Advertisement

Abstract

The lattice of theories of Sowa and the formal concept analysis of Wille each address certain formal aspects of concepts, though for different purposes and with different technical apparatus. Each is successful in part because it abstracts away from many difficulties of living human concepts. Among these difficulties are vagueness, ambiguity, flexibility, context dependence, and evolution. The purpose of this paper is first, to explore the nature of these difficulties, by drawing on ideas from contemporary cognitive science, sociology, computer science, and logic. Secondly, the paper suggests approaches for dealing with these difficulties, again drawing on diverse literatures, particularly ideas of Peirce and Latour. The main technical contribution is a unification of several formal theories of concepts, including the geometrical conceptual spaces of Gärdenfors, the symbolic conceptual spaces of Fauconnier, the information flow of Barwise and Seligman, the formal concept analysis of Wille, the lattice of theories of Sowa, and the conceptual integration of Fauconnier and Turner; this unification works over any formal logic at all, or even multiple logics. A number of examples are given illustrating the main new ideas. A final section draws implications for future research. One motivation is that better ways for computers to integrate and process concepts under various forms of heterogeneity, would help with many important applications, including database systems, search engines, ontologies, and making the web more semantic.

Keywords

Description Logic Category Theory Algebraic Theory Conceptual Space Formal Concept Analysis 
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.
    Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P. (eds.): Description Logic Handbook, Cambridge (2003)Google Scholar
  2. 2.
    Barr, M.: *-autonomous categories and linear logic. Mathematical Structures in Computer Science 1, 159–178 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Barwise, J., Seligman, J.: Information Flow: Logic of Distributed Systems. Tracts in Theoretical Computer Science, vol. 44, Cambridge (1997)Google Scholar
  4. 4.
    Bell, J.L.: Toposes and Local Set Theories: An Introduction. Oxford Logic Guides, vol. 14, Oxford (1988)Google Scholar
  5. 5.
    Bloor, D.: Knowledge and Social Imagery, 2nd edn., Chicago (1991)Google Scholar
  6. 6.
    Blumer, H.: Symbolic Interactionism: Perspective and Method, California (1986)Google Scholar
  7. 7.
    Burstall, R., Goguen, J.: Putting theories together to make specifications. In: Reddy, R. (ed.) Proceedings of Fifth International Joint Conference on Artificia Intelligence, pp. 1045–1058. Department of Computer Science, Carnegie-Mellon University (1977)Google Scholar
  8. 8.
    Clavel, M., Eker, S., Lincoln, P., Meseguer, J.: Principles of Maude. In: Meseguer, J. (ed.) Proceedings of First International Workshop on Rewriting Logic and its Applications. Electronic Notes in Theoretical Computer Science, vol. 4. Elsevier Science, Amsterdam (1996)Google Scholar
  9. 9.
    Cole, M.: Cultural Psychology: A once and future discipline. Harvard (1996)Google Scholar
  10. 10.
    Deacon, T.: Memes as signs in the dynamic logic of semiosis: Beyond molecular science and computation theory. In: Wolff, K.E., Pfeiffer, H.D., Delugach, H.S. (eds.) ICCS 2004. LNCS (LNAI), vol. 3127, pp. 17–30. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Diaconescu, R., Futatsugi, K.: CafeOBJ Report: The Language, Proof Techniques, and Methodologies for Object-Oriented Algebraic Specification. AMAST Series in Computing, vol. 6. World Scientific, Singapore (1998)zbMATHGoogle Scholar
  12. 12.
    Eilenberg, S., Lane, S.M.: General theory of natural equivalences. Transactions of the American Mathematical Society 58, 231–294 (1945)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Fauconnier, G.: Mental Spaces: Aspects of Meaning Construction in Natural Language. MIT, Bradford (1985)Google Scholar
  14. 14.
    Fauconnier, G., Turner, M.: The Way We Think. Basic (2002)Google Scholar
  15. 15.
    Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer, Heidelberg (1997)Google Scholar
  16. 16.
    Gärdenfors, P.: Conceptual Spaces: The Geometry of Thought, Bradford (2000)Google Scholar
  17. 17.
    Garfinkel, H.: Studies in Ethnomethodology. Prentice-Hall, Englewood Cliffs (1967)Google Scholar
  18. 18.
    Gibson, J.: An Ecological Approach to Visual Perception. Houghton Mifflin (1979)Google Scholar
  19. 19.
    Glaser, B., Strauss, A.: The Discovery of Grounded Theory: Strategies for qualitative research. Aldine de Gruyter, Berlin (1999)Google Scholar
  20. 20.
    Goguen, J.: The logic of inexact concepts. Synthese 19, 325–373 (1969)zbMATHCrossRefGoogle Scholar
  21. 21.
    Goguen, J.: A categorical manifesto. Mathematical Structures in Computer Science 1(1), 49–67 (March 1991)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Goguen, J.: Towards a social, ethical theory of information. In: Bowker, G., Star, L., Turner, W., Gasser, L. (eds.) Social Science, Technical Systems and Cooperative Work: Beyond the Great Divide, pp. 27–56. Erlbaum, Mahwah (1997)Google Scholar
  23. 23.
    Goguen, J.: An introduction to algebraic semiotics, with applications to user interface design. In: Nehaniv, C.L. (ed.) CMAA 1998. LNCS (LNAI), vol. 1562, pp. 242–291. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  24. 24.
    Goguen, J.: Semiotic morphisms, representations, and blending for interface design. In: Proceedings of AMAST Workshop on Algebraic Methods in Language Processing, pp. 1–15. AMAST Press (2003)Google Scholar
  25. 25.
    Goguen, J.: Data, schema and ontology integration. In: Proceedings of Workshop on Combinination of Logics, Center for Logic and Computation, Instituto Superior Tecnico, Lisbon, Portugal, pp. 21–31 (2004)Google Scholar
  26. 26.
    Goguen, J.: Information integration in instutions. In: Moss, L. (ed.) Memorial volume for Jon Barwise, Indiana (to appear)Google Scholar
  27. 27.
    Goguen, J., Burstall, R.: A study in the foundations of programming methodology: Specifications, institutions, charters and parchments. In: Pitt, D., Abramsky, S., Poigné, A., Rydeheard, D. (eds.) Proceedings of Conference on Category Theory and Computer Programming. LNCS, vol. 240, pp. 313–333. Springer, Heidelberg (1986); also Report CSLI-86-54, Center for the Study of Language and Information, Stanford University (June 1986)Google Scholar
  28. 28.
    Goguen, J., Burstall, R.: Institutions: Abstract model theory for specification and programming. Journal of the Association for Computing Machinery 39(1), 95–146 (January 1992)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Goguen, J., Harrell, F.: Style as a choice of blending principles. In: Argamon, S., Dubnov, S., Jupp, J. (eds.) Style and Meaning in Language, Art Music and Design, pp. 49–56. AAAI Press, Menlo Park (2004)Google Scholar
  30. 30.
    Goguen, J., Harrell, F.: Foundations for active multimedia narrative: Semiotic spaces and structural blending. To appear in Interaction Studies: Social Behaviour and Communication in Biological and Artificial Systems (2005)Google Scholar
  31. 31.
    Goguen, J., Lin, K.: Behavioral verification of distributed concurrent systems with BOBJ. In: Ehrich, H.-D., Tse, T.H. (eds.) Proceedings of Conference on Quality Software, pp. 216–235. IEEE Press, Los Alamitos (2003)CrossRefGoogle Scholar
  32. 32.
    Goguen, J., Meseguer, J.: Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science 105(2), 217–273 (1992); Drafts exist from as early as 1985zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Goguen, J., Roşu, G.: Institution morphisms. Formal Aspects of Computing 13, 274–307 (2002)zbMATHCrossRefGoogle Scholar
  34. 34.
    Goguen, J., Thatcher, J., Wagner, E., Wright, J.: Initial algebra semantics and continuous algebras. Journal of the Association for Computing Machinery 24(1), 68–95 (January 1977)zbMATHMathSciNetGoogle Scholar
  35. 35.
    Goguen, J., Tracz, W.: An implementation-oriented semantics for module composition. In: Leavens, G., Sitaraman, M. (eds.) Foundations of Component-based Systems, Cambridge, pp. 231–263 (2000)Google Scholar
  36. 36.
    Goguen, J., Winkler, T., Meseguer, J., Futatsugi, K., Jouannaud, J.-P.: Introducing OBJ. In: Goguen, J., Malcolm, G. (eds.) Software Engineering with OBJ: Algebraic Specification in Action, pp. 3–167. Kluwer, Dordrecht (2000)Google Scholar
  37. 37.
    Green, R.: Internally-structured conceptual models in cognitive semantics. In: Green, R., Bean, C., Myaeng, S.H. (eds.) The Semantics of Relationships, pp. 73–90. Kluwer, Dordrecht (2002)Google Scholar
  38. 38.
    Harnad, S.: The symbol grounding problem. Physica D 42, 335–346 (1990)CrossRefGoogle Scholar
  39. 39.
    Hatcher, W.S.: Foundations of Mathematics. W.B. Saunders, Philadelphia (1968)zbMATHGoogle Scholar
  40. 40.
    Hutchins, E.: Cognition in the Wild. MIT, Cambridge (1995)Google Scholar
  41. 41.
    Keeler, M.: Hegel in a strange costume. In: Ganter, B., de Moor, A., Lex, W. (eds.) ICCS 2003. LNCS, vol. 2746, pp. 37–53. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  42. 42.
    Kent, R.E.: Distributed conceptual structures. In: de Swart, H. (ed.) RelMiCS 2001. LNCS, vol. 2561, pp. 104–123. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  43. 43.
    Kent, R.: Formal or axiomatic semantics in the IFF (2003); Available at: http://www.suo.ieee.org/IFF/work-in-progress/
  44. 44.
    Kuhn, T.: The Structure of Scientific Revolutions, Chicago (1962)Google Scholar
  45. 45.
    Labov, W.: Language in the Inner City. University of Pennsylvania (1972)Google Scholar
  46. 46.
    Lakoff, G.: Women, Fire and Other Dangerous Things: What categories reveal about the mind, Chicago (1987)Google Scholar
  47. 47.
    Lakoff, G., Johnson, M.: Philosophy in the Flesh: The Embodied Mind and its Challenge to Western Thought. Basic (1999)Google Scholar
  48. 48.
    Latour, B.: We Have Never Been Modern. Harvard (1993); Translated by Catherine PorterGoogle Scholar
  49. 49.
    Latour, B., Woolgar, S.: Laboratory Life. Sage, Thousand Oaks (1979)Google Scholar
  50. 50.
    Livingston, E.: The Ethnomethodology of Mathematics. Routledge & Kegan Paul (1987)Google Scholar
  51. 51.
    Lane, S.M.: Categories for the Working Mathematician, 2nd edn. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  52. 52.
    MacKenzie, D.: Mechanizing Proof. MIT, Cambridge (2001)zbMATHGoogle Scholar
  53. 53.
    Mossakowski, T., Goguen, J., Diaconescu, R., Tarlecki, A.: What is a logic? In: Beziau, J.-Y. (ed.) Logica Universalis, Proceedings of First World Conference on Universal Logic. Birkhauser, Basel (2005)Google Scholar
  54. 54.
    Mosses, P. (ed.): CASL Reference Manual. LNCS, vol. 2960. Springer, Heidelberg (2004)zbMATHGoogle Scholar
  55. 55.
    Nagarjuna: Mulamadhyamika Karaka. Oxford (1995); Translated by Jay GarfieldGoogle Scholar
  56. 56.
    Peirce, C.S.: Collected Papers. Harvard (1965); In 6 volumes; see especially Volume 2: Elements of LogicGoogle Scholar
  57. 57.
    Pierce, B.C.: Basic Category Theory for Computer Scientists. MIT, Cambridge (1991)Google Scholar
  58. 58.
    Pribbenow, S.: Merenymic relationships: From classical mereology to complex part-whole relations. In: Green, R., Bean, C., Myaeng, S.H. (eds.) The Semantics of Relationships, pp. 35–50. Kluwer, Dordrecht (2002)Google Scholar
  59. 59.
    Sacks, H.: On the analyzability of stories by children. In: Gumpertz, J., Hymes, D. (eds.) Directions in Sociolinguistics, pp. 325–345. Holt, Rinehart and Winston (1972)Google Scholar
  60. 60.
    Schlorlemmer, M., Kalfoglou, Y.: A channel-theoretic foundation for ontology coordination. In: Proceedings of 18th European Workshop on Multi-Agent Systems (2004)Google Scholar
  61. 61.
    Sowa, J.: Knowledge Representation: Logical, Philosophical and Computational Foundations. Brooks/Coles (2000)Google Scholar
  62. 62.
    Star, S.L.: The structure of ill-structured solutions: Boundary objects and heterogeneous problem-solving. In: Gasser, L., Huhns, M. (eds.) Distributed Artificial Intelligence, vol. 2, pp. 37–54. Pitman (1989)Google Scholar
  63. 63.
    Tarlecki, A., Burstall, R., Goguen, J.: Some fundamental algebraic tools for the semantics of computation, part 3: Indexed categories. Theoretical Computer Science 91, 239–264 (1991); Also, Monograph PRG–77, Programming Research Group, Oxford University (August 1989)Google Scholar
  64. 64.
    Vygotsky, L.: Thought and Language. MIT, Cambridge (1962)CrossRefGoogle Scholar
  65. 65.
    Vygotsky, L.: Mind in Society. Harvard (1985)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Joseph Goguen
    • 1
  1. 1.Dept. Computer Science & EngineeringUniversity of California at San DiegoLa JollaUSA

Personalised recommendations