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HARMONIC INFERENTIALISM AND THE LOGIC OF IDENTITY

Published online by Cambridge University Press:  09 February 2016

STEPHEN READ
STEPHEN READ*
Affiliation:
University of St Andrews
*
*UNIVERSITY OF ST ANDREWS ARCHÉ RESEARCH CENTRE 17-19 COLLEGE ST. ST ANDREWS KY16 9AA SCOTLAND, UK E-mail: slr@st-andrews.ac.uk
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Abstract

Inferentialism claims that the rules for the use of an expression express its meaning without any need to invoke meanings or denotations for them. Logical inferentialism endorses inferentialism specifically for the logical constants. Harmonic inferentialism, as the term is introduced here, usually but not necessarily a subbranch of logical inferentialism, follows Gentzen in proposing that it is the introduction-rules which give expressions their meaning and the elimination-rules should accord harmoniously with the meaning so given. It is proposed here that the logical expressions are those which can be given schematic rules that lie in a specific sort of harmony, general-elimination (ge) harmony, resulting from applying a certain operation, the ge-procedure, to produce ge-rules in accord with the meaning defined by the I-rules. Griffiths (2014) claims that identity cannot be given such rules, concluding that logical inferentialists are committed to ruling identity a nonlogical expression. It is shown that the schematic rules for identity given in Read (2004), slightly amended, are indeed ge-harmonious, so confirming that identity is a logical notion.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 9 , Issue 2 , June 2016 , pp. 408 - 420
Copyright
Copyright © Association for Symbolic Logic 2016 

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References

BIBLIOGRAPHY

Basin, D., Matthews, S., & Viganò, L. (1997). Labelled propositional modal logics: theory and practice. Journal of Logic and Computation, 7, 685–717.Google Scholar
Brandom, R. (1994). Making it Explicit. Cambridge, Massachusetts: Harvard University Press.Google Scholar
Brandom, R. (2000). Articulating Reasons. Cambridge, Massachusetts: Harvard University Press.Google Scholar
Brandom, R. (2007). Inferentialism and some of its challenges. Philosophy and Phenomenological Research, 74, 651–676.CrossRefGoogle Scholar
Braüner, T. (2014). Hybrid logic. In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy (Winter 2014 ed.).Google Scholar
Buridan, J. (2014). Tractatus de Consequentiis. New York: Fordham University Press. English translation by Stephen Read.Google Scholar
Curry, H. (1950). A Theory of Formal Deducibility. Notre Dame: Indiana: University of Notre Dame Press.Google Scholar
da Costa, N., & Mortensen, C. (1980). Variable binding term operators in modal logic. In Prépublicações do Centro de Lógica, Epistemologia e História da Ciência—CLE, no. 4. Campinas-SP, Brazil: CLE.Google Scholar
Dummett, M. (1973). Frege: Philosophy of Language. London: Duckworth.Google Scholar
Dummett, M. (1977). Elements of Intuitionism. Oxford: Oxford University Press.Google Scholar
Dummett, M. (1991). Logical Basis of Metaphysics. London: Duckworth.Google Scholar
Etchemendy, J. (1990). The Concept of Logical Consequence. Cambridge, Massachusetts: Harvard University Press.Google Scholar
Fitch, F. (1952). Symbolic Logic: An Introduction. New York: The Ronald Press Co.Google Scholar
Francez, N., & Dyckhoff, R. (2012). A note on harmony. Journal of Philosophical Logic, 41, 613–628.Google Scholar
Gentzen, G. (1935). Untersuchungen über das logische Schliessen. Mathematische Zeitschrift, 39, 175–210, 405–31. English translation in Gentzen (1969).Google Scholar
Gentzen, G. (1969). Investigations concerning logical deduction. In Szabo, M., editor, The Collected Papers of Gerhard Gentzen, pp. 68–131. Amsterdam: North-Holland.Google Scholar
Griffiths, O. (2014). Harmonious rules for identity. Review of Symbolic Logic, 7, 499–510.Google Scholar
Hermes, H. (1959). Zum Inversionsprinzip der operativen Logik. In Heyting, A., editor, Constructivity in Mathematics, pp. 62–68. Amsterdam: North-Holland.Google Scholar
Kremer, M. (2007). Read on identity and harmony: a friendly correction and simplification. Analysis, 67, 157–59.Google Scholar
Lorenzen, P. (1950). Konstruktive Begründung der Mathematik. Mathematische Zeitschrift, 53, 162–202.Google Scholar
Lorenzen, P. (1955). Einführung in die operative Logik und Mathematik. Berlin, Göttingen, Heidelberg: Springer.Google Scholar
MacFarlane, J. (2015). Logical constants. In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy (Summer 2015 ed.).Google Scholar
Moriconi, E., & Tesconi, L. (2008). On inversion principles. History and Philosophy of Logic, 29, 103–13.CrossRefGoogle Scholar
Negri, S. (2005). Proof analysis in modal logic. Journal of Philosophical Logic, 34, 507–44.Google Scholar
Poggiolesi, F. (2011). Gentzen Calculi for Modal Propositional Logic. Dordrecht: Springer.CrossRefGoogle Scholar
Prawitz, D. (1965). Natural Deduction. Stockholm: Almqvist & Wiksell.Google Scholar
Prawitz, D. (1973). Towards the foundation of a general proof theory. In Suppes, P., Henkin, L., Joja, A., & Moisil, G., editors, Logic, Methodology and Philosophy of Science IV: Proceedings of the 1971 International Congress, pp. 225–50. Amsterdam: North-Holland.Google Scholar
Prawitz, D. (1979). Proofs and the meaning and completeness of the logical constants. In Hintikka, J., Niiniluoto, I., & Saarinen, E., editors, Essays on Mathematical and Philosophical Logic, pp. 25–40. Dordrecht: Reidel.Google Scholar
Prawitz, D. (1994). Review of Dummett (1991). Mind, 103, 373–76.Google Scholar
Prior, A. (1960). The runabout inference ticket. Analysis, 21, 38–39.Google Scholar
Read, S. (2000). Harmony and autonomy in classical logic. Journal of Philosophical Logic, 29, 123–154.Google Scholar
Read, S. (2004). Identity and harmony. Analysis, 64, 113–19.Google Scholar
Read, S. (2008). Harmony and modality. In Dégremont, C., Kieff, L., & Rückert, H., editors, Dialogues, Logics and Other Strange Things: Essays in Honour of Shahid Rahman, pp. 285–303. London: College Publications.Google Scholar
Read, S. (2010). General-elimination harmony and the meaning of the logical constants. Journal of Philosophical Logic, 39, 557–76.Google Scholar
Read, S. (2015). General-elimination harmony and higher-level rules. In Wansing, H., editor, Dag Prawitz on Proofs and Meaning, Studia Logica Library: Trends in Logic, pp. 293–312. Springer.CrossRefGoogle Scholar
Schroeder-Heister, P. (1984). A natural extension of natural deduction. Journal of Symbolic Logic, 49, 1284–1300.Google Scholar
Shapiro, S. (1991). Foundations without Foundationalism. Oxford: Clarendon Press.Google Scholar
Steinberger, F. (2011). What harmony could and could not be. Australasian Journal of Philosophy, 89, 617–639.CrossRefGoogle Scholar
Sundholm, G. (1998). Proofs as acts and proofs as objects: Some questions for Dag Prawitz. Theoria, 64, 187–216.Google Scholar
Tarski, A. (1986). What are logical notions? History and Philosophy of Logic, 7(2), 143–154.Google Scholar
Tarski, A. (2002). On the concept of following logically. History and Philosophy of Logic, 23, 155–96.CrossRefGoogle Scholar
von Plato, J. (2001). Natural deduction with general elimination rules. Archive for Mathematical Logic, 40, 521–47.Google Scholar
Wittgenstein, L. (1961). Tractatus Logico-Philosophicus. London: Routledge & Kegan Paul. Translated by Pears, D. F. & McGuinness, B. F..Google Scholar
Zucker, J., & Tragesser, R. (1978). The adequacy problem for inferential logic. Journal of Philosophical Logic, 7, 501–16.Google Scholar