Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-06-23T22:33:41.120Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  09 May 2024



Bochvar algebras consist of the quasivariety $\mathsf {BCA}$ playing the role of equivalent algebraic semantics for Bochvar (external) logic, a logical formalism introduced by Bochvar [4] in the realm of (weak) Kleene logics. In this paper, we provide an algebraic investigation of the structure of Bochvar algebras. In particular, we prove a representation theorem based on Płonka sums and investigate the lattice of subquasivarieties, showing that Bochvar (external) logic has only one proper extension (apart from classical logic), algebraized by the subquasivariety $\mathsf {NBCA}$ of $\mathsf {BCA}$. Furthermore, we address the problem of (passive) structural completeness ((P)SC) for each of them, showing that $\mathsf {NBCA}$ is SC, while $\mathsf {BCA}$ is not even PSC. Finally, we prove that both $\mathsf {BCA}$ and $\mathsf {NBCA}$ enjoy the amalgamation property (AP).

Research Article
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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.)


Aglianò, P., & Ugolini, S. (2024). Structural and universal completeness in algebra and logic. Annals of Pure and Applied Logic, 175(3), 103391.CrossRefGoogle Scholar
Bergman, C. (1988). Structural completeness in algebra and logic. Algebraic Logic (Budapest, 1988), 54, 5973.Google Scholar
Bergman, C. (2011). Universal Algebra: Fundamentals and Selected Topics. Boca Raton: CRC Press.CrossRefGoogle Scholar
Bochvar, D. (1981). On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus. History and Philosophy of Logic, 2(1–2), 87112. Translation of the original in Russian (Mathematicheskii Sbornik, 1938).CrossRefGoogle Scholar
Bonzio, S. (2018). Dualities for Płonka sums. Logica Universalis, 12, 327339.CrossRefGoogle Scholar
Bonzio, S., Fano, V., Graziani, P., & Pra Baldi, M. (2023). A logical modeling of severe ignorance. Journal of Philosophical Logic, 52, 10531080.CrossRefGoogle Scholar
Bonzio, S., Gil-Férez, J., Paoli, F., & Peruzzi, L. (2017). On paraconsistent weak Kleene logic: Axiomatization and algebraic analysis. Studia Logica, 105(2), 253297.CrossRefGoogle Scholar
Bonzio, S., Loi, A., & Peruzzi, L. (2019). A duality for involutive bisemilattices. Studia Logica, 107, 423444.CrossRefGoogle Scholar
Bonzio, S., Paoli, F., & Pra Bldi, M. (2022). Logics of Variable Inclusion, Trends in Logic. Cham: Springer.CrossRefGoogle Scholar
Burris, S., & Sankappanavar, H. P. (2012). A Course in Universal Algebra. Available from: Scholar
Campercholi, M., Stronkowski, M. M., & Vaggione, D. (2015). On structural completeness versus almost structural completeness problem: A discriminator varieties case study. Logic Journal of the IGPL, 23(2), 235246.CrossRefGoogle Scholar
Ciucci, D., & Dubois, D. (2013). A map of dependencies among three-valued logics. Information Sciences, 250, 162177.CrossRefGoogle Scholar
Czelakowski, J. (2001). Protoalgebraic Logics, Trends in Logic—Studia Logica Library, 10. Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
Dzik, W., & Stronkowski, M. M. (2016). Almost structural completeness; an algebraic approach. Annals of Pure and Applied Logic, 167(7), 525556.CrossRefGoogle Scholar
Ferguson, T. (2017). Meaning and Proscription in Formal Logic: Variations on the Propositional Logic of William T. Parry, Trends in Logic. Cham: Springer International Publishing.CrossRefGoogle Scholar
Finn, V., & Grigolia, R. (1980). Bochvar’s algebras and corresponding propositional calculi. Bulletin of the Section of Logic, 9(1), 3943.Google Scholar
Finn, V., & Grigolia, R. (1993). Nonsense logics and their algebraic properties. Theoria, 59(1–3), 207273.CrossRefGoogle Scholar
Font, J. (2016). Abstract Algebraic Logic: An Introductory Textbook. London: College Publications.Google Scholar
Gil-Férez, J., Jipsen, P., & Lodhia, S. (2023). The structure of locally integral involutive po-monoids and semirings. In Glück, R., Santocanale, L., and Winter, M., editors. Relational and Algebraic Methods in Computer Science. Cham: Springer International Publishing, pp. 6986.CrossRefGoogle Scholar
Grätzer, G. (1975). A note on the amalgamation property (abstract). Notices of the American Mathematical Society, 22.Google Scholar
Halldén, S. (1949). The Logic of Nonsense. Uppsala: Lundequista Bokhandeln.Google Scholar
Jenei, S. (2022). Group representation for even and odd involutive commutative residuated chains. Studia Logica, 110(4), 881922.CrossRefGoogle Scholar
Kleene, S. (1952). Introduction to Metamathematics. Amsterdam: North Holland.Google Scholar
Metcalfe, G., Montagna, F., & Tsinakis, C. (2014). Amalgamation and interpolation in ordered algebras. Journal of Algebra, 402, 2182.CrossRefGoogle Scholar
Moraschini, T., Raftery, J. G., & Wannenburg, J. J. (2020). Singly generated quasivarieties and residuated structures. Mathematical Logic Quarterly, 66(2), 150172.CrossRefGoogle Scholar
Paoli, F., & Pra Baldi, M. (2021). Extensions of paraconsistent weak Kleene logic. Logic Journal of the IGPL, 29(5), 798822.CrossRefGoogle Scholar
Płonka, J. (1967). On a method of construction of abstract algebras. Fundamenta Mathematicae, 61(2), 183189.CrossRefGoogle Scholar
Płonka, J. (1967). On distributive quasilattices. Fundamenta Mathematicae, 60, 191200.CrossRefGoogle Scholar
Płonka, J. (1969). On sums of direct systems of Boolean algebras. Colloquium Mathematicum, 21, 209214.CrossRefGoogle Scholar
Płonka, J. (1984). On the sum of a direct system of universal algebras with nullary polynomials. Algebra universalis, 19(2), 197207.CrossRefGoogle Scholar
Płonka, J., & Romanowska, A. (1992). Semilattice sums. In Romanowska, A., and Smith, J. D. H., editors. Universal Algebra and Quasigroup Theory. Berlin: Heldermann, pp. 123158.Google Scholar
Raftery, J. G. (2016). Admissible rules and the Leibniz hierarchy. Notre Dame Journal of Formal Logic, 57(4), 569606.CrossRefGoogle Scholar
Romanowska, A., & Smith, J. (2002). Modes. Singapore: World Scientific.CrossRefGoogle Scholar
Rybakov, V. V. (1997). Admissibility of Logical Inference Rules. Amsterdam: Elsevier.Google Scholar
Segerberg, K. (1965). A contribution to nonsense-logics. Theoria, 31(3), 199217.CrossRefGoogle Scholar
Szmuc, D., & Ferguson, T. M. (2021). Meaningless divisions. Notre Dame Journal of Formal Logic, 62(3), 399424.CrossRefGoogle Scholar
Urquhart, A. (2001). Basic many-valued logic . In Gabbay, D. M., and Guenthner, F., editors. Handbook of Philosophical Logic. Dordrecht: Springer, pp. 249295.CrossRefGoogle Scholar
Wroński, A. (2009). Overflow rules and a weakening of structural completeness. In J. Sytnik-Czetwertyn'ski, editor. Rozwazania o filozofii prawdziwej: Jerzemu Perzanowskiemu w darze. Kraków: Wydawnictwo Uniwersytetu Jagiellońskiego, pp. 6771.Google Scholar