Logica Universalis

, Volume 11, Issue 4, pp 421–437 | Cite as

Admissibility in Positive Logics

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Abstract

The paper studies admissibility of multiple-conclusion rules in positive logics. Using modification of a method employed by M. Wajsberg in the proof of the separation theorem, it is shown that the problem of admissibility of multiple-conclusion rules in the positive logics is equivalent to the problem of admissibility in intermediate logics defined by positive additional axioms. Moreover, a multiple-conclusion rule \(\mathsf {r}\) follows from a set of multiple-conclusion rules \(\mathsf {R}\) over a positive logic \(\mathsf {P}\) if and only if \(\mathsf {r}\) follows from \(\mathsf {R}\) over \(\mathbf {Int}+ \mathsf {P}\).

Keywords

Inference rule multiple-conclusion rule admissible rule positive logic intermediate logic Brouwerian algebra 

Mathematics Subject Classification

Primary 03B55 Secondary 03B60 03C05 

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References

  1. 1.
    Belnap Jr., N.D., Leblanc, H., Thomason, R.H.: On not strengthening intuitionistic logic. Notre Dame J. Form. Log. 4, 313–320 (1963)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Béziau, J.Y.: Rules, derived rules, permissible rules and the various types of systems of deduction. In: PRATICA, Pontif. Univ. Católica Rio de Janeiro, Rio de Janeiro, pp. 159–184 (1999)Google Scholar
  3. 3.
    Birkhoff, G.: Lattice Theory. American Mathematical Society, New York (1948)MATHGoogle Scholar
  4. 4.
    Cintula, P., Metcalfe, G.: Admissible rules in the implication-negation fragment of intuitionistic logic. Ann. Pure Appl. Log. 162(2), 162–171 (2010)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Citkin, A.: On admissible rules of intuitionistic propositional logic. Math. USSR Sb. 31, 279–288 (1977)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Citkin, A.: On admissibility of rules in positive logic. In: IX All-Union Conference for Mathematical Logic, pp. 171, Nauka (1988) (in Russian)Google Scholar
  7. 7.
    de Jongh D, Zhao Z.: Positive formulas in intuitionistic and minimal logic. In: Logic, Language and Computation, vol. 8984 Lecture Notes in Computer Science, pp. 175–189. Springer, 2015. 10th International Tbilisi Symposium on Logic, Language, and Computation, TbiLLC 2013, Gudauri, Georgia, 23–27 September 2013, Revised Selected PapersGoogle Scholar
  8. 8.
    Gorbunov, V.A.: Algebraic Theory of Quasivarieties. Siberian School of Algebra and Logic Consultants Bureau, New York (1998). (in Russian)MATHGoogle Scholar
  9. 9.
    Goudsmit, J.: Intuitionistic Rules Admissible Rules of Intermediate Logics. PhD thesis, Utrech University (2015)Google Scholar
  10. 10.
    Goudsmit, J.P., Iemhoff, R.: On unification and admissible rules in Gabbay-de Jongh logics. Ann. Pure Appl. Log. 165(2), 652–672 (2014)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Harrop, R.: Concerning formulas of the types \(A\rightarrow B\bigvee C,\, A\rightarrow (Ex)B(x)\) in intuitionistic formal systems. J. Symb. Log. 25, 27–32 (1960)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Jankov, V.A.: Calculus of the weak law of the excluded middle. Izv. Akad. Nauk SSSR Ser. Mat. 32(5), 1044–1051 (1968)MathSciNetGoogle Scholar
  13. 13.
    Jeřábek, E.: Independent bases of admissible rules. Log. J. IGPL 16(3), 249–267 (2008)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Jeřábek, E.: Canonical rules. J. Symb. Log. 74(4), 1171–1205 (2009)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Jeřábek, E.: Proof complexity of intuitionistic implicational formulas. Ann. Pure Appl. Log. 168(1), 150–190 (2017)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Köhler, P.: Varieties of Brouwerian algebras. Math. Semin. Giess. 116, iii+83 (1975)MATHMathSciNetGoogle Scholar
  17. 17.
    Kracht, M.: Book review of [30]. Notre Dame J. Form. Log. 40(4), 578–587 (1999)CrossRefGoogle Scholar
  18. 18.
    Kracht, M.: Modal consequence relations. In: Blackburn, P., et al. (eds.) Handbook of Modal Logic, pp. 491–545. Elsevier, Amsterdam (2007)CrossRefGoogle Scholar
  19. 19.
    Lorenzen, P.: Einführung in die Operative Logik und Mathematik. Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit Besonderer Berücksichtigung der Anwendungsgebiete, vol. LXXVIII. Springer, Berlin (1955)Google Scholar
  20. 20.
    Lorenzen, P.: Protologik. Ein Beitrag zum Begrndungsproblem der Logik. Kant-Studien, 47(1–4), 350–358 (1956) (Translated in Lorenzen , P.: Constructive Philosophy, pp. 59–70. Univerisity of Massachusettes Press, Amherst (1987)Google Scholar
  21. 21.
    Mints, G.E.: Derivability of admissible rules. J. Sov. Math. 6, 417–421 (1976) [Translated from Mints, G.E.: Derivability of admissible rules. (Russian) Investigations in constructive mathematics and mathematical logic, V. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 32 (1972), pp. 85 - 89Google Scholar
  22. 22.
    Monteiro, A.: Axiomes indépendants pour les algèbres de Brouwer. Rev. Un. Mat. Argent. 17(149–160), 1955 (1956)MATHGoogle Scholar
  23. 23.
    Novikov, P.S.: Constructive mathematical logic from the point of view of classical logic. In: Adjan, SI (ed.) Monographs in Mathematical Logic and Foundations of Mathematics (in Russian)Google Scholar
  24. 24.
    Odintsov, S., Rybakov, V.: Unification and admissible rules for paraconsistent minimal Johanssons’ logic \({\bf J}\) and positive intuitionistic logic \({\bf IPC}^+\). Ann. Pure Appl. Log. 164(7–8), 771–784 (2013)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Odintsov, S.P.: Constructive Negations and Paraconsistency. Trends in Logic-Studia Logica Library, vol. 26. Springer, New York (2008)CrossRefMATHGoogle Scholar
  26. 26.
    Prucnal, T.: On the structural completeness of some pure implicational propositional calculi. Stud. Log. 30, 45–52 (1972)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Rasiowa, H.: An algebraic approach to non-classical logics. In: Studies in Logic and the Foundations of Mathematics, vol. 78, North-Holland Publishing Co., Amsterdam (1974)Google Scholar
  28. 28.
    Rybakov, V.V.: A criterion for admissibility of rules in the modal system \({\rm S}4\) and intuitionistic logic. Algebra Log. 23(5), 546–572 (1984). 600CrossRefMathSciNetGoogle Scholar
  29. 29.
    Rybakov, V.V.: Bases of admissible rules of the logics S4 and Int. Algebra Log. 24(1), 87–107 (1985). 123CrossRefMATHGoogle Scholar
  30. 30.
    Rybakov, V.V.: Admissibility of Logical Inference Rules Volume 136 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam (1997)Google Scholar
  31. 31.
    Verhozina, M.: Intermediate positive logics. In: Algorithmic Problems of Algebraic Systems, pp. 13–25. Irkutsk State University, Irkutsk (1978) (in Russian)Google Scholar
  32. 32.
    Wajsberg, M.: On A. Heyting’s propositional calculus. In: Waysberg, M. (ed.) Logical Works Warszava, pp. 132–171 (1977) [Translated from: Untersuchungen uber den Aussagenkalkul von A. Heyting. Wiadomości matematyczne 46, 45–101 (1938)Google Scholar
  33. 33.
    Wroński, A.: On the degree of completeness of positive logic. Pol. Acad. Sci. Inst. Philos. Sociol. Bull. Sect. Log. 2(1), 65–70 (1973)MathSciNetGoogle Scholar
  34. 34.
    Wroński, A.: On reducts of intermediate logics. Pol. Acad. Sci. Inst. Philos. Sociol. Bull. Sect. Log. 9(4), 176–179 (1980)MATHMathSciNetGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Metropolitan TelecommunicationsNew YorkUSA

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