Section 1. Logic and applications PROBABILISTIC REASONING ABOUT SECOND ORDER UNCERTAINTY: A LOGICAL CHARACTERIZATION

Suppose a die is being rolled. One thing is to be uncertain about the facet that will eventually show up. One quite different thing, is to be uncertain about whether the die is fair or otherwise unbiased. We can rather naturally refer to first order and second order uncertainty, respectively. In the former case we are uncertain about some (presently unknown) state of affairs. In the latter we are uncertain about our uncertainty. How can we measure first and second order uncertainty? Reasoning under first order uncertainty, or simply uncertainty, has been modelled for over three centuries by the calculus of probability. However it was not until the late 1920’s that the key foundational questions about the meaning and the interpretation of probability were put directly under scrutiny. The work of Bruno de Finetti and Frank P. Ramsey is particularly important, in this respect. They put forward a view according to which probability can be justified as a mathematical measure of one individual’s uncertainty on essentially logical grounds, by referring to the concept of coherence. The emerging view is often referred to as bayesianism according to which probability is: