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Simulating polyadic modal logics by monadic ones

Published online by Cambridge University Press:  12 March 2014

George Goguadze ,
Carla Piazza  and
Yde Venema
George Goguadze
Affiliation:
Faculty of Computer Science, University of Saarland, IM Stadwald/36.1, Saarbrücken, Germany, E-mail: george@ags.uni-sb.de
Carla Piazza
Affiliation:
Dipartimento di Matematica e Infomatica, Università Degli Studi di Udine, Via Delle Scienze 206, I-33100 Udine, Italy Dipartimento di Informatica, Università ca' Foscari di Venezia, Via Torino 155, I-30172 Mestre, Italy, E-mail: piazza@dimi.uniud.it
Yde Venema
Affiliation:
Institute of Logic, Language and Computation, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands, E-mail: yde@science.uva.nl
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Abstract

We define an interpretation of modal languages with polyadic operators in modal languages that use monadic operators (diamonds) only. We also define a simulation operator which associates a logic Λsim in the diamond language with each logic Λ in the language with polyadic modal connectives. We prove that this simulation operator transfers several useful properties of modal logics, such as finite/recursive axiomatizability, frame completeness and the finite model property, canonicity and first-order definability.

Keywords

Modal logicpolyadic operatorsinterpretationsimulationtransfer results
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 2 , June 2003 , pp. 419 - 462
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[1]van Benthem, J., D'Agostino, G., Montanari, A., and Policriti, A., Modal deduction in second-order logic and set theory-I, Journal of Logic and Computation, vol. 7 (1997), no. 2, pp. 251–265.CrossRefGoogle Scholar
[2]van Benthem, J., D'Agostino, G., Montanari, A., and Policriti, A., Modal deduction in second-order logic and set theory-II, Studia Logica, vol. 60 (1998), no. 3, pp. 387–420.CrossRefGoogle Scholar
[3]Blackburn, P., de Rijke, M., and Venema, Y., Modal Logic, Cambridge Tracts in Computer Science, Cambridge University Press, Cambridge, 2001.CrossRefGoogle Scholar
[4]de Rijke, M. (editor), Diamonds and defaults, Synthese Library, vol. 229, Kluwer Academic Publishers, 1993.CrossRefGoogle Scholar
[5]Gödel, K., Eine Interpretation des intuitionistischen Aussagenkalküls, Ergebnisse eines Mathema¬tischen Kolloquiums, vol. 6 (1933), pp. 39–40.Google Scholar
[6]Goldblatt, R.I., Varieties of complex algebras, Annals of Pure and Applied Logic, vol. 38 (1989), pp. 173–241.CrossRefGoogle Scholar
[7]Hansen, M. R. and Zhou, C., Duration calculus: Logical foundations, Formal Aspects of Computing, vol. 9 (1997), pp. 283–330.CrossRefGoogle Scholar
[8]Hirsch, R. and Hodkinson, I., Relation Algebras by Games, Studies in Logic, vol. 147, Elsevier, Amsterdam, 2002.Google Scholar
[9]B.Jónsson, and Tarski, A., Boolean algebras with operators, Part I, American Journal of Mathematics, vol. 73 (1952), pp. 891–939.CrossRefGoogle Scholar
[10]Jónsson, B. and Tarski, A., Boolean algebras with operators, Part II, American Journal of Mathematics, vol. 74 (1952), pp. 127–162.CrossRefGoogle Scholar
[11]Kracht, M., Tools and Techniques in Modal Logic, Studies in Logic, vol. 142, Elsevier, Amsterdam, 1999.CrossRefGoogle Scholar
[12]Kracht, M. and Wolter, F., Simulation and transfer results in modal logic: A survey, Studia Logica, vol. 59 (1997), pp. 149–177.CrossRefGoogle Scholar
[13]Kracht, M. and Wolter, F., Normal modal logics can simulate all others, this Journal, (1999), pp. 99–138.Google Scholar
[14]Kurtonina, N., Categorial inference and modal logic, Journal of Logic, Language and Information, vol. 7 (1998), pp. 399–418.CrossRefGoogle Scholar
[15]Lambek, J., The mathematics of sentence structure, American Mathematical Monthly, vol. 65 (1958), pp. 154–170.CrossRefGoogle Scholar
[16]Marx, M., Pólos, L., and Masuch, M. (editors), Arrow Logic and Multi-Modal Logic, Studies in Logic, Language and Information, CSLI Publications, 1996.Google Scholar
[17]Roorda, D., Dyadic modalities and Lambek calculus, Diamonds and Defaults (de Rijke, M., editor), Kluwer, 1993, pp. 215–253.CrossRefGoogle Scholar
[18]Thomason, S. K., Reduction of second-order logic to modal logic, Zeitschrift fur Mathemathische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 107–114.CrossRefGoogle Scholar
[19]Thomason, S. K., Reduction of tense logic to modal logic II, Theoria, vol. 41 (1975), pp. 154–169.CrossRefGoogle Scholar
[20]Venema, Y., A modal logic for chopping intervals, Journal of Logic and Computation, vol. 1 (1993), pp. 453–476.CrossRefGoogle Scholar