 
Summary: Iterative Probability Kinematics
Horacio ArloCosta and Richmond H. Thomason
Abstract
ABSTRACT. Following the pioneer work of Bruno De Finetti [12], conditional prob
ability spaces (allowing for conditioning with events of measure zero) have been studied
since (at least) the 1950's. Perhaps the most salient axiomatizations are Karl Popper's
in [30], and Alfred Renyi's in [32]. Nonstandard probability spaces [33] are a well
know alternative to this approach. Vann McGee proposed in [29] a result relating both
approaches by showing that the standard values of innitesimal probability functions
are representable as Popper functions, and that every Popper function is representable
in terms of the standard real values of some innitesimal measure.
Our main goal in this article is to study the constraints on (qualitative and prob
abilistic) change imposed by an extended version of McGee's result. We focus on an
extension capable of allowing for iterated changes of view. Such extension, we argue,
seems to be needed in almost all considered applications. Since most of the available
axiomatizations stipulate (denitionally) important constraints on iterated change, we
propose a nonquestionbegging 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
