Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-02T12:13:57.028Z Has data issue: false hasContentIssue false
  • English
  • Français

THE LATTICE OF SUPER-BELNAP LOGICS

Part of: Model theory Algebraic logic

Published online by Cambridge University Press:  21 April 2021

ADAM PŘENOSIL
ADAM PŘENOSIL*
Affiliation:
INSTITUTE OF COMPUTER SCIENCE CZECH ACADEMY OF SCIENCES POD VODÁRENSKOU VĚŽÍ 271/1 PRAGUE, CZECHIA URL: https://sites.google.com/site/adamprenosil
*
E-mail: adam.prenosil@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the lattice of extensions of four-valued Belnap–Dunn logic, called super-Belnap logics by analogy with superintuitionistic logics. We describe the global structure of this lattice by splitting it into several subintervals, and prove some new completeness theorems for super-Belnap logics. The crucial technical tool for this purpose will be the so-called antiaxiomatic (or explosive) part operator. The antiaxiomatic (or explosive) extensions of Belnap–Dunn logic turn out to be of particular interest owing to their connection to graph theory: the lattice of finitary antiaxiomatic extensions of Belnap–Dunn logic is isomorphic to the lattice of upsets in the homomorphism order on finite graphs (with loops allowed). In particular, there is a continuum of finitary super-Belnap logics. Moreover, a non-finitary super-Belnap logic can be constructed with the help of this isomorphism. As algebraic corollaries we obtain the existence of a continuum of antivarieties of De Morgan algebras and the existence of a prevariety of De Morgan algebras which is not a quasivariety.

Keywords

Belnap-Dunn logicKleene logicLogic of Paradoxfour-valued logicparaconsistent logicabstract algebraic logic

MSC classification

Primary: 03G27: Abstract algebraic logic
Secondary: 03G10: Lattices and related structures 03C05: Equational classes, universal algebra
Type
Research Article
Information
The Review of Symbolic Logic , Volume 16 , Issue 1 , March 2023 , pp. 114 - 163
Check for updates
Copyright
© The Author(s), 2021. 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.)

References

BIBLIOGRAPHY

Adams, M. E., & Dziobiak, W. (1994). Lattices of quasivarieties of 3-element algebras. Journal of Algebra, 166, 181–210.CrossRefGoogle Scholar
Albuquerque, H., Přenosil, A., & Rivieccio, U. (2017). An algebraic view of super-Belnap logics. Studia Logica, 105(6), 1051–1086.CrossRefGoogle Scholar
Anderson, A. R., & Belnap, N. D. (1975). Entailment: The Logic of Relevance and Necessity, Vol. 1. Princeton, NJ: Princeton University Press.Google Scholar
Beall, J. C. (2013). LP+, K3+, FDE+, and their 'classical collapse'. The Review of Symbolic Logic, 6, 742–754.CrossRefGoogle Scholar
Belnap, N. D. (1977). How a computer should think. In Ryle, G., editor. Contemporary Aspects of Philosophy. Stocksfield: Oriel Press Ltd, pp. 30–55.Google Scholar
Belnap, N. D. (1977). A useful four-valued logic. In Dunn, J. M., and Epstein, G., editors. Modern Uses of Multiple-Valued Logic Episteme, Vol. 2. Dordrecht, Netherlands: Springer pp. 5–37.CrossRefGoogle Scholar
Bezhanishvili, N. (2006). Lattices of Intermediate and Cylindric Modal Logics. Ph.D. Thesis, Institute for Logic, Language and Computation.Google Scholar
Blok, W. J. (1980). The lattice of modal logics: An algebraic investigation. The Journal of Symbolic Logic, 45(2), 221–236.CrossRefGoogle Scholar
Burris, S., & Sankappanavar, H. P. (1981). A Course in Universal Algebra. Graduate Texts in Mathematics, Vol. 78. New York: Springer.CrossRefGoogle Scholar
Cintula, P., & Noguera, C. (2013). The proof by cases property and its variants in structural consequence relations. Studia Logica, 101(4), 713–747.CrossRefGoogle Scholar
Cornish, W. H., & Fowler, P. R. (1977). Coproducts of De Morgan algebras. Bulletin of the Australian Mathematical Society, 16, 1–13.CrossRefGoogle Scholar
Czelakowski, J. (1980). Reduced products of logical matrices. Studia Logica, 39(1), 19–43.CrossRefGoogle Scholar
Czelakowski, J. (2001). Protoalgebraic Logics. Trends in Logic: Studia Logica Library, Vol. 10. Dordrecht, Netherlands: Kluwer Academic Publishers.CrossRefGoogle Scholar
Dellunde, P., & Jansana, R. (1996). Some characterization theorems for infinitary universal Horn logic without equality. The Journal of Symbolic Logic, 61(4), 1242–1260.CrossRefGoogle Scholar
Dunn, J. M. (1966). The Algebra of Intensional Logics. Ph.D. Thesis, University of Pittsburgh.Google Scholar
Dunn, J. M. (1969). Natural Language Versus Formal Language. Unpublished manuscript.Google Scholar
Dunn, J. M. (1976). Intuitive semantics for first-degree entailments and ‘coupled trees’. Philosophical Studies, 29(3), 149–168.CrossRefGoogle Scholar
Dunn, J. M. (1976). A Kripke-style semantics for R-mingle using a binary accessibility relation. Studia Logica, 35, 163–172.CrossRefGoogle Scholar
Dunn, J. M. (2000). Partiality and its dual. Studia Logica, 66(1), 5–40.CrossRefGoogle Scholar
Dunn, J. M. (2010). Contradictory information: Too much of a good thing. Journal of Philosophical Logic, 39, 425–452.CrossRefGoogle Scholar
Dunn, J. M. (2016). An engineer in philosopher’s clothing . In Bimbó, K., editor. J. Michael Dunn on Information Based Logics. Outstanding Contributions to Logic, Vol. 8. Berlin: Springer International Publishing.Google Scholar
Erdős, P. (1959). Graph theory and probability. Canadian Journal of Mathematics, 11, 34–38.CrossRefGoogle Scholar
Font, J. M. (1997). Belnap’s four-valued logic and De Morgan lattices . Logic Journal of the IGPL, 5, 1–29.CrossRefGoogle Scholar
Font, J. M. (2016). Abstract Algebraic Logic—An Introductory Textbook. Studies in Logic, Vol. 60. London: College Publications.Google Scholar
Font, J. M., & Jansana, R. (2009). A General Algebraic Semantics for Sentential Logics (second edition). Lecture Notes in Logic, Vol. 7. New York: Springer.Google Scholar
Font, J. M., Jansana, R., & Pigozzi, D. (2003). A survey of abstract algebraic logic. Studia Logica, 74(1–2), 13–97.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. San Diego, CA: Elsevier Science.Google Scholar
Gorbunov, V. A. (1998). Algebraic Theory of Quasivarieties . Siberian School of Algebra and Logic. New York: Springer.Google Scholar
Grätzer, G., & Quackenbush, R. W. (2010). Positive universal classes in locally finite varieties. Algebra Universalis, 64(1), 1–13.CrossRefGoogle Scholar
Hell, P., & Nešetřil, J. (2004). Graphs and Homomorphisms . Oxford Lecture Series in Mathematics and Its Applications. Oxford: Oxford University Press.CrossRefGoogle Scholar
Kalman, J. A. (1958). Lattices with involution Transactions of the American Mathematical Society, 87(2), 485–491.CrossRefGoogle Scholar
Kleene, S. C. (1938). On notation for ordinal numbers. The Journal of Symbolic Logic, 3(4), 150–155.CrossRefGoogle Scholar
Kleene, S. C. (1952). Introduction to Metamathematics . Bibliotheca Mathematica, Vol. 1. Amsterdam, Netherlands: North-Holland Publishing Company.Google Scholar
Kripke, S. A. (1975). Outline of a theory of truth. Journal of Philosophy, 72(19), 690–716.CrossRefGoogle Scholar
Makinson, D. C. (1973). Topics in Modern Logic . London: Methuen.Google Scholar
Marcos, J. (2011). The value of the two values. In Coniglio, M. E., and Béziau, J. Y., editors. Logic Without Frontiers: Festschrift for Walter Alexandre Carnielli on the Occasion of His 60th Birthday. London: College Publications, pp. 277–294.Google Scholar
Pietz, A., & Rivieccio, U. (2013). Nothing but the truth. Journal of Philosophical Logic, 42(1), 125–135.CrossRefGoogle Scholar
Přenosil, A. (2018). Reasoning with Inconsistent Information. Ph.D. Thesis, Charles University.Google Scholar
Priest, G. (1979). The logic of paradox. Journal of Philosophical Logic, 8(1), 219–241.CrossRefGoogle Scholar
Pynko, A. P. (1995). Characterizing Belnap’s logic via De Morgan's laws. Mathematical Logic Quarterly, 41(4), 442–454.CrossRefGoogle Scholar
Pynko, A. P. (1999). Implicational classes of De Morgan lattices. Discrete Mathematics, 205(1–3), 171–181.CrossRefGoogle Scholar
Pynko, A. P. (2000). Subprevarieties versus extensions. Application to the Logic of Paradox. The Journal of Symbolic Logic, 65(2), 756–766.CrossRefGoogle Scholar
Rivieccio, U. (2011). On Extensions of the Belnap–Dunn Logic. Unpublished research notes.Google Scholar
Rivieccio, U. (2012). An infinity of super-Belnap logics. Journal of Applied Non-Classical Logics, 22(4), 319–335.CrossRefGoogle Scholar
Sankappanavar, H. P. (1980). A characterization of principal congruences of De Morgan algebras and its applications. In Arruda, A. I., Chuaqui, R., and Da Costa, N. C. A., editors. Mathematical Logic in Latin America: Proceedings of the IV Latin American Symposium on Mathematical Logic. Studies in Logic and the Foundations of Mathematics, Vol. 99. Santiago: Elsevier, pp. 341–349.CrossRefGoogle Scholar
Shoesmith, D. J., & Smiley, T. J. (1978). Multiple-Conclusion Logic. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Wójcicki, R. (1988). Theory of Logical Calculi: Basic Theory of Consequence Operations . Dordrecht, Netherlands: Kluwer Academic Publishers.CrossRefGoogle Scholar