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A Logic and Computation for Popper’s Conditional Probabilities

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Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2021)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12897))

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Abstract

A Popper function is one that gives a conditional probability of propositional formulae. This paper gives a finite set of axioms that defines the set of Popper functions in many-sorted monadic second-order logic, and proves the decidability of the validity problem for a practically important first-order fragment of the second-order language with respect to a set of Popper functions. Upon these logical foundations, we propose, with results on their time complexity, two algorithms that compute the range of values that designated conditional probabilities can take under given constraints.

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Notes

  1. 1.

    The numbers are just randomly chosen for the sake of a concrete example.

  2. 2.

    In this sequent calculus, the probability is assigned to a sequent but it can be proved equal to that of the corresponding material implication.

  3. 3.

    A Popper function has a salient characterisation as the standard part of a conditional probability of hyperreal-valued probabilities [24]. This result is extended to the representation theorem by non-Archimedean probabilities in [10]. Thanks to an anonymous referee for pointing this out.

  4. 4.

    However, our theorems and algorithms may apply to other cases, including those of \(P(A|B)=1\) or 0 for \(P(B)=0\).

  5. 5.

    For translation \(T_1\), we use the fact that, for any propositional formula x in \(\mathscr {L}\), its corresponding equivalence class [x] in the Lindenbaum-Tarski algebra \(\mathbb {L}\) can be calculated.

  6. 6.

    This assumption relieves the range of formulae applicable to the results on complexity in Sect. 5.2 (see also Theorem 11, infra). All results in Sect. 5 also hold in the original language.

  7. 7.

    This condition does not restrict the range of constraints \(\chi _1,\ldots ,\chi _n\) in practice, since our language now contains the constant symbols for all real numbers.

References

  1. Adams, E.: The logic of conditionals. Inquiry 8(1–4), 166–197 (1965). https://doi.org/10.1080/00201746508601430

    Article  Google Scholar 

  2. Adams, E.W.: Probability and the logic of conditionals. In: Hintikka, J., Suppes, P. (eds.) Aspects of Inductive Logic, Studies in Logic and the Foundations of Mathematics, vol. 43, pp. 265–316. Elsevier (1966). https://doi.org/10.1016/S0049-237X(08)71673-2, https://www.sciencedirect.com/science/article/pii/S0049237X08716732

  3. Adams, E.W.: The Logic of Conditionals: An Application of Probability to Deductive Logic, Synthese Library, vol. 86. Springer, Netherlands (1975). https://doi.org/10.1007/978-94-015-7622-2

  4. Aliseda, A.: The logic of abduction: an introduction. In: Magnani, L., Bertolotti, T. (eds.) Springer Handbook of Model-Based Science. SH, pp. 219–230. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-30526-4_10

    Chapter  Google Scholar 

  5. Anai, K., Yokoyama, K.: Algorithms of Quantifier Elimination and their Application: Optimization by Symbolic and Algebraic Methods [in Japanese]. University of Tokyo Press (2011)

    Google Scholar 

  6. Bacchus, F.: Representing and Reasoning with Probabilistic Knowledge: A Logical Approach to Probabilities. The MIT Press, Cambridge (1990)

    Google Scholar 

  7. Basu, S., Pollack, R., Roy, M.F.: On the combinatorial and algebraic complexity of quantifier elimination. J. ACM 43(6), 1002–1045 (1996). https://doi.org/10.1145/235809.235813

    Article  MathSciNet  MATH  Google Scholar 

  8. Boričić, M.: Inference rules for probability logic. PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE 100(114), 77–86 (2016). https://doi.org/10.2298/PIM1614077B

    Article  MathSciNet  MATH  Google Scholar 

  9. Boričić, M.: Suppes-style sequent calculus for probability logic. J. Logic Comput. 27(4), 1157–1168 (10 2015). https://doi.org/10.1093/logcom/exv068

  10. Brickhill, H., Horsten, L.: Triangulating non-archimedean probability. Rev. Symbolic Logic 11(3), 519–546 (2018). https://doi.org/10.1017/S1755020318000060

    Article  MathSciNet  MATH  Google Scholar 

  11. Buss, S.R.: An introduction to proof theory. In: Buss, S.R. (ed.) Handbook of Proof Theory, vol. 137, pp. 1–78. Elsevier, Amsterdam (1998)

    Chapter  Google Scholar 

  12. Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press (2002). https://doi.org/10.1017/CBO9780511809088

  13. Enderton, H.B.: A Mathematical Introduction to Logic, 2nd edn. Academic Press (2000)

    Google Scholar 

  14. Fagin, R., Halpern, J.Y., Megiddo, N.: A logic for reasoning about probabilities. Inf. Comput. 87(1), 78–128 (1990)

    Google Scholar 

  15. Gallier, J.H.: Logic for Computer Science: Foundations of Automatic Theorem Proving, 2nd edn. Dover Publications (2015)

    Google Scholar 

  16. Gilio, A.: Precise propagation of upper and lower probability bounds in system p. arXiv preprint math/0003046 (2000)

    Google Scholar 

  17. Hailperin, T.: Best possible inequalities for the probability of a logical function of events. Am. Math. Monthly 72(4), 343–359 (1965)

    Google Scholar 

  18. Hájek, A.: Probability, logic, and probability logic. In: Goble, L. (ed.) The Blackwell Guide to Philosophical Logic, chap. 16, pp. 362–384. Blackwell Publishing (2001)

    Google Scholar 

  19. Halpern, J.Y.: Reasoning About Uncertainty, 2nd edn. MIT Press (2017)

    Google Scholar 

  20. Hobbs, J.R., Stickel, M.E., Appelt, D.E., Martin, P.: Interpretation as abduction. Artif. Intell. 63(1), 69–142 (1993)

    Google Scholar 

  21. Ikodinović, N., Ognjanović, Z.: A logic with coherent conditional probabilities. In: Godo, L. (ed.) ECSQARU 2005. LNCS (LNAI), vol. 3571, pp. 726–736. Springer, Heidelberg (2005). https://doi.org/10.1007/11518655_61

    Chapter  MATH  Google Scholar 

  22. Kersting, K., De Raedt, L.: Bayesian logic programs. arXiv preprint cs.AI/0111058 (2001). https://arxiv.org/abs/cs/0111058

  23. Kersting, K., De Raedt, L.: Bayesian logic programming: theory and tool. In: Getoor, L., Taskar, B. (eds.) Introduction to Statistical Relational Learning, pp. 291–321. MIT Press (2007)

    Google Scholar 

  24. McGee, V.: Learning the impossible. In: Eells, E., Brian, S. (eds.) Probability and Conditionals: Belief Revision and Rational Decision, pp. 179–199. Cambridge University Press (1994)

    Google Scholar 

  25. Mishra, B.: Algorithmic Algebra. Monographs in Computer Science. Springer, New York (1993). https://doi.org/10.1007/978-1-4612-4344-1

  26. Mueller, E.T.: Commonsense reasoning: An Event Calculus Based Approach, 2nd edn. Morgan Kaufmann (2014)

    Google Scholar 

  27. Popper, K.: The Logic of Scientific Discovery, 2nd edn. Routledge (2002)

    Google Scholar 

  28. Richardson, M., Domingos, P.: Markov logic networks. Mach. Learn. 62(1), 107–136 (2006). https://doi.org/10.1007/s10994-006-5833-1

    Article  MATH  Google Scholar 

  29. Tabata, H.: On second-order Peano arithmetic [in Japanese]. Tottori University Journal of the Faculty of Education and Regional Sciences 4(1), 37–84 (2002)

    Google Scholar 

  30. Tarski, A.: A Decision Method for Elementary Algebra and Geometry: Prepared for Publication with the Assistance of J.C.C. McKinsey. RAND Corporation, Santa Monica, CA (1951)

    Google Scholar 

  31. Wagner, C.G.: Modus tollens probabilized. Brit. J. Phil. Sci. 55(4), 747–753 (2004). https://doi.org/10.1093/bjps/55.4.747

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to thank sincerely Takashi Maruyama, Takeshi Tsukada, Shun’ichi Yokoyama, Eiji Yumoto and Itaru Hosomi for discussion and many helpful comments. Thanks are also due to all anonymous referees who gave valuable comments on earlier versions of this article.

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  1. Data Science Research Laboratories, NEC Corporation, Kawasaki, Kanagawa, Japan

    Shota Motoura

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  1. Institute of Information Theory and Automation, Prague, Czech Republic

    Jiřina Vejnarová

  2. Insight Centre for Data Analytics, School of Computer Science and IT University College Cork, Cork, Ireland

    Nic Wilson

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Motoura, S. (2021). A Logic and Computation for Popper’s Conditional Probabilities. In: Vejnarová, J., Wilson, N. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2021. Lecture Notes in Computer Science(), vol 12897. Springer, Cham. https://doi.org/10.1007/978-3-030-86772-0_47

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  • DOI: https://doi.org/10.1007/978-3-030-86772-0_47

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