# Hailperin’s Probability Logic

T. Hailperin *Probability Logic* Notre Dame Journal of Formal Logic Volume 25, Number 3, July 1984 pp. 198-212.

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?*

## See Also

*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.

My notes on broader uncertainty and logic, e.g. work by Jack Good, with a fuller motivation, discussion of applications and linking to practice.

Hello, Dr. Marsay!

Though it’s been a while, I would like to inform you that Theodore Hailperin has published a book not too long ago. In the Preface, he also attributes his interests in probability theory and mathematical logic to reading John Maynard Keynes’s

A Treatise on Probability.Logic with a Probability Semantics

Also, as I have stated before, you ought to acquire a copies of these works by Theodore Hailperin:

http://inpress.sites.lehigh.edu/publications/details/Hailperin.html

Boole's Logic and Probability: A Critical Exposition from the Standpoint of Contemporary Algebra, Logic and Probability Theory

One last thing though – have you ever heard of the late David W. Miller? I believe he was a Professor of Management Science at Columbia University, and like you and Dr. Michael Emmett Brady, he had an interest in George Boole’s work on logic and probability. David W. Miller’s own book on the Last Challenge Problem has been released into paperback form recently, and you can acquire a copy over Amazon.com.

The Last Challenge Problem: George Boole's Theory of Probability

Thanks. Does Hailperin motivate his use of a ‘probability function’? I briefly link to Miller in my notes on Keynes’ Treatise, http://djmarsay.wordpress.com/?s=Miller. Both seem to preclude muddling, as in a real roulette wheel whose biases may change with wear.

Sorry for the belated response sir. I forget if he does. I would recommend reading his works though, especially that recent paper that I sent to you!