Preferential Structures for Comparative Probabilistic Reasoning

TL;DR

This paper demonstrates that a modified preferential structure aligns the qualitative reasoning system for comparative likelihood with a probabilistic approach based on sets of probabilities, bridging the gap between qualitative and quantitative methods.

Contribution

It introduces a natural modification to preferential structures that makes their logical system equivalent to that of a probabilistic approach using sets of probability measures.

Findings

Modified preferential structures match probabilistic reasoning with sets of probabilities.

Standard preferential approaches validate incorrect principles from a probabilistic perspective.

The same structures used in non-monotonic logic can be applied to comparative probabilistic reasoning.

Abstract

Qualitative and quantitative approaches to reasoning about uncertainty can lead to different logical systems for formalizing such reasoning, even when the language for expressing uncertainty is the same. In the case of reasoning about relative likelihood, with statements of the form $\varphi\succsim\psi$ expressing that $\varphi$ is at least as likely as $\psi$, a standard qualitative approach using preordered preferential structures yields a dramatically different logical system than a quantitative approach using probability measures. In fact, the standard preferential approach validates principles of reasoning that are incorrect from a probabilistic point of view. However, in this paper we show that a natural modification of the preferential approach yields exactly the same logical system as a probabilistic approach--not using single probability measures, but rather sets of probability measures. Thus, the same preferential structures used in the study of non-monotonic logics and belief revision may be used in the study of comparative probabilistic reasoning based on imprecise probabilities.