Abstract
This paper presents a recent formalization of a Henkin-style completeness proof for the propositional modal logic S5 using the Lean theorem prover. The proof formalized is close to that of Hughes and Cresswell [8], but the system, based on a different choice of axioms, is better described as a Mendelson system augmented with axiom schemes for K, T, S4, and B, and the necessitation rule as a rule of inference. The language has the false and implication as the only primitive logical connectives and necessity as the only primitive modal operator. The full source code is available online and has been typechecked with Lean 3.4.2.
The author thanks Jeremy Avigad, Mario Carneiro, Rajeev Goré, and Minchao Wu for helpful suggestions. An early version of this work was presented at the Lean Together 2019, Amsterdam, January 7–11, 2019. The source code described in this paper is publicly available online at: https://github.com/bbentzen/mpl/. An extended version is available at https://arxiv.org/abs/1910.01697.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In independent work done roughly at the same time the author first completed this formalization in 2018, Wu and Goré [14] have described a formalization in Lean of modal tableaux for modal logics K, KT, and S4 with decision procedures with proofs of soundness and completeness. Also in 2018, but unknown to the author, From [6] formalized a Henkin-style completeness proof for system K in Isabelle.
References
Carneiro, M.: The lean 3 mathematical library (mathlib) (2018). https://robertylewis.com/files/icms/Carneiro_mathlib.pdf. International Congress on Mathematical Software
Coquand, C.: A formalised proof of the soundness and completeness of a simply typed lambda-calculus with explicit substitutions. Higher-Order Symb. Comput. 15(1), 57–90 (2002)
Coquand, T., Huet, G.: The calculus of constructions. Inf. Comput. 76(2–3), 95–120 (1988)
Doczkal, C., Smolka, G.: Constructive completeness for modal logic with transitive closure. In: Hawblitzel, C., Miller, D. (eds.) CPP 2012. LNCS, vol. 7679, pp. 224–239. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-35308-6_18
Doczkal, C., Smolka, G.: Completeness and decidability results for CTL in coq. In: Klein, G., Gamboa, R. (eds.) ITP 2014. LNCS, vol. 8558, pp. 226–241. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08970-6_15
From, A.H.: Epistemic logic. Archive of Formal Proofs, October 2018. https://devel.isa-afp.org/entries/Epistemic_Logic.html. Formal proof development
Herbelin, H., Lee, G.: Forcing-based cut-elimination for Gentzen-style intuitionistic sequent calculus. In: Ono, H., Kanazawa, M., de Queiroz, R. (eds.) WoLLIC 2009. LNCS (LNAI), vol. 5514, pp. 209–217. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02261-6_17
Hughes, G.E., Cresswell, M.J.: A New Introduction to Modal Logic. Psychology Press (1996)
Negri, S.: Kripke completeness revisited. In: Acts of Knowledge: History, Philosophy and Logic: Essays Dedicated to Göran Sundholm, pp. 247–282 (2009)
Norell, U.: Dependently typed programming in Agda. In: Koopman, P., Plasmeijer, R., Swierstra, D. (eds.) AFP 2008. LNCS, vol. 5832, pp. 230–266. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04652-0_5
Pfenning, F., Paulin-Mohring, C.: Inductively defined types in the calculus of constructions. In: Main, M., Melton, A., Mislove, M., Schmidt, D. (eds.) MFPS 1989. LNCS, vol. 442, pp. 209–228. Springer, New York (1990). https://doi.org/10.1007/BFb0040259
The Coq project: The coq proof assistant. http://www.coq.inria.fr (2017)
Tews, H.: Formalizing cut elimination of coalgebraic logics in Coq. In: Galmiche, D., Larchey-Wendling, D. (eds.) TABLEAUX 2013. LNCS (LNAI), vol. 8123, pp. 257–272. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40537-2_22
Wu, M., Goré, R.: Verified decision procedures for modal logics. In: Harrison, J., O’Leary, J., Tolmach, A. (eds.) 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), vol. 141, pp. 31:1–31:19. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl (2019). https://doi.org/10.4230/LIPIcs.ITP.2019.31. http://drops.dagstuhl.de/opus/volltexte/2019/11086
Acknowledgements
Work supported in part by the AFOSR grant FA9550-18-1-0120. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the AFOSR.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Bentzen, B. (2021). A Henkin-Style Completeness Proof for the Modal Logic S5. In: Baroni, P., Benzmüller, C., Wáng, Y.N. (eds) Logic and Argumentation. CLAR 2021. Lecture Notes in Computer Science(), vol 13040. Springer, Cham. https://doi.org/10.1007/978-3-030-89391-0_25
Download citation
DOI: https://doi.org/10.1007/978-3-030-89391-0_25
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-89390-3
Online ISBN: 978-3-030-89391-0
eBook Packages: Computer ScienceComputer Science (R0)