The Fmla-Fmla Axiomatizations of the Exactly True and Non-falsity Logics and Some of Their Cousins

Abstract

In this paper we present a solution of the axiomatization problem for the Fmla-Fmla versions of the Pietz and Rivieccio exactly true logic and the non-falsity logic dual to it. To prove the completeness of the corresponding binary consequence systems we introduce a specific proof-theoretic formalism, which allows us to deal simultaneously with two consequence relations within one logical system. These relations are hierarchically organized, so that one of them is treated as the basic for the resulting logic, and the other is introduced as an extension of this basic relation. The proposed bi-consequences systems allow for a standard Henkin-style canonical model used in the completeness proof. The deductive equivalence of these bi-consequence systems to the corresponding binary consequence systems is proved. We also outline a family of the bi-consequence systems generated on the basis of the first-degree entailment logic up to the classic consequence.

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

References

  1. 1.

    Albuquerque, H., Přenosil, A., Rivieccio, U. (2017). An algebraic view of super-Belnap logics. Studia Logica, 105, 1051–1086.

    Article  Google Scholar 

  2. 2.

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

    Article  Google Scholar 

  3. 3.

    Anderson, A.R., & Belnap, N.D. (1975). Entailment: the logic of relevance and necessity (Vol. I). Princeton: Princeton University Press.

    Google Scholar 

  4. 4.

    Anderson, A.R., Belnap, N.D., Dunn, J.M. (1992). Entailment: the logic of relevance and necessity (Vol. II). Princeton: Princeton University Press.

    Google Scholar 

  5. 5.

    Avron, A., & E. Orłowska. (2003). Classical Gentzen-type methods in propositional many-valued logics. In M. Fitting (Ed.) , Beyond two: theory and applications of multiple-valued logic (pp. 117–155). Berlin–Heidelberg: Springer.

  6. 6.

    Belnap, N.D. (1977). A useful four-valued logic. In J.M. Dunn, & G. Epstein (Eds.) , Modern uses of multiple-valued logic (pp. 8–37). Dordrecht: D. Reidel Publishing Company.

  7. 7.

    Belnap, N.D. (1977). How a computer should think. In G. Ryle (Ed.) , Contemporary aspects of philosophy (pp. 30–55). London: Oriel Press.

  8. 8.

    Demri, S., & Orlowska, E. (2002). Incomplete information: structure, inference, complexity. Berlin: Springer.

    Google Scholar 

  9. 9.

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

    Article  Google Scholar 

  10. 10.

    Dunn, J.M. (1995). Positive modal logic. Studia Logica, 55, 301–317.

    Article  Google Scholar 

  11. 11.

    Dunn, J.M. (1999). A comparative study of various model-theoretic treatmets of negation: a history of formal negation. In D.M. Gabbay, & H. Wansing (Eds.) , What is negation? (pp. 23–51). Dordrecht/Boston/London: Kluwer Academic Publishers.

  12. 12.

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

    Article  Google Scholar 

  13. 13.

    Dunn, J.M., & Hardegree, G.M. (2001). Algebraic methods in philosophical logic. Oxford, New York: Clarendon Press, Oxford University Press.

    Google Scholar 

  14. 14.

    Fitting, M.C. (1989). Bilattices and the theory of truth. Journal of Philosophical Logic, 18, 225–256.

    Article  Google Scholar 

  15. 15.

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

    Article  Google Scholar 

  16. 16.

    Frankowski, S. (2004). Formalization of a plausible inference. Bulletin of the Section of Logic, 33, 41–52.

    Google Scholar 

  17. 17.

    Hartonas, C. (2017). Order-dual relational semantics for non-distributive propositional logics. Logic Journal of the IGPL, 25, 145–182.

    Google Scholar 

  18. 18.

    Humberstone, L. (2011). The connectives. Cambridge: MIT Press.

    Google Scholar 

  19. 19.

    Labuschagne, W., Heidema, J., Britz, K. (2013). Supraclassical consequence relations. In S. Cranefield, & A. Nayak (Eds.) , AI 2013: advances in artificial intelligence. AI 2013. Lecture Notes in Computer Science (Vol. 8272, pp. 326–337). Cham: Springer.

  20. 20.

    MacIntosh, J.J. (1991). Adverbially qualified truth values. Pacific Philosophical Quarterly, 72, 131–142.

    Article  Google Scholar 

  21. 21.

    Malinowski, G. (1990). Q-consequence operation. Reports on Mathematical Logic, 24, 49–59.

    Google Scholar 

  22. 22.

    Marcos, J. (2011). The value of the two values. In J.-Y. Beziau, M.E. Coniglio (Eds.), Logic without frontiers: festschrift for Walter Alexandre Carnielli on the occasion of his 60th birthday (pp. 277–294). (Tributes), College Publications.

  23. 23.

    Odintsov, S., & Wansing, H. (2015). The logic of generalized truth-values and the logic of bilattices. Studia Logica, 103, 91–112.

    Article  Google Scholar 

  24. 24.

    Omori, H., Wansing, H. (Eds.) (2017). 40 Years of FDE, Special Issue of Studia Logica, 105, Issue 6.

  25. 25.

    Pietz, A., & Rivieccio, U. (2013). Nothing but the truth. Journal of Philosophical Logic, 42, 125–135.

    Article  Google Scholar 

  26. 26.

    Priest, G. (2008). An introduction to non-classical logic (2nd edn). Cambridge: Cambridge University Press.

    Google Scholar 

  27. 27.

    Rescher, N. (1965). An intuitive interpretation of systems of four-valued logic. Notre Dame Journal of Formal Logic, 6, 154–156.

    Article  Google Scholar 

  28. 28.

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

    Article  Google Scholar 

  29. 29.

    Shramko, Y. (2016). 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, pp. 191–212). Berlin: Springer.

  30. 30.

    Shramko, Y. (2018). First-degree entailment and structural reasoning, to appear. In H. Omori, H. Wansing (Eds.), New Essays on Belnap-Dunn Logic, Synthese Library.

  31. 31.

    Shramko, Y. (2018). Dual-Belnal logic and anaything but falsehood, to appear in: Journal of Applied Logic.

  32. 32.

    Shramko, Y., Dunn, J.M., Takenaka, T. (2001). The trilaticce of constructive truth-values. Journal of Logic and Computation, 11, 761–788.

    Article  Google Scholar 

  33. 33.

    Shramko, Y., & Wansing, H. (2005). Some useful sixteen-valued logics: how a computer network should think. Journal of Philosophical Logic, 34, 121–153.

    Article  Google Scholar 

  34. 34.

    Shramko, Y., & Wansing, H. (2007). Entailment relations and/as truth values. Bulletin of the Section of Logic, 36, 131–143.

    Google Scholar 

  35. 35.

    Shramko, Y., Zaitsev, D., Belikov, A. (2017). First-degree entailment and its relatives. Studia Logica, 105, 1291–1317.

    Article  Google Scholar 

  36. 36.

    Vakarelov, D. (1995). A duality between Pawlak’s knowledge representation systems and bi-consequence systems. Studia Logica, 55, 205–228.

    Article  Google Scholar 

  37. 37.

    van Benthem, J. (2008). Logic and reasoning: do the facts matter? Studia Logica, 88, 67–84.

    Article  Google Scholar 

  38. 38.

    Wansing, H., & Shramko, Y. (2008). A note on two ways of defining a many-valued logic. In M. Pelis (Ed.) The logica yearbook 2007 (pp. 255–266). Filosofia: Prague.

  39. 39.

    Wintein, S., & Muskens, R. (2016). A Gentzen calculus for nothing but the truth. Journal of Philosophical Logic, 45, 451–465.

    Article  Google Scholar 

  40. 40.

    Zaitsev, D. (2009). A few more useful 8-valued logics for reasoning with tetralattice E I G H T 4. Studia Logica, 92, 265–280.

    Article  Google Scholar 

Download references

Acknowledgments

We thank two anonymous reviewers for their valuable comments that greatly contributed to improving the final version of the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yaroslav Shramko.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shramko, Y., Zaitsev, D. & Belikov, A. The Fmla-Fmla Axiomatizations of the Exactly True and Non-falsity Logics and Some of Their Cousins. J Philos Logic 48, 787–808 (2019). https://doi.org/10.1007/s10992-018-9494-x

Download citation

Keywords

  • First-degree entailment
  • Exactly true logic
  • Non-falsity logic
  • Fmla-Fmla logical framework
  • Bi-consequence system