Hailperin’s Probability Logic
Hailperin develops Boole‘s ideas on bounds on possible conventional probability assignments. It considers a pair <B,P>where B is a Boolean algebra and P is a ‘probability function’ for which Hailperin develops axioms and theorem, most importantly about the bounds on probabilities P(B|A).
My impression is that interval probabilities are much less commonly used than supposedly precise probabilities. But, informally, it is common to estimate the worst-case probabilities for ‘cost’ and ‘benefit’ factors. The explicit use of intervals is indicated when one has factors that are sometimes good, sometimes bad, but even here it is common to want to explore different possible probability assignments (scenarios) explicitly, because Hailperin’s bounds are necessarily imprecise. Many clients also find it easier to grasp a ‘spanning’ collection of precise scenarios as against an imprecise probability (which can invoke the scorn of some). So in practice the use of even well-founded imprecise probabilities tend to be relegated to behind the scenes, with precise probabilities – even if logically flawed – taking centre stage. Problems seem to arise when people reason ‘pragmatically’; as if a flawed theory could necessarily be trusted.
I would like to know if his later work is of broader significance. Reading suggestions?
Logic with a probability semantics – has a good summary. Makes it clear that we suppose that P(X)=p for some p in an interval, so that one has the usual law of large numbers. Muddling is not allowed.