Abstract
Boolean-valued models for first-order languages generalize two-valued models, in that the value range is allowed to be any complete Boolean algebra instead of just the Boolean algebra 2. Boolean-valued models are interesting in multiple aspects: philosophical, logical, and mathematical. The primary goal of this paper is to extend a number of critical model-theoretic notions and to generalize a number of important model-theoretic results based on these notions to Boolean-valued models. For instance, we will investigate (first-order) Boolean valuations, which are natural generalizations of (first-order) theories, and prove that Boolean-valued models are sound and complete with respect to Boolean valuations. With the help of Boolean valuations, we will also discuss the Löwenheim-Skolem theorems on Boolean-valued models.
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Acknowledgements
For their helpful feedback on early drafts of this paper, I would like to thank Vann McGee, Stephen Yablo, and Agustín Rayo. I am also grateful to Johannes Stern and other members of Foundational Studies Bristol at the University of Bristol. Finally, I’d like to thank the editor and especially the anonymous referee for his invaluable comments.
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Wu, X. Boolean Valued Models, Boolean Valuations, and Löwenheim-Skolem Theorems. J Philos Logic 53, 293–330 (2024). https://doi.org/10.1007/s10992-023-09732-5
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DOI: https://doi.org/10.1007/s10992-023-09732-5