Abstract
Weakly Aggregative Modal Logic (\(\textsf {WAML}\)) is a collection of disguised polyadic modal logics with n-ary modalities whose arguments are all the same. \(\textsf {WAML}\) has some interesting applications on epistemic logic and logic of games, so we study some basic model theoretical aspects of \(\textsf {WAML}\) in this paper. Specifically, we give a van Benthem-Rosen characterization theorem of \(\textsf {WAML}\) based on an intuitive notion of bisimulation and show that each basic \(\textsf {WAML}\) system \(\mathbb {K}_n\) lacks Craig Interpolation.
The main work of the first author was completed during his Ph.D. at Peking University.
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Notes
- 1.
This is not to be confused with the non-contingency operator, which is also denoted as \(\nabla \) in non-contingency or knowing whether logics [14].
- 2.
One can find a model theoretical survey on \(\textsf {PML}\) in [22].
- 3.
Name mentioned by Yde Venema via personal communications.
- 4.
Other connections between WAML and graph coloring problems can be found in [24] where the four-color problem is coded by the validity of some formulas in the WAML language.
- 5.
This rule can be simplified by the axiom \(\Box \top \) since we have \(\mathtt {RM}\) here.
- 6.
We have another proof for the Characterization theorem over arbitrary n-models, using tailored notions of saturation and ultrafilter extension for \(\textsf {WAML}^n\), due to the space limit we only present the proof which also works for finite models.
References
Allen, M.: Complexity results for logics of local reasoning and inconsistent belief. In: Proceedings of the 10th Conference on Theoretical Aspects of Rationality and Knowledge, pp. 92–108. National University of Singapore (2005)
Andréka, H., Németi, I., van Benthem, J.: Modal languages and bounded fragments of predicate logic. J. Philos. Logic 27(3), 217–274 (1998)
Andréka, H., Van Benthem, J., Németi, I.: Back and forth between modal logic and classical logic. Logic J. IGPL 3(5), 685–720 (1995)
Apostoli, P.: On the completeness of first degree weakly aggregative modal logics. J. Philos. Logic 26(2), 169–180 (1997)
Apostoli, P., Brown, B.: A solution to the completeness problem for weakly aggregative modal logic. J. Symbolic Logic 60(3), 832–842 (1995)
Arló Costa, H.: Non-adjunctive inference and classical modalities. J. Philos. Logic 34(5), 581–605 (2005)
Baader, F., Horrocks, I., Sattler, U.: Description logics. Found. Artif. Intell. 3, 135–179 (2008)
Beall, J., et al.: On the ternary relation and conditionality. J. Philos. Logic 41(3), 595–612 (2012)
van Benthem, J., van Eijck, J., Kooi, B.: Logics of communication and change. Inf. Comput. 204(11), 1620–1662 (2006)
Blackburn, P., De Rijke, M., Venema, Y.: Modal Logic, vol. 53. Cambridge University Press, Cambridge (2002)
Chellas, B.F.: Modal Logic: An Introduction. Cambridge University Press, Cambridge (1980)
De Caro, F.: Graded modalities, ii (canonical models). Studia Logica 47(1), 1–10 (1988). https://doi.org/10.1007/BF00374047
Fagin, R., Halpern, J., Moses, Y., Vardi, M.: Reasoning About Knowledge. MIT Press, Cambridge (1995)
Fan, J., Wang, Y., van Ditmarsch, H.: Contingency and knowing whether. Rev. Symbol. Logic 8, 75–107 (2015)
Fattorosi-Barnaba, M., De Caro, F.: Graded modalities. I. Studia Logica 44(2), 197–221 (1985). https://doi.org/10.1007/BF00379767
Fervari, R., Herzig, A., Li, Y., Wang, Y.: Strategically knowing how. In: Proceedings of IJCAI 2017, pp. 1031–1038 (2017)
Fine, K., et al.: In so many possible worlds. Notre Dame J. Formal Logic 13(4), 516–520 (1972)
Gu, T., Wang, Y.: “Knowing value” logic as a normal modal logic. In: Proceedings of AiML, vol. 11, pp. 362–381 (2016)
Hansen, H., Kupke, C., Pacuit, E.: Neighbourhood structures: bisimilarity and basic model theory. Logical Meth. Comput. Sci. 5(2) (2009). https://doi.org/10.2168/LMCS-5(2:2)2009
Jennings, R.E., Schotch, P.K.: Some remarks on (weakly) weak modal logics. Notre Dame J. Formal Logic 22(4), 309–314 (1981)
Kamp, H.: Tense logic and the theory of linear order. Ph.D. thesis, UCLA (1968)
Liu, J.: Model theoretical aspects of polyadic modal logic: an exposition. Stud. Logic 12(3), 79–101 (2019)
Lutz, C., Wolter, F.: Foundations for uniform interpolation and forgetting in expressive description logics. In: Twenty-Second International Joint Conference on Artificial Intelligence (2011)
Nicholson, T., Allen, M.: Aggregative combinatorics: an introduction. In: Student Session of 2nd North American Summer School in Language, Logic, and Information, pp. 15–25 (2003)
Nicholson, T., Allen, M.: Aggregative combinatorics: an introduction. In: Proceedings of the Student Session, 2nd North American Summer School in Logic, Language, and Information (NASSLLI-03), pp. 15–25 (2003)
Nicholson, T., Jennings, R.E., Sarenac, D.: Revisiting completeness for the \({K}_{n}\) modal logics: a new proof. Logic J. IGPL 8(1), 101–105 (2000)
Otto, M.: Elementary proof of the van Benthem-Rosen characterisation theorem. Technical report 2342 (2004)
de Rijke, M.: Extending modal logic. Ph.D. thesis, ILLC, University of Amsterdam (1993)
Routley, R., Meyer, R.K.: The semantics of entailment – II. J. Philos. Logic 1(1), 53–73 (1972)
Routley, R., Meyer, R.K.: The semantics of entailment – III. J. Philos. Logic 1(2), 192–208 (1972)
Santocanale, L., Venema, Y., et al.: Uniform interpolation for monotone modal logic. Adv. Modal Logic 8, 350–370 (2010)
Schotch, P., Jennings, R.: Modal logic and the theory of modal aggregation. Philosophia 9(2), 265–278 (1980)
Segerberg, K.: An Essay in Classical Modal Logic. Filosofiska Föreningen Och Filosofiska Institutionen Vid Uppsala Universitet, Uppsala (1971)
Van Benthem, J., Bezhanishvili, N., Enqvist, S., Yu, J.: Instantial neighbourhood logic. Rev. Symbolic Logic 10(1), 116–144 (2017)
Wang, Y.: A logic of goal-directed knowing how. Synthese 195(10), 4419–4439 (2018)
Wang, Y., Fan, J.: Conditionally knowing what. In: Proceedings of AiML, vol. 10, pp. 569–587 (2014). www.aiml.net/volumes/volume10/Wang-Fan.pdf
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Liu, J., Wang, Y., Ding, Y. (2019). Weakly Aggregative Modal Logic: Characterization and Interpolation. In: Blackburn, P., Lorini, E., Guo, M. (eds) Logic, Rationality, and Interaction. LORI 2019. Lecture Notes in Computer Science(), vol 11813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-60292-8_12
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