The Teridentity and Peircean Algebraic Logic

  • Joachim Hereth Correia
  • Reinhard Pöschel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4068)

Abstract

A main source of inspiration for the work on Conceptual Graphs by John Sowa and on Contextual Logic by Rudolf Wille has been the Philosophy of Charles S. Peirce and his logic system of Existential Graphs invented at the end of the 19th century. Although Peirce has described the system in much detail, there is no formal definition which suits the requirements of contemporary mathematics.

In his book A Peircean Reduction Thesis: The Foundations of topological Logic, Robert Burch has presented the Peircean Algebraic Logic (PAL) which aims to reconstruct in an algebraic precise manner Peirce’s logic system.

Using a restriction on the allowed constructions, he is able to prove the Peircean Reduction Thesis, that in PAL all relations can be constructed from ternary relations, but not from unary and binary relations alone. This is a mathematical version of Peirce’s central claim that the category of thirdness cannot be decomposed into the categories of firstness and secondness.

Removing Burch’s restriction from PAL makes the system very similar to the system of Existential Graphs, but the proof of the Reduction Thesis becomes extremely complicated. In this paper, we prove that the teridentity relation is – as also elaborated by Burch – irreducible, but we prove this without the additional restriction on PAL. This leads to a proof of the Peircean Reduction Thesis.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Arn01]
    Arnold, M.: Einführung in die kontextuelle Relationenlogik. Diplomarbeit, Technische Universität Darmstadt (2001)Google Scholar
  2. [Bur91]
    Burch, R.W.: A Peircean Reduction Thesis: The Foundations of Topological Logic. Texas Tech University Press (1991)Google Scholar
  3. [DHC06]
    Dau, F., Correia, J.H.: Two Instances of Peirce’s Reduction Thesis. In: Missaoui, R., Schmidt, J. (eds.) Formal Concept Analysis. LNCS (LNAI), vol. 3874, pp. 105–118. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. [DaK05]
    Dau, F., Klinger, J.: From Formal Concept Analysis to Contextual Logic. In: Ganter, B., Stumme, G., Wille, R. (eds.) Formal Concept Analysis. LNCS (LNAI), vol. 3626, pp. 81–100. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  5. [HCP04]
    Hereth Correia, J., Pöschel, R.: The Power of Peircean Algebraic Logic (PAL). In: Eklund, P. (ed.) ICFCA 2004. LNCS (LNAI), vol. 2961, pp. 337–351. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. [Her81]
    Herzberger, H.G.: Peirce’s Remarkable Theorem. In: Sumner, L.W., Slater, J.G., Wilson, F. (eds.) Pragmatism and Purpose: Essays Presented to Thomas A. Goudge. University of Toronto Press, Toronto (1981)Google Scholar
  7. [PR67]
    Peirce, C.S.: The Peirce Papers in the Houghton Library, Harvard University. Catalogue by R. Robin: Annotated Catalogue of the Papers of Charles S. Peirce. University of Massachusetts Press, Massachusetts (1967)Google Scholar
  8. [Sow84]
    Sowa, J.F.: Conceptual structures: Information processing in mind and machine. Addison-Wesley, Reading (1984)MATHGoogle Scholar
  9. [Sow92]
    Sowa, J.F.: Conceptual graphs summary. In: Nagle, T., Nagle, J., Gerholz, L., Eklund, P. (eds.) Conceptual structures: Current Research and Practice, Ellis Horwood, pp. 3–52 (1992)Google Scholar
  10. [Sow00]
    Sowa, J.F.: Knowledge Representation: Logical, Philosophical, and Computational Foundations. Brooks Cole Publishing Co., Pacific Grove (1999)Google Scholar
  11. [Wil00b]
    Wille, R.: Contextual Logic summary. In: Stumme, G. (ed.) Working with conceptual structures. Contributions to ICCS 2000, pp. 265–276. Shaker-Verlag, Aachen (2000)Google Scholar
  12. [Wil00]
    Wille, R.: Lecture notes on Contextual Logic of Relations, Darmstadt University of Technology (2000) (preprint)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Joachim Hereth Correia
    • 1
  • Reinhard Pöschel
    • 1
  1. 1.Institut für AlgebraTechnische Universität Dresden, Fakultät Mathematik und NaturwissenschaftenDresdenGermany

Personalised recommendations