Skip to main content
SpringerLink
Log in

Containment Logics: Algebraic Completeness and Axiomatization

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

The paper studies the containment companion (or, right variable inclusion companion) of a logic \(\vdash \). This consists of the consequence relation \(\vdash ^{r}\) which satisfies all the inferences of \(\vdash \), where the variables of the conclusion are contained into those of the set of premises, in case this is not inconsistent. In accordance with the work started in [10], we show that a different generalization of the Płonka sum construction, adapted from algebras to logical matrices, allows to provide a matrix-based semantics for containment logics. In particular, we provide an appropriate completeness theorem for a wide family of containment logics, and we show how to produce a complete Hilbert style axiomatization.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Albuquerque, H., A. Přenosil, and U. Rivieccio, An Algebraic View of Super-Belnap Logics, Studia Logica 105:1051–1086, 2017.

    Article  Google Scholar 

  2. Anderson, A., and N. Belnap, Tautological Entailments, Philosophical Studies 13:9–24, 1961.

    Article  Google Scholar 

  3. Beall, J.C., Off-Topic: A New Interpretation of Weak-Kleene Logic, The Australasian Journal of Logic 13(6), 2016.

  4. Belikov, A., and Y. Petrukhin, Exactly true and non-falsity logics meeting infectious ones, Journal of Applied Non-Classical Logics 30(2):93–122, 2020.

  5. Belnap, N., A Useful Four-Valued Logic, in J. M. Dunn, and G. Epstein, (eds.), Modern Uses of Multiple-Valued Logic, Springer Netherlands, Dordrecht, 1977, pp. 5–37.

  6. Bergman, C., Universal Algebra: Fundamentals and Selected Topics, Chapman and Hall/CRC, 2011.

  7. Blackburn, P., M. de Rijke, and Y. Venema, Modal logic, Cambridge University Press, Cambridge, 2001.

  8. Bochvar, D., and M. Bergmann, On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus (1938), History and Philosophy of Logic 2(1–2):87–112, 1981.

  9. Bonzio, S., J. Gil-Ferez, F. Paoli, and L. Peruzzi, On Paraconsistent Weak Kleene Logic: Axiomatization and Algebraic Analysis, Studia Logica 105(2):253–297, 2017.

    Article  Google Scholar 

  10. Bonzio, S., T. Moraschini, and M. Pra Baldi, Logics of left variable inclusion and Płonka sums of matrices, Archive for Mathematical Logic, forthcoming.

  11. Burris, S., and H.P. Sankappanavar, A course in Universal Algebra, The Millennium Edition, https://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html, 2012.

  12. Caleiro, C., S. Marcelino, and U. Rivieccio, Characterizing finite-valuedness, Fuzzy Sets and Systems 345:113–125, 2018.

    Article  Google Scholar 

  13. Campercholi, M.A., and J.G. Raftery, Relative congruence formulas and decompositions in quasivarieties, Algebra Universalis 78(3):407–425, 2017.

    Article  Google Scholar 

  14. Ciuni, R., and M. Carrara, Characterizing Logical Consequence in Paraconsistent Weak Kleene, in L. Felline, A. Ledda, F. Paoli, and E. Rossanese, (eds.), New Developments in Logic and the Philosophy of Science, College Publications, London, 2016, pp. 165–176.

    Google Scholar 

  15. Ciuni, R., T.M. Ferguson, and D. Szmuc, Relevant Logics obeying Component Homogeneity, Australasian Journal of Logic 15(2):301–361, 2018.

    Article  Google Scholar 

  16. Ciuni, R., T.M. Ferguson, and D. Szmuc, Logics based on linear orders of contaminating values, Journal of Logic and Computation 29(5):631–663, 06 2019.

  17. Daniels, C., A note on negation, Erkenntnis 32:423–429, 1990.

    Article  Google Scholar 

  18. Deutsch, H., Relevant analytic entailment, The Relevance Logic Newsletter 1(2):26–44, 1977.

    Google Scholar 

  19. Diego, A., Sobre álgebras de Hilbert, volume 12 of Notas de Lógica Matemática. Universidad Nacional del Sur, Bahía Blanca (Argentina), 1965.

  20. Epstein, R.L., The semantic foundations of logic, in The Semantic Foundations of Logic Volume 1: Propositional Logics, Springer, 1990, pp. 315–321.

  21. Ferguson, T.M., A computational interpretation of conceptivism, Journal of Applied Non-Classical Logics 24(4):333–367, 2014.

    Article  Google Scholar 

  22. Ferguson, T.M., Faulty Belnap computers and subsystems of FDE, Journal of Logic and Computation 26(5):1617–1636, 2014.

    Article  Google Scholar 

  23. Ferguson, T.M., Logics of Nonsense and Parry Systems, Journal of Philosophical Logic 44:65–80, 2015.

    Article  Google Scholar 

  24. Ferguson, T.M., Meaning and Proscription in Formal Logic: Variations on the Propositional Logic of William T. Parry. Springer, 2017.

  25. Finn, V., and R. Grigolia, Nonsense logics and their algebraic properties, Theoria 59(1-3):207–273, 1993.

    Google Scholar 

  26. Font, J.M., Belnap’s Four-Valued Logic and De Morgan Lattices, Logic Journal of IGPL 5(3):1–29, 1997.

    Article  Google Scholar 

  27. Font, J.M., Abstract Algebraic Logic: An Introductory Textbook. College Publications, 2016.

  28. Galatos, N., P. Jipsen, T. Kowalski, and H. Ono, Residuated Lattices: an algebraic glimpse at substructural logics. Elsevier, Amsterdam, 2007.

    Google Scholar 

  29. Halldén, S., The Logic of Nonsense, Lundequistska Bokhandeln, Uppsala, 1949.

    Google Scholar 

  30. Iseki, K., BCK-algebras, Mathematical Seminar Notes 4:77–86, 1976.

  31. Kalman, J., Lattices with involution, Transactions of the AMS 87:485–491, 1958.

    Article  Google Scholar 

  32. Karpenko, A., and N. Tomova, Bochvar’s Three-Valued Logic and Literal Paralogics: Their Lattice and Functional Equivalence, Logic and Logical Philosophy 26(2):207–235, 2016.

    Google Scholar 

  33. Lávička, T., An Abstract Study of Completeness in Infinitary Logics, PhD Thesis, Charles University, 2018.

  34. Ledda, A., F. Paoli, and M. Pra Baldi, Algebraic Analysis of Demodalised Analytic Implication, Journal of Philosophical Logic 48:957–979, 2019.

  35. Moisil, G., Recherches sur l’algèbre de la logique, Annales Scientifiques de l’Université de Jassy 22:1–117, 1935.

  36. Parry, W.T., Implication, PhD Thesis, Harvard University, 1932.

  37. Petrukhin, Y., Natural Deduction for Four-Valued both Regular and Monotonic Logics, Logic and Logical Philosophy 27:53–66, 2018.

    Google Scholar 

  38. Płonka, J., On a method of construction of abstract algebras, Fundamenta Mathematicae 61(2):183–189, 1967.

    Article  Google Scholar 

  39. Płonka, J., On distributive quasilattices, Fundamenta Mathematicae 60:191–200, 1967.

    Article  Google Scholar 

  40. Płonka, J., On the sum of a direct system of universal algebras with nullary polynomials, Algebra Universalis 19(2):197–207, 1984.

  41. Płonka, J., and A. Romanowska, Semilattice sums, in A. Romanowska and J. D. H. Smith, (eds.), Universal Algebra and Quasigroup Theory, Heldermann, 1992, pp. 123–158.

  42. Pra Baldi, M., Logics of variable inclusion and the lattice of consequence relations, Journal of Applied Non-Classical Logics 30(4):367–381, 2020.

  43. Priest, G., The Logic of Paradox, Journal of Philosophical Logic 8:219–241, 1979.

    Article  Google Scholar 

  44. Priest, G., The Logic of Catuskoti, Comparative Philosophy 1(2):24–54, 2010.

  45. Pynko, A., Characterizing Belnap’s Logic via De Morgan’s Laws, Mathematical Logic Quarterly 41(4):442–454, 1995.

    Article  Google Scholar 

  46. Pynko, A., On Priest’s logic of paradox, Journal of Applied Non-Classical Logics, 5(2):219–225, 1995.

    Article  Google Scholar 

  47. Raftery, J.G., Inconsistency lemmas in algebraic logic, Mathematical Logic Quarterly 59(6):393–406, 2013.

    Article  Google Scholar 

  48. Szmuc, D., Defining LFIs and LFUs in extensions of infectious logics, Journal of Applied Non-Classical Logics 26(4):286–314, 2016.

    Article  Google Scholar 

  49. Tomova, N., About four-valued regular logics, Logical Investigations 15:223–228, 2009, (in Russian).

    Google Scholar 

  50. Urquhart, A., Basic Many-Valued Logic, in D. M. Gabbay and F. Guenthner, (eds.), Handbook of Philosophical Logic, Springer Netherlands, Dordrecht, 2001, pp. 249–295.

Download references

Acknowledgements

The first author gratefully acknowledges funding from the PRIN 2017 project “From models to decisions” funded by the Italian Ministry for University, Education and Research (MIUR) and the ERC project ‘PhilPharm’ (Grant 639276). The second author gratefully acknowledges the MIUR, within the projects PRIN 2017 “Theory and applications of resource sensitive logics”, CUP: 20173WKCM5, and “Logic and cognition. Theory, experiments, and applications”, CUP: 2013YP4N3. Finally, we thank Francesco Paoli and three anonymous referees for their useful comments and suggestions on a previous version of the paper.

Author information

Authors and Affiliations

  1. Department of Philosophy and Education Science, University of Turin, Turin, Italy

    Stefano Bonzio

  2. Department of Pedagogy, Psychology and Philosophy, University of Cagliari, Cagliari, Italy

    Michele Pra Baldi

Authors
  1. Stefano Bonzio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michele Pra Baldi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefano Bonzio.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bonzio, S., Pra Baldi, M. Containment Logics: Algebraic Completeness and Axiomatization. Stud Logica 109, 969–994 (2021). https://doi.org/10.1007/s11225-020-09930-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-020-09930-1

Keywords

Mathematics Subject Classification

Search

Navigation