Studia Logica

, Volume 105, Issue 6, pp 1319–1347 | Cite as

Interpolation Methods for Dunn Logics and Their Extensions

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Abstract

The semantic valuations of classical logic, strong Kleene logic, the logic of paradox and the logic of first-degree entailment, all respect the Dunn conditions: we call them Dunn logics. In this paper, we study the interpolation properties of the Dunn logics and extensions of these logics to more expressive languages. We do so by relying on the \(\mathbf{Dunn}\)calculus, a signed tableau calculus whose rules mirror the Dunn conditions syntactically and which characterizes the Dunn logics in a uniform way. In terms of the \(\mathbf{Dunn}\) calculus, we first introduce two different interpolation methods, each of which uniformly shows that the Dunn logics have the interpolation property. One of the methods is closely related to Maehara’s method but the other method, which we call the pruned tableau method, is novel to this paper. We provide various reasons to prefer the pruned tableau method to the Maehara-style method. We then turn our attention to extensions of Dunn logics with so-called appropriate implication connectives. Although these logics have been considered at various places in the literature, a study of the interpolation properties of these logics is lacking. We use the pruned tableau method to uniformly show that these extended Dunn logics have the interpolation property and explain that the same result cannot be obtained via the Maehara-style method. Finally, we show how the pruned tableau method constructs interpolants for functionally complete extensions of the Dunn logics.

Keywords

Interpolation methods Dunn logics First degree entailment Logic of paradox Strong Kleene logic Exactly true logic Tableau calculus 

References

  1. 1.
    Anderson, A.R., and N.D. Belnap, Entailment: The Logic of Relevance and Necessity, Volume I, Princeton University Press, 1975.Google Scholar
  2. 2.
    Arieli, O., and A. Avron, Reasoning with Logical Bilattices, Journal of Logic Language and Information 5: 25–63, 1996.CrossRefGoogle Scholar
  3. 3.
    Arieli, O., and A. Avron, The Value of the Four Values, Artificial Intelligence 102: 97–141, 1998.CrossRefGoogle Scholar
  4. 4.
    Avron, A., Natural 3-Valued Logics–Characterization and Proof Theory, Symbolic Logic 56(1): 276–294, 1991.Google Scholar
  5. 5.
    Avron, A., On the expressive power of three-valued and four-valued languages, Journal of Logic and Computation 9: 977–994, 1999.CrossRefGoogle Scholar
  6. 6.
    Batens, D., Paraconsistent Extensional Propositional Logics, Logique et Analyse 90: 195–234, 1980.Google Scholar
  7. 7.
    Batens, D., and K. de Clerq, A rich paraconsistent eextension of full positive logic, Logique et Analyse 47: 185–188, 220–257, 2004.Google Scholar
  8. 8.
    Belnap, N.D., How a Computer Should Think, in G. Ryle, (ed.), Contemporary Aspects of Philosophy, Oriel Press, Stocksfield 1976, pp. 30–56.Google Scholar
  9. 9.
    Belnap, N.D., A useful four-valued logic, in J.M. Dunn, and G. Epstein, (eds.), Modern Uses of Multiple-Valued Logic Springer, Dordrecht, 1977.Google Scholar
  10. 10.
    Bendova, K., Interpolation and three-valued logics, Reports on Mathematical Logic 39: 127–131, 2005.Google Scholar
  11. 11.
    Dunn, J.M., Intuitive Semantics for First-Degree Entailments and ‘Coupled Trees’, Philosophical Studies 29: 149–168, 1976.CrossRefGoogle Scholar
  12. 12.
    Fitting, M., Bilattices and the Semantics of Logic Programming, Journal of Logic Programming 11: 91–116, 1991.CrossRefGoogle Scholar
  13. 13.
    Kleene, S.C., Introduction to Metamathematics, D. Van Nostrand Company, New York and Toronto, 1952.Google Scholar
  14. 14.
    Milne, P., A non-classical refinement of the interpolation property for classical propositional logic, Logique and Analyse 59(235): 273–281, 2016.Google Scholar
  15. 15.
    Muskens, R.A., On Partial and Paraconsistent Logics, Notre Dame Journal of Formal Logic 40: 352–373, 1999.CrossRefGoogle Scholar
  16. 16.
    Omori, H., and H. Sano, Generalizing Functional Completeness in Belnap-Dunn Logic, Studia Logica 103(5): 883–917, 2015.CrossRefGoogle Scholar
  17. 17.
    Pietz, A., and U. Rivieccio, Nothing but the Truth, Journal of Philosophical Logic 42: 125–135, 2013.CrossRefGoogle Scholar
  18. 18.
    Priest, G., The Logic of Paradox, Journal of Philosophical Logic 8: 219–241, 1979.CrossRefGoogle Scholar
  19. 19.
    Pynko, A.P., Functional Completeness and Axiomatizability within Belnap’s Logic, Journal of Applied Non-classical Logics 9(1): 61–105, 1999.CrossRefGoogle Scholar
  20. 20.
    Ruet, P., Complete sets of connectives and complete sequent calculus for Belnap’s Logic, Tech. rep., Ecole Normal Supérieure, Logic Colloquium 96, 1996.Google Scholar
  21. 21.
    Smullyan, R., First-order Logic, Dover, New York, 1995.Google Scholar
  22. 22.
    Takano, M., Interpolation in many-valued logics with designated values, Kodai Mathematical Journal 12: 125–131, 1989.CrossRefGoogle Scholar
  23. 23.
    Takeuti, G., Proof Theory, North-Holland, Amsterdam, 1987.Google Scholar
  24. 24.
    Wintein, S., On all Strong Kleene generalizations of classical logic, Studia Logica 104(3): 503–545, 2016.CrossRefGoogle Scholar
  25. 25.
    Wintein, S., and R.A. Muskens, A gentzen calculus for nothing but the truth, Journal of Philosophical Logic 45(4): 451–465, 2016.CrossRefGoogle Scholar

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of PhilosophyErasmus University RotterdamRotterdamThe Netherlands
  2. 2.Tilburg Center for Logic, Ethics, and Philosophy of Science (TiLPS)Tilburg UniversityTilburgThe Netherlands

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