Advertisement

Journal of Philosophical Logic

, Volume 39, Issue 5, pp 557–576 | Cite as

General-Elimination Harmony and the Meaning of the Logical Constants

  • Stephen Read
Article

Abstract

Inferentialism claims that expressions are meaningful by virtue of rules governing their use. In particular, logical expressions are autonomous if given meaning by their introduction-rules, rules specifying the grounds for assertion of propositions containing them. If the elimination-rules do no more, and no less, than is justified by the introduction-rules, the rules satisfy what Prawitz, following Lorenzen, called an inversion principle. This connection between rules leads to a general form of elimination-rule, and when the rules have this form, they may be said to exhibit “general-elimination” harmony. Ge-harmony ensures that the meaning of a logical expression is clearly visible in its I-rule, and that the I- and E-rules are coherent, in encapsulating the same meaning. However, it does not ensure that the resulting logical system is normalizable, nor that it satisfies the conservative extension property, nor that it is consistent. Thus harmony should not be identified with any of these notions.

Keywords

Harmony Inferentialism Autonomy Validity Tonk Dummett Gentzen Prawitz Lorenzen 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brandom, R. B. (2000). Articulating reasons. Cambridge, MA: Harvard University Press.Google Scholar
  2. 2.
    Dummett, M. (1973). Frege: Philosophy of language. London: Duckworth.Google Scholar
  3. 3.
    Dummett, M. (1991). Logical basis of metaphysics. London: Duckworth.Google Scholar
  4. 4.
    Dummett, M. (1993). Language and truth. In The seas of language (pp. 117–165). Oxford: Clarendon Press.Google Scholar
  5. 5.
    Dyckhoff, R., & Francez, N. (2007). A note on harmony. Typescript, August 2007.Google Scholar
  6. 6.
    Gentzen, G. (1969). Investigations concerning logical deduction. In M. Szabo (Ed.), The collected papers of Gerhard Gentzen (pp. 68–131). Amsterdam: North-Holland.Google Scholar
  7. 7.
    Lorenzen, P. (1955). Einführung in die operative Logik und Mathematik. Berlin, Göttingen, Heidelberg: Springer.Google Scholar
  8. 8.
    Meyer, R. K., & Sylvan, R. (2003). Extensional reduction II. In R. Brady, et al. (Eds.), Relevant logics and their rivals (Vol. II, pp. 352–407). Aldershot: Ashgate.Google Scholar
  9. 9.
    Milne, P. (2002). Harmony, purity, simplicitty and a ‘seemingly magical fact’. The Monist, 85, 498–534.Google Scholar
  10. 10.
    Moriconi, E., & Tesconi, L. (2008). On inversion principles. History and Philosophy of Logic, 29, 103–113.CrossRefGoogle Scholar
  11. 11.
    Negri, S., & von Plato, J. (2001). Structural proof theory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  12. 12.
    Prawitz, D. (1965). Natural deduction. Stockholm: Almqvist & Wiksell.Google Scholar
  13. 13.
    Prawitz, D. (1979). Proofs and the meaning and completeness of the logical constants. In J. Hintikka, I. Niiniluoto, & E. Saarinen (Eds.), Essays on mathematical and philosophical logic (pp. 25–40). Dordrecht: Reidel.Google Scholar
  14. 14.
    Read, S. (2000). Harmony and autonomy in classical logic. Journal of Philosophical Logic, 29, 123–154.CrossRefGoogle Scholar
  15. 15.
    Read, S. (2005). The unity of the fact. Philosophy, 80, 317–342.CrossRefGoogle Scholar
  16. 16.
    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.Google Scholar
  17. 17.
    Restall, G. (2005). Multiple conclusions. In P. Hajek, L. Valdes-Villanueva, & D. Westerstahl (Eds.), Logic, methodology and philosophy of science: Proceedings of the twelfth international congress (pp. 189–205). London: Kings College Publications.Google Scholar
  18. 18.
    Schroeder-Heister, P. (1984). A natural extension of natural deduction. Journal of Symbolic Logic, 49, 1284–1300.CrossRefGoogle Scholar
  19. 19.
    Schroeder-Heister, P. (2004). On the notion of assumption in logical systems. In Philosophy-science-scientific philosophy: Selected papers contributed to the sections of the 5th international congress of the Society for Analytical Philosophy, Bielefeld, 22–26 September 2003 (pp. 27–48). Paderborn: Mentis.Google Scholar
  20. 20.
    von Plato, J. (2001). Natural deduction with general elimination rules. Archive for Mathematical Logic, 40, 521–547.Google Scholar
  21. 21.
    von Plato, J. (2008). Gentzen’s proof of normalization for natural deduction. Bulletin of Symbolic Logic, 14, 245–257.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of St AndrewsScotlandUK

Personalised recommendations