What is the plausibility of probability?(revised 2003, 2015)

Abstract: We present and examine a result related to uncertainty reasoning, namely that a certain plausibility space of Cox's type can be uniquely embedded in a minimal ordered field. This, although a purely mathematical result, can be claimed to imply that every rational method to reason with uncertainty must be based on sets of extended probability distributions, where extended probability is standard probability extended with infinitesimals. This claim must be supported by some argumentation of non-mathematical type, however, since pure mathematics does not tell us anything about the world. We propose one such argumentation, and relate it to results from the literature of uncertainty and statistics. In an added retrospective section we discuss some developments in the area regarding countable additivity, partially ordered domains and robustness, and philosophical stances on the Cox/Jaynes approach since 2003. We also show that the most general partially ordered plausibility calculus embeddable in a ring can be represented as a set of extended probability distributions or, in algebraic terms, is a subdirect sum of ordered fields. In other words, the robust Bayesian approach is universal. This result is exemplified by relating Dempster-Shafer's evidence theory to robust Bayesian analysis.