The structure of this article is as follows: First, we describe the limitations of the orthodox approach to probability theory (Section 2). In particular, we will show how NAP can provide a diagnosis of where the objections against appealing to infinitesimals in probability theory go wrong. We shall argue that the mathematical details of the non-Archimedean probability theory matter in this discussion.

The main purpose of the present article is to defend non-standard probability against these critiques. But in recent years, philosophical arguments have been developed by Williamson ( ) Easwaran ( ), and others that purport to show on conceptual grounds that an appeal to infinitesimal probability values is inherently problematic. Non-Archimedean probability theories have been developed that are unobjectionable from a mathematical point of view. That is, the theory should allow for a mathematical representation of any probabilistic situation that is conceptually possible (from a pretheoretic standpoint). 3 Fourth, вЂweak LaplacianismвЂ™ is the requirement that a probability theory should allow for a uniform probability distribution on sample spaces of any cardinality as well as many other probability ratios between the atomic events. Third, вЂperfect additivityвЂ™ is the requirement that the probability of an arbitrary union of mutually disjoint events is equal to the sum of the probabilities of the separate events, where вЂsumвЂ™ has to be defined in an appropriate way in the infinite case. In other words, all sets should be measurable. Second, вЂtotalityвЂ™ is the desideratum that all subsets of the sample space must be assigned a probability value. 2 More generally, we want our probability function to be maximally sensitive to differences in this partial order (inclusion) between events. It is a special case of the Euclidean principle, which requires that any set should be given a strictly larger probability than each of its strict subsets. First, вЂregularityвЂ™ is the constraint that the probability of a possible event (that is, a non-empty subset of the sample space) should be strictly larger than that of the impossible event (that is, the empty set).

NAP is motivated by four desiderata for a theory of probability: regularity, totality, perfect additivity, and weak Laplacianism. Moreover, we think that NAP can be useful in the context of physics, where similar methods have found applications already (see Albeverio et al. As such, NAP is of relevance both to scholars who are interested in objective probability (or вЂchanceвЂ™) and to those interested in subjective probability (and in particular in the rational kind thereof, вЂcredenceвЂ™). 1 Like classical probability theory, NAP is applicable in a wide range of situations and can be employed to model different sources of uncertainty. Examples of such events include the random or biased selection of an element from the set of the natural numbers or the integers, or from an interval of the rational or real numbers. We have proposed a specific non-Archimedean probability theory (henceforth called NAP), which allows the assignment of non-zero probabilities to infinitely unlikely events ( Benci et al.

Infinitesimals