Probabilistic Logic Revisited

Before beginning the research that led to "Probabilistic logic" [11 ], I had participated with Richard Duda, Peter Hart, and Georgia Sutherland on the PROSPECTOR project [3]. There, we used Bayes' rule (with some assumptions about conditional independence) to deduce the probabilities of hypotheses about ore deposits given (sometimes uncertain) geologic evidence collected in the field [4]. At that time, I was also familiar with the use of "certainty factors" by Shortliffe [18], the use of "fuzzy logic" by Zadeh [20], and the Dempster/Shafer formalism [16]. All of these methods made (sometimes implicit and unacknowledged) assumptions about underlying joint probability distributions, and I wanted to know how the mathematics would work out if no such assumptions were made. I began by asking how modus ponens would generalize when one assigned probabilities (instead of binary truth values) to P and P D Q. As can be verified by simple calculations using a Venn diagram, the probability of Q is under-determined in this case but can be bounded as follows: