Belief and probability: A general theory of probability cores

This paper considers varieties of probabilism capable of distilling paradox-free qualitative doxastic notions (e.g., full belief, expectation, and plain belief) from a notion of probability taken as a primitive. We show that core systems, collections of nested propositions expressible in the underlying algebra, can play a crucial role in these derivations. We demonstrate how the notion of a probability core can be naturally generalized to high probability, giving rise to what we call a high probability core, a notion that when formulated in terms of classical monadic probability coincides with the notion of stability proposed by Hannes Leitgeb [2010]. Our work continues work done by one of us in collaboration with Rohit Parikh [Arló-Costa and Parikh, 2005]. In turn, the latter work was inspired by the seminal work of Bas van Fraassen [1995]. We argue that the adoption of dyadic probability as a primitive (as articulated by van Fraassen [1995]) admits a smoother connection with the standard theory of probability cores as well as a better model in which to situate doxastic notions like full belief. We also illustrate how the the basic structure underlying a system of cores naturally leads to alternative probabilistic acceptance rules, like the so-called ratio rule initially proposed by Isaac Levi [1996]. Core systems in their various guises are ubiquitous in many areas of formal epistemology (e.g., belief revision, the semantics of conditionals, modal logic, etc.). We argue that core systems can also play a natural and important role in Bayesian epistemology and decision theory. In fact, the final part of the article shows that probabilistic core systems are naturally derivable from basic decisiontheoretic axioms which incorporate only qualitative aspects of core systems; that the qualitative aspects of core systems alone can be naturally integrated in the articulation of coherence of primitive conditional probability; and that the guiding idea behind the primary qualitative features of a core system gives rise to the formulation of lexicographic decision rules. We dedicate this paper to the memory of Henry E. Kyburg, Jr., whose ideas inspired some of the main results presented here. One of us (Horacio Arló-Costa) had the good fortune to know Henry personally and to correspond with him, something which always was extremely insightful and helpful. Although he was a man of deep convictions, he was also capable of understanding and articulating in non-trivial ways alternative ideas. His views of the lottery changed during the last years of his life, as we explain below, and we can only speculate how he would have reacted to recent responses to his paradox. What is true is that his ideas have lasting value and the paradoxes he addressed continue to inspire innovative work in formal epistemology. We are happy to contribute to his legacy by offering new responses to his many insightful questions. We still have much to learn from Henry’s work, and we look forward to many years of fruitful discussion of his work. This paper is also dedicated to the memory of Horacio Arló-Costa.