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Erdős Graphs Resolve Fine's Canonicity Problem

Published online by Cambridge University Press:  15 January 2014

Robert Goldblatt ,
Ian Hodkinson  and
Yde Venema
Robert Goldblatt
Affiliation:
Centre for Logic, Language and Computation, Victoria University, Po Box 600, Wellington, New ZealandE-mail: , [email protected]
Ian Hodkinson
Affiliation:
Office for Logic, Language and Computation, 426 Huxley Building, Department of Computing, Imperial College London, South Kensington Campus, London SW7 2AZ, UKE-mail: , [email protected]
Yde Venema
Affiliation:
Institute for Logic, Language and Computation, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, NetherlandsE-mail: , [email protected]
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Abstract

We show that there exist 2ℵ0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order definable class of Kripke frames (a monomodal example can then be obtained using simulation results of Thomason). The constructions use the result of Erdos that there are finite graphs with arbitrarily large chromatic number and girth.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 10 , Issue 2 , June 2004 , pp. 186 - 208
Copyright
Copyright © Association for Symbolic Logic 2004

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References

REFERENCES

[1] Birkhoff, G., Subdirect unions in universal algebra, Bulletin of the American Mathematical Society, vol. 50 (1944), pp. 764–768.CrossRefGoogle Scholar
[2] Blackburn, Patrick, de Rijke, Maarten, and Venema, Yde, Modal logic, Cambridge University Press, 2001.Google Scholar
[3] Blok, W.J., The lattice of modal logics: an algebraic investigation, The Journal of Symbolic Logic, vol. 45 (1980), pp. 221–236.Google Scholar
[4] Burris, S. and Sankappanavar, H.P., A course in universal algebra, Graduate Texts in Mathematics, vol. 78, Springer-Verlag, New York, 1981, available at www.thoralf.uwaterloo.ca/htdocs/ualg.html.Google Scholar
[5] Chagrov, Alexander and Zakharyascnev, Michael, Modal logic, Oxford University Press, 1997.Google Scholar
[6] Cresswell, M.J., A Henkin completeness theorem for T, Notre Dame Journal of Formal Logic, vol. 8 (1967), pp. 186–190.Google Scholar
[7] de Rijke, Maarten and Venema, Yde, Sahlqvist's theorem for Boolean algebras with operators with an application to cylindric algebras, Studia Logica, vol. 54 (1995), pp. 61–78.CrossRefGoogle Scholar
[8] Diestel, R., Graph theory, Graduate Texts in Mathematics, vol. 173, Springer-Verlag, Berlin, 1997.Google Scholar
[9] Erdős, Paul, Graph theory and probability, Canadian Journal of Mathematics, vol. 11 (1959), pp. 34–38.CrossRefGoogle Scholar
[10] Fine, Kit, Some connections between elementary and modal logic, In Kanger [39], pp. 15–31.Google Scholar
[11] Fine, Kit, Logics containing K4. Part II , The Journal of Symbolic Logic, vol. 50 (1985), no. 3, pp. 619–651.CrossRefGoogle Scholar
[12] Gehrke, Mai, The order structure of Stone spaces and the TD -separation axiom, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 37 (1991), pp. 515.Google Scholar
[13] Gehrke, Mai, Harding, John, and Venema, Yde, MacNeille completions and canonical extensions , 2003, ILLC beta preprint PP-2004-05. Manuscript available at staff.science.uva.nl/~yde.Google Scholar
[14] Gehrke, Mai and Jónsson, Bjarni, Monotone bounded distributive lattice expansions, Mathematica Japonica, vol. 52 (2000), no. 2, pp. 197–213.Google Scholar
[15] Givant, Steven, Universal classes of simple relation algebras, The Journal of Symbolic Logic, vol. 64 (1999), no. 2, pp. 575–589.CrossRefGoogle Scholar
[16] Goldblatt, Robert, Elementary logics are canonical andpseudo-equational, In [18], pp. 243–257.Google Scholar
[17] Goldblatt, Robert, Varieties of complex algebras, Annals of Pure and Applied Logic, vol. 44 (1989), pp. 173–242.Google Scholar
[18] Goldblatt, Robert, Mathematics of modality, CSLI Lecture Notes, vol. 43, CSLI Publications, Stanford, California, 1993, distributed by Cambridge University Press.Google Scholar
[19] Goldblatt, Robert, Elementary generation and canonicity for varieties of Boolean algebras with operators, Algebra Universalis, vol. 34 (1995), pp. 551–607.Google Scholar
[20] Goldblatt, Robert, Algebraic polymodal logic: a survey, Logic Journal of the IGPL. Interest Group in Pure and Applied Logics Special Issue on Algebraic Logic edited by Németi, István and Sain, Ildikó, vol. 8 (2000), no. 4, pp. 393–450, electronically available at: www3.oup.co.uk/igpl.Google Scholar
[21] Goldblatt, Robert, Persistence and atomic generation for varieties of Boolean algebras with operators, Studia Logica, vol. 68 (2001), no. 2, pp. 155–171.Google Scholar
[22] Goldblatt, Robert, Mathematical modal logic: a view of its evolution, Journal of Applied Logic, vol. 1 (2003), no. 5-6, pp. 309–392.Google Scholar
[23] Goldblatt, Robert, Questions of canonicity, Trends in logic — 50 years of Studia Logica (Hendricks, Vincent F. and Malinowski, Jacek, editors), Kluwer Academic Publishers, 2003, pp. 93–128.Google Scholar
[24] Goldblatt, Robert, Hodkinson, Ian, and Venema, Yde, On canonical modal logics that are not elementarily determined, Logique et Analyse, to appear.Google Scholar
[25] Henkin, L., Monk, J.D., Tarski, A., Andréka, H., and Németi, I., Cylindric set algebras, Lecture Notes in Mathematics, vol. 883, Springer-Verlag, Berlin, 1981.CrossRefGoogle Scholar
[26] Henkin, Leon, The completeness of the first-order functional calculus, The Journal of Symbolic Logic, vol. 14 (1949), pp. 159–166.Google Scholar
[27] Henkin, Leon, Monk, J. Donald, and Tarski, Alfred, Cylindric algebras I, Studies in Logic and the Foundations of Mathematics, vol. 64, North-Holland, Amsterdam, 1971.Google Scholar
[28] Henkin, Leon, Monk, J. Donald, and Tarski, Alfred, Cylindric algebras II, Studies in Logic and the Foundations of Mathematics, vol. 115, North-Holland, Amsterdam, 1985.Google Scholar
[29] Hirsch, Robin and Hodkinson, Ian, Relation algebras by games, Studies in Logic and the Foundations of Mathematics, vol. 147, North-Holland, Amsterdam, 2002.Google Scholar
[30] Hirsch, Robin and Hodkinson, Ian, Strongly representable atom structures of relation algebras, Proceedings of the American Mathematical Society, vol. 130 (2002), pp. 1819–1831.Google Scholar
[31] Hodoes, W., Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.Google Scholar
[32] Hodkinson, Ian and Venema, Yde, Canonical varieties with no canonical axiomatisation, Transactions of the American Mathematical Society, to appear. ILLC beta preprint PP-2003-13. Manuscript available at staff.science.uva.nl/~yde.Google Scholar
[33] Hughes, G.E., Every world can see a reflexive world, Studia Logica, vol. 49 (1990), pp. 175–181.Google Scholar
[34] Jónsson, Bjarni, Varieties of relation algebras, Algebra Universalis, vol. 15 (1982), pp. 273–298.Google Scholar
[35] Jónsson, Bjarni, On the canonicity of Sahlqvist identities, Studia Logica, vol. 53 (1994), pp. 473–491.CrossRefGoogle Scholar
[36] Jónsson, Bjarni and Tarski, Alfred, Boolean algebras with operators, Bulletin of the American Mathematical Society, vol. 54 (1948), pp. 79–80.Google Scholar
[37] Jónsson, Bjarni and Tarski, Alfred, Boolean algebras with operators, part I, American Journal of Mathematics, vol. 73 (1951), pp. 891–939.CrossRefGoogle Scholar
[38] Jónsson, Bjarni and Tarski, Alfred, Boolean algebras with operators, part II, American Journal of Mathematics, vol. 74 (1952), pp. 127–162.Google Scholar
[39] Kanger, Stig (editor), Proceedings of the Third Scandinavian Logic Symposium, Studies in Logic and the Foundations of Mathematics, vol. 82, North-Holland, Amsterdam, 1975.Google Scholar
[40] Kracht, M., Tools and techniques in modal logic, Studies in Logic and the Foundations of Mathematics, vol. 142, North-Holland, Amsterdam, 1999.Google Scholar
[41] Lachlan, A.H., A note on Thomason's refined structures for tense logics, Theoria, vol. 40 (1974), pp. 117–120.Google Scholar
[42] Lemmon, E.J. and Scott, D., Intensional logic, preliminary draft of initial chapters by Lemmon, E.J., Stanford University (later published as An Introduction to Modal Logic, American Philosophical Quarterly Monograph Series, no. 11 (ed. by Krister Segerberg), Basil Blackwell, Oxford, 1977), July 1966.Google Scholar
[43] Maddux, R., Some varieties containing relation algebras, Transactions of the American Mathematical Society, vol. 272 (1982), no. 2, pp. 501–526.Google Scholar
[44] Makinson, D.C., On some completeness theorems in modal logic, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 379–384.Google Scholar
[45] McKinsey, J. C. C. and Tarski, Alfred, Some theorems about the sentential calculi of Lewis and Heyting, The Journal of Symbolic Logic, vol. 13 (1948), pp. 1–15.Google Scholar
[46] Sahlqvist, Henrik, Completeness and correspondence in the first and second order semantics for modal logic, In Kanger [39], pp. 110–143.Google Scholar
[47] Thomason, S.K., Reduction of second-order logic to modal logic, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 107–114.Google Scholar
[48] Thomason, S.K., Reduction of tense logic to modal logic. II, Theoria, vol. 41 (1975), pp. 154–169.Google Scholar
[49] Venema, Yde, Canonical pseudo-correspondence, Advances in modal logic, vol. 2 (Zakharyaschev, Michael, Segerberg, Krister, de Rijke, Maarten, and Wansing, Heinrich, editors), vol. 119, CSLI Publications, 2001, pp. 421–430.Google Scholar
[50] Wolter, Frank, Properties of tense logics, Mathematical Logic Quarterly, vol. 42 (1996), pp. 481–500.Google Scholar
[51] Wolter, Frank, The structure of lattices of subframe logics, Annals of Pure and Applied Logic, vol. 86 (1997), pp. 47–100.Google Scholar
[52] Zakharyaschev, Michael, Canonical formulas for K4. Part II. Cofinal subframe logics, The Journal of Symbolic Logic, vol. 61 (1996), no. 2, pp. 421–449.Google Scholar