Truth, Possibility and Probability: New Logical Foundations of Probability and Statistical Inference (Volume 166) (North-Holland Mathematics Studies, Volume 166)

Anyone involved in the philosophy of science is naturally drawn into the study of the foundations of probability. Different interpretations of probability, based on competing philosophical ideas, lead to different statistical techniques, and frequently to mutually contradictory consequences. This unique book presents a new interpretation of probability, rooted in the traditional interpretation that was current in the 17th and 18th centuries. Mathematical models are constructed based on this interpretation, and statistical inference and decision theory are applied, including some examples in artificial intelligence, solving the main foundational problems. Nonstandard analysis is extensively developed for the construction of the models and in some of the proofs. Many nonstandard theorems are proved, some of them new, in particular, a representation theorem that asserts that any stochastic process can be approximated by a process defined over a space with equiprobable outcomes. Content: Editor Page ii Edited by Page iii Copyright page Page iv Dedication Page ivA Preface Pages v-viii List of Figures Page xix Preliminaries Pages 1-14 Chapter I The Problem of Foundations Pages 17-40 Chapter II Truth, Possibility, and Probability Pages 41-55 Chapter III Probability Models Pages 57-84 Chapter IV The Basic Principles of Inference and Decision Pages 85-116 Chapter V A Medical Example Pages 117-125 Chapter VI Introduction to Probability Theory Pages 129-157 Chapter VI Disjunctive Probability Spaces and Invariant Measures Pages 159-170 Chapter VIII Elements of Infinitesimal Analysis Pages 171-195 Chapter IX Integration Pages 197-226 Chapter X Probability Distributions Pages 227-251 Chapter XI Hyperfinite Random Processes Pages 253-276 Chapter XII Simple Probability Structures Pages 279-291 Chapter XIII The structure of chance Pages 293-317 Chapter XIV Equiprobability structures Pages 319-341 Chapter XV Probability structures Pages 343-351 Chapter XVI Examples of probability structures Pages 353-364 Chapter XVII Logical probability Pages 365-376 Chapter XVIII Classical statistical inference Pages 379-411 Chapter XIX Problems in statistical inference Pages 413-430 Chapter XX Decision theory Pages 431-442 Appendix A Foundations of nonstandard analysis Pages 443-459 Appendix B Extensions of integrals and measures Pages 461-468 Bibliography Pages 469-473 Index Pages 474-484 Anyone involved in the philosophy of science is naturally drawn into the study of the foundations of probability. Different interpretations of probability, based on competing philosophical ideas, lead to different statistical techniques, and frequently to mutually contradictory consequences.

This unique book presents a new interpretation of probability, rooted in the traditional interpretation that was current in the 17th and 18th centuries. Mathematical models are constructed based on this interpretation, and statistical inference and decision theory are applied, including some examples in artificial intelligence, solving the main foundational problems. Nonstandard analysis is extensively developed for the construction of the models and in some of the proofs. Many nonstandard theorems are proved, some of them new, in particular, a representation theorem that asserts that any stochastic process can be approximated by a process defined over a space with equiprobable outcomes.