Non-Classical Probabilities Invariant Under Symmetries [corrected]

TL;DR

This paper investigates conditions under which non-classical probability frameworks can preserve symmetries, addressing philosophical issues with classical probabilities in infinite contexts and providing technical characterizations.

Contribution

It offers a systematic analysis of symmetry preservation in non-classical probabilities, filling a gap in understanding their philosophical and mathematical properties.

Findings

Characterizes when symmetries are preserved in full conditional probabilities

Provides conditions for qualitative probabilities to maintain symmetries

Offers technical results supporting the plausibility of strong symmetry notions

Abstract

Classical countably additive real-valued probabilities come at a philosophical cost: in many infinite situations, they assign the same probability value -- namely, zero -- to cases that are impossible as well as to cases that are possible. There are three non-classical approaches to probability that can avoid this drawback: full conditional probabilities, qualitative probabilities and hyperreal probabilities. These approaches have been criticized for failing to preserve intuitive symmetries that can easily be preserved by the classical probability framework, but there has not been a systematic study of the conditions under which these symmetries can and cannot be preserved. This paper fills that gap by giving complete characterizations under which symmetries understood in a certain "strong" way can be preserved by these non-classical probabilities, as well as by offering some results to make it plausible that the strong notion of symmetry here may be the right one. Philosophical implications are briefly discussed, but the main purpose of the paper is to offer technical results to inform more sophisticated further philosophical discussion.