Transfinite Modal Logic: a Semi-quantitative Explanation for Bayesian Reasoning

TL;DR

This paper introduces transfinite modal logic, combining modal logic with ordinal arithmetic, to formalize Bayesian reasoning in a semi-quantitative manner, bridging mathematical rigor with practical interpretability.

Contribution

It presents a novel transfinite modal logic framework that integrates ordinal arithmetic into modal logic, providing a semi-quantitative formalization of Bayesian reasoning.

Findings

Properties of ordinal arithmetic are explored and utilized.

The semantics of normal modal logic are extended to transfinite modal logic.

A finite model property theorem for the new logic is proved.

Abstract

Bayesian reasoning plays a significant role both in human rationality and in machine learning. In this paper, we introduce transfinite modal logic, which combines modal logic with ordinal arithmetic, in order to formalize Bayesian reasoning semi-quantitatively. Technically, we first investigate some nontrivial properties of ordinal arithmetic, which then enable us to expand normal modal logic's semantics naturally and elegantly onto the novel transfinite modal logic, while still keeping the ordinary definition of Kripke models totally intact. Despite all the transfinite mathematical definition, we argue that in practice, this logic can actually fit into a completely finite interpretation as well. We suggest that transfinite modal logic captures the essence of Bayesian reasoning in a rather clear and simple form, in particular, it provides a perfect explanation for Sherlock Holmes' famous saying, "When you have eliminated the impossible, whatever remains, however improbable, must be the truth." We also prove a counterpart of finite model property theorem for our logic.