Probability as the degree of rational belief
We would all agree that Newton's law of gravitation4 is more confirmed than the stars as an influence on our character and behaviour. In spite of this it is not easy to quantify such a difference as a degree of confirmation. Similarly, given the relative frequencies of fatal car and train accidents, it seems rational to believe that one is more likely to die in a car accident than a train accident. Nevertheless, the difference in probability is difficult to establish with exactitude.
Ever since the pioneering work of Keynes (TP), scholars have attempted to find a logic of rational beliefs; these are in most cases neither certainly true nor certainly false. Beliefs are to be expressed by means of sentences, so our first problem is to establish what it means to say that a certain sentence has a certain probability of being true. After Kolmogorov, we know that a probability measure on a set E is a function p, which ascribes a real number belonging to [0,1] to every subset e of E, in such a way that p(E) = 1, р(ф) = 0 and if a,b belong to E and their intersection is empty, then p(a и b) = p(a) + p(b). If E is a set of sentences, then, if e is a logical truth, p(e) = 1, if it is a contradiction, p(e) = 0, if a and b are incompatible, p(a v b) = p(a) + p(b). But, as we saw in the preceding examples, it is quite difficult to assign a quantitative probability to our rational beliefs when they concern the truth of both scientific and common sentences. In the following we will come back to this point.
A second problem is that rational beliefs are always referred to as a set of already available knowledge. This knowledge is of two kinds: 'foreground evidence' and 'background knowledge'. We are usually interested in finding out the probability of a certain belief with respect to the truth of one or more evidences, keeping the background knowledge unmodified. For instance, if we want to know the probability of experiencing an aircraft accident, given the statistics about the accidents of last year, we implicitly assume that the general situation of the flight we are going to take has undergone no essential changes with respect to last year. Thereafter we can define the conditioned probability of a belief h with respect to one or more evidences e in the following way:
It is possible to prove that, if p(h) is a probability measure, then p(h/e) is a probability measure as well.
If we consider probability as a measure of the degree of rational belief, it seems more sensible to maintain that the conditioned probability is epistemologically primary with respect to absolute probability, because our beliefs are always based on one or more pieces of background knowledge that we are making use of for our cognitive evaluation. However, from the logical point of view, if we move from conditioned probability it is easy to define the absolute probability of a sentence as the conditioned probability with respect to a logical truth. We will return to this point as well in the following.
At this point we emphasize that degrees of rational belief are intended as probabilities that the sentence being considered is true, not as situations of objective indeterminacy. The probabilities we are investigating are, as it were, de dicto, not de re.
A further problem concerning probabilities as degrees of belief is that of updating them on the basis of new evidences. Indeed, it often happens that the set of relevant evidences for a certain hypothesis is modified, that is, that we acquire new evidences. We usually make use of Bayes's theorem, which is easily derivable from Kolmogorov's third axiom and the definition of conditioned probability:
In this equation e is a new evidence and h is the hypothesis we are investigating. In this context p(h) is already a conditioned probability with respect to the old evidence.
On the other hand, the application of Bayes's theorem presupposes that the initial probabilities are already known, that is, first of all the probability of h given the old evidences must be known. Thus we return to the problem we posed at the beginning, namely, that of establishing a probability given certain evidences.
From the examples we have proposed it seems that it is often possible to establish a comparison between probabilities but not to determine their quantitative value. Indeed, there are few cases where the probability can be evaluated quantitatively: gambles, some very simple empirical situations and little else. In general, as regards the confirmation of scientific hypotheses, it seems unreasonable to ascribe a probability measure to it, whereas it is often possible to establish that a hypothesis is more probable than another. The same holds for evaluations concerning the common world. In the second chapter of his TP, Keynes provides a series of arguments favouring the qualitative character of probability evaluations. He observes that, even for brokers, who determine insurance premiums quantitatively on the basis of statistics, it is enough for the premium to exceed the probability of the accident occurring multiplied by the amount to be paid by the insurance company (TP, p. 23). Therefore, they have to establish only that the probability of the disaster happening is lower than a certain value. Furthermore, he continues, although it is true that a favourable evidence increases the probability of a certain hypothesis, it is difficult to determine by how much it increases (TP, pp. 30-1).
In order to portray the relation between probabilities, Keynes presents an interesting picture (TP, p. 42) in which the impossibility (probability = 0) and certitude (probability = 1) are two points on the plane connected by different lines, of which one is straight and represents a probability measure, whereas the others are curves, which sometimes intersect one another. A quantitative probability (straight line) can be ascribed only in a few cases. In general probabilities are only comparative, that is, curved lines. Furthermore, a comparison is possible only between probabilities that lie on the same curved line. Therefore, in general, probabilities are neither measurable nor comparable. They are measurable in only a very few cases and in only slightly more cases are they comparable.
According to Keynes (TP, p. 70), it is certainly possible to compare probability only in the following cases:
That is: I. when the two probabilities have the same evidence but the hypothesis is enlarged; II. when the two probabilities have the same hypothesis and the evidence is augmented. This limitation seems excessive because, although it is not possible to compare probabilities from completely different realms, it is possible to compare probabilities that cannot be traced back to the aforementioned patterns. For instance, given the relative frequencies of car and aircraft accidents, we can reasonably maintain that we are more likely to die in the former than in the latter. This comparison is of neither the first nor the second kind, nor can it be traced back to them. We will return to this problem as well.