Iterative probability kinematics

Following the pioneer work of Bruno De Finetti [12], conditional probabilit y spaces (allowing for conditioning with events of measure zero) have been studied since (at least) the 1950's. Perhaps the most salient axiomatizatio ns are Karl Popper's in [31], and Alfred Renyi's in [33]. Nonstandard proba bility space [34] are a well known alternative to this approach. Vann McGee proposed in [30] a result relating both approaches by showing that the sta ndard values of infinitesimal probability functions are representable as Po pper functions, and that every Popper function is representable in terms of the standard real values of some infinitesimal measure. Our main goal in t his article is to study the constraints on (qualitative and probabilistic) change imposed by an extended version of McGee's result. We focus on an ext ension capable of allowing for iterated changes of view. Such extension, we argue, seems to be needed in almost all considered applications. Since mos t of the available axiomatizations stipulated (definitionally) important co nstraints on iterated change, we propose a non-question-begging framework, Iterative Probability Systems (IPS) and we show that every Popper function can be regarded as a Bayesian IPS. A generalized version of McGee's result is then proved and several of its consequences considered. In particular we note that our proof requires the imposition of Cumulativity, i.e. the prin ciple that a proposition that is accepted at any stage of an iterative proc ess of acceptance will continue to be accepted at any later stage. The plau sibility and range of applicability of Cumulativity is then studied. In par ticular we appeal to a method for defining belief from conditional probabil ity (first proposed in [42] and then slightly modified in [6] and [3]) in o rder to characterize the notion of qualitative change induced by Cumulative models of probability kinematics. The resulting cumulative notion is then compared with existing axiomatizations of belief change and probabilistic s upposition. We also consider applications in the probabilistic accounts of conditionals [1] and [30].