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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.12502 |
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| _version_ | 1866908492842926080 |
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| author | Esteves, Marta |
| author_facet | Esteves, Marta |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12502 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Logic of Stability: Formalizing Similarity in Counterfactual Reasoning Esteves, Marta Logic 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. |
| title | A Logic of Stability: Formalizing Similarity in Counterfactual Reasoning |
| topic | Logic |
| url | https://arxiv.org/abs/2508.12502 |