Studia Logica

, Volume 101, Issue 1, pp 157–191 | Cite as

Proof Theory for Reasoning with Euler Diagrams: A Logic Translation and Normalization

Article

Abstract

Proof-theoretical notions and techniques, developed on the basis of sentential/symbolic representations of formal proofs, are applied to Euler diagrams. A translation of an Euler diagrammatic system into a natural deduction system is given, and the soundness and faithfulness of the translation are proved. Some consequences of the translation are discussed in view of the notion of free ride, which is mainly discussed in the literature of cognitive science as an account of inferential efficacy of diagrams. The translation enables us to formalize and analyze free ride in terms of proof theory. The notion of normal form of Euler diagrammatic proofs is investigated, and a normalization theorem is proved. Some consequences of the theorem are further discussed: in particular, an analysis of the structure of normal diagrammatic proofs; a diagrammatic counterpart of the usual subformula property; and a characterization of diagrammatic proofs compared with natural deduction proofs.

Keywords

Proof theory Natural deduction Diagrammatic reasoning Euler diagrams 

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References

  1. 1.
    Allwein, G., and J. Barwise (eds.), Logical Reasoning with Diagrams, Oxford Studies in Logic and Computation Series, 1996.Google Scholar
  2. 2.
    Barwise, J., and J. Seligman, Information Flow: The Logic of Distributed Systems, Cambridge University Press, 1997.Google Scholar
  3. 3.
    Buss, S. R., Propositional proof complexity: an introduction, in U. Berger and H. Schwichtenberg (eds.), Computational Logic, Springer, Berlin, 1999 pp. 127–178.Google Scholar
  4. 4.
    Gentzen, G., Untersuchungen über das logische Schließen, Mathematische Zeitschrift 39:176–210, 405–431, 1934. English Translation: Investigations into logical deduction, in M. E. Szabo (ed.), The collected Papers of Gerhard Gentzen, 1969.Google Scholar
  5. 5.
    Howse, J., G. Stapleton, and J. Taylor, Spider diagrams, LMS Journal of Computation and Mathematics 8:145–194, 2005, London Mathematical Society.Google Scholar
  6. 6.
    Mineshima, K., M. Okada, and R. Takemura, Conservativity for a hierarchy of Euler and Venn reasoning systems, Proceedings of Visual Languages and Logic 2009, CEUR Series 510:37–61, 2009.Google Scholar
  7. 7.
    Mineshima, K., M. Okada, and R. Takemura, A diagrammatic inference system with Euler circles, accepted for publication in Journal of Logic, Language and Information.Google Scholar
  8. 8.
    Mineshima, K., M. Okada, and R. Takemura, Two types of diagrammatic inference systems: natural deduction style and resolution style, in Diagrammatic Representation and Inference: 6th International Conference, Diagrams 2010, Lecture Notes In Artificial Intelligence, Springer, 2010, pp. 99–114.Google Scholar
  9. 9.
    Mossakowski T., Diaconescu R., Tarlecki A.: What is a logic translation?. Logica Universalis 3(1), 95–124 (2009)CrossRefGoogle Scholar
  10. 10.
    Negri, S., and J. von Plato, Structural Proof Theory, Cambridge, UK, 2001.Google Scholar
  11. 11.
    von Plato, J., Proof theory of classical and intuitionistic logic, in L. Haaparanta (ed.), History of Modern Logic, Oxford University Press, 2009, pp. 499–515.Google Scholar
  12. 12.
    Prawitz, D., Natural Deduction, Almqvist & Wiksell, 1965 (Dover, 2006).Google Scholar
  13. 13.
    Prawitz, D., Ideas and results in proof theory, in Proceedings 2nd Scandinavian Logic Symposium, 1971, pp. 237–309.Google Scholar
  14. 14.
    Shimojima, A., On the efficacy of representation, Ph.D thesis, Indiana University, 1996.Google Scholar
  15. 15.
    Shin, S.-J., The Logical Status of Diagrams, Cambridge University Press, 1994.Google Scholar
  16. 16.
    Stapleton G.: A survey of reasoning systems based on Euler diagrams, in Proceedings of Euler 2004. Electronic Notes in Theoretical Computer Science 134(1), 127–151 (2005)CrossRefGoogle Scholar
  17. 17.
    Stapleton G., Howse J., Rodgers P., Zhang L.: ZhangGenerating Euler Diagrams from existing layouts, Layout of (Software) Engineering Diagrams 2008. Electronic Communications of the EASST 13, 16–31 (2008)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.College of CommerceNihon UniversityTokyoJapan

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