Foundational Issues in the Measurement of Belief and Uncertainty

Abstract

Abstract : New methods from mathematics and mathematical logic were employed to generalize the event algebra of probability theory while retaining the numerical, algebraic structure of its probability functions. One form of generalization developed consisted of extending the event space of standard probability theory to include events corresponding to counterfactuals. Another form of generalization researched extended the probability concept to arbitrary lattices (with maximal and minimal elements) and characterized how the probability concept structurally restricted a lattice to which it is applied. Because lattices can be viewed as propositional logics (either classical or non-classical) or as event spaces (either boolean or non-boolean), this is equivalent to understanding how the probability concept restricted the underlying logic or event space on which it was defined. The generalizations were applied to the empirical literature about probability judgments and to game theory. For probability judgments, a new foundation was given for the most prominent theory of probability judgments in psychology, Support Theory,'' and a new method was developed for the analysis of several puzzling empirical phenomena involving probability judgments. For game theory, a new approach to deterrence based on counterfactual reasoning was developed.