Natural Deduction for Post’s Logics and their Duals

Abstract

In this paper, we introduce the notion of dual Post’s negation and an infinite class of Dual Post’s finitely-valued logics which differ from Post’s ones with respect to the definitions of negation and the sets of designated truth values. We present adequate natural deduction systems for all Post’s k-valued (\(k\geqslant 3\)) logics as well as for all Dual Post’s k-valued logics.

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References

  1. 1.

    Bolotov, A., Shangin, V.: Natural deduction system in paraconsistent setting: proof search for PCont. J. Intell. Syst. 21, 1–24 (2012)

    Article  Google Scholar 

  2. 2.

    Copi, I.M., Cohen, C., McMahon, K.: Introduction to Logic, 14th edn. Routledge, London (2011)

    Google Scholar 

  3. 3.

    Dwinger, P.: Notes on post algebras. Indag. Math. 28, 462–478 (1966)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Dwinger, P.: Generalized post algebras. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16, 559–563 (1968)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Dwinger, P.: A survey of the theory of Post algebras and their generalizations. In: Dunn, J.M., Epstein, G. (eds.) Modern Uses of Multiple-Valued Logic, pp. 53–75. Reidel, Dordrecht (1977)

    Google Scholar 

  6. 6.

    Epstein, G., Rasiowa, H.: Theory and uses of Post algebras of order \( \omega +\omega ^\ast \). Part II. In: Proceedings of 21st International Symposium on Multiple-Valued Logic, Victoria/B.C., 1991, IEEE Computer Society, New York, pp. 248–254 (1991)

  7. 7.

    Epstein, G., Rasiowa, H.: Theory and uses of post algebras of order \( \omega +\omega ^\ast \) I. In: Proceedings of 20th International Symposium on Multiple-Valued Logic, Charlotte/NC, 1990, IEEE Computer Society, New York, pp. 42–47 (1990)

  8. 8.

    Epstein, G.: The lattice theory of post algebras. Trans. Am. Math. Soc. 95, 300–317 (1960)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Gottwald, S.: A Treatise on Many-Valued Logics. Research Studies Press, Baldock (2001)

    Google Scholar 

  10. 10.

    Hazen, A., Pelletier, F.J.: Gentzen and Jaśkowski natural deduction: fundamentally similar but importantly different. Stud. Log. 102, 1103–1142 (2014)

    Article  MATH  Google Scholar 

  11. 11.

    Kirin, V.: Gentzen’s method for the many-valued propositional calculi. Z. Math. Log. Grundl. Math. 12, 317–332 (1966)

    Article  MATH  Google Scholar 

  12. 12.

    Kirin, V.: Post algebras as semantic bases of some many-valued logics. Fund. Math. 63, 279–294 (1968)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Kooi, B., Tamminga, A.: Completeness via correspondence for extensions of the logic of paradox. Rev. Symb. Logic 5, 720–730 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Petrukhin, Y., Shangin, V.: Automated correspondence analysis for the binary extensions of the logic of paradox. Rev. Symb. Logic 10, 756–781 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Post, E.: Introduction to a general theory of elementary propositions. Am. J. Math. 43, 163–185 (1921)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Rasiowa, H.: On generalised Post algebras of order \( \omega ^+ \) and \( \omega ^+ \)-valued predicate calculi. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21, 209–219 (1973)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Rasiowa, H.: An Algebraic Approach to Non-Classical Logics. PWN, Warsaw and North-Holland Publishing Company, Amsterdam (1974)

    Google Scholar 

  18. 18.

    Rasiowa, H.: Post algebras as a semantic foundation of m-valued logic. Stud. Math. 9, 92–142 (1974)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Rescher, N.: Many-Valued Logic. McGraw Hill, New York (1969)

    Google Scholar 

  20. 20.

    Rosenberg, I.: The number of maximal closed classes in the set of functions over a finite domain. J. Comb. Theory 14, 1–7 (1973)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Rosenbloom, P.C.: Post algebras I, Postulates and general theory. Am. J. Math. 64, 167–188 (1942)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Rousseau, G.: Sequents in many-valued logic I. Fund. Math. 60, 23–33 (1967)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Rousseau, G.: Logical systems with finitely many truth-values. Bull. Polon. Sci., Sér. Sci. Math. Astronom. Phys. 17, 189–194 (1969)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Rousseau, G.: Post algebras and pseudo-post algebras. Fund. Math. 67, 133–145 (1970)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Rousseau, G.: Sequents in many-valued logic II. Fund. Math. 67, 125–131 (1970)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Saloni, Z.: The sequent Gentzen system for m-valued logic. Bull. Sect. Logic 2, 30–37 (1973)

    MathSciNet  Google Scholar 

  27. 27.

    Tamminga, A.: Correspondence analysis for strong three-valued logic. Log. Investig. 20, 255–268 (2014)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Traczyk, T.: Axioms and some properties of Post algebras. Colloq. Math. 10, 193–209 (1963)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Traczyk, T.: An equational definition of a class of post algebras. Bull. Acad. Polon. Sci. 12, 147–149 (1964)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Traczyk, T.: On Post algebras with uncountable chain of constants. Algebras and homomorphisms. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 15, 673–680 (1967)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

My special thanks go to Dmitry Zaitsev, Andrzej Pietruszczak, and Mateusz Klonowski. I also extend my thanks to an anonymous referee for valuable remarks.

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Correspondence to Yaroslav Petrukhin.

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To Sabrina Skoryukina

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Petrukhin, Y. Natural Deduction for Post’s Logics and their Duals. Log. Univers. 12, 83–100 (2018). https://doi.org/10.1007/s11787-018-0190-y

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Mathematics Subject Classification

  • Primary 03B50
  • Secondary 03B22
  • 03F03

Keywords

  • Natural deduction
  • Post’s logic
  • Post’s negation
  • cyclic negation
  • Dual Post’s logic
  • many-valued logic
  • proof theory