On rules of induction and the raven paradox in Bayesian confirmation theory

Confirmation theory is studying how one can confirm a universal statement like "All ravens are black". Early authors discussed how one's degree of belief in such a statement should change with new evidence and suggested various rules of induction. Nicod's Condition (NC) says that the claim that all F are G is supported by observing a previously unseen object that is both F and G. Hempel pointed out that NC implies the paradoxical conclusion that observing a white sock supports that all ravens are black. In our time, confirmation is studied by using subjective conditional probability as degrees of belief with Kolmogorov's axioms as the main rules of induction. The old rules and problems of induction are, however, still studied within the probabilistic framework. We consider a setting where the number of individuals having a particular property is given and find that NC can contradict a simpler principle, namely projectability (PJ) which says that if we observe an object with property ψ then other objects a...