Common probability theories only allow the deduction of probabilities by using previously known or presupposed probabilities. They do not, however, allow the derivation of probabilities from observed data alone. The question thus arises as to how probabilities in the empirical sciences, especially in medicine, may be arrived at. Carnap hoped to be able to answer this question byhis theory of inductive probabilities. In the first four sections of the present paper the above mentioned problem is discussed in general. After a (...) short presentation of Carnap''s theory it is then shown that this theory cannot claim validity for arbitrary random processes. It is suggested that the theory be only applied to binomial and multinomial experiments. By application of de Finetti''s theorem Carnap''s inductive probabilities are interpreted as consecutive probabilities of the Bayesian kind. Through the introduction of a new axiom the decision parameter can be determined even if no a priori knowledge is given. Finally, it is demonstrated that the fundamental problem of Wald''s decision theory, i.e., the determination of a plausible criterion where no a priori knowledge is available, can be solved for the cases of binomial and multinomial experiments. (shrink)

In this paper we describe in some detail a formal computer model of inferential discourse based on a belief system. The key issue is that a logical model in a computer, based on rational sets, can usefully model a human situation based on irrational sets. The background of this work is explained elsewhere, as is the issue of rational and irrational sets (Billinge and Addis, in: Magnani and Dossena (eds.), Computing, philosophy and cognition, 2004; Stepney et al., Journey: Non-classical philosophy—socially (...) sensitive computing in journeys non-classical computation: A grand challenge for computing research, 2004). The model is based on the Belief System (Addis and Gooding, Proceedings of the AISB’99 Symposium on Scientific Creativity, 1999) and it provides a mechanism for choosing queries based on a range of belief. We explain how it provides a way to update the belief based on query results, thus modelling others’ experience by inference. We also demonstrate that for the same internal experience, different models can be built for different actors. (shrink)