Deduction and Inference Using Conditional Logic and Probability

Abstract

Abstract : In contrast to the author's 1987 paper, which presented an algebraic synthesis of conditional logic and conditional probability starting with an initial Boolean algebra of propositions,this paper starts with an initial probability space of events and generates the associated propositions as measurable indicator functions (a la the approach of B. De Finetti). Conditional propositions are generated as measureable indicator functions restricted to subset of positive probability measure. The operations of and, or, not, and given are defined for arbitrary conditional propositions. The representation of the resulting conditional event algebra as a 3-valued logic (always possible according to a new theorem due to I. R. Goodman) is given in terms of 3-valued truth tables. Formulas for the conditional probability of complex conditional propositions such as (q/p) v (s/r) are proved. A second major theme of the paper concerns deductions in the realm of conditional propositions. It turns out that there are varieties of logical deduction for conditional propositions depending on the particular entailment relation (< or +) chosen. These relations are explored including their lattice properties and properties and properties of non-monotonicity. Computational aspects for Artificial Intelligence are also discussed.