Skip to main content
SpringerLink
Log in

Some Notes on Proofs with Alpha Graphs

  • Conference paper
Conceptual Structures: Inspiration and Application (ICCS 2006)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4068))

Included in the following conference series:

Abstract

It is well-known that Peirce’s Alpha graphs correspond to propositional logic (PL). Nonetheless, Peirce’s calculus for Alpha graphs differs to a large extent to the common calculi for PL. In this paper, some aspects of Peirce’s calculus are exploited. First of all, it is shown that the erasure-rule of Peirce’s calculus, which is the only rule which does not enjoy the finite choice property, is admissible. Then it is shown that this calculus is faster than the common cut-free calculi for propositional logic by providing formal derivations with polynomial lengths of Statman’s formulas. Finally a natural generalization of Peirce’s calculus (including the erasure-rule) is provided such that we can find proofs linear in the number of propositional variables used in the formular, depending on the number of propositional variables in the formula.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Brünnler, K.: Deep Inference and Symmetry in Classical Proofs. PhD thesis, Technische Universität Dresden (2003)

    Google Scholar 

  2. Burch, R.W.: A Peircean Reduction Thesis: The Foundation of Topological Logic. Texas Tech. University Press, Texas (1991)

    Google Scholar 

  3. Baaz, M., Zach, R.: Short proofs of tautologies using the schema of equivalence. In: Meinke, K., Börger, E., Gurevich, Y. (eds.) CSL 1993. LNCS, vol. 832, pp. 33–35. Springer, Heidelberg (1994)

    Chapter  Google Scholar 

  4. Dau, F.: An embedding of existential graphs into concept graphs with negations. In: Priss, U., Corbett, D.R., Angelova, G. (eds.) ICCS 2002. LNCS (LNAI), vol. 2393, pp. 326–340. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  5. Dau, F.: Types and tokens for logic with diagrams. In: Wolff, K.E., Pfeiffer, H.D., Delugach, H.S. (eds.) ICCS 2004. LNCS (LNAI), vol. 3127, pp. 62–93. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  6. Dau, F.: Mathematical logic with diagrams, based on the existential graphs of peirce. Habilitation thesis (to be published, 2006), available at: http://www.dr-dau.net

  7. Dau, F., Mugnier, M.-L., Stumme, G. (eds.): Common Semantics for Sharing Knowledge: Contributions to ICCS 2005. Kassel University Press, Kassel (2005)

    Google Scholar 

  8. Hammer, E.M.: Logic and Visual Information. CSLI Publications, Stanford (1995)

    MATH  Google Scholar 

  9. Hartshorne, W., Burks (eds.): Collected Papers of Charles Sanders Peirce. Harvard University Press, Cambridge (1931-1935)

    Google Scholar 

  10. Hodgins, D.P., Kocura, P.: Propositional theorem prover for peirce-logik. In: Dau, et al. (eds.) [DMS 2005], pp. 203–204

    Google Scholar 

  11. Liu, X.-W.: An axiomatic system for peirce’s alpha graphs. In: Dau, et al. (eds.) [DMN 2005], pp. 122–131

    Google Scholar 

  12. Pape, H.: Charles S. Peirce: Phänomen und Logik der Zeichen. Suhrkamp Verlag Wissenschaft, Frankfurt am Main, Germany (1983); German translation of Peirce’s Syllabus of Certain Topics of Logic

    Google Scholar 

  13. Peirce, C.S.: MS 478: Existential Graphs. Harvard University Press, (1931–1935); Partly published in of [HB35] (4.394-417). Complete german translation in [Pap83]

    Google Scholar 

  14. Peirce, C.S.: Reasoning and the logic of things. In: Kremer, K.L., Putnam, H. (eds.) The Cambridge Conferences Lectures of 1898. Harvard Univ. Press, Cambridge (1992)

    Google Scholar 

  15. Peirce, C.S., Sowa, J.F.: Existential Graphs: MS 514 by Charles Sanders Peirce with commentary J. Sowa (1908) (2000),available at: http://www.jfsowa.com/peirce/ms514.htm

  16. Roberts, D.D.: The Existential Graphs of Charles S. Peirce. Mouton, The Hague (1973)

    Google Scholar 

  17. Roberts, D.D.: The existential graphs. Computers Math. Appl. 23(6–9), 639–663 (1992)

    Article  MATH  Google Scholar 

  18. Schütte, K.: Beweistheorie. Springer, Heidelberg (1960)

    MATH  Google Scholar 

  19. Shin, S.-J.: The Iconic Logic of Peirce’s Graphs. Bradford Book, Massachusetts (2002)

    MATH  Google Scholar 

  20. Shin, S.-J.: Multiple readings in peirce’s alpha graphs. In: Anderson, M., Meyer, B., Olivier, P. (eds.) Diagrammatic Representation and Reasoning. Springer, Heidelberg (2002)

    Google Scholar 

  21. Sowa, J.F.: Conceptual structures: information processing in mind and machine. Addison-Wesley, Reading (1984)

    MATH  Google Scholar 

  22. Sowa, J.F.: Logic: Graphical and algebraic, Croton-on-Hudson (1997) (manuscript)

    Google Scholar 

  23. Statman, R.: Bounds for proof-search and speed-up in predicate calculus. Annals of Mathematical Logic 15, 225–287 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  24. van Heuveln, B.: Existential graphs. (2003), Presentations and Applications at: http://www.rpi.edu/~heuveb/research/EG/eg.html

  25. Yukami, T.: Some results on speed-up. Ann. Japan Assoc. Philos. Sci. 6, 195–205 (1984)

    MATH  MathSciNet  Google Scholar 

  26. Zeman, J.J.: The Graphical Logic of C. S. Peirce. PhD thesis, University of Chicago (1964), available at: http://www.clas.ufl.edu/users/jzeman/

Download references

Author information

Authors and Affiliations

  1. Technische Universität Dresden, Dresden, Germany

    Frithjof Dau

Authors
  1. Frithjof Dau
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Communication and Psychology, Aalborg University,  

    Henrik Schärfe

  2. AIFB Institute, Universität Karlsruhe (TH), D-76128, Karlsruhe, Germany

    Pascal Hitzler

  3. Department of Communication, Aalborg University, Kroghstraede 3, 9220, Aalborg, Denmark

    Peter Øhrstrøm

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Dau, F. (2006). Some Notes on Proofs with Alpha Graphs. In: Schärfe, H., Hitzler, P., Øhrstrøm, P. (eds) Conceptual Structures: Inspiration and Application. ICCS 2006. Lecture Notes in Computer Science(), vol 4068. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11787181_13

Download citation

  • DOI: https://doi.org/10.1007/11787181_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35893-0

  • Online ISBN: 978-3-540-35902-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation