APPROXIMATION OF CARNAP'S OPTIMUM ESTIMATION METHOD
Carnap's optimum estimation method is not applicable in a direct way for trivial reasons. In this article (originating from 1972), a limit procedure will be defined that leads to an approximation of this method and that is applicable in the inductive situation.
The Carnapian inductive probability function is defined on the set of sentences of a first-order monadic predicate language with tl, t2, ... as individuals, called trials, and P~, P2 .... P~ as monadic predicates, which are exhaustive and mutually exclusive relative to any trial. Let e k be a formulation of the result of the first k trials, such that e k includes the information, for every P~, how many trials, k~, have resulted in P~. Let hi be the hypothesis that the next trial will result in P~.
In his 'The Continuum of Inductive Methods' Carnap obtains as the special values of the inductive probability function: the inductive probability of hi, given e k, inp(h~ I ek), is ki + A/a/(k + A), with A as a parameter for a positive real number. With these special values the inp-function is completely determined by the requirement that inp must be a probability function.
Carnap has proposed inductive probability as explicatum for several explicanda of which estimate of relative frequency is central in this article. He has given a sound base for this explicandum in his 'Logical Foundations of Probability', ยง41D. In our particular language inp(hi [ e k) can be interpreted as an estimate, after the k-th trial, of the relative frequency of the P~ among the total set of trials. Carnap has proved in his Continuum, ยง ยง19-22, that there is a value of A, A ~, which gives the best estimations in the sense that the mean expectation of the square of the error in each estimate is minimal.
Let rj be the unknown objective relative frequency of Pj among the trials, then we have ~7=1 ri = 1 and 1/c~ <~ R =dfY~)'=l (r/) 2 ~ 1, and according to Carnap's calculation: Aa=(1-R)/(R-l/a). Carnap does not see any possibility to use A a in an inductive situation, i.e., after the k-th trial, for A ~ is expressed in the unknown ri's. But after the k-th