Abstract. A subjective probability function is broadminded to the extent that it assigns positive probabilities to conjectures that can be possibly true. Assigning to such a conjecture the value 0 amounts to a priori ruling out the possibility of confirming the conjecture to any extent by the growing evidence. A positive value leaves, in principle, the possibility of learning from the evidence. In general, broadmindedness is not an absolute notion, but a graded one, and there is a price for it: the more broadminded the probability, the more complicated it is, because it has to assign non-zero values to more complicated conjectures. The framework which is suggested in the old Gaifman-Snir paper is suitable for phrasing this claim in a precise way and proving it. The technique by which this claim is established is to assume a definable probability function, and to state within the same language a conjecture that can be possibly true, whose probability is 0.
The complexity of the conjecture depends on the complexity of the probability, i.e., the complexity of the formulas that are used in defining it. In the Gaifman-Snir paper we used the arithmetical hierarchy as a measure of complexity. It is possible however to establish similar results with respect to a more “down to earth” measures, defined in terms of the time that it takes to calculate the probabilities, with given precisions.
A claim of this form, for a rather simple setup, was first proven by Hilary Putnam in his paper ““Degree of Confirmation” and inductive logic”, which was published in the 1963 Schilpp volume dedicated to Carnap. The proof uses in a probabilistic context, a diagonalization technique, of the kind used in set theory and in computer science. In the talk I shall present Putnam’s argument and show how diagonalization can be applied in considerably richer setups.
The second part of the talk is rather speculative. I shall point out the possibility that there might be epistemic limitations to what human science can achieve, which are imposed by certain pragmatic factors ‒ such as the criterion of repeatable experiments. All of which would recommend a skeptic attitude.