the sense and reference of a logical constant,” Philosophical Quarterly , vol. 54 (2004), pp. 134-65.
[5] Prawitz, D., “Meaning and proofs: On the conflict between classical and intuitionistic logic,” Theoria , vol. 43 (1977), pp. 2-40. · Zbl 0361.02008
[6] Prawitz, D., “Proofs and the meaning and completeness of the logical constants,” pp. 25-40 in Essays on Mathematical and Philosophical Logic , edited by J. Hintikka, vol. 122 of Synthese Library , Reidel, Dordrecht, 1979. · Zbl 0406.03069
[7] Prawitz, D., Natural Deduction: A Proof-Theoretical Study , Dover, Mineola, N. Y., 2006. · Zbl 0173.00205
[8] Prior, A., “The runabout inference ticket,” Analysis , vol. 21 (1960), pp. 38-39.
[9] Steinberger, F., “Not so stable,” Analysis , vol. 69 (2009), pp. 655-61. · Zbl 1284.03105
[10] Steinberger, F., “Harmony in a sequent setting: A reply to Tennant,” Analysis , vol. 71 (2011), pp. 273-80. · Zbl 1284.03104
[11] Steinberger, F., “What harmony could and could not be,” Australasian Journal of Philosophy , vol. 89 (2011), pp. 617-39.
[12] Tennant, N., Anti-realism and Logic , Oxford University Press, Oxford, 1987. · Zbl 0652.03010
[13] Tennant N., The Taming of the True , Clarendon Press, Oxford, 1997. · Zbl 0929.03001
[14] Tennant, N., “Inferentialism, logicism, harmony and a counterpoint,” forthcoming in Essays for Crispin Wright: Logic, Language and Mathematics , edited by A. Miller, Oxford University Press, Oxford.
[15] Tennant, N., “Harmony in a sequent setting,” Analysis , vol. 70 (2011), pp. 462-68. · Zbl 1284.03107
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