Skip to main content
SpringerLink
Log in

Multiple Conclusion Rules in Logics with the Disjunction Property

  • Conference paper
  • First Online:
Logical Foundations of Computer Science (LFCS 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9537))

Included in the following conference series:

Abstract

We prove that for the intermediate logics with the disjunction property any basis of admissible rules can be reduced to a basis of admissible m-rules (multiple-conclusion rules), and every basis of admissible m-rules can be reduced to a basis of admissible rules. These results can be generalized to a broad class of logics including positive logic and its extensions, Johansson logic, normal extensions of \(\mathsf {S4}\), n-transitive logics and intuitionistic modal logics.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    This book was published in 1977, but it is based on the notes of a course that P.S. Novikov taught in 1950th; A.V. Kuznetsov was recalling that P.S. Novikov had used the notion of derivable rule much earlier, in this lectures in 1940th.

  2. 2.

    Using idea from [8], it is not hard to show that if an intermediate logic has a recursively enumerable explicit basis of admissible rules, it has a recursive basis.

  3. 3.

    In [6] A. Chagrov has constructed a decidable modal logic having undecidable admissibility problem, and gave a negative answer to V.Rybakov’s question [35, Problem (1)]. The problem whether there exists a decidable intermediate logic with undecidable admissibility problem remains open.

References

  1. Belnap Jr., N.D., Leblanc, H., Thomason, R.H.: On not strengthening intuitionistic logic. Notre Dame J. Formal Log. 4, 313–320 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bezhanishvili, G.: Varieties of monadic heyting algebras. I. Stud. Logica 61(3), 367–402 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cabrer, L., Metcalfe, G.: Admissibility via natural dualities. J. Pure Appl. Algebra 219(9), 4229–4253 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chagrov, A., Zakharyaschev, M.: Modal Logic. Oxford Logic Guides. The Clarendon Press, vol. 35. Oxford University Press, Oxford Science Publications, New York (1997)

    MATH  Google Scholar 

  5. Chagrov, A., Zakharyashchev, M.: The disjunction property of intermediate propositional logics. Stud. Log. 50(2), 189–216 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chagrov, A.V.: A decidable modal logic for which the admissibility of inference rules is an undecidable problem. Algebra i Logika 31(1), 83–93 (1992)

    Article  MathSciNet  Google Scholar 

  7. Citkin, A.: A note on admissible rules and the disjunction property in intermediate logics. Arch. Math. Logic 51(1–2), 1–14 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Craig, W.: On axiomatizability within a system. J. Symb. Log. 18, 30–32 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fedorishin, B.R.: An explicit basis for the admissible inference rules in the Gödel-Löb logic GL. Sibirsk. Mat. Zh. 48, 423–430 (2007). (In Russian)

    MathSciNet  MATH  Google Scholar 

  10. Friedman, H.: One hundred and two problems in mathematical logic. J. Symb. Log. 40, 113–129 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  11. Goudsmit, J.: A note on extensions: admissible rules via semantics. In: Artemov, S., Nerode, A. (eds.) LFCS 2013. LNCS, vol. 7734, pp. 206–218. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  12. Goudsmit, J.: Intuitionistic Rules Admissible Rules of Intermediate Logics. Ph.D. thesis, Utrech University (2015)

    Google Scholar 

  13. Goudsmit, J.P., Iemhoff, R.: On unification and admissible rules in Gabbay-de Jongh logics. Ann. Pure Appl. Log. 165(2), 652–672 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Iemhoff, R.: On the admissible rules of intuitionistic propositional logic. J. Symb. Log. 66(1), 281–294 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Iemhoff, R.: Intermediate logics and Visser’s rules. Notre Dame J. Formal Log. 46(1), 65–81 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Iemhoff, R.: On the rules of intermediate logics. Arch. Math. Log. 45(5), 581–599 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jeřábek, E.: Admissible rules of modal logics. J. Log. Comput. 15(4), 411–431 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jeřábek, E.: Independent bases of admissible rules. Log. J. IGPL 16(3), 249–267 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jeřábek, E.: Canonical rules. J. Symb. Log. 74(4), 1171–1205 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kleene, S.C.: Introduction to Metamathematics. D. Van Nostrand Co. Inc., New York (1952)

    MATH  Google Scholar 

  22. Kracht, M.: Book review of [37]. Notre Dame J. Form. Log. 40(4), 578–587 (1999)

    Article  MathSciNet  Google Scholar 

  23. Kracht, M.: Modal consequence relations, chap. 8. In: Blackburn, P., et al. (eds.) Handbook of Modal Logic. Studies in Logic and Practical Reasonong, vol. 3, pp. 491–545. Elsevier, New York (2007)

    Chapter  Google Scholar 

  24. Lorenzen, P.: Einführung in die operative Logik und Mathematik. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd. LXXVIII. Springer, Berlin-Göttingen-Heidelberg (1955)

    Google Scholar 

  25. Lorenzen, P.: Protologik. Ein Beitrag zum Begrndungsproblem der Logik. Kant-Studien 47, 1–4, Januay 1956, pp. 350–358. Translated in Lorenzen, P., Constructive Philosophy, 59–70. Univerisity of Massachusettes Press, Amherst (1987)

    Google Scholar 

  26. Mints, G.: Derivability of admissible rules. J. Sov. Math. 6 (1976), 417–421. 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, pp. 85–89 (1972)

    Google Scholar 

  27. Muravitsky, A.: Logic KM: a biography. In: Bezhanishvili, G. (ed.) Leo Esakia on Duality in Modal and Intuitionistic Logics. Outstanding Contributions to Logic, vol. 4, pp. 155–185. Springer, Netherlands (2014)

    Google Scholar 

  28. Novikov, P. S.: Konstruktivnaya matematicheskaya logika s tochki zreniya klassicheskoi [Constructive mathematical logic from the point of view of classical logic]. Izdat. “Nauka”, Moscow (1977) With a preface by S. I. Adjan, Matematicheskaya Logika i Osnovaniya Matematiki. [Monographs in Mathematical Logic and Foundations of Mathematics] (in Russian)

    Google Scholar 

  29. Odintsov, S., Rybakov, V.: Unification and admissible rules for paraconsistent minimal Johanssons’ logic \(\mathbf{J}\) and positive intuitionistic logic \(\mathbf{IPC}^+\). Ann. Pure Appl. Logic 164(7–8), 771–784 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Pogorzelski, W.A.: Structural completeness of the propositional calculus. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19, 349–351 (1971)

    MathSciNet  MATH  Google Scholar 

  31. Prucnal, T.: Structural completeness of Medvedev’s propositional calculus. Rep. Math. Logic 6, 103–105 (1976)

    MathSciNet  MATH  Google Scholar 

  32. Roziére, P.: Régles admissibles en calcul propositionnel intuitionniste. Ph.D. thesis, Université Paris VII (1992)

    Google Scholar 

  33. Rybakov, V.V.: A criterion for admissibility of rules in the modal system S4 and intuitionistic logic. Algebra i Logika 23(5), 546–572 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rybakov, V. V.: Bases of admissible rules of the modal system Grz and intuitionistic logic. Mat. Sb. (N.S.) vol. 128(170), 3, pp. 321–338, 446 (1985)

    Google Scholar 

  35. Rybakov, V.V.: Problems of admissibility and substitution, logical equations and restricted theories of free algebras. In: Logic, methodology and philosophy of science, VIII (Moscow, 1987), vol. 126. Stud. Logic Found. Math., pp. 121–139. North-Holland, Amsterdam (1989)

    Google Scholar 

  36. Rybakov, V.V.: Admissibility of rules of inference, and logical equations in modal logics that axiomatize provability. Izv. Akad. Nauk SSSR Ser. Mat. 54, 357–377 (1990). (In Russian)

    Google Scholar 

  37. Rybakov, V.V.: Admissibility of Logical Inference Rules. Studies in Logic and the Foundations of Mathematics, vol. 136. North-Holland Publishing Co., Amsterdam (1997)

    Book  MATH  Google Scholar 

  38. Rybakov, V.V., Terziler, M., Gencer, C.: Unification and passive inference rules for modal logics. J. Appl. Non-Class. Log. 10(3–4), 369–377 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rybakov, V.V.: Construction of an explicit basis for rules admissible in modal system S4. MLQ Math. Log. Q. 47(4), 441–446 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Metropolitan Telecommunications, New York, NY, 10041, USA

    Alex Citkin

Authors
  1. Alex Citkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alex Citkin .

Editor information

Editors and Affiliations

  1. City University of New York , New York, New York, USA

    Sergei Artemov

  2. Cornell University , Ithaca, New York, USA

    Anil Nerode

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Citkin, A. (2016). Multiple Conclusion Rules in Logics with the Disjunction Property. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2016. Lecture Notes in Computer Science(), vol 9537. Springer, Cham. https://doi.org/10.1007/978-3-319-27683-0_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-27683-0_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-27682-3

  • Online ISBN: 978-3-319-27683-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation