Abstract
The present paper studies formal properties of so-called modal information logics (MILs)—modal logics first proposed in (van Benthem 1996) as a way of using possible-worlds semantics to model a theory of information. They do so by extending the language of propositional logic with a binary modality defined in terms of being the supremum of two states. First proposed in 1996, MILs have been around for some time, yet not much is known: (van Benthem 2017, 2019) pose two central open problems, namely (1) axiomatizing the two basic MILs of suprema on preorders and posets, respectively, and (2) proving (un)decidability. The main results of the first part of this paper are solving these two problems: (1) by providing an axiomatization [with a completeness proof entailing the two logics to be the same], and (2) by proving decidability. In the proof of the latter, an emphasis is put on the method applied as a heuristic for proving decidability ‘via completeness’ for semantically introduced logics; the logics lack the FMP w.r.t. their classes of definition, but not w.r.t. a generalized class. These results are build upon to axiomatize and prove decidable the MILs attained by endowing the language with an ‘informational implication’—in doing so a link is also made to the work of (Buszkowski 2021) on the Lambek Calculus.
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Evidently, this technique can be adapted to quickly show many more ‘absurdities’; for instance, an infinite strictly ascending chain of MCSs all claiming to be the supremum of two MCSs below. Even if amusing, by now, this becomes too much of a sidetrack (even for an appendix), so we leave this as an activity for the reader.
References
Adriaans, P. (2020). Information. In E.N. Zalta, (Ed.) The Stanford Encyclopedia of Philosophy (Fall 2020 ed.), Metaphysics Research Lab, Stanford University
Aloni, M. (2022). Logic and conversation: The case of free choice. Semantics and Pragmatics
Anderson, A. R., Belnap, N. D., & Dunn, J. M. (1992). Entailment, Vol. II: The Logic of Relevance and Necessity. Princeton University Press.
Blackburn, P., M.d. Rijke, & Y. Venema. (2001). Modal Logic. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press
Burgess, J. (1982). 10. Axioms for tense logic. i. “since” and “until”. Notre Dame Journal of Formal Logic 23. https://doi.org/10.1305/ndjfl/1093870149
Buszkowski, W. (2021) 03. Lambek Calculus with Classical Logic, pp. 1–36
Buszkowski, W., & Farulewski, M. (2009). Nonassociative Lambek Calculus with Additives and Context-Free Languages (Vol. 5533). Berlin, Heidelberg: Springer.
Fine, K. (2017). Truthmaker Semantics, Chapter 22, pp. 556–577. John Wiley & Sons, Ltd
Jongh, D. & F. Veltman (1999). Intensional logics. Available at:https://staff.fnwi.uva.nl/f.j.m.m.veltman/papers/FVeltman-intlog.pdf
Kaminski, M., & Francez, N. (2014). Relational semantics of the lambek calculus extended with classical propositional logic. Stud Logica, 102, 479–497. https://doi.org/10.1007/s11225-013-9474-7
Knudstorp, S.B. (2022). Modal information logics. Master’s thesis, Institute for Logic, Language and Computation. University of Amsterdam
Lambek, J. (1958). The mathematics of sentence structure. American Mathematical Monthly, 65, 154–170.
Urquhart, A. (1972). Semantics for relevant logics. Journal of Symbolic Logic, 37, 159–169.
Urquhart, A. (1973). The Semantics of Entailment. Ph. D. thesis, University of Pittsburgh
van Benthem, J. (1996). Modal logic as a theory of information. In J. Copeland (Ed.) Logic and Reality. Essays on the Legacy of Arthur Prior, pp. 135–168. Clarendon Press, Oxford
van Benthem, J. (2017) 10. Constructive agents.Indagationes Mathematicae 29. https://doi.org/10.1016/j.indag.2017.10.004
van Benthem, J. (2019). Implicit and explicit stances in logic. Journal of Philosophical Logic, 48(3), 571–601. https://doi.org/10.1007/s10992-018-9485-y
van Benthem, J. (2021) Forthcoming Relational patterns, partiality, and set lifting in modal semantics. In Y. Weiss (Ed.) Kripke volume in the series ‘Outstanding Contributions to Logic’, Springer. Preprint available at: https://eprints.illc.uva.nl/id/eprint/1773/
van Fraassen, B. C. (1969). Facts and tautological entailments. The Journal of Philosophy, 66, 477–487.
Wang, X., & Wang, Y. (2022). Tense logics over lattices. Cham: Springer.
Yang, F., & Väänänen, J. (2016). Propositional logics of dependence. Annals of Pure and Applied Logic, 167(7), 557–589. https://doi.org/10.1016/j.apal.2016.03.003
Yang, F., & Väänänen, J. (2017). 07). Propositional team logics. Annals of Pure and Applied Logic, 168, 1406–1441. https://doi.org/10.1016/j.apal.2017.01.007
Acknowledgements
This paper is based on Chapters 1 to 4 from my Master’s thesis [11], supervised by Johan van Benthem and Nick Bezhanishvili. I greatly appreciate their guidance during the thesis and manuscript preparation. My sincere thanks to the editor and referees for their valuable comments
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Appendix A Wildness of the canonical frame
Appendix A Wildness of the canonical frame
As an informal addendum to Section 2, we briefly remark that the canonical relation \(C_\textbf{Pre}\) of the canonical frame for \(\mathbf {MIL_{Pre}}\) is not the supremum relation of \(\le _\textbf{Pre}\).
Remark A.0.1. The following hold:
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1.
There are MCSs \(\Gamma , \Delta \) s.t. \(C_\textbf{Pre}\Gamma \Delta \Delta \) even if \(\Gamma \nleq _\textbf{Pre}\Delta \). In other words, although \(\Gamma \) and \(\Delta \) aren’t in the same cluster (\(\Gamma \nleq _\textbf{Pre}\Delta \)), \(\Gamma \) ‘claims’ to be the ‘supremum’ of \(\Delta \).
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2.
In fact, there are continuum many such MCSs \(\Gamma _i\) all claiming to be the supremum of \(\Delta \).
Proof
Consider the model depicted below where the worlds satisfy all and only the proposition letters shown.
Then \(||v_1||,||v_2||,||w||\) are MCSs where \(||x||\mathrel {:=}\{\varphi \in \mathcal {L}_M\mid x\Vdash \varphi \}\). Moreover, \(Pp\notin ||v_1||=||v_2||\), so (a) since \(p\in ||w||\) we have that \(\Gamma \mathrel {:=}||w||\nleq _\textbf{Pre}||v_1||\mathrel {=:}\Delta \), and (b) since \(w=\sup \{v_1,v_2\}\) we also have \(C_\textbf{Pre}\Gamma \Delta \Delta \), which proves the first claim.
For the second, simply change the valuation of w for proposition letters \(q\ne p\) to get the same results for different MCSs \(\Gamma _i\). Since there are countably many proposition letters (so continuum many subsets of proposition letters), we get continuum many MCSs claiming to be supremum of \(\Delta =||v_1||= ||v_2||\).Footnote 1\(\square \)
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Knudstorp, S.B. Modal Information Logics: Axiomatizations and Decidability. J Philos Logic 52, 1723–1766 (2023). https://doi.org/10.1007/s10992-023-09724-5
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DOI: https://doi.org/10.1007/s10992-023-09724-5