Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-04T19:07:14.360Z Has data issue: false hasContentIssue false

HEREDITARILY STRUCTURALLY COMPLETE POSITIVE LOGICS

Published online by Cambridge University Press:  23 April 2019

ALEX CITKIN*
Affiliation:
Metropolitan Telecommunications
*
*INFORMATION TECHNOLOGY METROPOLITAN COMMUNICATIONS 55 WATER STREET, 32 FLOOR NEW YORK, NY 10041, USA E-mail: [email protected]

Abstract

Positive logics are $\{ \wedge , \vee , \to \}$-fragments of intermediate logics. It is clear that the positive fragment of $Int$ is not structurally complete. We give a description of all hereditarily structurally complete positive logics, while the question whether there is a structurally complete positive logic which is not hereditarily structurally complete, remains open.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Bahturin, Y. & Ol’shanskij, A. (1991). Identities. In Kostrikin, A. I. and Shafarevich, I., editors. Algebra, Vol. 18. Encyclopedia of Mathematical Sciences. Berlin: Springer, pp. 107221.Google Scholar
Balbes, R. & Horn, A. (1970). Injective and projective Heyting algebras. Transactions of the American Mathematical Society, 148, 549559.CrossRefGoogle Scholar
Belnap, N. D. Jr., Leblanc, H., & Thomason, R. H. (1963). On not strengthening intuitionistic logic. Notre Dame Journal of Formal Logic, 4, 313320.CrossRefGoogle Scholar
Bergman, C. (2012). Universal Algebra. Fundamentals and Selected Topics. Pure and Applied Mathematics (Boca Raton), Vol. 301. Boca Raton, FL: CRC Press.Google Scholar
Bezhanishvili, G. & Grigolia, R. (2005). Locally finite varieties of Heyting algebras. Algebra Universalis, 54(4), 465473.CrossRefGoogle Scholar
Bezhanishvili, N. (2006). Lattices of Intermediate and Cylindric Modal Logics. Ph.D. Thesis, Institute for Logic, Language and Computation University of Amsterdam.Google Scholar
Burris, S. & Sankappanavar, H. P. (1981). A Course in Universal Algebra. Graduate Texts in Mathematics, Vol. 78. New York: Springer-Verlag.CrossRefGoogle Scholar
Cintula, P. & Metcalfe, G. (2010). Admissible rules in the implication-negation fragment of intuitionistic logic. Annals of Pure and Applied Logic. 162(2), 162171.CrossRefGoogle Scholar
Citkin, A. (1987). Structurally complete superintuitionistic logics and primitive varieties of pseudo-Boolean algebras. Matematicheskie Issledovaniya, 98, 134151. (A. Tsitkin).Google Scholar
Citkin, A. I. (1978). Structurally complete superintuitionistic logics. Doklady Akademii Nauk SSSR, 241(1), 4043.Google Scholar
Citkin, A. I. (1986). Finite axiomatizability of locally tabular superintuitionistic logics. Matematicheskie Zametki, 40(3), 407413, 430.Google Scholar
Day, A. (1973). Splitting algebras and a weak notion of projectivity. In Fajtlowicz, S., and Kaiser, K., editors. Proceedings of the University of Houston Lattice Theory Conference (Houston, Tex., 1973). University of Houston, Houston, Texas: Department of Mathematics, pp. 466485.Google Scholar
Dzik, W. (2004). Chains of structurally complete predicate logics with the application of Prucnal’s substitution. Reports on Mathematical Logic, 38, 3748.Google Scholar
Dzik, W. & Wroński, A. (1973). Structural completeness of Gödel’s and Dummett’s propositional calculi. Studia Logica, 32, 6975.CrossRefGoogle Scholar
Galatos, N., Jipsen, P., Kowalski, T., & Ono, H. (2007). Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Studies in Logic and the Foundations of Mathematics, Vol. 151. Amsterdam: Elsevier B. V.Google Scholar
Gorbunov, V. A. (1998). Algebraic Theory of Quasivarieties. Siberian School of Algebra and Logic. New York: Consultants Bureau. Translated from the Russian.Google Scholar
Grätzer, G. & Quackenbush, R. W. (2010). Positive universal classes in locally finite varieties. Algebra Universalis, 64(1–2), 113.CrossRefGoogle Scholar
Harrop, R. (1960). Concerning formulas of the types $A \to B \vee C,\,A \to \left( {Ex} \right)B\left( x\right)$ in intuitionistic formal systems. Journal of Symbolic Logic, 25, 2732.CrossRefGoogle Scholar
Huczynska, S. & Ruškuc, N. (2015). Well quasi-order in combinatorics: Embeddings and homomorphisms. In Czumaj, A., editor. Surveys in Combinatorics 2015. London Mathematical Society Lecture Note Series, Vol. 424. Cambridge: Cambridge Universtiy Press, pp. 261293.CrossRefGoogle Scholar
Jankov, V. A. (1968). Calculus of the weak law of the excluded middle. Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya, 32, 10441051. English translation in Mathematics of the USSR-Izvestiya, 2(5), 997–1004 (1968).Google Scholar
Jankov, V. A. (1969). Conjunctively irresolvable formulae in propositional calculi. Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya, 33, 1838. English translation in Mathematics of the USSR-Izvestiya, 3(1), 17–35 (1969).Google Scholar
Jónsson, B. (1967). Algebras whose congruence lattices are distributive. Mathematica Scandinavica, 21, 110121 (1968).CrossRefGoogle Scholar
Kuznetsov, A. V. (1973). On finitely generated pseudo-Boolean algebras and finitely approximable varieties. In Starostin, A. I., editor. Proceedings of the 12th USSR Algebraic Colloquium. Sverdlovsk, p. 281 (in Russian).Google Scholar
Kuznetsov, A. V. & Gerčiu, V. J. (1970). The superintuitionistic logics and finite approximability. Doklady Akademii Nauk SSSR, 195, 10291032. (in Russian).Google Scholar
Lorenzen, P. (1955). Einführung in die operative Logik und Mathematik. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd. LXXVIII. Springer-Verlag, Berlin-Göttingen-HeidtemCrossRefGoogle Scholar
Maksimova, L. L., Skvorcov, D. P., & Šehtman, V. B. (1979). Impossibility of finite axiomatization of Medvedev’s logic of finite problems. Doklady Akademii Nauk SSSR, 245(5), 10511054.Google Scholar
Mal’cev, A. (1973). Algebraic systems. Die Grundlehren der mathematischen Wissenschaften. Band 192. Berlin-Heidelberg-New York: Springer-Verlag; Berlin: Akademie-Verlag. p. XII, 317.CrossRefGoogle Scholar
Mints, G. (1976). Derivability of admissible rules. Journal of Soviet Mathematics, 6, 417421. Translated from Mints, G. E. Derivability of admissible rules. (Russian) Investigations in constructive mathematics and mathematical logic, V. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 32 (1972), pp. 85–89.CrossRefGoogle Scholar
Nemitz, W. & Whaley, T. (1973). Varieties of implicative semilattices II. Pacific Journal of Mathematics, 45, 303311.CrossRefGoogle Scholar
Novikov, P. S. (1977). Konstruktivnaya matematicheskaya logika s tochki zreniya klassicheskoi [Constructive mathematical logic from the point of view of classical logic]. Izdat. “Nauka”, Moscow. With a preface by S. I. Adjan, Matematicheskaya Logika i Osnovaniya Matematiki. [Monographs in Mathematical Logic and Foundations of Mathematics] (in Russian).Google Scholar
Odintsov, S. P. (2008). Constructive Negations and Paraconsistency. Trends in Logic—Studia Logica Library, Vol. 26. New York: Springer.CrossRefGoogle Scholar
Olson, J. S., Raftery, J. G., & van Alten, C. J. (2008). Structural completeness in substructural logics. Logic Journal of the IGPL, 16(5), 455495.CrossRefGoogle Scholar
Pogorzelski, W. A. (1974). Concerning the notion of completeness of invariant propositional calculi. Studia Logica, 33, 6972. (errata insert).CrossRefGoogle Scholar
Prucnal, T. (1972a). On the structural completeness of some pure implicational propositional calculi. Studia Logica, 30, 4552.CrossRefGoogle Scholar
Prucnal, T. (1972b). Structural completeness of Lewis’s system S5. Bulletin de l’Academie Polonaise des Sciences, Serie des Sciences, Mathematiques, Astronomiques et Physiques, 20, 101103.Google Scholar
Prucnal, T. (1976). Structural completeness of Medvedev’s propositional calculus. Reports on Mathematical Logic, 6, 103105.Google Scholar
Raftery, J. G. (2016). Admissible rules and the Leibniz hierarchy. Notre Dame Journal of Formal Logic, 57(4), 569606.CrossRefGoogle Scholar
Rasiowa, H. & Sikorski, R. (1970). The Mathematics of Metamathematics (third edition). Warsaw: PWN—Polish Scientific Publishers. Monografie Matematyczne, Tom 41.Google Scholar
Rybakov, V. V. (1995). Hereditarily structurally complete modal logics. Journal of Symbolic Logic, 60(1), 266288.CrossRefGoogle Scholar
Rybakov, V. V. (1997). Admissibility of Logical Inference Rules. Studies in Logic and the Foundations of Mathematics, Vol. 136. Amsterdam: North-Holland.CrossRefGoogle Scholar
Skura, T. (1991). On decision procedures for sentential logics. Studia Logica, 50(2), 173179.CrossRefGoogle Scholar
Troelstra, A. S. (1965). On intermediate propositional logics. Nederlandse Akademie van Wetenschappen: Proceedings Series A, 68=Indagationes Mathematicae 27, 141152.Google Scholar
Wroński, A. (1986). On factoring by compact congruences in algebras of certain varieties related to the intuitionistic logic. Institute of the Polish Academy of Sciences, Sociology Bulletin of the Section of Logic 15(2), 4851.Google Scholar