Skip to main content
SpringerLink
Log in

Boolean Valued Models, Boolean Valuations, and Löwenheim-Skolem Theorems

  • Published:
Journal of Philosophical Logic Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aratake, H. (2021). Sheaves of structures, heyting-valued structures, and a generalization of Łoś’s theorem. Mathematical Logic Quarterly, 67(4), 445–468.

    Article  Google Scholar 

  2. Bell, J. L. (2011). Set theory: Boolean-valued models and independence proofs, (vol. 47). Oxford University Press.

  3. Button, T., & Walsh, S. (2018). Philosophy and model theory. Oxford University Press.

    Book  Google Scholar 

  4. Chang, C. C., & Keisler, J. (1990). Model theory. Elsevier.

    Google Scholar 

  5. Fine, K. (1975). Vagueness, truth and logic. Synthese, 265–300.

  6. Givant, S., & Halmos, P. (2008). Introduction to Boolean algebras. Springer Science & Business Media.

  7. Hamkins, J. D. (2021). A gentle introduction to boolean-valued model theory. Mathematics and philosophy of the infinite.

  8. Hamkins, J. D., & Seabold, D. E. (2012). Well-founded boolean ultrapowers as large cardinal embeddings. arXiv:1206.6075.

  9. Hodges, W. (1993). Model theory. Cambridge University Press.

    Book  Google Scholar 

  10. Jech, T. (2013). Set theory. Springer Science & Business Media.

  11. Jech, T. J. (1985). Abstract theory of abelian operator algebras: an application of forcing. Transactions of the American Mathematical Society, 133–162.

  12. Mansfield, R. (1971). The theory of boolean ultrapowers. Annals of Mathematical Logic, 2(3), 297–323.

    Article  Google Scholar 

  13. McGee, V., & McLaughlin, B. (Forthcoming). Terrestrial logic. Oxford University Press.

  14. Pierobon, M., & Viale, M. (2020). Boolean valued models, presheaves, and étalé spaces. arXiv:2006.14852.

  15. Rasiowa, H. (1963). The mathematics of metamathematics.

  16. Smith, N. (2008). Vagueness and degrees of truth. OUP Oxford.

  17. Takeuti, G. (2015). Two applications of logic to mathematics. In two applications of logic to mathematics: Princeton University Press.

  18. Vaccaro, A., & Viale, M. (2017). Generic absoluteness and boolean names for elements of a polish space. Bollettino dell’Unione Matematica Italiana, 10(3), 293–319.

    Article  Google Scholar 

  19. Viale, M. (2014). Notes on forcing. manuscript.

  20. Wu, X. (2021). Boolean-valued models and indeterminate identity. manucript.

  21. Wu, X. (2022). Boolean-valued models and their applications. PhD thesis, Massachusetts Institute of Technology.

  22. Wu, X. (Forthcoming). Boolean mereology. Journal of Philosophical Logic.

Download references

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.

Author information

Authors and Affiliations

  1. Department of Philosophy and Religious Studies, North Carolina State University, Raleigh, NC, 27695, USA

    Xinhe Wu

Authors
  1. Xinhe Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinhe Wu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10992-023-09732-5

Keywords

Search

Navigation