Over the years there have been a large number of papers by many different individuals on these matters. Essentially all of them have formulated conditions in terms of events, with the underlying structure being that of the Boolean algebra of events and the ordering relation being a binary relation of one event being at least as probable as another. The conditions have turned out not to be simple. The important aspect of the paper with Zanotti is our recognition that events are the wrong objects to order. To each event there is a corresponding indicator function for that event, with the indicator function having the value one when a possible outcome lies in the event and the outcome zero otherwise—as is apparent, in this standard formulation events are sets of possible outcomes, that is, sets of points in the probability space. We obtain what Zanotti and I have baptized as extended indicator functions by closing the set of indicator functions under the operation of functional addition. Using results already known in the theory of extensive measurement it is then easy to give quite simple necessary and sufficient axioms on the ordering of extended indicator functions to obtain a numerical probability representation.

Recently we have found correspondingly simple necessary and sufficient qualitative axioms for conditional probability. The qualitative formulations of this theory beginning with the early work of B. O. Koopman (1940a, l940b) have been especially complex. We have been able drastically to simplify the axioms by using not only extended indicator functions, but the restriction of such functions to a given event to express conditionalization. In the ordinary logic of events, when we have a conditional probability P(A|B), there is no conditional event A|B, and thus it is not possible to define operations on conditional or restricted events. However, if we replace the event A by its indicator function Ac, then Ac|B is just the indicator function restricted to the set B, and we can express in a simple and natural way the operation of function addition of two such partial functions having the same domain. The analysis of conditional probability requires considerably more deviation from the theory of extensive measurement than does the unconditional case: for example, addition as just indicated is partial rather than total. More importantly, a way has to be found to express the conceptual content of the theorem on total probability. The solution to this problem is the most interesting aspect of the axiomatization.

The move from events to extended indicator functions is especially interesting philosophically, because the choice of the right objects to consider in formulating a given theory is, more often than I originally thought, mistaken in first efforts and, as these first efforts become crystallized and familiar, difficult to move away from.

Apart from technical matters of formulation and axiomatic niceties, there are, it seems to me, three fundamental concepts underlying probability theory. One is the addition of probabilities for mutually exclusive events, the second is the concept of independence of events or random variables, and the third is the concept of randomness. I have not said much here about either independence or randomness. A conceptually adequate formulation of the foundations of probability should deal with both of these concepts in a transparent and intuitively satisfactory way. For any serious applications there is a fourth notion of equal importance. This is the notion of conditionalization, or the appropriate conceptual method for absorbing new information and changing the given probabilities. I have ideas, some of which are surely wrong, about how to deal with these matters and hope to be able to spend time on them in the future. However, rather than try to sketch what is still quite premature, I want to end with some general comments about the foundations of probability and decision theory.

It has been remarked by many people that logic is now becoming a mathematical subject and that philosophers are no longer main contributors to the subject. Because of its more advanced mathematical character this has really been true of probability from the beginning. The great contributions to the foundations of probability have been made by mathematicians—de Moivre, Laplace, von Mises, and Kolmogorov come quickly to mind. Although there is a tradition of these matters in philosophy—and here one thinks of Reichenbach and Carnap—it is still certainly true that philosophers have not had a strong influence on the mainstream of probability theory, even in the formulation of its foundations. On the other hand, I strongly believe in the proposition that there is important and significant work in the foundations of probability that is more likely to be done by philosophers than by anyone else. The various interpretations of foundations, ranging from the subjective view of the classical period through the relative frequency theory of the first part of this century to propensity and other views of late, have probably been discussed more thoroughly and more carefully by philosophers than by anyone else. I see no reason to think that this tradition will come to an end. The closely related problems of decision theory are just beginning to receive equal attention from philosophers after their rapid developnnent by mathematical statisticians in the two decades after World War II.