Skip to main content
SpringerLink
Log in

Model Theoretical Aspects of Weakly Aggregative Modal Logic

  • Published:
Journal of Logic, Language and Information Aims and scope Submit manuscript

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 interesting applications on epistemic logic, deontic logic, and the logic of belief. In this paper, we study some basic model theoretical aspects of \(\textsf {WAML}\). Specifically, we first give a van Benthem–Rosen characterization theorem of \(\textsf {WAML}\) based on an intuitive notion of bisimulation. Then, in contrast to many well known normal or non-normal modal logics, we show that each basic \(\textsf {WAML}\) system \({\mathbb {K}}_n\) lacks Craig interpolation. Finally, by model theoretical techniques, we show that an extension of \({\mathbb {K}}_2\) does have Craig interpolation, as an example of amending the interpolation problem of \(\textsf {WAML}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

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 (Fan et al. 2015).

  2. One can find a model theoretical survey on \(\textsf {PML}\) in Liu (2019).

  3. Name mentioned by Yde Venema via personal communications.

  4. It is worth noting that the parameter n appears only in the semantics and not in the syntax. This suggests a natural generalization of the semantics where we drop the requirement that all successor tuples are of the same length; they do not even need to be tuples, since how worlds are ordered in those tuples is also not important. The resulting semantics is then based on hypergraphs. See Ding et al. (2021) and Nicholson (2009) for developments along this line.

  5. Other connections between WAML and graph coloring problems can be found in Nicholson and Allen (2003), where the four-color problem is coded by the validity of some rules in the WAML language.

  6. We include the propositional constant \(\top \) here because otherwise logics in this language trivially cannot have the Craig Interpolation Property.

  7. The rule \(\mathtt {N}\) can be replaced by the axiom \(\Box \top \) since we have \(\mathtt {RM}\) here.

  8. An infinitary proof for the Characterization Theorem over arbitrary n-models using \(\omega \)-saturated ultrapowers is also possible. but due to the space limit we only present the proof which also works for finite models.

  9. To avoid too many subscripts, we use square brackets to index vectors: for any tuple \(\vec {v}\) of length n and natural number i such that \(1 \le i \le n\), \(\vec {v}[i]\) is the i’th element of \(\vec {v}\).

  10. See Libkin (2013, Chap 4), for a thorough introduction to Gaifman graph and related results, though we do not rely on any of them in this paper; the graph is introduced only to define the distance function.

  11. In textbooks such as Libkin (2013) using Gaifman graphs, the distance to be preserved is usually \(3^{q-m}\) instead of \(2^{q-m}\). However, in the literature of modal logic, the \(2^{q-m}\) bound is more common (see Goranko et al. 2007 for example), and even in a general setting in first-order logic, an \(O(2^n)\) bound is achievable. See Exercise 4.10 of Libkin (2013).

  12. A formula \(\varphi \) is rank-1 if all propositional letters occur in the scope of exactly 1 modal operator. A rank-1 monotone logic is the smallest set of formulas containing a set of rank-1 formulas and all propositional tautologies and is closed under the rules RM, modus ponens, and uniform substitution.

  13. Let \(P_1\) and \(P_2\) be the two parts and assume that \(n > 2\). Then for each \(u \in P_1\) pick an edge \(e_u\) covering u and thus forming a function \(f: P_1 \rightarrow P_2\) such that f(u) is the vertex of \(e_u\) in \(P_2\). If the range of f has more than 1 vertex so that \(|P_2 {\setminus } {\textit{range}}(f)| \le n-2\), then we are done as we just need to pick one edge for each one in \(P_2 {\setminus } {\textit{range}}(f)\), resulting in a total of at most \(2n-2\) edges. If \({\textit{range}}(f)\) is a singleton, say \(\{b_1\}\), we still pick for each \(v \in P_2 {\setminus } \{b_1\}\) an edge \(e_v\) covering v and thus forming similarly a function \(g:P_2{\setminus } \{b_1\} \rightarrow P_1\) such that g(v) is the vertex of \(e_v\) in \(P_1\). Now we may have picked \(2n-1\) edges. However, pick \(b_2 \in P_2 {\setminus } \{b_1\}\), and then we can omit the edge \(e_{g(b_2)}\) we picked. This is because the two vertices \(e_{g(b_2)}\) covers, namely \(g(b_2)\) and \(b_1\), are both covered by other edges. \(g(b_2)\) of course is covered by \(e_{b_2}\). \(b_1\) is covered by \(e_u\) for any \(u \in P_1 {\setminus } \{g(b_2)\}\). Hence we still need no more than \(2n-2\) edges.

References

  • Allen, M. (2005). 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.

  • Andréka, H., Németi, I., & van Benthem, J. (1998). Modal languages and bounded fragments of predicate logic. Journal of Philosophical Logic, 27(3), 217–274.

    Article  Google Scholar 

  • Andréka, H., Van Benthem, J., & Németi, I. (1995). Back and forth between modal logic and classical logic. Logic Journal of the IGPL, 3(5), 685–720.

    Article  Google Scholar 

  • Apostoli, P. (1997). On the completeness of first degree weakly aggregative modal logics. Journal of Philosophical Logic, 26(2), 169–180.

    Article  Google Scholar 

  • Apostoli, P., & Brown, B. (1995). A solution to the completeness problem for weakly aggregative modal logic. The Journal of Symbolic Logic, 60(3), 832–842.

    Article  Google Scholar 

  • Arló Costa, H. (2005). Non-adjunctive inference and classical modalities. Journal of Philosophical Logic, 34(5), 581–605.

    Article  Google Scholar 

  • Baader, F., Horrocks, I., & Sattler, U. (2008). Description logics. Foundations of Artificial Intelligence, 3, 135–179.

    Article  Google Scholar 

  • Beall, J., Brady, R., Dunn, J. M., Hazen, A. P., Mares, E., Meyer, R. K., Priest, G., Restall, G., Ripley, D., Slaney, J., & Sylvan, R. (2012). On the ternary relation and conditionality. Journal of Philosophical Logic, 41(3), 595–612.

    Article  Google Scholar 

  • Blackburn, P., De Rijke, M., & Venema, Y. (2002). Modal logic (Vol. 53). Cambridge University Press.

  • Blackburn, P., van Benthem, J. F., & Wolter, F. (2006). Handbook of modal logic (Vol. 3). Elsevier.

  • Chellas, B. F. (1980). Modal logic: An introduction. Cambridge University Press.

  • De Caro, F. (1988). Graded modalities, ii (canonical models). Studia Logica, 47(1), 1–10.

    Article  Google Scholar 

  • de Rijke, M. (1993). Extending modal logic. Ph.D. thesis, ILLC, University of Amsterdam.

  • Ding, Y., Liu, J., & Wang, Y. (2021). Hypergraphs, local reasoning, and weakly aggregative modal logic. In International workshop on logic, rationality and interaction (pp. 58–72). Springer.

  • Fagin, R., Halpern, J., Moses, Y., & Vardi, M. (1995). Reasoning about knowledge. MIT Press.

  • Fan, J., Wang, Y., & van Ditmarsch, H. (2015). Contingency and knowing whether. The Review of Symbolic Logic, 8, 75–107.

    Article  Google Scholar 

  • Fattorosi-Barnaba, M., & De Caro, F. (1985). Graded modalities. I. Studia Logica, 44(2), 197–221.

    Article  Google Scholar 

  • Feferman, S. (2008). Harmonious logic: Craig’s interpolation theorem and its descendants. Synthese, 164(3), 341–357.

    Article  Google Scholar 

  • Fervari, R., Herzig, A., Li, Y., & Wang, Y. (2017). Strategically knowing how. In Proceedings of IJCAI’17 (pp. 1031–1038).

  • Fine, K., et al. (1972). In so many possible worlds. Notre Dame Journal of formal logic, 13(4), 516–520.

    Google Scholar 

  • Fusco, M. (2015). Deontic modality and the semantics of choice. Philosophers’ Imprint, 15.

  • Goranko, V., Otto, M., et al. (2007). Model theory of modal logic. Handbook of Modal Logic, 3, 249–329.

    Article  Google Scholar 

  • Gu, T., & Wang, Y. (2016). “knowing value” logic as a normal modal logic. In L. Beklemishev, S. Demri, & A. Máté (Eds.), Advances in modal logic (Vol. 11, pp. 362–381). College Publications.

  • Hansen, H., Kupke, C., & Pacuit, E. (2009). Neighbourhood structures: Bisimilarity and basic model theory. Logical Methods in Computer Science, 5(2)

  • Jennings, R. E., Johnston, D., & Schotch, P. K. (1980). Universal first-order definability in modal logic. Mathematical Logic Quarterly, 26(19–21), 327–330.

    Article  Google Scholar 

  • Jennings, R. E., & Schotch, P. K. (1981). Some remarks on (weakly) weak modal logics. Notre Dame Journal of Formal Logic, 22(4), 309–314.

    Google Scholar 

  • Kamp, H. (1968). Tense logic and the theory of linear order. Ph.D. thesis, UCLA

  • Kracht, M. (1999). Tools and techniques in modal logic (Vol. 97). Elsevier.

  • Lewis, D. (1974). Intensional logics without interative axioms. Journal of Philosophical Logic, 3, 457–466.

    Article  Google Scholar 

  • Libkin, L. (2013). Elements of finite model theory. Springer.

  • Liu, J. (2019). Model theoretical aspects of normal polyadic modal logic: An exposition. Studies in Logic, 12(3), 79–101.

    Google Scholar 

  • Liu, J., Wang, Y., & Ding, Y. (2019). Weakly aggregative modal logic: Characterization and interpolation. In Proceedings of the 7th international workshop on logic, rationality, and interaction (pp. 153–167). Springer.

  • Lutz, C., & Wolter, F. (2011). Foundations for uniform interpolation and forgetting in expressive description logics. In Twenty-second international joint conference on artificial intelligence.

  • Nicholson, D. (2009). A dualization of neighbourhood structures. In R. E. Jennings, P. K. Schotch, & B. Brown (Eds.), On Preserving: Essays on preservationism and paraconsistent logic (pp. 49–60). University of Toronto Press.

  • Nicholson, T., & Allen, M. (2003). Aggregative combinatorics: An introduction. In Proc. student session, 2nd North American summer school in logic, language, and information (NASSLLI-03) (pp. 15–25).

  • Nicholson, T., Jennings, R. E., & Sarenac, D. (2000). Revisiting completeness for the \({K}_{n}\) modal logics: A new proof. Logic Journal of IGPL, 8(1), 101–105.

    Article  Google Scholar 

  • Otto, M. (2004). Elementary proof of the van Benthem–Rosen characterisation theorem. Technical Report 2342.

  • Pattinson, D. (2013). The logic of exact covers: Completeness and uniform interpolation. In Proceedings of the 2013 28th annual ACM/IEEE symposium on logic in computer science (pp. 418–427). IEEE Computer Society.

  • Rosen, E. (1997). Modal logic over finite structures. Journal of Logic, Language and Information, 6(4), 427–439.

    Article  Google Scholar 

  • Routley, R., & Meyer, R. K. (1972). The semantics of entailment-II. Journal of Philosophical Logic, 1(1), 53–73.

    Article  Google Scholar 

  • Routley, R., & Meyer, R. K. (1972). The semantics of entailment-III. Journal of Philosophical Logic, 1(2), 192–208.

    Article  Google Scholar 

  • Santocanale, L., Venema, Y., et al. (2010). Uniform interpolation for monotone modal logic. In L. Beklemishev, V. Goranko, & V. Shehtman (Eds.), Advances in Modal Logic (Vol. 8, pp. 350–370). College Publications.

  • Schotch, P., & Jennings, R. (1980). Modal logic and the theory of modal aggregation. Philosophia, 9(2), 265–278.

    Article  Google Scholar 

  • Segerberg, K. (1971). An essay in classical modal logic. Filosofiska Föreningen Och Filosofiska Institutionen Vid Uppsala Universitet.

  • Seifan, F., Schröder, L., & Pattinson, D. (2017). Uniform interpolation in coalgebraic modal logic. In 7th Conference on algebra and coalgebra in computer science (CALCO 2017). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.

  • Urquhart, A. (2009). Weakly additive algebras and a completeness problem. In R. E. Jennings, P. K. Schotch, & B. Brown (Eds.), On preserving: Essays on preservationism and paraconsistent logic (pp. 33–48). University of Toronto Press.

  • Van Benthem, J. (2008). The many faces of interpolation. Synthese, 451–460.

  • Van Benthem, J., Bezhanishvili, N., Enqvist, S., & Yu, J. (2017). Instantial neighbourhood logic. The Review of Symbolic Logic, 10(1), 116–144.

    Article  Google Scholar 

  • van Benthem, J., van Eijck, J., & Kooi, B. (2006). Logics of communication and change. Information and Computation, 204(11), 1620–1662.

    Article  Google Scholar 

  • Wang, Y. (2018). A logic of goal-directed knowing how. Synthese, 195(10), 4419–4439.

    Article  Google Scholar 

  • Wang, Y., & Fan, J. (2014). Conditionally knowing what. In R. Goré, B. P. Kooi, & A. Kurucz (Eds.), Advances in modal logic (Vol. 10, pp. 569–587). College Publications.

Download references

Author information

Authors and Affiliations

  1. Department of Philosophy, Sichuan University, Chengdu, China

    Jixin Liu

  2. Department of Philosophy, Peking University, Beijing, China

    Yifeng Ding & Yanjing Wang

Authors
  1. Jixin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yifeng Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yanjing Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yifeng Ding.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This paper is an extended journal version of  Liu et al. (2019). In this extended version, a detailed proof of Theorem 2 is provided using updated techniques, and a new result on recovering Craig interpolation (Theorem 6) is added. Jixin Liu thanks China Postdoctoral Science Foundation (2020M683344) and Fundamental Research Funds of SCU (2021-Philosophy-02) for support. Yanjing Wang gratefully acknowledges the support from NSSF (Grant 19BZX135).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, J., Ding, Y. & Wang, Y. Model Theoretical Aspects of Weakly Aggregative Modal Logic. J of Log Lang and Inf 31, 261–286 (2022). https://doi.org/10.1007/s10849-022-09366-x

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10849-022-09366-x

Keywords

Search

Navigation