Logical fiberings and polycontextural systems

  • J. Pfalzgraf
Part II Selected Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 535)


Based on the notion of abstract fiber spaces the concept of a logical fibering is developed. This was motivated by a project where so-called polycontextural logics were discussed. The fiber space approach provides a rather general framework for the modeling of such non classical logics. It gives the possibility to construct a variety of new logical spaces from a given (indexed) system of logics. These spaces are in some sense parallel (inference) systems. We can give a straight forward definition and classification of the so-called transjunctions arising in polycontextural logics. These are bivariate operations having values distributed over different logical subsystems. Univariate, bivariate operations are introduced in functional notation. The group generated by the generalized negation operations and system changes is investigated. We make some remarks on aspects of applicability and links to other work.


Base Space Logical Space Fiber Space Input Pair Valuation Versus 
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10 References

Selected Literature on Polycontextural Logic

  1. [G1]
    G. Günther. Beiträge zur Grundlegung einer operationsfähigen Dialektik, 3 volumes. Felix Meiner Verlag, Hamburg 1980Google Scholar
  2. [G2]
    G. Günther. Cybernetic Ontology and Transjunctional Operations. Biological Computer Lab. Publ. vol.68 (Urbana, Ill.), published in 'self-organizing Systems 1962', Spartan Books, Washington, D.C., pp..313–392 (1962).Google Scholar
  3. [K1]
    R.Kaehr. Materialien zur Formalisierung der dialektischen Logik und der Morphogrammatik 1973–1975. In: Idee und Grundriß einer nicht-aristotelischen Logik, 2.Auflage, Hamburg 1978.Google Scholar
  4. [K2]
    R.Kaehr. Excurs zu Logica. In: Die Logik des Wissens und das Problem der Erziehung, Hamburg 1981.Google Scholar
  5. [R1]
    READER 1. Texte G. Günther's zur Polykontexturalen Logik und Arithmetik, Morphogrammatik und Kenogrammatik, Proemial-Relation, Strukturtypentheorie, Cybernetic Ontology. Collected by R.Kaehr, University Witten/Herdecke 1988.Google Scholar
  6. [R2]
    READER 2. Texte R.Kaehr's zur Formalisierung der Polykontexturalen Logik. University Witten/Herdecke 1988.Google Scholar

Literature on Categories, Logics, Fiberings

  1. [vD]
    D. van Dalen. Logic and Stucture, 2nd ed. Springer Universitext, 1983.Google Scholar
  2. [DO]
    C.T.J.Dodson. Categories, Bundles and Spacetime Topology. Kluwer Academic Publ. 1988.Google Scholar
  3. [E]
    P.Érdi. On the Ultrametric Structure of Semantic Memory: Scope and Limits. In: R. Trappl (ed.), Cybernetics and Systems '88, pp.329–336.Google Scholar
  4. [GA]
    D. Gabbay. LDS-Labelled Deductive Systems. MEDLAR Milestone 1 Deliverables, Oct 1990. (Chapter 1 of the draft of a book). To be published.Google Scholar
  5. [GO]
    R.Goldblatt. Topoi. Studies in Logic and the Foundations of Mathematics, vol.98, North-Holland 1986.Google Scholar
  6. [LNCS]
    D.Pitt, S.Abramsky, A.Poigné and D.Rydeheard (eds.). Category Theory and Computer Programming. Springer Lecture Notes in Computer Science, vol.240, 1986.Google Scholar
  7. [LN M1]
    P.T.Johnstone, R.Paré (eds.). Indexed Categories and Their Applications. Springer Lecture Notes in Mathematics, vol.661, 1978.Google Scholar
  8. [LN M2]
    M.P.Fourman, C.J.Mulvey and D.S.Scott. Applications of Sheaves. Springer Lecture Notes in Mathematics, vol.753, 1979.Google Scholar
  9. [LS]
    J.Lambek and P.J.Scott. Introduction to Higher Order Categorical Logic. Cambridge Studies in Advanced Mathematics, vol.7, Cambridge University Press 1986.Google Scholar
  10. [ML]
    S.MacLane. Categories for the Working Mathematician. Springer Graduate Text in Mathematics, vol.5, 1971.Google Scholar
  11. [PF1]
    J.Pfalzgraf. Reasoning on a Möbius Strip. MEDLAR Newsletter No.1, Sept-Nov 1990, J.Cunningham, D.Gabbay, R. de Queiroz (eds.), Imperial College London.Google Scholar
  12. [PF2]
    J. Pfalzgraf. Representation of geometric spaces as Fibered Structures. Results in Math. Vol.12 (1987), 172–190 (in German).Google Scholar
  13. [PF3]
    J.Pfalzgraf. On Logical Fiberings and Polycontextural Systems. A First Approach. RISC-Linz Publ. Series No. 91–13.0 (1991).Google Scholar
  14. [RB]
    D.E.Rydeheard and R.M.Burstall. Computational Category Theory. Prentice Hall 1988.Google Scholar
  15. [S]
    N.Steenrod. The Topology of Fibre Bundles. Princeton University Press 1951.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • J. Pfalzgraf
    • 1
  1. 1.RISC-Linz Johannes Kepler UniversityLinzAustria

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