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TWO-VARIABLE LOGIC HAS WEAK, BUT NOT STRONG, BETH DEFINABILITY

Part of: Model theory General logic

Published online by Cambridge University Press:  01 February 2021

HAJNAL ANDRÉKA  and
ISTVÁN NÉMETI
HAJNAL ANDRÉKA
Affiliation:
SET THEORY, LOGIC AND TOPOLOGY DEPARTMENT ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS BUDAPEST, REÁLTANODA ST. 13–15, H-1053, HUNGARYE-mail:[email protected]:[email protected]
ISTVÁN NÉMETI
Affiliation:
SET THEORY, LOGIC AND TOPOLOGY DEPARTMENT ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS BUDAPEST, REÁLTANODA ST. 13–15, H-1053, HUNGARYE-mail:[email protected]:[email protected]
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Abstract

We prove that the two-variable fragment of first-order logic has the weak Beth definability property. This makes the two-variable fragment a natural logic separating the weak and the strong Beth properties since it does not have the strong Beth definability property.

Keywords

weak Beth definabilityabstract model theorytwo-variable logichomogeneous model

MSC classification

Primary: 03C40: Interpolation, preservation, definability
Secondary: 03B20: Subsystems of classical logic (including intuitionistic logic) 03C95: Abstract model theory
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 2 , June 2021 , pp. 785 - 800
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Copyright
© The Association for Symbolic Logic 2021

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References

REFERENCES

Andréka, H., Comer, S. D., Madarász, J. X., Németi, I., and Sayed-Ahmed, T., Epimorphisms in cylindric algebras and definability in finite variable logic . Algebra Universalis , vol. 61 (2009), no. 3–4, pp. 261–282.CrossRefGoogle Scholar
Andréka, H., Düntsch, I., and Németi, I., A nonpermutational integral relation algebra . Michigan Mathematical Journal , vol. 39 (1992), pp. 371–384.CrossRefGoogle Scholar
Andréka, H. and Németi, I., Finite-variable logics do not have weak Beth definability property , The Road to Universal Logic (Festschrift for the 50th Birthday of Jean-Yves Beziau, Vol II) (Koslow, A. and Buchsbaum, A., editors), Studies in Universal Logic, Birkhäuser, Basel, 2015, pp. 125–133, Chapter 4.Google Scholar
Andréka, H., Németi, I., and Sain, I., Algebraic logic, Handbook of Philosophical Logic , vol. 2, second ed. (Gabbay, D. M. and Guenthner, F., editors), Kluwer Academic, Dordrecht, 2001, pp. 133–247.CrossRefGoogle Scholar
Barany, V., Benedikt, M., and Cate, B. T., Some model theory of guarded negation , this Journal, vol. 83 (2018), no. 4, pp. 1307–1344.Google Scholar
Barwise, J. and Feferman, S. (eds.), Model-Theoretic Logics , Springer-Verlag, New York, 1985.Google Scholar
Benthem, J. F. A. K., Language in Action: Categories, Lamdas and Dynamic Logic , Elsevier, Amsterdam, 1991.Google Scholar
Blok, W. J. and Pigozzi, D., Algebraizable logics. Memoirs of the American Mathematical Society, vol. 77 (1989), p. 396.CrossRefGoogle Scholar
Chang, C. C. and Keisler, H. J., Model Theory , third ed., North-Holland, Amsterdam, 1990, first published in 1973.Google Scholar
Comer, S. D., Galois-theory of cylindric algebras and its applications . Transactions of the American Mathematical Society , vol. 286 (1984), pp. 771–785.CrossRefGoogle Scholar
Ferenczi, M., Probabilities defined on standard and non-standard cylindric set algebras. Synthese, vol. 192 (2015), no. 7, pp. 2025–2033.CrossRefGoogle Scholar
Friedman, H., Beth’s theorem in cardinality logics . Israel Journal of Mathematics , vol. 14 (1973), pp. 205–212.CrossRefGoogle Scholar
Gyenis, Z., On atomicity of free algebras in certain cylindric-like varieties . Logic Journal of the IGPL , vol. 19 (2011), no. 1, pp. 44–52.CrossRefGoogle Scholar
Henkin, L., Monk, J. D., and Tarski, A., Cylindric Algebras. Parts I–II, North-Holland, Amsterdam, 1971 and 1985.Google Scholar
Hirsch, R. and Hodkinson, I., Relation Algebras by Games, North-Holland, Amsterdam, 2002.Google Scholar
Hodkinson, I., Finite variable logics. Bulletin of the European Association for Theoretical Computer Science, vol. 51 (1993), pp. 111–140, updated version of paper appeared in this journal. With addendum in Vol. 52, http://www.doc.ic.ac.uk/~imh/papers/fvl_revised.pdf, p. 37.Google Scholar
Hoogland, E., Algebraic characterizations of various Beth definability properties . Studia Logica , vol. 65 (2000), pp. 91–112.CrossRefGoogle Scholar
Hoogland, E., Marx, M., and Otto, M., Beth definability for the guarded fragment , Logic for Programming and Automated Reasoning. LPAR 1999 (Ganzinger, H., McAllester, D., and Voronkov, A., editors), Lecture Notes in Computer Science, 1705, Springer, Berlin, 1999, pp. 273–285.CrossRefGoogle Scholar
Jung, J. and Wolter, F., Interpolant existence in the guarded fragment. Semantic Scholar Corpus ID:221018030, 2020.Google Scholar
Khaled, M. and Sayed Ahmed, T., Vaught’s theorem holds for L2 but fails for Ln when n > 2. Bulletin of the Section of Logic, vol. 39 (2010), no. 3–4, pp. 107–122.Google Scholar
Khaled, M., Székely, G., Lefever, K., and Friend, M., Distances between formal theories . The Review of Symbolic Logic , vol. 13 (2020), no. 3, pp. 633–654.CrossRefGoogle Scholar
Madarász, J. and Sayed Ahmed, T., Amalgamation, interpolation and epimorphisms. Algebra Universalis, vol. 56 (2007), no. 2, pp. 179–210.CrossRefGoogle Scholar
Madarász, J. X., Interpolation and amalgamation; pushing the limits. Parts I and II . Studia Logica , vol. 61 (1998), no. 3, pp. 311–345 and vol. 62 (1999), no. 1, pp. 1–19.CrossRefGoogle Scholar
Makowsky, J. A., Shelah, S., and Stavi, J., Delta-logics and generalized quantifiers . Annals of Mathematical Logic, vol. 10 (1976), no. 2, pp. 155–192.CrossRefGoogle Scholar
Maksimova, L., Amalgamation and interpolation in normal modal logics . Studia Logica , vol. 50 (1991), pp. 457–471.CrossRefGoogle Scholar
Mekler, A. H. and Shelah, S., Stationary logic and its friends I . Notre Dame Journal of Formal Logic , vol. 26 (1985), no. 2, pp. 129–137.CrossRefGoogle Scholar
Pigozzi, D., Amalgamation, congruence extension and interpolation properties in algebras . Algebra Universalis , vol. 1 (1972), no. 3, pp. 269–349.CrossRefGoogle Scholar
Sági, G. and Shelah, S., Weak and strong interpolation for algebraic logics, this Journal, vol. 71 (2006), pp. 104–118.Google Scholar
Sain, I., Beth’s and Craig’s properties via epimorphisms and amalgamation in algebraic logic , Algebraic Logic and Universal Algebra in Computer Science (Bergman, D. H., Maddux, R. D., and Pigozzi, D. L., editors), Lecture Notes in Computer Science, vol. 425, Springer Verlag, Berlin, 1990, pp. 209–226.CrossRefGoogle Scholar
Sain, I. and Simon, A., Beth properties of finite variable fragments of first order logic, preprint, 1992, Mathematical Institute of the Hungarian Academy of Sciences, Budapest.Google Scholar
Sayed Ahmed, T., Varying interpolation and amalgamation for MV polyadic algebras . Journal of Applied Non-Classical Logics , vol. 25 (2015), no. 2, pp. 140–192.CrossRefGoogle Scholar
Tarski, A. and Givant, S. R., A Formalization of Set Theory Without Variables, AMS Colloquium Publications, 41, American Mathematical Society, Providence, RI, 1987.CrossRefGoogle Scholar