Notre Dame Journal of Formal Logic

Speech Acts, Categoricity, and the Meanings of Logical Connectives

Ole Thomassen Hjortland

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In bilateral systems for classical logic, assertion and denial occur as primitive signs on formulas. Such systems lend themselves to an inferentialist story about how truth-conditional content of connectives can be determined by inference rules. In particular, for classical logic there is a bilateral proof system which has a property that Carnap in 1943 called categoricity. We show that categorical systems can be given for any finite many-valued logic using n-sided sequent calculus. These systems are understood as a further development of bilateralism—call it multilateralism. The overarching idea is that multilateral proof systems can incorporate the logic of a variety of denial speech acts. So against Frege we say that denial is not the negation of assertion and, with Mark Twain, that denial is more than a river in Egypt.

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Notre Dame J. Formal Logic, Volume 55, Number 4 (2014), 445-467.

First available in Project Euclid: 7 November 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B50: Many-valued logic 03B20: Subsystems of classical logic (including intuitionistic logic)

logical inferentialism logical constants categoricity proof theory sequent calculus


Thomassen Hjortland, Ole. Speech Acts, Categoricity, and the Meanings of Logical Connectives. Notre Dame J. Formal Logic 55 (2014), no. 4, 445--467. doi:10.1215/00294527-2798700.

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  • [1] Baaz, M., C. Fermüller, and R. Zach, “Systematic construction of natural deduction systems for many-valued logics,” pp. 208–13 in Proceedings of the 23rd International Symposium on Multiple-Valued Logic, IEEE Press, New York, 1993.
  • [2] Belnap, N. D., Jr., and G. J. Massey, “Semantic holism,” Studia Logica, vol. 49 (1990), pp. 67–82.
  • [3] Bonnay, D., and D. Westerståhl, “Logical consequence inside out,” pp. 193–202 in Logic, Language and Meaning, Springer, Berlin, 2010.
  • [4] Boričić, B. R., “On sequence-conclusion natural deduction systems,” Journal of Philosophical Logic, vol. 14 (1985), pp. 359–77.
  • [5] Carnap, R., Formalization of Logic, Harvard Univ. Press, Cambridge, Mass., 1943.
  • [6] Dummett, M., The Logical Basis of Metaphysics, Harvard Univ. Press, Cambridge, Mass., 1991.
  • [7] Dummett, M., “‘Yes,’ ‘no’ and ‘can’t say,’” Mind, vol. 111 (2002), pp. 289–95.
  • [8] Dunn, J. M., and G. M. Hardegree, Algebraic Methods in Philosophical Logic, vol. 41 of Oxford Logic Guides, Oxford Univ. Press, New York, 2001.
  • [9] Ferreira, F., “The co-ordination principles: A problem for bilateralism,” Mind, vol. 117 (2008), pp. 1051–57.
  • [10] Field, H., Saving Truth from Paradox, Oxford Univ. Press, Oxford, 2008.
  • [11] Garson, J. W., “Natural semantics: Why natural deduction is intuitionistic,” Theoria (Stockholm), vol. 67 (2001), pp. 114–39.
  • [12] Garson, J. W., “Expressive power and incompleteness of propositional logics,” Journal of Philosophical Logic, vol. 39 (2010), pp. 159–71.
  • [13] Gibbard, P., “Price and Rumfitt on rejective negation and classical logic,” Mind, vol. 111 (2002), pp. 297–303.
  • [14] Hardegree, G. M., “Completeness and super-valuations,” Journal of Philosophical Logic, vol. 34 (2005), pp. 81–95.
  • [15] Hjortland, O. T., “Logical pluralism, meaning-variance, and verbal disputes,” Australasian Journal of Philosophy, vol. 91 (2013), pp. 355–73.
  • [16] Hodes, H., “On the sense and reference of a logical constant: The foundations of mathematics and logic,” The Philosophical Quarterly, vol. 54 (2004), pp. 134–65.
  • [17] Kripke, S., “Outline of a theory of truth,” Journal of Philosophy, vol. 72 (1975), pp. 690–716.
  • [18] Milne, P., “Classical harmony: Rules of inference and the meaning of the logical constants,” Synthese, vol. 100 (1994), pp. 49–94.
  • [19] Milne, P., “Harmony, purity, simplicity and a ‘seemingly magical fact,’” The Monist, vol. 85 (2002), pp. 498–534.
  • [20] Prawitz, D., “Ideas and results in proof theory,” pp. 235–307 in Proceedings of the 2nd Scandinavian Logic Symposium (Oslo, 1970), edited by J. Fenstad, vol. 63 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1971.
  • [21] Priest, G., An Introduction to Non-classical Logic: From If to Is, 2nd ed., Cambridge Univ. Press, Cambridge, 2008.
  • [22] Prior, A. N., “The runabout inference ticket,” Analysis, vol. 21 (1961), pp. 38–39.
  • [23] Read, S., “Harmony and autonomy in classical logic,” Journal of Philosophical Logic, vol. 29 (2000), pp. 123–54.
  • [24] Restall, G., “Multiple conclusions,” pp. 189–205 in Logic, Methodology and Philosophy of Science (Oviedo, Spain, 2003), College Publications, London, 2005.
  • [25] Restall, G., “Truth values and proof theory,” Studia Logica, vol. 92 (2009), pp. 241–64.
  • [26] Rousseau, G., “Sequents in many valued logic, I,” Fundamenta Mathematicae, vol. 60 (1967), pp. 23–33.
  • [27] Rumfitt, I., “The categoricity problem and truth-value gaps,” Analysis (Oxford), vol. 57 (1997), pp. 223–35.
  • [28] Rumfitt, I., “`Yes’ and ‘No,”' Mind, vol. 109 (2000), pp. 781–823.
  • [29] Rumfitt, I., “Unilateralism disarmed: A reply to Dummett and Gibbard,” Mind, vol. 111 (2002), pp. 305–21.
  • [30] Rumfitt, I., “Co-ordination Principles: A Reply,” Mind, vol. 117 (2008), pp. 1059–63.
  • [31] Rumfitt, I., “Knowledge by deduction,” Grazer Philosophische Studien, vol. 77 (2008), pp. 61–84.
  • [32] Schroeder-Heister, P., “Generalized definitional reflection and the inversion principle,” Logica Universalis, vol. 1 (2007), pp. 355–76.
  • [33] Schröter, K., “Methoden zur Axiomatisierung beliebiger Aussagen- und Prädikatenkalküle,” Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 1 (1955), pp. 241–51.
  • [34] Shoesmith, D. J., and T. J. Smiley, Multiple-conclusion Logic, Cambridge Univ. Press, Cambridge, 1978.
  • [35] Smiley, T., “Rejection,” Analysis (Oxford), vol. 56 (1996), pp. 1–9.
  • [36] Stevenson, J. T., “Roundabout the runabout inference-ticket,” Analysis, vol. 21 (1961), pp. 124–28.
  • [37] Tennant, N., Anti-realism and Logic: Truth as Eternal, Oxford Univ. Press, Oxford, 1987.
  • [38] Tennant, N., The Taming of the True, Oxford Univ. Press, Oxford, 1997.
  • [39] Tennant, N., “Inferentialism, logicism, harmony and a counterpoint,” in Essays for Crispin Wright: Logic, Language and Mathematics 2vols. edited by A. Miller, Oxford Univ. Press, Oxford, forthcoming.
  • [40] Troelstra, A. S., and H. Schwichtenberg, Basic Proof Theory, 2nd ed., vol. 43 of Cambridge Tracts in Theoretical Computer Science, Cambridge Univ. Press, Cambridge, 2000.
  • [41] Ungar, A. M., Normalization, Cut-elimination and the Theory of Proofs, vol. 28 of CSLI Lecture Notes, Stanford University Center for the Study of Language and Information, Stanford, 1992.
  • [42] Wagner, S., “Tonk,” Notre Dame Journal of Formal Logic, vol. 22 (1981), pp. 289–300.
  • [43] Weir, A., “Classical harmony,” Notre Dame Journal of Formal Logic, vol. 27 (1986), pp. 459–82.