Skip to main content

Tense Logics over Lattices

Part of the Lecture Notes in Computer Science book series (LNCS,volume 13468)


Lattice theory has close connections with modal logic via algebraic semantics and lattices of modal logics. However, one less explored direction is to view lattices as relational structures based on partial orders, and study the modal logic over them. In this paper, following the earlier steps of Burgess and van Benthem in the 1980s, we use the basic tense logic and its nominal extensions with binary modalities of infimum and supremum to talk about lattices via standard Kripke semantics. As the main results, we obtain a series of complete axiomatizations of lattices, (un)bounded lattices over partial orders or strict preorders. In particular, we solve an axiomatization problem left open by Burgess (1984).


  • tense logic
  • lattice
  • completeness
  • step-by-step
  • hybrid logic
  • modal logic
  • polyadic modal logic

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions


  1. 1.

    Burgess calls the strict order satisfying \(\texttt{FOASym}\) and \(\texttt{FOSTrans}\) partial order.

  2. 2.

    Here “unbounded” means there are no greatest and least elements.

  3. 3.

    Note that the canonicity is based on the canonical model of normal tense logics.

  4. 4.

    E.g., by Proposition 1 (2), a frame validating \(\mathtt {CR_\textsf {G}}\) also has the property \(\forall x\forall y (xRy \rightarrow \exists t(yRt))\) (obtained by taking \(y=z\) in the original corresponding first-order property).

  5. 5.

    Two formulas \(\textsf {F}p \wedge \textsf {F}q \rightarrow \textsf {F}(\textsf {P}p \wedge \textsf {P}q)\), \(\textsf {P}p \wedge \textsf {P}q \rightarrow \textsf {P}(\textsf {F}p \wedge \textsf {F}q)\) are mentioned in [10] as alternatives for \(\mathtt {CR_\textsf {G}}\) and \(\mathtt {CR_\textsf {H}}\), but they are also not valid over \(\mathfrak {L}_{<}\).

  6. 6.

    This condition is to break the potential symmetry and reflexivity in \(\mathcal {M}^c_\mathbb{S}\mathbb{L}\).

  7. 7.

    Note that this can indeed happen, e.g., when \(\textsf {G}\bot \wedge \textsf {H}\bot \in \varSigma _0\) thus no \(\textsf {F}\varphi \) or \(\textsf {P}\varphi \) is in \(\varSigma _0\).


  1. van Benthem, J.: Semantic parallels in natural language and computation. In: Studies in Logic and the Foundations of Mathematics, vol. 129 (1989)

    Google Scholar 

  2. van Benthem, J.: Modal logic as a theory of information. In: Copeland, J. (ed.) Logic and Reality: Essays on the legacy of Arthur Prior, pp. 135–158. Clarendon Press, Oxford (1996)

    Google Scholar 

  3. van Benthem, J.: Implicit and explicit stances in logic. J. Philos. Log. 48(3), 571–601 (2019)

    CrossRef  Google Scholar 

  4. Birkhoff, G., Neumann, J.V.: The Logic of Quantum Mechanics. Springer, Dordrecht (1975)

    Google Scholar 

  5. Birkhoff, G.T.: Lattice Theory. American Mathematical Society (1967)

    Google Scholar 

  6. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)

    CrossRef  Google Scholar 

  7. Blackburn, P.: Nominal tense logic. Notre Dame J. Formal Logic 34(1), 56–83 (1993)

    Google Scholar 

  8. Blok, W.J.: The lattice of modal logics: an algebraic investigation. J. Symb. Log. 45(2), 221–236 (1980)

    CrossRef  Google Scholar 

  9. Boole, G.: The mathematical analysis of logic. Bull. Math. Biophys. 12(5), 107–107 (2009)

    Google Scholar 

  10. Burgess, J.P.: Basic tense logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic: Volume II: Extensions of Classical Logic, pp. 89–133. Springer, Dordrecht (1984).

    CrossRef  Google Scholar 

  11. Fine, K.: Truth-maker semantics for intuitionistic logic. J. Philos. Log. 43(2–3), 549–577 (2014)

    CrossRef  Google Scholar 

  12. Gargov, G., Goranko, V.: Modal logic with names. J. Philos. Log. 22(6), 607–636 (1993)

    CrossRef  Google Scholar 

  13. Goldblatt, R., Hodkinson, I.: The McKinsey-Lemmon logic is barely canonical. Australas. J. Logic 5 (2007)

    Google Scholar 

  14. Hamkins, J.D., Kikuchi, M.: Set-theoretic mereology. Logic Logical Philos. Special issue “Mereology and beyond, Part II” 25(3), 285–308 (2016)

    Google Scholar 

  15. Holliday, W.H.: On the modal logic of subset and superset: tense logic over Medvedev frames. Stud. Logica. 105(1), 13–35 (2017)

    CrossRef  Google Scholar 

  16. Li, D., Wang, Y.: Mereological bimodal logics. Rev. Symb. Logic 1–36 (2022).

  17. Medvedev, Y.T.: Degrees of difficulty of the mass problems. Dokl. Akad. Nauk SSSR 104, 501–504 (1955)

    Google Scholar 

  18. Prior, A.: Time and Modality. Oxford University Press, Clarendon Press (1957)

    Google Scholar 

  19. Rose, A.: Systems of logic whose truth-values form lattices. Math. Ann. 123, 152–165 (1951)

    CrossRef  Google Scholar 

  20. Sadrzadeh, M., Dyckhoff, R.: Positive logic with adjoint modalities: proof theory, semantics, and reasoning about information. Rev. Symb. Logic 3(3), 351–373 (2010)

    CrossRef  Google Scholar 

Download references


The authors thank John Burgess, Johan van Benthem, and the two anonymous reviewers of WoLLIC2022 for relevant pointers and suggestions. The research is supported by NSSF grant 19BZX135.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Yanjing Wang .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Wang, X., Wang, Y. (2022). Tense Logics over Lattices. In: Ciabattoni, A., Pimentel, E., de Queiroz, R.J.G.B. (eds) Logic, Language, Information, and Computation. WoLLIC 2022. Lecture Notes in Computer Science, vol 13468. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-15297-9

  • Online ISBN: 978-3-031-15298-6

  • eBook Packages: Computer ScienceComputer Science (R0)