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Interpolation and the projective Beth property in well-composed logics. (English. Russian original) Zbl 1285.03028
Algebra Logic 51, No. 2, 163-184 (2012); translation from Algebra Logika 51, No. 2, 244-275 (2012).
Summary: We study the interpolation and Beth definability problems in propositional extensions of minimal logic J. Previously, all J-logics with the weak interpolation property (WIP) were described, and it was proved that WIP is decidable over J. In this paper, we deal with so-called well-composed J-logics, i.e., J-logics satisfying an axiom \((\perp \to A) \vee (A \to\perp)\). Representation theorems are proved for well-composed logics possessing Craig’s interpolation property (CIP) and the restricted interpolation property (IPR). As a consequence, we show that only finitely many well-composed logics share these properties and that IPR is equivalent to the projective Beth property (PBP) on the class of well-composed J-logics.

MSC:
03C40 Interpolation, preservation, definability
03B53 Paraconsistent logics
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[1] I. Johansson, ”Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus,” Comp. Math., 4, 119–136 (1937). · Zbl 0015.24102
[2] W. Craig, ”Three uses of Herbrand–Gentzen theorem in relating model theory and proof theory,” J. Symb. Log., 22, 269–285 (1957). · Zbl 0079.24502
[3] J. Barwise and S. Feferman (eds.), Model-Theoretic Logics, Springer, New York (1985). · Zbl 0587.03001
[4] D. M. Gabbay and L. Maksimova, Interpolation and Definability: Modal and Intuitionistic Logics, Oxford Univ. Press, Oxford (2005). · Zbl 1091.03001
[5] K. Schütte, ”Der Interpolationssatz der intuitionistischen Prädikatenlogik,” Math. Ann., 148, No. 3, 192–200 (1962). · Zbl 0108.00301
[6] D. M. Gabbay, Semantical Investigations in Heyting’s Intuitionistic Logic, Synthese Library, 148, Reidel, Dordrecht (1981). · Zbl 0453.03001
[7] E. W. Beth, ”On Padoa’s method in the theory of definitions,” Indag. Math., 15, No. 4, 330–339 (1953). · Zbl 0053.34402
[8] L. L. Maksimova, ”Craig’s theorem in superintuitionistic logics and amalgamable varieties of pseudo-Boolean algebras,” Algebra Logika, 16, No. 6, 643–681 (1977). · Zbl 0413.03018
[9] L. L. Maksimova, ”Intuitionistic logic and implicit definability,” Ann. Pure Appl. Log., 105, Nos. 1–3, 83–102 (2000). · Zbl 0963.03044
[10] L. L. Maksimova, ”Decidability of the projective Beth property in varieties of Heyting algebras,” Algebra Logika, 40, No. 3, 290–301 (2001). · Zbl 0989.03025
[11] L. L. Maksimova, ”Implicit definability and positive logics,” Algebra Logika, 42, No. 1, 65–93 (2003). · Zbl 1034.03008
[12] L. Maksimova, ”Problem of restricted interpolation in superintuitionistic and some modal logics,” Log. J. IGPL, 18, No. 3, 367–380 (2010). · Zbl 1203.03035
[13] L. L. Maksimova, ”The projective Beth property and interpolation in positive and related logics,” Algebra Logika, 45, No. 1, 85–113 (2006). · Zbl 1119.03026
[14] L. L. Maksimova, ”Decidability of the weak interpolation property over the minimal logic,” Algebra Logika, 50, No. 2, 152–188 (2011). · Zbl 1285.03029
[15] L. Maksimova, ”Interpolation and joint consistency,” in We Will Show Them! Essays in Honour of D. Gabbay. Vol. 2, S. Artemov, H. Barringer, A. d’Avila Garcez, et al. (eds.) King’s Coll. Publ., London (2005), pp. 293–305. · Zbl 1260.03064
[16] L. L. Maksimova, ”Joint consistency in extensions of the minimal logic,” Sib. Mat. Zh., 51, No. 3, 604–619 (2010). · Zbl 1208.03031
[17] L. Maksimova, ”Interpolation and definability over the logic Gl,” Stud. Log., 99, Nos. 1–3, 249–267 (2011). · Zbl 1254.03069
[18] L. L. Maksimova, ”Interpolation and definability in extensions of the minimal logic,” Algebra Logika, 44, No. 6, 726–750 (2005). · Zbl 1106.03023
[19] L. L. Maksimova, ”Projective Beth properties in modal and superintuitionistic logics,” Algebra Logika, 38, No. 3, 316–333 (1999). · Zbl 0930.03018
[20] G. Kreisel, ”Explicit definability in intuitionistic logic,” J. Symb. Log., 25, No. 4, 389–390 (1960).
[21] L. Maksimova, ”Restricted interpolation in modal logics,” in Adv. Modal Log., Vol. 4, King’s Coll. Publ., London (2003), pp. 297–311. · Zbl 1082.03019
[22] A. I. Mal’tsev, Algebraic Systems [in Russian], Nauka, Moscow (1970).
[23] S. Miura, ”A remark on the intersection of two logics,” Nagoya Math. J., 26, 167–171 (1966). · Zbl 0202.29502
[24] S. Odintsov, ”Logic of classic refutability and class of extensions of minimal logic,” Log. Log. Phil., 9, 91–107 (2001). · Zbl 1034.03027
[25] L. L. Maksimova, ”A weak form of interpolation in equational logic,” Algebra Logika, 47, No. 1, 94–107 (2008). · Zbl 1164.03319
[26] B. Jonsson, ”Algebras whose congruence lattices are distributive,” Math. Scand., 21, 110–121 (1967). · Zbl 0167.28401
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