What is the plausibility of probability

We present and examine a result related to uncertainty reasoning, namely that a certain plausibility measure of Cox’s type can be uniquely embedded in a minimal ordered field. This, although a purely mathematical result, can be claimed to imply that every rational method to reason with uncertainty must be based on sets of extended probability distributions, where extended probability is standard probability extended with infinitesimals. This claim must be supported by some argumentation of non-mathematical type, however, since pure mathematics does not tell us anything about the world. We propose one such argumentation, and relate it to results from the literature of uncertainty and statistics.

[1]  J. Hartigan.  Theories of Probability , 1983 .

[2]  R. T. Cox.  Probability, frequency and reasonable expectation , 1990 .

[3]  D. Ellsberg.  Decision, probability, and utility: Risk, ambiguity, and the Savage axioms , 1961 .

[4]  Leonard J. Seligman,et al.  Decision-Centric Information Monitoring , 2000, Journal of Intelligent Information Systems.

[5]  Michael Smithson,et al.  Human Judgment under Sample Space Ignorance , 2000, ISIPTA.

[6]  Stefan Arnborg,et al.  Information awareness in command and control: precision, quality, utility , 2000, Proceedings of the Third International Conference on Information Fusion.

[7]  D. Freedman,et al.  BAYES' METHOD FOR BOOKIES , 1969 .

[8]  Stefan Arnborg,et al.  Bayes Rules in Finite Models , 2000, ECAI.

[9]  P. Walley.  Statistical Reasoning with Imprecise Probabilities , 1990 .

[10]  J. Aczél,et al.  Lectures on Functional Equations and Their Applications , 1968 .

[11]  H. Brachinger,et al.  Decision analysis , 1997 .

[12]  Edward Nelson.  Radically Elementary Probability Theory. , 1987 .

[13]  Stefan Arnborg,et al.  On the foundations of Bayesianism , 2001 .

[14]  Lotfi A. Zadeh,et al.  The Roles of Fuzzy Logic and Soft Computing in the Conception, Design and Deployment of Intelligent Systems , 1997, Software Agents and Soft Computing.

[15]  Didier Dubois,et al.  Nonmonotonic Reasoning, Conditional Objects and Possibility Theory , 1997, Artif. Intell..

[16]  Paul Snow.  On the Correctness and Reasonableness of Cox's Theorem for Finite Domains , 1998, Comput. Intell..

[17]  P. Walley.  Measures of Uncertainty in Expert Systems , 1996, Artificial Intelligence.

[18]  R. Tennant.  Algebra , 1941, Nature.

[19]  J. Berger.  Statistical Decision Theory and Bayesian Analysis , 1988 .

[20]  J. Conway.  On Numbers and Games , 1976 .

[21]  José Sanmartín Esplugues.  Introduction to model theory and to the metamathematics of algebra , 1971 .

[22]  Nic Wilson.  A Logic of Extended Probability , 1999, ISIPTA.

[23]  Paul Snow.  The Vulnerability of the Transferable Belief Model to Dutch Books , 1998, Artif. Intell..

[24]  J. B. Paris,et al.  The Uncertain Reasoner's Companion: Bibliography , 1995 .

[25]  G. Fricchione.  Descartes’ Error: Emotion, Reason and the Human Brain , 1995 .

[26]  Frans Voorbraak,et al.  Decision analysis using partial probability theory , 1997 .

[27]  David Lindley.  Scoring rules and the inevitability of probability , 1982 .

[28]  Stefan Arnborg,et al.  Learning in prevision Space , 1999, ISIPTA.

[29]  Irwin R. Goodman,et al.  A Decision-Aid For Nodes in Command and Control Systems Based on Cognitive Probability Logic , 1999 .

[30]  Joseph Y. Halpern.  A Counterexample to Theorems of Cox and Fine , 1996, AAAI/IAAI, Vol. 2.

[31]  E. W. Adams,et al.  Probability and the Logic of Conditionals , 1966 .

[32]  P. Dangerfield.  Logic , 1996, Aristotle and the Stoics.

[33]  Joseph Y. Halpern.  Cox's Theorem Revisited , 1999, ArXiv.

[34]  C. Peirce,et al.  Philosophical Writings of Peirce , 1955 .

[35]  Alessandro Saffiotti,et al.  Belief functions and default reasoning , 1995, Artif. Intell..

[36]  Eyke Hüllermeier.  Toward a Probabilistic Formalization of Case-Based Inference , 1999, IJCAI.