The Quest for Uncertainty

  • Jörg Zimmermann
  • Armin B. Cremers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6570)


The question of how to represent and process uncertainty is of fundamental importance to the scientific process, but also in everyday life. Currently there exist a lot of different calculi for managing uncertainty, each having its own advantages and disadvantages. Especially, almost all are defining the domain and structure of uncertainty values a priori, e.g., one real number, two real numbers, a finite domain, and so on, but maybe uncertainty is best measured by complex numbers, matrices or still another mathematical structure. Here we investigate the notion of uncertainty from a foundational point of view, provide an ontology and axiomatic core system for uncertainty, derive and not define the structure of uncertainty, and review the historical development of approaches to uncertainty which have led to the results presented here.


Uncertainty Measure Boolean Algebra Axiom System Inductive Logic Truth Bearer 
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.


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  1. 1.
    Arnborg, S., Sjödin, G.: What is the plausibility of probability? Preprint, Nada, KTH (2001)Google Scholar
  2. 2.
    Cox, R.T.: Probability, frequency, and reasonable expectation. Am. Jour. Phys. 14, 1–13 (1946)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dubois, D.: Possibility theory and statistical reasoning. Computational Statistics & Data Analysis 51(1), 47–69 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gärdenfors, P. (ed.): Belief Revision. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  5. 5.
    Ginsberg, M. (ed.): Readings in Nonmonotonic Reasoning. Morgan Kaufmann, Los Altos (1987)Google Scholar
  6. 6.
    Halpern, J.: A counterexample to theorems of Cox and Fine. Journal of A.I. Research 10, 76–85 (1999)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Halpern, J.: Reasoning about Uncertainty. MIT Press, Cambridge (2003)zbMATHGoogle Scholar
  8. 8.
    Hutter, M.: Universal Artificial Intelligence. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  9. 9.
    Jaynes, E.T.: Probability Theory: The Logic of Science. Cambridge University Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  10. 10.
    Knight, F.: Risk, Uncertainty, and Profit. Houghton Mifflin (1921)Google Scholar
  11. 11.
    Lehman, D., Magidor, M.: What does a conditional knowledge base entail? Artificial Intelligence 55(1), 1–60 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    MacLane, S., Birkhoff, G.: Algebra. The MacMillan Company, Basingstoke (1967)zbMATHGoogle Scholar
  13. 13.
    Paris, J.B.: The Uncertain Reasoner’s Companion. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  14. 14.
    Pearl, J.: Causality: Models, Reasoning, and Inference. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  15. 15.
    Schmidhuber, J.: Ultimate cognition à la Gödel. Cognitive Computation 1(2), 177–193 (2009)CrossRefGoogle Scholar
  16. 16.
    Shafer, G.: Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)zbMATHGoogle Scholar
  17. 17.
    Solomonoff, R.: A formal theory of inductive inference, part I. Information and Control 7(1), 1–22 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Solomonoff, R.: A formal theory of inductive inference, part II. Information and Control 7(2), 224–254 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Spohn, W.: A survey of ranking theory. In: Huber, F., Schmidt-Petri, C. (eds.) Degrees of Belief. Springer, Heidelberg (2009)Google Scholar
  20. 20.
    Zimmermann, J.: A proof outline for the ring conjecture (2010),

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Jörg Zimmermann
    • 1
  • Armin B. Cremers
    • 1
  1. 1.Institute of Computer ScienceUniversity of BonnGermany

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