The Logic of Uncertainty and Geometry of the Worlds

Part of the Studies in Computational Intelligence book series (SCI, volume 407)

Abstract

The classical logic is built in an axiomatic way without containing any uncertainty. Over the years, many attempts have been made to extend the classical logic framework to incorporate uncertainties. This work has produced different and apparently conflicting definitions of uncertainties. Previously we have argued that an extended modal logic framework can provide a unifying formal language for many formalizations of uncertainty such as probability, approximate probability, evidence theory, fuzzy sets, and rough sets.

In this chapter, we have shown how such an extended modal of logic framework provides a unifying formal language for formalizations of probability and fuzzy sets theories. We also show that this originally pure theoretical approach can be used to describe a mathematical model for linguistic uncertainty in a unified framework. The approach combines concepts of modal logic together the linguistic context space previously proposed. This combined approach is illustrated with an application to the question of economical preference between customers and goods.

A common language able to represent different types of uncertainties is useful because it allows us to define the uncertainties using simple entities and to compare different types of uncertainties as being different aspects of the same fundamental structure. The justification of different theories of uncertainty using a unified language is beyond the scope of this paper.

Our goal is to rebuild a foundation of the uncertainty concept by using a new interpretation of the modal logic structure. Using this new foundation we discover that disparate types of uncertainties and the idea of uncertainty itself can be understood. This new foundation opens opportunities to discover hidden connection between different types of uncertainties.

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References

  1. 1.
    Carnap, R., Jeffrey, R.: Studies in Inductive Logics and Probability, vol. 1, pp. 35–165. University of California Press, Berkeley (1971)Google Scholar
  2. 2.
    Fagin, R., Halpern, J.: Reasoning about Knowledge and Probability. Journal of the ACM 41(2), 340–367 (1994)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Edmonds, B.: Review of Reasoning about Rational Agents by Michael Wooldridge. Journal of Artificial Societies and Social Simulation 5(1) (2002), http://jasss.soc.surrey.ac.uk/5/1/reviews/edmonds.html
  4. 4.
    Ferber, J.: Multi Agent Systems. Addison Wesley (1999)Google Scholar
  5. 5.
    Gigerenzer, G., Selten, R.: Bounded Rationality. The MIT Press, Cambridge (2002)Google Scholar
  6. 6.
    Halpern, J.: Reasoning about uncertainty. MIT Press (2005)Google Scholar
  7. 7.
    Hisdal, E.: Logical Structures for Representation of Knowledge and Uncertainty. Springer, Heidelberg (1998)MATHGoogle Scholar
  8. 8.
    Resconi, G., Jain, L.: Intelligent agents. Springer, Heidelberg (2004)MATHGoogle Scholar
  9. 9.
    Resconi, G., Kovalerchuk, B.: The Logic of Uncertainty with Irrational Agents. In: Proc. of JCIS-2006 Advances in Intelligent Systems Research. Atlantis Press, Taiwan (2006)Google Scholar
  10. 10.
    Kahneman, D.: Maps of Bounded Rationality: Psychology for Behavioral Economics. The American Economic Review 93(5), 1449–1475 (2003)CrossRefGoogle Scholar
  11. 11.
    Kovalerchuk, B.: Analysis of Gaines’ logic of uncertainty. In: Turksen, I.B. (ed.) Proceeding of NAFIPS 1990, Toronto, Canada, vol. 2, pp. 293–295 (1990)Google Scholar
  12. 12.
    Kovalerchuk, B.: Context spaces as necessary frames for correct approximate reasoning. International Journal of General Systems 25(1), 61–80 (1996)MATHCrossRefGoogle Scholar
  13. 13.
    Kovalerchuk, B., Vityaev, E.: Data mining in finance: advances in relational and hybrid methods. Kluwer (2000)Google Scholar
  14. 14.
    Wooldridge, M.: Reasoning about Rational Agents. The MIT Press, Cambridge (2000)MATHGoogle Scholar
  15. 15.
    Montero, J., Gomez, D., Bustine, H.: On the relevance of some families of fuzzy sets. Fuzzy Sets and Systems (2007) (in print)Google Scholar
  16. 16.
    Atanassov, K.T.: Intuitionistic Fuzzy Sets. Physica Verlag, Springer, Heidelberg (1999)MATHGoogle Scholar
  17. 17.
    Flament, C.: Applications of graphs theory to group structure. Prentice Hall, London (1963)Google Scholar
  18. 18.
    Ruspini, E.H.: A new approach to clustering. Information and Control 15, 22–32 (1969)MATHCrossRefGoogle Scholar
  19. 19.
    Priest, G., Tanaka, K.: Paraconsistent Logic. Stanford Encyclopedia of Philosophy (2004), http://plato.stanford.edu/entries/logic-paraconsistent/

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Dept. Mathematics and PhysicsCatholic UniversityBresciaItaly

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