Conditional Probability in the Light of Qualitative Belief Change

We explore ways in which purely qualitative belief change in the AGM tradition throws light on options in the treatment of conditional probability. First, by helping see why it can be useful to go beyond the ratio rule defining conditional from one-place probability. Second, by clarifying what is at stake in different ways of doing that. Third, by suggesting novel forms of conditional probability corresponding to familiar variants of qualitative belief change, and conversely. Likewise, we explain how recent work on the qualitative part of probabilistic inference leads to a very broad class of ‘proto-probability’ functions.

[1]  Peter Gärdenfors,et al.  Knowledge in Flux: Modeling the Dynamics of Epistemic States , 2008 .

[2]  Peter Gärdenfors,et al.  Belief Revision , 1995 .

[3]  William Harper,et al.  Rational belief change, popper functions and counterfactuals , 1975, Synthese.

[4]  Bart Jacobs.  Orthomodular lattices, Foulis Semigroups and Dagger Kernel Categories , 2010, Log. Methods Comput. Sci..

[5]  B. O. Koopman.  The bases of probability , 1940 .

[6]  Tomasz Jarmużek.  David Makinson, "Bridges from Classical to Nonmonotonic Logic", King’s College Publications, London, 2005 , 2007 .

[7]  Towarzystwo Naukowe Warszawskie.  Sprawozdania z Posiedzeń Towarzystwa Naukowego Warszawskiego , 1912 .

[8]  Theodore Hailperin,et al.  The development of probability logic from leibniz to maccoll , 1988 .

[9]  Peter Giirdenfors.  The Dynamics of Belief: Contractions and Revisions of Probability Functions , 1986 .

[10]  William Harper,et al.  Rational Conceptual Change , 1976, PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association.

[11]  David Makinson,et al.  Completeness Theorems, Representation Theorems: What's the Difference? , 2007 .

[12]  Alfréd Rényi,et al.  Probability Theory , 1970 .

[13]  Didier Dubois,et al.  Possibility Theory: Qualitative and Quantitative Aspects , 1998 .

[14]  Peter Gärdenfors,et al.  On the logic of theory change: Partial meet contraction and revision functions , 1985, Journal of Symbolic Logic.

[15]  Craig Boutilier,et al.  On the Revision of Probabilistic Belief States , 1995, Notre Dame J. Formal Log..

[16]  C. Bicchieri,et al.  PSA: proceedings of the biennial meeting of the Philosophy of Science Association , 2002 .

[17]  Emil Weydert,et al.  General Belief Measures , 1994, UAI.

[18]  David Makinson,et al.  Logical questions behind the lottery and preface paradoxes: lossy rules for uncertain inference , 2012, Synthese.

[19]  Vann McGee,et al.  Learning the impossible , 1994 .

[20]  Bas C. van Fraassen,et al.  Fine-grained opinion, probability, and the logic of full belief , 1995, J. Philos. Log..

[21]  William Harper.  Rational Belief Change, Popper Functions and Counterfactuals , 1975 .

[22]  A. Rényi.  On a new axiomatic theory of probability , 1955 .

[23]  Joseph Y. Halpern.  Lexicographic probability, conditional probability, and nonstandard probability , 2001, Games Econ. Behav..

[24]  Peter Gärdenfors,et al.  The Dynamics of Belief as a Basis for Logic , 1984, The British Journal for the Philosophy of Science.

[25]  Hans Reichenbach,et al.  Axiomatik der Wahrscheinlichkeitsrechnung , 1932 .

[26]  Wolfgang Spohn,et al.  A Survey of Ranking Theory , 2009 .

[27]  T. Hailperin,et al.  Probability logic in the twentieth century , 1991 .

[28]  Richard C. Jeffrey,et al.  Studies in inductive logic and probability , 1971 .

[29]  Hugues Leblanc,et al.  Probability Theory and Probability Logic , 1999 .

[30]  Dov M. Gabbay,et al.  Handbook of defeasible reasoning and uncertainty management systems: volume 2: reasoning with actual and potential contradictions , 1998 .

[31]  Bas C. van Fraassen,et al.  Representational of conditional probabilities , 1976, J. Philos. Log..

[32]  H. Geiringer.  On the Foundations of Probability Theory , 1967 .

[33]  H. Jeffreys.  Logical Foundations of Probability , 1952, Nature.

[34]  Robert Stalnaker.  Probability and Conditionals , 1970, Philosophy of Science.

[35]  Wolfgang Spohn,et al.  The representation of Popper measures , 1986 .

[36]  B. V. Fraassen.  Fine-grained opinion, probability, and the logic of full belief , 1995, Journal of Philosophical Logic.

[37]  David Miller.  Probability and Conditionals. Belief Revision and Rational Decision , 1996 .

[38]  José L. Balcázar,et al.  Transforming Outermost into Context-Sensitive Rewriting , 2010, Log. Methods Comput. Sci..

[39]  James Hawthorne,et al.  On the logic of nonmonotonic conditionals and conditional probabilities , 1996, J. Philos. Log..

[40]  Paul Pukite,et al.  Foundations of Probability Theory , 1998 .

[41]  Joseph Y. Halpern.  Plausibility Measures: A General Approach For Representing Uncertainty , 2001, IJCAI.

[42]  David Makinson,et al.  Bridges from classical to nonmonotonic logic , 2005, Texts in computing.

[43]  Horacio Arló-Costa.  Bayesian Epistemology and Epistemic Conditionals: On the Status of the Export-Import Laws , 2001 .

[44]  F. Huber,et al.  Degrees of Belief , 2009 .

[45]  B. O. Koopman.  The Axioms and Algebra of Intuitive Probability , 1940 .

[46]  L. Dubins.  Finitely Additive Conditional Probabilities, Conglomerability and Disintegrations , 1975 .

[47]  Joseph B. Kadane,et al.  Non-Conglomerability for Finite-Valued, Finitely Additive Probability , 1998 .

[48]  Rudolf Carnap,et al.  The continuum of inductive methods , 1952 .

[49]  Wlodzimierz Rabinowicz,et al.  On probabilistic representation of non-probabilistic belief revision , 1989, J. Philos. Log..

[50]  Rohit Parikh,et al.  Conditional Probability and Defeasible Inference , 2005, J. Philos. Log..

[51]  James Hawthorne,et al.  The Quantitative/Qualitative Watershed for Rules of Uncertain Inference , 2007, Stud Logica.

[52]  P G rdenfors,et al.  Knowledge in flux: modeling the dynamics of epistemic states , 1988 .

[53]  Janina Hosiasson-Lindenbaum,et al.  On confirmation , 1940, Journal of Symbolic Logic (JSL).

[54]  Jeff B. Paris,et al.  O IS NOT ENOUGH , 2009, The Review of Symbolic Logic.

[55]  B. M. Hill,et al.  Theory of Probability , 1990 .

[56]  M. Kendall,et al.  The Logic of Scientific Discovery. , 1959 .

[57]  L. M. M.-T..  Theory of Probability , 1929, Nature.

[58]  Hugues Leblanc,et al.  On Relativizing Kolmogorov's Absolute Probability Functions , 1989, Notre Dame J. Formal Log..

[59]  Alan Hájek,et al.  What Conditional Probability Could Not Be , 2003, Synthese.