Advertisement

Syntactic Variety in Boundary Logic

  • William Bricken
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4045)

Abstract

Boundary logic is a formal diagrammatic system that combines Peirce’s Entitative Graphs with Spencer Brown’s Laws of Form. Its conceptual basis includes boundary forms composed of non-intersecting closed curves, void-substitution (deletion of irrelevant structure) as the primary mechanism of reduction, and spatial pattern-equations that define valid transformations. Pure boundary algebra, free of interpretation, is first briefly described, followed by a description of boundary logic. Then several new diagrammatic notations for logic derived from geometrical and topological transformation of boundary forms are presented. The algebra and an example proof of modus ponens is provided for textual, enclosure, graph, map, path and block based forms. These new diagrammatic languages for logic convert connectives into configurations of containment, connectivity, contact, conveyance, and concreteness.

Keywords

Boundary Form Distinction Path Permeable Boundary Topological Transformation Conventional Logic 
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. 1.
    Kneale, W., Kneale, M.: The Development of Logic. Oxford Univ. Press, Oxford (1962)MATHGoogle Scholar
  2. 2.
    Peirce, C.S.: Collected Papers of Charles Sanders Peirce. In: Hartshorne, C., Weiss, P., Burks, A. (eds.). Harvard Univ. Press, Cambridge (1931–1958)Google Scholar
  3. 3.
    Spencer Brown, G.: Laws of Form. George Allen and Unwin (1969)Google Scholar
  4. 4.
    Bricken, W.: The Mathematics of Boundaries: A Beginning. In: Barker-Plummer, D., Cox, R., Swoboda, N. (eds.) Diagrams 2006. LNCS, vol. 4045, pp. 70–72. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Bricken, W., Gullichsen, E.: Introduction to Boundary Logic. Future Computing Systems 2:4, 1–77 (1989)Google Scholar
  6. 6.
    Bricken, W.: Distinction Networks. In: Wachsmuth, I., Brauer, W., Rollinger, C.-R. (eds.) KI 1995. LNCS, vol. 981, pp. 35–48. Springer, Heidelberg (1995)Google Scholar
  7. 7.
    James, J., Bricken, W.: A Boundary Notation for Visual Mathematics. In: 1992 IEEE Workshop on isual Languages, Seattle, pp. 267–269. IEEE Press, Los Alamitos (1992)CrossRefGoogle Scholar
  8. 8.
    Shin, S.: The Logical Status of Diagrams. Cambridge Univ. Press, Cambridge (1994)MATHGoogle Scholar
  9. 9.
    Kauffman, L.H., Varela, F.J.: Form Dynamics. J. Soc. Biol. Structures 3, 171–206 (1980)CrossRefGoogle Scholar
  10. 10.
    Barwise, J., Etchemendy, J.: Heterogeneous Logic. In: Allwein, G., Barwise, J. (eds.) Logical Reasoning with Diagrams. Oxford Univ. Press, Oxford (1996)Google Scholar
  11. 11.
    Shin, S.: The Iconic Logic of Peirce’s Graphs. MIT Press, Cambridge (2002)MATHGoogle Scholar
  12. 12.
    Hammer, E.: Logic and Visual Information. CSLI Publications, Stanford (1995)MATHGoogle Scholar
  13. 13.
    Halmos, P., Givant, S.: Logic as Algebra. Mathematical Assoc. of America (1998)Google Scholar
  14. 14.
    Birkoff, G.: On the Structure of Abstract Algebras. Proc. Cambridge Phil. Soc. 31, 417–429 (1935)Google Scholar
  15. 15.
    Stern, A.: Matrix Logic. North-Holland/Elsevier, Amsterdam (1988)MATHGoogle Scholar
  16. 16.
    Kauffman, L.H.: Knots and Physics, 2nd edn. World Scientific, Singapore (1993)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • William Bricken
    • 1
  1. 1.Boundary InstituteSaratogaUSA

Personalised recommendations