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Undecidability of the Problem of Recognizing Axiomatizations of Superintuitionistic Propositional Calculi

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Abstract

We give a new proof of the following result (originally due to Linial and Post): it is undecidable whether a given calculus, that is a finite set of propositional formulas together with the rules of modus ponens and substitution, axiomatizes the classical logic. Moreover, we prove the same for every superintuitionistic calculus. As a corollary, it is undecidable whether a given calculus is consistent, whether it is superintuitionistic, whether two given calculi have the same theorems, whether a given formula is derivable in a given calculus. The proof is by reduction from the undecidable halting problem for the so-called tag systems introduced by Post. We also give a historical survey of related results.

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References

  1. Ariola, Z. M., and H. Herbelin, Minimal classical logic and control operators, in J. C. M. Baeten, J. K. Lenstra, J. Parrow, and G. J. Woeginger (eds.), Proceedings of the 30th International Colloquium on Automata, Languages and Programming (ICALP 2003), Vol. 2719 of Lecture Notes in Computer Science, Springer, Berlin, 2003, pp. 871–885.

  2. Bokov, G. V., Completeness problem in the propositional calculus, Intellectual Systems 13(1–4):165–182, 2009, (Russian), http://www.intsys.msu.ru/magazine/archive/v13(1-4)/bokov-165-182.pdf, (A correction in 17(1-4):543–544, 2013).

  3. Chagrov, A, Undecidable properties of superintuitionistic logics, in S. V. Jablonskij, (ed.), Mathematical Problems of Cybernetics, Physmatlit, Moscow, Vol. 5, 1994, pp. 62–108, (Russian).

  4. Chagrov A., Zakharyaschev M.: The undecidability of the disjunction property of propositional logics and other related problems. The Journal of Symbolic Logic 58(3), 967–1002 (1993)

    Article  Google Scholar 

  5. Chagrov A., Zakharyaschev M.: Modal Logic, Vol. 35 of Oxford Logic Guides. Oxford University Press, New York (1997)

    Google Scholar 

  6. Chagrova L.A.: undecidable problem in correspondence theory. The Journal of Symbolic Logic 56(4), 1261–1272 (1991)

    Article  Google Scholar 

  7. Davis M., Computability and Unsolvability. McGraw-Hill, New York (1958)

    Google Scholar 

  8. Fel’dman, N. I., A. V. Kuznecov and J. I. Manin et al., Twelve Papers on Logic and Algebra, Vol. 59 of American Mathematical Society Translations, series 2, American Mathematical Society, USA, 1966.

  9. Gladstone M.D.: Some ways of constructing a propositional calculus of any required degree of unsolvability. Transactions of the American Mathematical Society 118, 192–210 (1965)

    Article  Google Scholar 

  10. Harrop R.: On the existence of finite models and decision procedures for propositional calculi. Mathematical Proceedings of the Cambridge Philosophical Society 54(1), 1–13 (1958)

    Article  Google Scholar 

  11. Harrop R.: A relativization procedure for propositional calculi, with an application to a generalized form of Post’s theorem. Proceedings of the London Mathematical Society s3 1(14), 595–617 (1964)

    Article  Google Scholar 

  12. Ihrig A.H.: Post–Linial theorems for arbitrary recursively enumerable degrees of unsolvability. Notre Dame Journal of Formal Logic 6, 54–72 (1965)

    Article  Google Scholar 

  13. Kuznetsov, A. V., Undecidability of the general problems of completeness, decidability and equivalence for propositional calculi, Algebra and Logic 2(4):47–66, (Russian), 1963.

    Google Scholar 

  14. Linial, S., and E. L. Post, Recursive unsolvability of the deducibility, Tarski’s completeness, and independence of axioms problems of propositional calculus, Bulletin of the American Mathematical Society 55:50, 1949, (Abstract).

    Google Scholar 

  15. Łukasiewicz, J., The shortest axiom of the implicational calculus of propositions, Proceedings of the Royal Irish Academy. Section A 52(3):25–33, 1948.

  16. Łukasiewicz, J., The shortest axiom of the implicational calculus of propositions, in L. Borowski, and J. Łukasiewicz, (eds.), Selected Works, North-Holland Publishing Company, London, 1970, pp. 295–305.

  17. Maksimova L.L.: Pretabular superintuitionist logic. Algebra and Logic 11(5), 308–314 (1972)

    Article  Google Scholar 

  18. Maksimova L.L.: Craig’s theorem in superintuitionistic logics and amalgamable varieties of pseudo-Boolean algebras. Algebra and Logic 16(6), 427–455 (1977)

    Article  Google Scholar 

  19. Marcinkowski, J., A Horn clause that implies an undecidable set of Horn clauses, in Selected papers of the 7th Workshop on Computer Science Logic (CSL ’93), 832:223–237, 1994.

  20. Meredith C.: A single axiom of positive logic. Journal of Computing Systems 1, 169–170 (1953)

    Google Scholar 

  21. Minsky M.L.: Recursive unsolvability of Post’s problem of “tag” and other topics in theory of Turing machines. Annals of Mathematics 74(3), 437–455 (1961)

    Article  Google Scholar 

  22. Minsky, M. L., Computation: Finite and Infinite Machines, Prentice-Hall, Englewood Cliffs, NJ, 1967.

  23. Post E.L.: Formal reduction of the general combinatorial decision problem. American Journal of Mathematics 65, 197–215 (1943)

    Article  Google Scholar 

  24. Shehtman, V. B., An undecidable superintuitionistic propositional calculus, Soviet Mathematics Doklady 19:656–660, 1978, (Russian).

  25. Shehtman, V. B., Undecidable propositional calculi, in Problems of Cybernetics. Non-Classical Logics and Their Applications, Vol. 75, Nauka, Moscow, 1982, pp. 74–116, (Russian).

  26. Singletary, W. E., A complex of problems proposed by Post, Bulletin of the American Mathematical Society (70)1:105–109, 1964, (A correction in 70(6):826).

  27. Singletary W.E.: A note on finite axiomatization of partial propositional calculi. The Journal of Symbolic Logic 32(3), 352–354 (1967)

    Article  Google Scholar 

  28. Singletary W.E.: Results regarding the axiomatization of partial propositional calculi. Notre Dame Journal of Formal Logic 9, 193–221 (1968)

    Article  Google Scholar 

  29. Urquhart, A., Review: V. B. Šehtman (translated by B. F. Wells), An undecidable superintuitionistic propositional calculus, The Journal of Symbolic Logic 50(4):1081–1083, 1985.

  30. Urquhart, A., Emil Post, in D. M. Gabbay, and J. Woods, (eds.), Handbook of the History of Logic. Vol. 5: Logic from Russell to Church, Elsevier, Amsterdam, 2009, pp. 617–666.

  31. Wang H.: Tag systems and lag systems. Mathematische Annalen 152(1), 65–74 (1963)

    Article  Google Scholar 

  32. Yntema M.K.: A detailed argument for the Post–Linial theorems. Notre Dame Journal of Formal Logic 5(1), 37–50 (1964)

    Article  Google Scholar 

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  1. Department of Mathematical Logic and Theory of Algorithms, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, Russia

    Evgeny Zolin

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  1. Evgeny Zolin
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Zolin, E. Undecidability of the Problem of Recognizing Axiomatizations of Superintuitionistic Propositional Calculi. Stud Logica 102, 1021–1039 (2014). https://doi.org/10.1007/s11225-013-9520-5

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