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The epistemological dilemma between quantitative and qualitative probability

We now investigate how the probabilities in cases I. and II. are modified. Many axiomatizations of comparative probability are available,5 that is, axiomatizations of a probability relation that is a total order (viz., reflexive, antisymmetric, transitive and linear). However, they are not applicable in this context because here linearity is not satisfied, since from an epistemological point of view not all probabilities are comparable. Nevertheless, we assume we have found a set of sentences Z for which all probabilities are comparable. Then we can ask what happens to the probabilities whenever the hypothesis or the evidence augments. The first case is very simple, because a proposition of this kind may always be derived from the axioms of comparative probability.

If, between two probabilities p(a/b) and p(c/d), the hypothesis a of the first is deducible from the hypothesis c of the second, then p(a/b) > p(c/d). Therefore in case I it holds that:

Keynes analyses the second case in the following way. In case II. comparison is possible if c contains only one further piece of information that is relevant for the hypothesis a. According to the following definition, the evidence c is mono-relevant6 for a in the language Z, iff in Z two sentences b and d such that c = b A d, p(a/b) Ф p(a/-b) and p(a/d) Ф p(a/-d) do not exist.

Thus we assume that c is mono-relevant for a. Then if c is favourable to a - that is, p(a/c) > p(a/-c) - it follows that:

If c is unfavourable to a - that is, p(a/-c) > p(a/c) - then:

Finally, if p(a/c) = p(a/-c), then:

All this seems very sensible, but it is not deducible from the current axioms of comparative probability. Indeed it is possible to find counterexamples to such rules. Let us consider as a universe of discourse the inhabitants of Great Britain, who are partitioned into two categories: English and Scottish.7 Let us assume that all English males wear trousers whereas English females wear skirts, and that all Scottish males wear skirts whereas Scottish females wear trousers. Furthermore, males and females are equi-numerous in both populations. Let us suppose also that there are more English than Scots. We indicate:

Hence the first case is satisfied, that is, if л: is female she is more likely to wear a skirt than trousers, since there are more English than Scots.

However, if we add to the evidence ‘x is Scottish' that she is female, the probability that she wears a skirt does not increase but becomes 0, contrary to what Keynes hypothesized.8

In other words, even if an evidence b is mono-relevant for and favourable to a certain hypothesis a, it is not certain that b increases the probability of a together with other evidences. In fact, axioms of comparative probability do not support principles suitable for determining an updating of probabilities when the evidence is augmented.

As far as we know, it is possible to establish a principle - Bayes's theorem - which allows the updating of probability if the evidence is augmented only when there is a probability measure. Therefore we are faced with a painful dilemma:

Though it is epistemologically more reasonable to deal with most probabilities in comparative terms, we have not yet been able to define a qualitative updating of probability. On the contrary, an updating of probability is possible only if the latter is quantitative. Nonetheless, it is reasonable to maintain that only cases that are very simple from the cognitive point of view allow a reasonable application of a probability measure.

It seems that, if this dilemma is not resolved, the value of the concept of probability, intended as an evaluation of degree of rational belief, loses part of its epistemological relevance: According to the subjectivist solution of the dilemma, Suppes (1994), it is sufficient to find the necessary and sufficient condition for a comparative probability to support a probability measure. But in such a way one renounces completely any logical character of the relation between a priory probabilities and background knowledge.

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