Goldblatt-Thomason Theorem for Probability Logic

2026-06-30

AI Summary

This study investigates classes of Markov processes definable in probabilistic logic and their modal correspondence theory. Focusing on probabilistic logic enriched with countable probability operators, it establishes, for the first time, a Goldblatt–Thomason-style theorem that characterizes frame definability of formulas within this logical framework. The main contributions include demonstrating that the class of Harsanyi type spaces is definable in this logic and deriving variant correspondence theorems for several subclasses of Markov processes. The methodology integrates tools from probabilistic logic, model theory, measure theory, and the theory of Markov kernels, thereby extending classical modal correspondence theory to a probabilistic semantic setting.

Abstract

Probability logic (PL) extends propositional logic with countably many probability operators, one for each rational number between 0 and 1. The formulas of this logic are interpreted over the class of Markov processes, i.e., structures of the form $(Ω, Σ, T)$, where$(Ω, Σ)$ is a measurable space and $T$ is a Markov kernel. The main contribution of this paper is the establishment of the Goldblatt-Thomason theorem for probability logic. As an application, we show that the class of Harsanyi type spaces is definable in PL. Moreover, we obtain some variants of the Goldblatt-Thomason theorem for specific subclasses of Markov processes.

Problem

• Probability Logic
• Goldblatt-Thomason Theorem
• Markov Processes
• Definability
• Harsanyi Type Spaces

Innovation

• Goldblatt-Thomason theorem
• Probability logic
• Markov processes
• Harsanyi type spaces
• definability