Abstract
In the topological semantics for modal logic, S4 is well known to be complete for the rational line and for the real line: these are special cases of S4’s completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete but strongly complete, for the rational line. But no similarly easy amendment is available for the real line. In an earlier paper, we proved a general theorem: S4 is strongly complete for any dense-in-itself metric space. Strong completeness for the real line is a special case. In the current paper, we give a proof of strong completeness tailored to the special case of the real line: the current proof is simpler and more accessible than the proof of the more general result and involves slightly different techniques. We proceed in two steps: first, we show that S4 is strongly complete for the space of finite and infinite binary sequences, equipped with a natural topology; and then we show that there is an interior map from the real line onto this space.
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R. Goldblatt
Victoria University of Wellington
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References
Aiello, M., van Benthem, J., & Bezhanishvili, G. (2003). Reasoning about space: The modal way. Journal of Logic and Computation, 13(6), 889–920.
Bezhanishvili, G., & Gehrke, M. (2005). Completeness of S4 with respect to the real line: revisited. Annals of Pure and Applied Logic, 131(1–3), 287–301.
Freyd, P. J., & Scedov, A. (1990). Categories, allegories. Amsterdam: North-Holland.
Goldblatt, R. (1980). Diodorean modality in Minkowski spacetime. Studia Logica, 39(2–3), 219–236.
Goldblatt, R., & Hodkinson, I. (2019). Strong completeness of modal logics over 0-dimensional metric spaces. Review of Symbolic Logic (pp. 1–22). Published online, 24 October 2019.
Kremer, P. (2013). Strong completeness of S4 for any dense-in-itself metric space. Review of Symbolic Logic, 6(3), 545–570.
Kremer, P. (2014). Quantified modal logic on the rational line. Review of Symbolic Logic, 7(3), 439–454.
Lando, T. (2012). Completeness of \(S4\) for the Lebesgue measure algebra. Journal of Philosophical Logic, 41(2), 287–316.
McKinsey, J. C. C. (1941). A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology. Journal of Symbolic Logic, 6, 117–134.
McKinsey, J. C. C., & Tarski, A. (1944). The algebra of topology. Annals of Mathematics, 2(45), 141–191.
Mints, G., & Zhang, T. (2005). A proof of topological completeness for \(S4\) in \((0,1)\). Annals of Pure and Applied Logic, 133(1–3), 231–245.
Rasiowa, H., & Sikorski, R. (1963). The mathematics of metamathematics. Monografie Matematyczne, Tom 41. Państwowe Wydawnictwo Naukowe, Warsaw.
Tarski, A. (1938). Der Aussagenkalkül und die Topologie. Fundamenta Mathematicae, 31, 103–134.
van Benthem, J., Bezhanishvili, G., ten Cate, B., & Sarenac, D. (2006). Multimodal logics of products of topologies. Studia Logica, 84(3), 369–392.
Vickers, S. (1989). Topology via logic. Cambridge tracts in theoretical computer science (Vol. 5). Cambridge: Cambridge University Press.
Acknowledgements
Thanks to the audience at the Ninth International Tbilisi Symposium on Language, Logic and Computation (2011) in Kutaisi, Georgia, for listening to me present the more general paper, Kremer (2013). Special thanks to each of David Gabelaia, Nick Bezhanishvili, Roman Kontchakov, and Mamuka Jibladze, for indulging me by letting me explain the proof in detail in the case considered by this paper, \(\mathbb {R}\). Also, a big thanks to Robert Goldblatt, for carefully reading a draft of this paper and for very useful comments.
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Kremer, P. (2022). Strong Completeness of S4 for the Real Line. In: Düntsch, I., Mares, E. (eds) Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs. Outstanding Contributions to Logic, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-71430-7_10
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