A Logic of Stability: Formalizing Similarity in Counterfactual Reasoning

Counterfactual reasoning is a foundational topic in both philosophical and logical studies \cite{Stalnaker1968-STAATO-5, Lewis1973-LEWC-2}. A pivotal component of counterfactual analysis is the concept of similarity between possible worlds \cite{CORR_A_2022, ESTEVA1997235, Lewis1979-LEWCDA, Makinson94, Pollock1976-POLTPW}. In this paper, we propose the introdutcion of a metric to quantify the degree of similarity between possible worlds, where two worlds are the more similar the longer they share a common history, drawing on a similarity framework influenced by \cite{Lewis1979-LEWCDA}. We prove that this metric satisfies the properties of an ultra-metric, offering a mathematically robust foundation for a corresponding graded notion of hierarchical similarity. We develop and axiomatize a multi-modal logic of similarity, $L_{\square_\varepsilon}$, and demonstrate its soundness and completeness with respect to the class of ultra-metric spaces. Finally, we explore modal definability, establishing connections between ultra-metric semantics and the broader theory of counterfactuals.