One of the main criticisms of Carnap's inductive logic that Hilary Putnam has raised – alluded to in the epigraph – is that it fails in situations where inductive inference ought to go beyond relative frequencies. It is a little ironic that Carnap and his collaborators could have immediately countered this criticism if they had been more familiar with the work of Bruno de Finetti, who had introduced a formal framework that could be used for solving Putnam's problem already in the late 1930s. De Finetti's central innovation was to use symmetries that generalize exchangeability to various notions of partial exchangeability for inductive inference of patterns.

The goal of this chapter is to show how generalized symmetries can be used to overcome the inherent limitations of order invariant learning models, such as the Johnson–Carnap continuum of inductive methods or the basic model of reinforcement learning. What we shall see is that learning procedures can be modified so as to be able to recognize in principle any finite pattern.

Taking Turns

Order invariant learning rules collapse when confronted with the problem of learning how to take turns. Taking turns is important whenever a learning environment is periodic.