Advertisement

Studia Logica

, Volume 105, Issue 1, pp 121–152 | Cite as

Equivocation Axiom on First Order Languages

  • Soroush Rafiee Rad
Open Access
Article
  • 208 Downloads

Abstract

In this paper we investigate some mathematical consequences of the Equivocation Principle, and the Maximum Entropy models arising from that, for first order languages. We study the existence of Maximum Entropy models for these theories in terms of the quantifier complexity of the theory and will investigate some invariance and structural properties of such models.

Keywords

Maximum Entropy models Probabilistic reasoning Equivocation principle 

References

  1. 1.
    Bacchus, F., A. J. Grove, J. Y. Halpern, and D. Koller, Generating new beliefs from old, Proceedings of the Tenth Annual Conference on Uncertainty in Artificial Intelligence, (UAI-94), 1994, pp. 37–45.Google Scholar
  2. 2.
    Barnett O. W., J. B. Paris.: Maximum Entropy inference with qualified knowledge. Logic Journal of the IGPL 16(1), 85–98 (2008)CrossRefGoogle Scholar
  3. 3.
    Berger A., Della Pietra S., Della Pietra V.: A maximum entropy approach to natural language processing. Computational Linguistics 22(1), 39–71 (1996)Google Scholar
  4. 4.
    Chen, C. H., Maximum entropy analysis for pattern recognition, in P. F. Fougere (ed.), Maximum Entropy and Bayesian Methods, Kluwer Academic Publisher, London, 1990.Google Scholar
  5. 5.
    Gaifman H.: Concerning measures in first order calculi. Israel Journal of Mathematics 24, 1–18 (1964)CrossRefGoogle Scholar
  6. 6.
    Grotenhuis, M. G., An Overview of the Maximum Entropy Method of Image Deconvolution, A University of Minnesota Twin Cities Plan B Masters paper.Google Scholar
  7. 7.
    Grove A. J., J. Y. Halpern, and D. Koller, Asymptotic conditional probabilities: the unary case, SIAM Journal of Computing 25(1):1–51, 1996.Google Scholar
  8. 8.
    Jaynes, E. T., Information theory and statistical mechanics, Physical Reviews 106:620–630, 108:171–190, 1957.Google Scholar
  9. 9.
    Jaynes, E. T., Notes on present status and future prospects, in W. T. Grandy and L. H. Schick (eds.), Maximum Entropy and Bayesian Methods, Kluwer, London, 1990, pp. 1–13.Google Scholar
  10. 10.
    Jaynes, E. T., How Should We Use Entropy in Economics? 1991, manuscript available at: http://www.leibniz.imag.fr/LAPLACE/Jaynes/prob.html.
  11. 11.
    Kapur J. N.: Twenty five years of maximum entropy. Journal of Mathematical and Physical Sciences 17(2), 103–156 (1983)Google Scholar
  12. 12.
    Kapur J. N.: Non-additive measures of entropy and distributions of statistical mechanics. Indian Journal of Pure and Applied Mathematics 14(11), 1372–1384 (1983)Google Scholar
  13. 13.
    Landes J., Williamson J.: Objective Bayesianism and the maximum entropy principle. Entropy 15(9), 3528–3591 (2013)CrossRefGoogle Scholar
  14. 14.
    Landes J., Williamson J.: Justifying objective bayesianism on predicate languages. Entropy 17, 2459–2543 (2015)CrossRefGoogle Scholar
  15. 15.
    Paris J. B.: The Uncertain Reasoner’s Companion. Cambridge University Press, Cambridge (1994)Google Scholar
  16. 16.
    Paris J. B., Vencovská A.: A note on the inevitability of maximum entropy. International Journal of Approximate Reasoning 4(3), 183–224 (1990)CrossRefGoogle Scholar
  17. 17.
    Paris J.B., Vencovská A.: In defense of the maximum entropy inference process. International Journal of Approximate Reasoning 17(1), 77–103 (1997)CrossRefGoogle Scholar
  18. 18.
    Paris, J. B., and S. Rafiee Rad, A note on the least informative model of a theory, in F. Ferreira, B. Löwe, E. Mayordomo, and L. Mendes Gomes (eds.), Programs Proofs Processes, CiE 2010, Springer LNCS 6158, 2010, pp. 342–351.Google Scholar
  19. 19.
    Rafiee Rad, S., Inference Processes For Probabilistic First Order Languages. PhD Thesis, University of Manchester, 2009. http://www.maths.manchester.ac.uk/~jeff/
  20. 20.
    Rosenkrantz, R. D., Inference, Method and Decision: Towards a Bayesian Philosophy of Science, Reidel, Dordrecht, 1977.Google Scholar
  21. 21.
    Shannon C. E., Weaver W.: The Mathematical Theory of Communication. University of Illinois Press, Champaign (1949)Google Scholar
  22. 22.
    Williamson J.: Bayesian nets and causality: philosophical and computational foundations. Oxford University Press, Oxford (2005)Google Scholar
  23. 23.
    Williamson J.: Objective Bayesian probabilistic logic. Journal of Algorithms in Cognition, Informatics and Logic 63, 167–183 (2008)Google Scholar
  24. 24.
    Williamson, J., In Defence of Objective Bayesianism, Oxford University Press, Oxford, 2010, pp. 167–183.Google Scholar
  25. 25.
    Zellner, A., Bayesian methods and entropy in economics and econometrics, in W. T. Grandy and L. H. Schick (eds.), Maximum Entropy and Bayesian Methods, 1991.Google Scholar

Copyright information

© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute for Logic, Language and Computation, UvAAmsterdamNetherlands

Personalised recommendations