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Invariant probabilities

The last condition we consider concerns the relevance quotient. This condition is stated for exchangeable probabilities. Let n and n' be occupation vectors whose evidence sizes are n, while j Ф g and к Ф m cells, then

We call invariant a probability function for which C3 holds. Invariance means that, apart from a parameter that we specify below, the relevance quotient is unaffected by both the considered cells and the occupation vector, but it depends only upon the evidence size. For this reason we shall write Q(n) instead of Qg (n).

A first trivial consequence of C3 concerns the relevance quotient at V. As we have just seen, there is no need to specify the individuals and the cells involved in the relevance quotient. Moreover, in Qg (V) the evidence size is 0. If we take this into account, it becomes natural to use for this notion a symbol q, following Carnap, that is, we write

Another immediate consequence of C3 is that the probability (2), which obviously depends upon j, only depends upon the occupation number of the cell j, that is n, and the evidence size, n. In fact, for C3 we have

Hence P( j | ng ) = P( j | nk ). This means that the probability of the cell j does not vary, changing the occupation number of cells different from j. In other words,

Carnap called (13) the А-principle. Thus we have proved that the А-principle follows from invariance, and we are in a position to say something about the problem Carnap left open. Essentially Carnap's confirmation theory amounts to a search for a well-defined predictive probability, that is to say, to a search for the value of (2). This was done by stating some conditions Carnap regarded as plausible. Thus, besides the probability axioms, he assumed C1 and C2. But these conditions are not enough to determine the value of (2). To arrive at this value he suggested the А-principle (Carnap, 1980, p. 84), that is, (13). Exchangeability ensures that the probability of j does not depend upon the full evidence but upon its occupation vector. That is, exchangeability cuts down the influence of the evidence to its statistical distribution. The А-principle further cuts down the influence of the statistical distribution to nj, the occupation number of the cell, and n, the size of the system. In other words, the А-principle turns all families of attributes into dicotho- mies: the cell j and the cell of all individuals not belonging to j.

 
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