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MEREOLOGICAL BIMODAL LOGICS

Part of: General logic Model theory Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  27 January 2022

DAZHU LI Open the ORCID record for DAZHU LI [Opens in a new window]  and
YANJING WANG Open the ORCID record for YANJING WANG [Opens in a new window]
DAZHU LI
Affiliation:
INSTITUTE OF PHILOSOPHY CHINESE ACADEMY OF SCIENCES BEIJING, CHINA and DEPARTMENT OF PHILOSOPHY UNIVERSITY OF CHINESE ACADEMY OF SCIENCES BEIJING, CHINA E-mail lidazhu@ucas.ac.cn
YANJING WANG
Affiliation:
INSTITUTE OF FOREIGN PHILOSOPHY AND DEPARTMENT OF PHILOSOPHY PEKING UNIVERSITY BEIJING, CHINA E-mail: y.wang@pku.edu.cn
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Abstract

In this paper, using a propositional modal language extended with the window modality, we capture the first-order properties of various mereological theories. In this setting, $\Box \varphi $ reads all the parts (of the current object) are $\varphi $, interpreted on the models with a whole-part binary relation under various constraints. We show that all the usual mereological theories can be captured by modal formulas in our language via frame correspondence. We also correct a mistake in the existing completeness proof for a basic system of mereology by providing a new construction of the canonical model.

Keywords

mereologymodal logicwindow modalitycorrespondence

MSC classification

Primary: 03B45: Modal logic (including the logic of norms)
Secondary: 03A05: Philosophical and critical 03C80: Logic with extra quantifiers and operators 03C07: Basic properties of first-order languages and structures

Information

Type
Research Article
Information
The Review of Symbolic Logic , Volume 15 , Issue 4 , December 2022 , pp. 823 - 858
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

BIBLIOGRAPHY

Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal Logic. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Casati, R., & Varzi, A. (1999). Parts and Places. MIT Press, Cambridge, MA.Google Scholar
Cotnoir, A. J. (2010). Anti-symmetry and non-extensional mereology. The Philosophical Quarterly, 60, 396–405.CrossRefGoogle Scholar
Cotnoir, A. J. (2016). Does universalism entail extensionalism? Noûs, 50, 121–132.CrossRefGoogle Scholar
Eberle, R. A. (1970). Nominalistic Systems. Springer, Dordrecht.CrossRefGoogle Scholar
Gargov, G., Passy, S., & Tinchev, T. (1987). Modal environment for Boolean speculations. In Skordev, D. G., editor. Mathematical Logic and Its Applications. Springer, Cham, pp. 253–263.CrossRefGoogle Scholar
Goodman, N. (1951). The Structure of Appearance. Harvard University Press, Dordrecht.Google Scholar
Goranko, V. (1990a). Completeness and incompleteness in the bimodal base $\mathcal{L} $ (R, − R). In Petkov, P. P., editor. Mathematical Logic. Springer, Cham, pp. 311–326.CrossRefGoogle Scholar
Goranko, V. (1990b). Modal definability in enriched language. Notre Dame Journal of Formal Logic, 31, 81–105.Google Scholar
Goranko, V., & Passy, S. (1992). Using the universal modality: Gains and questions. Journal of Logic and Computation, 2, 5–30.CrossRefGoogle Scholar
Gruszczyński, R., & Pietruszczak, A. (2014). The relations of supremum and mereological sum in partially ordered sets. In Calosi, C., & Graziani, P., editors. Mereology and the Sciences. Springer, Cham, pp. 123–140.CrossRefGoogle Scholar
Gruszczyński, R., & Varzi, A. C. (2015). Mereology then and now. Logic and Logical Philosophy, 24, 409–427.CrossRefGoogle Scholar
Grzegorczyk, A. (1955). The systems of Leśniewski in relation to contemporary logical research. Studia Logica, 3, 77–97.CrossRefGoogle Scholar
Hodges, W. (1997). A Shorter Model Theory. Cambridge University Press, Cambridge.Google Scholar
Holliday, W. H. (2017). On the modal logic of subset and superset: Tense logic over Medvedev frames. Studia Logica, 105, 13–35.CrossRefGoogle Scholar
Hovda, P. (2009). What is classical mereology? Journal of Philosophical Logic, 38, 55–82.CrossRefGoogle Scholar
Humberstone, I. L. (1983). Inaccessible worlds. Notre Dame Journal of Formal Logic, 24, 346–352.CrossRefGoogle Scholar
Humberstone, I. L. (1987). The modal logic of ‘all and only’. Notre Dame Journal of Formal Logic, 28, 177–188.CrossRefGoogle Scholar
Leonard, H. S., & Goodman, N. (1940). The calculus of individuals and its uses. Journal of Symbolic Logic, 5, 45–55.CrossRefGoogle Scholar
Lewis, D. K. (1991). Parts of Classes. Wiley-Blackwell, Oxford.Google Scholar
Lutz, C., & Wolter, F. (2006). Modal logics of topological relations. Logical Methods in Computer Science, 2, 1–41.Google Scholar
Medvedev, Y. T. (1962). Finite problems. Soviet Mathematics Doklady, 3, 227–230.Google Scholar
Nenov, Y., & Vakarelov, D. (2008). Modal logics for mereotopological relations. Advances in Modal Logic, 7, 249–272.Google Scholar
Niebergall, K.-G. (2009). Calculi of individuals and some extensions: An overview. In Alexander, H., & Hannes, L., editors. Reduction, Abstraction, Analysis. De Gruyter, Frankfurt, pp. 335–354.Google Scholar
Niebergall, K.-G. (2011). Mereology. In Pettigrew, R., & Horsten, L., editors. The Continuum Companion to Philosophical Logic. Continuum, London, pp. 271–298.Google Scholar
Pietruszczak, A. (2018). Metamereology. Toruń: The Nicolaus Copernicus University Scientific Publishing House.Google Scholar
Pietruszczak, A. (2020). Foundations of the theory of parthood . A Study of Mereology. Trends in Logic, Vol. 54. Springer.Google Scholar
Sharvy, R. (1983). Mixtures. Philosophy and Phenomenological Research, 44, 227–239.CrossRefGoogle Scholar
Simons, P. (1987). Parts: A Study in Ontology. Clarendon Press, Oxford.Google Scholar
Simons, P. (1992). The formalization of Husserl’s theory of wholes and parts. In Philosophy and Logic in Central Europe from Bolzano to Tarski. Nijhoff International Philosophy Series. Dordrecht: Springer, pp. 71–116.CrossRefGoogle Scholar
Simons, P. (2015). Stanisław Leśniewski. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.Google Scholar
Tang, T. (1938). Algebraic postulates and a geometric interpretation for the Lewis calculus of strict implication. Bulletin of the American Mathematical Society, 44, 737–744.Google Scholar
Tarski, A. (1938). Der aussagenkalkül und die topologie. Fundamenta Mathematicae, 31, 103–134.CrossRefGoogle Scholar
Tarski, A. (1956a). Foundations of the geometry of solids. In Logic, Semantics, Metamathematics. Oxford University Press, pp. 24–29. Translated by J. H. Woodger.Google Scholar
Tarski, A. (1956b). On the foundations of the Boolean algebra. In Logic, Semantics, Metamathematics. Oxford University Press, pp. 320–341. Translated by J. H. Woodger.Google Scholar
Torrini, P., Stell, J. G., & Bennett, B. (2002). Mereotopology in 2nd-order and modal extensions of intuitionistic propositional logic. Journal of Applied Non-Classical Logics, 12, 495–525.CrossRefGoogle Scholar
Tsai, H. (2009). Decidability of mereological theories. Logic and Logical Philosophy, 18, 45–63.CrossRefGoogle Scholar
Tsai, H. (2013a). A comprehensive picture of the decidability of mereological theories. Studia Logica, 101, 987–1012.CrossRefGoogle Scholar
Tsai, H. (2013b). Decidability of general extensional mereology. Studia Logica, 101, 619–636.CrossRefGoogle Scholar
Tsai, H. (2018). General extensional mereology is finitely axiomatizable. Studia Logica, 106, 809–826.CrossRefGoogle Scholar
Uzquiano, G. (2014). Mereology and modality. In Kleinschmidt, S., editor. Mereology and Location. Oxford University Press, Oxford, pp. 33–56.CrossRefGoogle Scholar
van Benthem, J. (1979). Minimal deontic logics. Bulletin of the Section of Logic, 8, 36–41.Google Scholar
van Benthem, J. (1984). Correspondence theory. In Gabbay, D., & Guenthner, F., editors. Handbook of Philosophical Logic. Synthese Library (Studies in Epistemology, Logic, Methodology, and Philosophy of Science), Vol. 165. Springer, Dordrecht, pp. 167–247.CrossRefGoogle Scholar
van Benthem, J. (1985). Modal Logic and Classical Logic. Napoli: Bibliopolis.Google Scholar
van Benthem, J. (1999). The range of modal logic. Journal of Applied Non-Classical Logics, 9, 407–442.CrossRefGoogle Scholar
van Benthem, J., & Bezhanishvili, G. (2007). Modal logics of space. In Aiello, M., Pratt-Hartmann, I., & van Benthem, J., editors. Handbook of Spatial Logics. Springer, Dordrecht, pp. 217–298.CrossRefGoogle Scholar
Varzi, A. C. (2016). Mereology. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.Google Scholar
Walsh, S. D. (2012). Modal mereology and modal supervenience. Philosophical Studies, 159, 1–20.CrossRefGoogle Scholar