Figures in a Probability Landscape

Abstract

In general, our opinion is certainly not expressible in precise numerical probabilities. But in some cases it is, and in the case of a small field of propositions—e.g. a field generated by a single proposition—my opinion may be exactly represented by a single probability function. Therefore I shall begin with the fiction that it is always so.

I am very glad to have this opportunity to honor my teacher, and eventual colleague and friend, Nuel Belnap. In my first year as a graduate student I had Nuel’s seminar on the logic of questions, which took us on an exploration of the riches of formal semantics—greater than the fabled treasures of the Indies and the Spanish Main to my eyes. Questions, it turned out, needed a great deal for their understanding: semantics, the technique of consistency and completeness proofs, modal logic, and most of all, the enlargement of philosophical logic beyond the realm of declarative factual statements. That was the beginning; tautological entailment, relevance, algebraic techniques, and much more were to follow. It is easy and pleasant to recall those days, and perhaps most of all Nuel’s gentle and unpolemical spirit, always ready to laugh a little at his own and our shared excitement.