Coherent Prevision as a Linear Functional without an Underlying Measure Space: The Purely Arithmetic Structure of Logical Relations Among Conditional Quantities

Abstract

My first two presentations at this workshop were oriented toward the theoretical construction and a substantive application of the operational subjective statistical outlook to numerically specified systems of uncertain knowledge. In this concluding discussion I shall emphasize the formal aspects of these activities that appear critically different from the structures associated with recent work on conditional events that derives its outlook at least implicitly from a measure theoretic tradition in probability theory. Specifically, our focus will be on the structure of conditional quantities and conditional prevision assertions which are crucial to implementing the fundamental theorem of prevision as an “inference engine”, foreseen in principle and constructed in simple forms for more than a century. I shall argue that conditional quantities do entail a minimal logical structure as required by the principle of coherency. But attempts to identify a more extensive structure of a logic of conditional events via many-valued logical functions are misdirected. The equivalent roles of arithmetic and of many-valued logics in generating a function space of object quantities shall be made apparent. I shall again presume the reader’s familiarity with the detailed definitions and conceptual exposition appearing in the articles by Lad, Dickey, and Rahman (1990, 1992) which were discussed in my first session at this workshop. A brief review of relevant notation, definitions, and theorems appears in an Appendix to the article of Lad and Coope in the present volume.