Some New Probability Operators

Abstract

This chapter is dedicated to the formalization of the reasoning about probability involving qualitative statements (e.g. the probability of p is greater than the probability of q), probabilistic quantification (e.g. the probability of p is equal to 3n+1/4n+7 for some positive integer n), conditional statements (e.g., the conditional probability of p given q is at least r), confirmation (e.g., p confirms q), and independence. The common restriction is that iterations and nesting of probabilistic operators are not allowed. In other words, admissible statements are Boolean combinations of the atomic probabilistic assessments. Beside the presentation of complete axiomatizations, we shall also discuss the hierarchical structure of the introduced logics in terms of their expressiveness.