MODAL LOGICS WITH THE INTERSECTION MODALITY
- Authors: Zolin E.E1
-
Affiliations:
- Lomonosov Moscow State University
- Issue: Vol 521, No 1 (2025)
- Pages: 107-123
- Section: MATHEMATICS
- URL: https://rjsvd.com/2686-9543/article/view/683157
- DOI: https://doi.org/10.31857/S2686954325010139
- EDN: https://elibrary.ru/BRIGLT
- ID: 683157
Cite item
Abstract
We give a simple proof of a recently obtained in [12] result on the completeness of modal logics with the modality that corresponds to the intersection of accessibility relations in a Kripke model. In epistemic logic, this is the so-called distributed knowledge operator. We prove completeness for the logics in the modal languages of two types: one has the modalities □1,...,□n for the relations R1,...,Rn that satisfy a unimodal logic L, and the modality □n+1 for the intersection Rn+1=R1 ∩...∩ Rn; the other language has the modalities □i (i ∈ Σ) for the relations Ri that satisfy the logic L, and, for every nonempty subset of indices I ⊆ Σ, the modality □I for the intersection ∩i∈I Ri. While in [12] the completeness is proved only for the logics over K, KD, KT, K4, S4, and S5, here we give a "uniform" construction that enables us to obtain completeness for the logics with intersection over the 15 so-called "traditional" modal logics KΛ, for Λ ⊆ {D, T, B, 4, 5}. The proof method is based on unravelling of a frame and then taking the Horn closure of the resulting frame.
About the authors
E. E Zolin
Lomonosov Moscow State University
Email: vshehtman@gmail.com
Moscow, Russia
References
- Chagrov A., Zakharyaschev M. Modal Logic. Clarendon Press, 1997. (Oxford logic guides). ISBN 9780198537793.
- Gabbay D., Shehtman V., Skvortsov D. Quantification in nonclassical logic, volume 1. Elsevier, 2009. ISBN 9780444520128.
- Goldblatt R. Logics of Time and Computation. Center for the Study of Language, Information, 1987. (CSLI lecture notes). ISBN 9780937073124.
- Goranko V., Passy S. Using the Universal Modality: Gains and Questions // Journal of Logic and Computation. 1992. V2, N.4. P. 5—30.
- Handbook of Epistemic Logic / ed. by H. van Ditmarsch [et al.]. College Publications, 2015. ISBN 978-1-84890-158-2.
- Kikot S., Shapirovsky I., Zolin E. Modal Logics with Transitive Closure: Completeness, Decidability, Filtration In Advances in Modal Logic, v.13, p. 369—388, College Publications, 2020.
- Kozen D., Parikh R. An elementary proof of the completeness of PDL // Theoretical Computer Science. 1981. V 14, N. 1. P. 113-118.
- Modal Logic / Stanford Encyclopedia of Philosophy. 2018. URL: https://plato.stanford.edu/entries/logic-modal/.
- Segerberg K. A completeness theorem in the modal logic of programs // Banach Center Publications. 1982. V. 9, N. 1. P. 31-46.
- Segerberg K. A Model Existence Theorem in Infinitary Propositional Modal Logic // Journal of Philosophical Logic. 1994. V. 23, N. 4. P. 337-367.
- Sundholm G. A Completeness Proof for an Infinitary Tense-Logic // Theoria. 1977. V. 43, N. 1. P. 47-51.
- Wang J.N., Agotnes T. Simpler completeness proofs for modal logics with intersection. ArXiv:2004.02120 [cs.LO].
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