Abstract: Among the many approaches to combining logic and probability, an important one has been to assign probabilities to formulas of a classical logic, instantiated from some fixed domain, in a manner that respects logical structure. A natural additional condition is to require that the distribution satisfy the symmetry property known as exchangeability. In this talk I will trace some of the history of this line of investigation, viewing exchangeability from a logical perspective, and report on the current status of a joint program of Ackerman, Freer and myself on countable exchangeable structures. The talk will be accessible to logicians and non-logicians alike.