Advertisement

Abstract

The aim of this paper is to show how a Boolean Concept Logic may be elaborated as a mathematical theory based on Formal Concept Analysis  [GW96]. For this purpose, concept lattices are extended by further operations, mainly negation and opposition. Two extensions are discussed which lead, on the one hand, to algebras of protoconcepts equationally equivalent to double Boolean algebras and, on the other hand, to concept algebras quasi-equationally equivalent to dicomplemented lattices. In both cases, basic representation theorems are proved. These results are not only basic for Contextual Concept Logic but also for Contextual Judgment Logic with its theory of concept graphs.

Keywords

Boolean Algebra Formal Concept Equational Theory Conceptual Structure Concept Lattice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BM74.
    Beierwaltes, W., Menne, A.: Gegensatz. In: Ritter†, J., Gründer, K. (eds.) Historisches Wörterbuch der Philosophie, Bd. 3, pp. 105–119. Schwabe & Co, Basel (1974)Google Scholar
  2. Bo54.
    Boole, G.: An investigation of the laws of thought, on which are founded the mathematical theories of logic and probabilities. Macmillan, Basingstoke (1854); Reprinted by Dover Publ., New York (1958)Google Scholar
  3. DP92.
    Davey, B.A., Priestley, H.: Introduction to lattices and order. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  4. GW96.
    Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer, Heidelberg (1999); German version: Springer, Heidelberg (1996)Google Scholar
  5. GW99 .
    Ganter, B., Wille, R.: Contextual attribute logic. In: Tepfenhart, W., Cyre, W. (eds.) ICCS 1999. LNCS (LNAI), vol. 1640, pp. 377–388. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  6. HLSW99.
    Herrman, C., Luksch, P., Skorsky, M., Wille, R.: Algebras of semiconcepts and double Boolean algebras. Contributions to General Algebra 13 (to appear)Google Scholar
  7. KL67.
    Kamlah, W., Lorenzen, P.: Logische Propädeutik. Vorschule des vernünftigen Redens. B.I.-HTB 227, Mannheim (1967)Google Scholar
  8. Ka88.
    Kant, I.: Logic. Dover, New York (1988)Google Scholar
  9. LW91.
    Luksch, P., Wille, R.: A mathematical model for conceptual knowledge systems. In: Bock, H.-H., Ihm, P. (eds.) Classification, data analysis, and knowledge organization, pp. 156–162. Springer, Heidelberg (1991)Google Scholar
  10. Me84.
    Menne, A.: Negation. In: Ritter†, J., Gründer, K., (eds.): Historisches Wörterbuch der Philosophie, 6th Bd., pp. 666–670. Schwabe & Co, Basel (1984)Google Scholar
  11. MSW99.
    Mineau, G., Stumme, G., Wille, R.: Conceptual structures represented by conceptual graphs and formal concept analysis. In: Tepfenhart, W., Cyre, W. (eds.) ICCS 1999. LNCS, vol. 1640, pp. 423–441. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  12. Pr98 .
    Prediger, S.: Kontextuelle Urteilslogik mit Begriffsgraphen. Ein Beitrag zur Restrukturierung der mathematischen Logik. Dissertation, TU Darmstadt. Shaker Verlag, Aachen (1998)Google Scholar
  13. PW99.
    Prediger, S., Wille, R.: The lattice of concept graphs of a relationally scaled context. In: Tepfenhart, W.M. (ed.) ICCS 1999. LNCS, vol. 1640, pp. 401–414. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  14. Wi92.
    Wille, R.: Concept lattices and conceptual knowledge systems. Computers & Mathematics with Applications 23, 493–515 (1992)zbMATHCrossRefGoogle Scholar
  15. Wi96.
    Wille, R.: Restructuring mathematical logic: an approach based on Peirce’s pragmatism. In: Ursini, A., Agliano, P. (eds.) Logic and Algebra, pp. 267–281. Marcel Dekker, New York (1996)Google Scholar
  16. Wi97a .
    Wille, R.: Conceptual Graphs and Formal Concept Analysis. In: Lukose, D., Delugach, H., Keeler, M., Searle, L., Sowa, J.F. (eds.) ICCS 1997. LNCS (LNAI), vol. 1257, pp. 290–303. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  17. Wi97b.
    Wille, R.: Conceptual landscapes of knowledge: a pragmatic paradigm for knowledge processing. In: Gaul, W., Locarek-Junge, H. (eds.) Classification in the Information Age, pp. 344–356. Springer, Heidelberg (1999); already printed In: G. Mineau, A. Fall (eds.): Proc. 2nd Intl. KRUSE. Simon Fraser University, Vancouver, pp. 2–13 (1997)Google Scholar
  18. Wi98.
    Wille, R.: Triadic Concept Graphs. In: Mugnier, M.-L., Chein, M. (eds.) ICCS 1998. LNCS (LNAI), vol. 1453, pp. 194–208. Springer, Heidelberg (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Rudolf Wille
    • 1
  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadt

Personalised recommendations