Many-Valued Coalgebraic Logic: From Boolean Algebras to Primal Varieties

Authors Alexander Kurz, Wolfgang Poiger

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Author Details

Alexander Kurz
  • Fowler School of Engineering, Chapman University, Orange, CA, USA
Wolfgang Poiger
  • Department of Mathematics, University of Luxembourg, Esch-sur-Alzette, Luxembourg

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Alexander Kurz and Wolfgang Poiger. Many-Valued Coalgebraic Logic: From Boolean Algebras to Primal Varieties. In 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 270, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We study many-valued coalgebraic logics with primal algebras of truth-degrees. We describe a way to lift algebraic semantics of classical coalgebraic logics, given by an endofunctor on the variety of Boolean algebras, to this many-valued setting, and we show that many important properties of the original logic are inherited by its lifting. Then, we deal with the problem of obtaining a concrete axiomatic presentation of the variety of algebras for this lifted logic, given that we know one for the original one. We solve this problem for a class of presentations which behaves well with respect to a lattice structure on the algebra of truth-degrees.

Subject Classification

ACM Subject Classification
  • Theory of computation → Modal and temporal logics
  • Theory of computation → Categorical semantics
  • Theory of computation → Algebraic semantics
  • coalgebraic modal logic
  • many-valued logic
  • primal algebras
  • algebraic semantics
  • presenting functors


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  1. M. Bílková and M. Dostál. Expressivity of many-valued modal logics, coalgebraically. In J. Väänänen, Å. Hirvonen, and R. de Queiroz, editors, Logic, Language, Information, and Computation, pages 109-124. Springer Berlin Heidelberg, 2016. Google Scholar
  2. M. Bílková, A. Kurz, D. Petrişan, and J. Velebil. Relation lifting, with an application to the many-valued cover modality. Logical Methods in Computer Science, 9(4):739-790, 2013. Google Scholar
  3. P. Blackburn, M. de Rijke, and Y. Venema. Modal Logic. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 2001. Google Scholar
  4. M. M. Bonsangue and A. Kurz. Presenting functors by operations and equations. In L. Aceto and A. Ingólfsdóttir, editors, Foundations of Software Science and Computation Structures, pages 172-186. Springer Berlin Heidelberg, 2006. Google Scholar
  5. F. Bou, F. Esteva, L. Godo, and R. O. Rodríguez. On the minimum many-valued modal logic over a finite residuated lattice. Journal of Logic and Computation, 21(5):739-790, 2011. Google Scholar
  6. S. Burris. Boolean powers. Algebra Universalis, 5:341-360, 1975. Google Scholar
  7. S. Burris and H.P. Sankappanavar. A Course in Universal Algebra. Graduate Texts in Mathematics. Springer, 1981. Google Scholar
  8. X. Caicedo and R. Rodriguez. Standard Gödel modal logics. Studia Logica, 94:189-214, 2010. Google Scholar
  9. C. Cîrstea and D. Pattinson. Modular construction of modal logics. In P. Gardner and N. Yoshida, editors, CONCUR 2004 - Concurrency Theory, pages 258-275. Springer Berlin Heidelberg, 2004. Google Scholar
  10. M. C. Fitting. Many-valued modal logics. Fundamenta Informaticae, 15(3-4):35-254, 1991. Google Scholar
  11. A. L. Foster. Generalized "Boolean" theory of universal algebras. Part i. Mathematische Zeitschrift, 58:306-336, 1953. Google Scholar
  12. G. Hansoul and B. Teheux. Extending Łukasiewicz logics with a modality: Algebraic approach to relational semantics. Studia Logica, 101:505-545, 2013. Google Scholar
  13. T.-K. Hu. Stone duality for primal algebra theory. Mathematische Zeitschrift, 110:180-198, 1969. Google Scholar
  14. T.-K. Hu. On the topological duality for primal algebra theory. Algebra Universalis, 1:152-154, 1971. Google Scholar
  15. B. Jacobs and A. Sokolova. Exemplaric expressivity of modal logics. Journal of Logic and Computation, 20(5):1041-1068, 2009. Google Scholar
  16. B. Klin. Coalgebraic modal logic beyond sets. Electronic Notes in Theoretical Computer Science, 173:177-201, 2007. Proceedings of the 23rd Conference on the Mathematical Foundations of Programming Semantics (MFPS XXIII). Google Scholar
  17. C. Kupke, A. Kurz, and D. Pattinson. Algebraic semantics for coalgebraic logics. Electronic Notes in Theoretical Computer Science, 106:219-241, 2004. Google Scholar
  18. C. Kupke and D. Pattinson. Coalgebraic semantics of modal logics: An overview. Theoretical Computer Science, 412(38):5070-5094, 2011. Google Scholar
  19. A. Kurz and R. Leal. Modalities in the Stone age: A comparison of coalgebraic logics. Theoretical Computer Science, 430:88-116, 2012. Mathematical Foundations of Programming Semantics (MFPS XXV). Google Scholar
  20. A. Kurz and D. Petrişan. Presenting functors on many-sorted varieties and applications. Information and Computation, 208(12):1421-1446, 2010. Google Scholar
  21. A. Kurz, W. Poiger, and B. Teheux. New perspectives on semi-primal varieties. Preprint available at, 2023.
  22. A. Kurz and J. Rosický. Strongly complete logics for coalgebras. Logical Methods in Computer Science, 8(3):1-32, 2012. Google Scholar
  23. C.-Y. Lin and C.-J. Liau. Many-valued coalgebraic modal logic: One-step completeness and finite model property. Preprint available at, 2022.
  24. M. Marti and G. Metcalfe. Expressivity in chain-based modal logics. Archive for Mathematical Logic, 57:361-380, 2018. Google Scholar
  25. Y. Maruyama. Algebraic study of lattice-valued logic and lattice-valued modal logic. In R. Ramanujam and S. Sarukkai, editors, Logic and Its Applications. ICLA, pages 170-184. Springer Berlin Heidelberg, 2009. Google Scholar
  26. Y. Maruyama. Natural duality, modality, and coalgebra. Journal of Pure and Applied Algebra, 216(3):565-580, 2012. Google Scholar
  27. L. S. Moss. Coalgebraic logic. Annals of Pure and Applied Logic, 96(1):277-317, 1999. Google Scholar
  28. E. Pacuit. Neighborhood Semantics for Modal Logic. Short Textbooks in Logic. Springer, 2017. Google Scholar
  29. D. Pattinson. Coalgebraic modal logic: Soundness, completeness and decidability of local consequence. Theoretical Computer Science, 309(1):177-193, 2003. Google Scholar
  30. D. Pattinson. Expressive logics for coalgebras via terminal sequence induction. Notre Dame Journal of Formal Logic, 45(1):19-33, 2004. Google Scholar
  31. R. W. Quackenbush. Primality: The influence of Boolean algebras in universal algebra. In Georg Grätzer. Universal Algebra. Second Edition, pages 401-416. Springer, New York, 1979. Google Scholar
  32. U. Rivieccio, A. Jung, and R. Jansana. Four-valued modal logic: Kripke semantics and duality. Journal of Logic and Computation, 27(1):155-199, 2017. Google Scholar
  33. A. Salibra, A. Bucciarelli, A. Ledda, and F. Paoli. Classical logic with n truth values as a symmetric many-valued logic. Foundations of Science, 28:115-142, 2023. Google Scholar
  34. A. Schiendorfer, A. Knapp, G. Anders, and W. Reif. MiniBrass: Soft constraints for MiniZinc. Constraints, 23(4):403-450, 2018. Google Scholar
  35. L. Schröder. Expressivity of coalgebraic modal logic: The limits and beyond. Theoretical Computer Science, 390(2):230-247, 2008. Foundations of Software Science and Computational Structures. Google Scholar
  36. A. Vidal, F. Esteva, and L. Godo. On modal extensions of product fuzzy logic. Journal of Logic and Computation, 27(1):299-336, 2017. Google Scholar