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First-Degree Entailment and its Relatives

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Abstract

We consider a family of logical systems for representing entailment relations of various kinds. This family has its root in the logic of first-degree entailment formulated as a binary consequence system, i.e. a proof system dealing with the expressions of the form \(\varphi \vdash \psi \), where both \(\varphi \) and \(\psi \) are single formulas. We generalize this approach by constructing consequence systems that allow manipulating with sets of formulas, either to the right or left (or both) of the turnstile. In this way, it is possible to capture proof-theoretically not only the entailment relation of the standard four-valued Belnap’s logic, but also its dual version, as well as some of their interesting extensions. The proof systems we propose are, in a sense, of a hybrid Hilbert–Gentzen nature. We examine some important properties of these systems and establish their completeness with respect to the corresponding entailment relations.

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References

  1. Ackermann, W., Begründung einer strengen Implikation, Journal of Symbolic Logic 21:113–128, 1956.

    Article  Google Scholar 

  2. Anderson, A. R., and N. D. Belnap, Entailment: The Logic of Relevance and Necessity, vol. I, Princeton University Press, Princeton, NJ, 1975.

    Google Scholar 

  3. Anderson, A. R., Belnap, N. D., and J. M. Dunn, Entailment: The Logic of Relevance and Necessity, vol. II, Princeton University Press, Princeton, NJ, 1992.

    Google Scholar 

  4. Belnap, N. D. Tautological entailments (abstract), Journal of Symbolic Logic 24:316, 1959.

    Google Scholar 

  5. Anderson, A. R., and N. D. Belnap, Tautological entailments, Philosophical Studies 13:9–24, 1962.

    Article  Google Scholar 

  6. Belnap, N. D., A useful four-valued logic, in J. M. Dunn and G. Epstein (eds.), Modern Uses of Multiple-Valued Logic, D. Reidel Publishing Company, Dordrecht, 1977, pp. 8–37.

    Google Scholar 

  7. Belnap, N. D., How a computer should think, in G. Ryle (ed.), Contemporary Aspects of Philosophy, Oriel Press, 1977, pp. 30–55.

  8. Belnap, N. D., and J. M. Dunn, Entailment and the disjunctive syllogism, in G. Fløistad and G. H. Von Wright (eds.), Contemporary Philosophy: A New Survey, Vol. 1, Philosophy of language, Philosophical logic, Martinus Nijhoff Publishers, Dordrecht, 1981, pp. 337–366.

    Google Scholar 

  9. Buss, S. R., An introduction to proof theory, in S. R. Buss (ed.), Handbook of Proof Theory, Elsevier, Amsterdam, 1998, pp. 1–78.

    Google Scholar 

  10. Dunn, J. M., The Algebra of Intensional Logics, Doctoral Dissertation, University of Pittsburgh, Ann Arbor, (University Microfilms), 1966.

  11. Dunn, J. M., Intuitive semantics for first-degree entailment and coupled trees, Philosophical Studies 29:149–168, 1976.

    Article  Google Scholar 

  12. Dunn, J. M., Relevance logic and entailment, in F. Guenthner and D. Gabbay (eds.), Handbook of Philosophical Logic, vol. 3, Dordrecht, Reidel, 1986, pp. 117–24.

    Chapter  Google Scholar 

  13. Dunn, J. M., Positive modal logic, Studia Logica 55:301–317, 1995.

    Article  Google Scholar 

  14. Dunn, J. M., Partiality and its dual, Studia Logica 66:5–40, 2000.

    Article  Google Scholar 

  15. Fitting, M., Kleene’s three-valued logics and their children, Fundamenta Informaticae 20:113–131, 1994.

    Google Scholar 

  16. Font, J. M., Belnap’s four-valued logic and De Morgan lattices, Logic Journal of the IGPL, 5:413–440, 1997.

    Article  Google Scholar 

  17. Font, J. M., Guzmán, F., and V. Verdú, Characterization of the reduced matrices for the \(\{\wedge , \vee \}\)-fragment of classical logic, Bulletin of the Section of Logic 20:124–128, 1991.

    Google Scholar 

  18. Kleene, S. C., Introduction to Metamathematics, Amsterdam: North-Holland, 1952.

  19. Kleene, S. C., Mathematical Logic, Wiley, New York, 1967.

    Google Scholar 

  20. Nelson, D., Constructible falsity, Journal of Symbolic Logic 14:16–26, 1949.

    Article  Google Scholar 

  21. Marcos, J., The value of the two values, in J. -Y. Beziau and M. E. Coniglio (eds.), Logic Without Frontiers: Festschrift for Walter Alexandre Carnielli on the Occasion of his 60th Birthday (Tributes), College Publications, 2011, pp. 277–294.

  22. Pietz, A., and U. Rivieccio, Nothing but the Truth, Journal of Philosophical Logic 42:125–135, 2013.

    Article  Google Scholar 

  23. Priest, G., An Introduction to Non-Classical Logic, 2nd edn, Cambridge University Press, Cambridge, 2008.

    Book  Google Scholar 

  24. Rivieccio, U., An infinity of super-Belnap logics, Journal of Applied Non-Classical Logics 22:319–335, 2012.

    Article  Google Scholar 

  25. Shramko, Y., A philosophically plausible modified Grzegorczyk semantics for first-degree intuitionistic entailment, Logique et Analyse 161-162-163:167–188, 1998.

  26. Shramko, Y., Truth, falsehood, information and beyond: the American plan generalized, in K. Bimbo (ed.), J. Michael Dunn on Information Based Logics, Outstanding Contributions to Logic, vol. 8, Springer, 2016, pp. 191–212.

  27. Wansing, H., Proofs, disproofs, and their duals, in V. Goranko, L. Beklemishev and V. Shehtman (eds.), Advances in Modal Logic, vol. 8, College Publications, London, 2010, pp. 483–505.

    Google Scholar 

  28. Wansing, H., On split negation, strong negation, information, falsification, and verification, in K. Bimbo (ed.), J. Michael Dunn on Information Based Logics, Outstanding Contributions to Logic, vol. 8, Springer, 2016, pp. 161–189.

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Correspondence to Yaroslav Shramko.

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Shramko, Y., Zaitsev, D. & Belikov, A. First-Degree Entailment and its Relatives. Stud Logica 105, 1291–1317 (2017). https://doi.org/10.1007/s11225-017-9747-7

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