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Tutorial on Admissible Rules in Gudauri

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Logic, Language, and Computation (TbiLLC 2013)

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Abstract

Most theorems have more than one proof and most theories have more than one axiomatization. Certain proofs or axiomatizations are preferable to others because they are shorter or more transparent or for some other reason. Our aim is to describe or study the possible proofs of a theorem or the possible axiomatizations of a theory. As the former is a special instance of the latter, by considering a theory consisting of one theorem, it suffices to consider theories.

R. Iemhoff—Support by the Netherlands Organisation for Scientific Research under grant 639.032.918 is gratefully acknowledged.

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References

  1. Avron, A.: Simple consequence relations. Inf. Comput. 92(1), 105–139 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  2. Babenyshev, S., Rybakov, V.V.: Linear temporal logic LTL: basis for admissible rules. J. Log. Comput. 21(2), 157–177 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chagrov, A.: A decidable modal logic with undecidable admissibility problem. Algebra Log. 31(1), 53–61 (1992). In Russian

    Article  MATH  MathSciNet  Google Scholar 

  4. Cintula, P., Metcalfe, G.: Admissible rules in the implication-negation fragment of intuitionistic logic. Ann. Pure Appl. Log. 162(2), 162–171 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dzik, W., Wroński, A.: Structural completeness of Gödel and Dummett’s propositional calculi. Stud. Log. 32, 69–73 (1973)

    Article  MATH  Google Scholar 

  6. Ghilardi, S.: Unification in intuitionistic logic. J. Symb. Log. 64(2), 859–880 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ghilardi, S.: A resolution/tableaux algorithm for projective approximations in IPC. Log. J. IGPL 10(3), 227–241 (2002)

    Article  MathSciNet  Google Scholar 

  8. Ghilardi, S.: Unification, finite duality and projectivity in varieties of Heyting algebras. Ann. Pure Appl. Log. 127(1–3), 99–115 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Iemhoff, R., Metcalfe, G.: Proof theory for admissible rules. Ann. Pure Appl. Log. 159(1–2), 171–186 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Iemhoff, R., Metcalfe, G.: Hypersequent systems for the admissible rules of modal and intermediate logics. In: Artemov, S., Nerode, A. (eds.) LFCS 2009. LNCS, vol. 5407, pp. 230–245. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  13. Iemhoff, R., Rozière, P.: Unification in fragments of intermediate logics. J. Symb. Logic (to appear)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  15. Jeřábek, E.: Complexity of admissible rules. Arch. Math. Log. 46, 73–92 (2007)

    Article  MATH  Google Scholar 

  16. Jeřábek, E.: Blending margins: the modal logic K has nullary unification type. J. Logic Comput. (to appear)

    Google Scholar 

  17. Mints, G.: Derivability of admissible rules. In: Studies in Constructive Mathematics and Mathematical Logic. Zap. Nauchn. Sem. LOMI, part V, vol. 32, pp. 85–89. Nauka, Leningrad (1972)

    Google Scholar 

  18. Odintsov, S., Rybakov, V.V.: Unification and admissible rules for paraconsistent minimal Johansson’s logic J and positive intuitionistic logic IPC\(^+\). Ann. Pure Appl. Log. 164(7–8), 771–784 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  19. Pogorzelski, W.A.: Structural completeness of the propositional calculus. Bulletin de l’Académie Polonaise des Sciences, Série des sciences mathématiques, astronomiques et physiques 19, 349–351 (1971)

    MATH  MathSciNet  Google Scholar 

  20. Pogorzelski, W.A., Prucnal, T.: Structural completeness of the first-order predicate calculus. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 21(1), 315–320 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  21. Prucnal, T.: On the structural completeness of some pure implicational propositional calculi. Stud. Log. 32(1), 45–50 (1973)

    MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  23. Rozière, P.: Règles admissibles en calcul propositionnel intuitionniste. Ph.D. thesis, Université Paris VII (1992)

    Google Scholar 

  24. Rybakov, V.: Admissibility of Logical Inference Rules. Elsevier, Amsterdam (1997)

    MATH  Google Scholar 

  25. Rybakov, V.: Rules admissible in transitive temporal logic \(T_{S4}\), sufficient condition. Theoret. Comput. Sci. 411(50), 4323–4332 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  26. Rybakov, V.: Writing out unifiers in linear temporal logic. J. Log. Comput. 22(5), 1199–1206 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  27. Skura, R.: A complete syntactical characterization of the intuitionistic logic. Rep. Math. Log. 23, 75–80 (1989)

    MATH  MathSciNet  Google Scholar 

  28. Slaney, J., Meyer, R.: A structurally complete fragment of relevant logic. Notre Dame J. Formal Log. 33(4), 561–566 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  29. Słomczyńska, K.: Algebraic semantics for the \((\leftrightarrow,\lnot \lnot )\)-fragment of IPC. Math. Log. Q. 58(12), 29–37 (2012)

    MATH  Google Scholar 

  30. Visser, A.: Rules and arithmetics. Notre Dame J. Formal Log. 40(1), 116–140 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  31. Visser, A.: Substitutions of \(\Sigma _1^0\)-sentences: explorations between intuitionistic propositional logic and intuitionistic arithmetic. Ann. Pure Appl. Log. 114(1–3), 227–271 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  32. Williamson, T.: An alternative rule of disjunction in modal logic. Notre Dame J. Formal Log. 33(1), 89–100 (1992)

    Article  MATH  Google Scholar 

  33. Wojtylak, P.: On a problem of H. Friedman and its solution by T. Prucnal. Rep. Math. Log. 38, 69–86 (2004)

    MATH  Google Scholar 

  34. Wolter, F., Zakharyaschev, M.: Undecidability of the unification and admissibility problems for modal and description logics. ACM Trans. Comput. Log. 9(4), 1–20 (2008). Article 25

    Article  MathSciNet  Google Scholar 

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Iemhoff, R. (2015). Tutorial on Admissible Rules in Gudauri. In: Aher, M., Hole, D., Jeřábek, E., Kupke, C. (eds) Logic, Language, and Computation. TbiLLC 2013. Lecture Notes in Computer Science(), vol 8984. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46906-4_2

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  • DOI: https://doi.org/10.1007/978-3-662-46906-4_2

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