Abstract
The propositional modal logic is obtained by adding the necessity operator □ to the propositional logic. Each formula in the propositional logic is equivalent to a formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom \(\Box\varphi \leftrightarrow\Diamond\varphi \) to K and get a new system K + . Each formula in such a logic is equivalent to a formula in the disjunctive normal form, where □k(k ≥ 0) only occurs before an atomic formula p, and \(\lnot\) only occurs before a pseudo-atomic formula of form □k p. Maximally consistent sets of K + have a property holding in the propositional logic: a set of pseudo-atom-complete formulas uniquely determines a maximally consistent set. When a pseudo-atomic formula □k p i (k,i ≥ 0) is corresponding to a propositional variable q ki , each formula in K + then can be corresponding to a formula in the propositional logic P + . We can also get the correspondence of models between K + and P + . Then we get correspondences of theorems and valid formulas between them. So, the soundness theorem and the completeness theorem of K + follow directly from those of P + .
Chapter PDF
Similar content being viewed by others
Keywords
References
Van Benthem, J.: Modal correspondence theory [Ph.D. Thesis]. University of Amsterdam, Netherlands (1976)
Blackburn, P., Van Benthem, J., Wolter, F.: Handbook of Modal Logic. Elsevier Science Ltd (2006)
Van Benthem, J.: Correspondence Theory. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, pp. 325–408. Kluwer Academic Publishers (2001)
Aloni, M.: Quantification under Conceptual Covers [Ph.D. Thesis]. University of Amsterdam, Amsterdam (2001)
Aloni, M.: Individual Concepts in Modal Predicate Logic. Journal of Philosophical Logic 34(1), 1–64 (2005)
Hazen, A., Rin, B., Wehmeier, K.: Actuality in Propositional Modal Logic. Studia Logica 101(3), 487–503 (2013)
Hughes, G.E., Cresswell, M.J.: A New introduction to Modal Logic. Routledge, Lodon (1996)
Hamilton, A.G.: Logic for Mathematicians. Cambridge University Press (1988)
Fitting, M., Mendelsohn, R.: First-order Modal Logic. Kluwer Academic Publishers, The Netherlands (1998)
Kripke, S.: Semantical analysis of modal logic I: Normal modal propositional calculi. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 9, 67–96 (1963)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 IFIP International Federation for Information Processing
About this paper
Cite this paper
Sun, M., Deng, S., Sui, Y. (2014). The Correspondence between Propositional Modal Logic with Axiom \(\Box\varphi \leftrightarrow \Diamond \varphi \) and the Propositional Logic. In: Shi, Z., Wu, Z., Leake, D., Sattler, U. (eds) Intelligent Information Processing VII. IIP 2014. IFIP Advances in Information and Communication Technology, vol 432. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44980-6_16
Download citation
DOI: https://doi.org/10.1007/978-3-662-44980-6_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-44979-0
Online ISBN: 978-3-662-44980-6
eBook Packages: Computer ScienceComputer Science (R0)