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

Abstract

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 are also more likely to have property ψ. We find that intuition can side with either one depending on the situation. We suggest that a more appropriate formalization of the intuition behind NC is the weaker principle of reasoning by analogy (RA). RA says that if we see an object that is F and G and we know that another object is F, then it is more likely to also be G. Projectability might still be considered valid for relatively uninformed a priori beliefs. If one decides that a principle like projectability is valid for confirmation in an uninformed situation, it provides a test that an a priori distribution must satisfy. Hence, decreasing the arbitrariness of the choice of measure. Further, by considering background knowledge saying only how many ravens there are in the world we conclude that if someone accepts the projectability principle, an agent will not increment the belief that all ravens are black when having observed a white sock. Most Bayesian approaches have so far derived a small increment in confirmation by relying on some particular a priori measure. A conclusion rejected by common sense. We here resolve the contradiction by formally identifying the natural background knowledge considered and the inductive rule that the a priori belief should comply with in the situation at hand. The original paradox is dispelled by rejecting NC.