Henry E. Kyburg Jr.

Several full professors of philosophy today were once undergraduates of Henry Kyburg, including Daniel Dennett, Robert Stalnaker, Rich Thomason, Teddy Seidenfeld, and William L. Harper.

His AI dissertation students were Ronald Loui, Bulent Murtezaoglu, and Choh Man Teng, and postdoctoral visitor Fahiem Bacchus. His philosophy students included daughter Alice Kyburg, Mariam Thalos, Gregory Wheeler, William Harper, Abhaya Nayak, Prashanta Bandyopadhaya, in addition to those listed above.

Several ideas distinguish Kyburg's Kyburgian or epistemological interpretation of probability:

• Probability is measured by an interval (some mistake this as an affinity to Dempster–Shafer theory, but Kyburg firmly rejects their rule of combination; his work remained closer to confidence intervals, and was often interpreted by Bayesians as a commitment to a set of distributions, which Kyburg did not repudiate)
• All probability statements can be traced to direct inference of frequency in a reference class (there can be Bayes-rule calculations upon direct-inference conclusions, but there is nothing like a prior distribution in Kyburg's theory)
• The reference class is the most specific class with suitable frequency knowledge (this is the Reichenbach rule, which Kyburg made precise; his framework was later reinterpreted as a defeasible reasoning system by John L. Pollock, but Kyburg never intended the calculation of objective probabilities to be shortcut by bounded rationality due to computational imperfection)
• All probability inferences are based on knowledge of frequencies and properties, not ignorance of frequencies; however, randomness is essentially the lack of knowledge of bias (Kyburg especially rejects the maximum entropist methods of Harold Jeffreys, E.T. Jaynes and other uses of the Principle of Indifference here; and Kyburg disagrees here with Isaac Levi who believes that chance must be positively asserted upon knowledge of relevant physical symmetries)
• There is no disagreement over the probability once there is agreement on the relevant knowledge; this is an objectivism relativized to an evidential state (i.e., relativized to a set of observed frequencies of properties in a class, and a set of asserted properties of events)

Example: Suppose a corpus of Knowledge at a level of acceptance. Contained in this corpus are statements,

 e is a T1 and e is a T2.

The observed

 frequency of P among T1 is .9.

The observed

 frequency of P among T2 is .4.

What is the probability that e is a P?

Here, there are two conflicting reference classes, so the probability is either [0, 1], or some interval combining the .4 and .9, which sometimes is just [.4, .9] (but often a different conclusion will be warranted). Adding the knowledge

 All T1's are T2's

now makes T1 the most specific relevant reference class and a dominator of all interfering reference classes. With this universal statement of class inclusion,

 the probability is [.9, .9], by direct inference from T1.

Kyburg's rules apply to conflict and subsumption in complicated partial orders.

Kyburg's inferences are always relativized to a level of acceptance that defines a corpus of morally certain statements. This is like a level of confidence, except that Neyman–Pearson theory is prohibited from retrospective calculation and post-observational acceptance, while Kyburg's epistemological interpretation of probability licenses both. At a level of acceptance, any statement that is more probable than the level of acceptance can be adopted as if it were a certainty. This can create logical inconsistency, which Kyburg illustrated in his famous lottery paradox.

In the example above, the calculation that e is a P with probability .9 permits the acceptance of the statement e is a P categorically, at any level of acceptance lower than .9 (assuming also that the calculation was performed at an acceptance level above .9). The interesting tension is that very high levels of acceptance contain few evidentiary statements. They do not even include raw observations of the senses if those senses have often been fooled in the past. Similarly, if a measurement device reports within an interval of error at a rate of .95, then no measurable statements are acceptable at a level above .95, unless the interval of error is widened. Meanwhile, at lower levels of acceptance, so many contradictory statements are acceptable that nothing useful can be derived without inconsistency.

Kyburg's treatment of universally quantified sentences is to add them to the Ur-corpus or meaning postulates of the language. There, a statement like F = ma or preference is transitive provides additional inferences at all acceptance levels. In some cases, the addition of an axiom produces predictions that are not refuted by experience. These are the adoptable theoretical postulates (and they must still be ordered by some kind of simplicity). In other cases, the theoretical postulate is in conflict with the evidence and measurement-based observations, so the postulate must be rejected. In this way, Kyburg provides a probability-mediated model of predictive power, scientific theory-formation, the Web of Belief, and linguistic variation. The theory of acceptance mediates the tension between linguistic categorical assertion and probability-based epistemology.

• Official Obituary