Skip to main content
Log in

General-Elimination Harmony and the Meaning of the Logical Constants

  • Published:
Journal of Philosophical Logic Aims and scope Submit manuscript


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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Brandom, R. B. (2000). Articulating reasons. Cambridge, MA: Harvard University Press.

    Google Scholar 

  2. Dummett, M. (1973). Frege: Philosophy of language. London: Duckworth.

    Google Scholar 

  3. Dummett, M. (1991). Logical basis of metaphysics. London: Duckworth.

    Google Scholar 

  4. Dummett, M. (1993). Language and truth. In The seas of language (pp. 117–165). Oxford: Clarendon Press.

    Google Scholar 

  5. Dyckhoff, R., & Francez, N. (2007). A note on harmony. Typescript, August 2007.

  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. Lorenzen, P. (1955). Einführung in die operative Logik und Mathematik. Berlin, Göttingen, Heidelberg: Springer.

    Google Scholar 

  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. Milne, P. (2002). Harmony, purity, simplicitty and a ‘seemingly magical fact’. The Monist, 85, 498–534.

    Google Scholar 

  10. Moriconi, E., & Tesconi, L. (2008). On inversion principles. History and Philosophy of Logic, 29, 103–113.

    Article  Google Scholar 

  11. Negri, S., & von Plato, J. (2001). Structural proof theory. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  12. Prawitz, D. (1965). Natural deduction. Stockholm: Almqvist & Wiksell.

    Google Scholar 

  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. Read, S. (2000). Harmony and autonomy in classical logic. Journal of Philosophical Logic, 29, 123–154.

    Article  Google Scholar 

  15. Read, S. (2005). The unity of the fact. Philosophy, 80, 317–342.

    Article  Google Scholar 

  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. 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. Schroeder-Heister, P. (1984). A natural extension of natural deduction. Journal of Symbolic Logic, 49, 1284–1300.

    Article  Google Scholar 

  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.

  20. von Plato, J. (2001). Natural deduction with general elimination rules. Archive for Mathematical Logic, 40, 521–547.

    Google Scholar 

  21. von Plato, J. (2008). Gentzen’s proof of normalization for natural deduction. Bulletin of Symbolic Logic, 14, 245–257.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Stephen Read.

Additional information

This work is supported by Research Grant AH/F018398/1 (Foundations of Logical Consequence) from the Arts and Humanities Research Council, UK.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Read, S. General-Elimination Harmony and the Meaning of the Logical Constants. J Philos Logic 39, 557–576 (2010).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: