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Normal monomodal logics can simulate all others

Published online by Cambridge University Press:  12 March 2014

Marcus Kracht  and
Frank Wolter
Marcus Kracht
Affiliation:
II. Mathematisches Institut, Freie Universität Berlin, Arnimallee 3, D-14195 Berlin, Germany, E-mail: [email protected]
Frank Wolter
Affiliation:
Japan Advanced Institute of Science and Technology (JAIST), Ishikawa 923-12., Japan, E-mail: [email protected]
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Abstract

This paper shows that non-normal modal logics can be simulated by certain polymodal normal logics and that polymodal normal logics can be simulated by monomodal (normal) logics. Many properties of logics are shown to be reflected and preserved by such simulations. As a consequence many old and new results in modal logic can be derived in a straightforward way, sheding new light on the power of normal monomodal logic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 1 , March 1999 , pp. 99 - 138
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[1]Bellissima, Fabio, On the lattice of extensions of the modal logics K.Altn, Archive of Mathematical Logic, vol. 27 (1988), pp. 107–114.CrossRefGoogle Scholar
[2]Birkhoff, Gareth, Lattice theory, American Mathematical Society Colloquium Publications, Rhode Island, 1973.Google Scholar
[3]Burgess, John P., Basic tense logic, Handbook of philosophical logic (Gabbay, Dov M. and Guenthner, Franz, editors), vol. 2, Reidel, 1984, pp. 89–133.Google Scholar
[4]Chagrov, Alexander, The undecidability of the tabularity problem for modal logic, Logic colloquiumxs'94, 1994, p. 34.Google Scholar
[5]Chagrov, Alexander and Zakharyaschev, Michael, Modal companions of intermediate propo-sitional logics, Studio Logica, vol. 51 (1992), pp. 49–82.CrossRefGoogle Scholar
[6]Chagrov, Alexander and Zakharyaschev, Michael, The undecidability of the disjunction property and other related problems, this Journal, vol. 58 (1993), pp. 967–1002.Google Scholar
[7]Chellas, Brian F., Modal logic: An introduction, Cambridge University Press, 1980.CrossRefGoogle Scholar
[8]Došen, Kosta, Duality between modal algebras and neighbourhood frames, Studia Logica, vol. 48 (1989), pp. 219–234.CrossRefGoogle Scholar
[9]Freese, R., Free modular lattices, Transactions of the American Mathematical Society, vol. 261 (1980), pp. 81–91.CrossRefGoogle Scholar
[10]Gerson, Martin, A neighbourhood frame for T with no equivalent relational frame, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 22 (1976), pp. 29–34.CrossRefGoogle Scholar
[11]Gödel, Kurt, Eine Interpretation des intuitionistischen Aussagenkalküls, Ergebnisse eines mathe-matischen Kolloquiums, vol. 6 (1933), pp. 39–40.Google Scholar
[12]Goldblatt, Robert I., Logics of Time and Computation, CSLI Lecture Notes, no. 7, CSLI, 1987.Google Scholar
[13]Goranko, Valentin and Gargov, George, Modal logic with names, Journal of Philosophical Logic, vol. 22 (1993), pp. 607–636.Google Scholar
[14]Goranko, Valentin and Passy, Solomon, Using the universal modality: Gains and Questions, Journal of Logic and Computation, vol. 2 (1992), pp. 5–30.CrossRefGoogle Scholar
[15]Grefe, Carsten, Module Logiken funktionaler Frames, Master's thesis, Dept. of Mathematics, FU Berlin, 1994.Google Scholar
[16]Kracht, Marcus, Tools and Techniques in Modal Logic, Habilitationschrift, Freie Universität Berlin, 1997, to appear.Google Scholar
[17]Kracht, Marcus, How completeness and correspondence theory got married, Diamonds and defaults (de Rijke, Maarten, editor), Synthese, Kluwer, 1993, pp. 175–214.Google Scholar
[18]Kracht, Marcus, Highway to the danger zone, Journal of Logic and Computation, vol. 5 (1995), pp. 93–109.CrossRefGoogle Scholar
[19]Kracht, Marcus and Wolter, Frank, Properties of independently axiomatizable bimodal logics, this Journal, vol. 56 (1991), pp. 1469–1485.Google Scholar
[20]Maksimova, Larisa L., Interpolation theorems in modal logic and and amalgamated varieties of topoboolean algebras, Algebra and Logic (1979).Google Scholar
[21]Markov, A. A., Impossibility of Certain Algorithms in the Theory of Associative Systems (Russian), Doklady Akademia Nauk SSSR, vol. 77 (1951), pp. 953–956.Google Scholar
[22]Post, Emil L., Recursive Undecidability of a Problem of Thue, this Journal, vol. 12 (1947), pp. 1–11.Google Scholar
[23]Rabin, Michael O., Recursive Unsolvability of Group Theoretic Problems, Annals of Mathematics, vol. 67 (1958), pp. 172–194.CrossRefGoogle Scholar
[24]Rautenberg, Wolfgang, Klassische und nichtklassische aussagenlogik, Vieweg, Braunschweig/Wiesbaden, 1979.CrossRefGoogle Scholar
[25]Sahlqvist, Hendrik, First and second order semantics for modal logic, Proceedings of the third Scandinavian logic symposium (Kanger, Stig, editor), North-Holland, 1975, pp. 15–31.Google Scholar
[26]Sambin, Giovanni and Vaccaro, V., Topology and duality in modal logic, Annals of Pure and Applied Logic, vol. 37 (1988), pp. 249–296.CrossRefGoogle Scholar
[27]Segerberg, Krister, Modal logics with functional alternative relations, Notre Dame Journal of Formal Logic, vol. 27 (1986), pp. 504–522.CrossRefGoogle Scholar
[28]Thomason, S. K., Reduction of tense logic to modal logic I, this Journal, vol. 39 (1974), pp. 549–551.Google Scholar
[29]Thomason, S. K., Reduction of tense logic to modal logic II, Theoria, vol. 41 (1975), pp. 154–169.CrossRefGoogle Scholar
[30]Thomason, S. K., Undecidability of the completeness problem of modal logic, Universal algebra and applications Banach center publications vol. 9 (Warsaw), PNW–Polish Scientific Publishers, 1982.Google Scholar
[31]van Benthem, Johan, Modal and classical logic, Bibliopolis, 1983.Google Scholar
[32]Wolter, Frank, Lattices of modal logics, Ph.D. thesis, FU Berlin, 1993.Google Scholar
[33]Wolter, Frank, Solution to a problem of Goranko and Passy, Journal of Logic and Computation, vol. 4 (1994), pp. 21–22.CrossRefGoogle Scholar
[34]Wolter, Frank, The finite model property in tense logic, this Journal, vol. 60 (1995), pp. 757–774.Google Scholar