Abstract
With any structural inference rule A/B, we associate the rule \({(A \lor p)/(B \lor p)}\) , providing that formulas A and B do not contain the variable p. We call the latter rule a join-extension (\({\lor}\) -extension, for short) of the former. Obviously, for any intermediate logic with disjunction property, a \({\lor}\) -extension of any admissible rule is also admissible in this logic. We investigate intermediate logics, in which the \({\lor}\) -extension of each admissible rule is admissible. We prove that any structural finitary consequence operator (for intermediate logic) can be defined by a set of \({\lor}\) -extended rules if and only if it can be defined through a set of well-connected Heyting algebras of a corresponding quasivariety. As we exemplify, the latter condition is satisfied for a broad class of algebraizable logics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bezhanishvili, G.: Varieties of monadic Heyting algebras. I. Studia Logica 61(3), 367–402 (1998) doi:10.1023/A:1005073905902, http://dx.doi.org/10.1023/A:1005073905902
Chagrov A., Zakharyaschev M.: Modal Logic, Oxford Logic Guides. The Clarendon Press Oxford University Press, Oxford Science Publications, New York (1997)
Citkin A.: On admissible rules of intuitionistic propositional logic. Math. USSR, Sb 31, 279–288 (1977) (A. Tsitkin)
Citkin A.: On structurally complete superintuitionistic logics. Sov. Math. Dokl. 19, 816–819 (1978) (A. Tsitkin)
Iemhoff R.: On the admissible rules of intuitionistic propositional logic. J. Symbol. Log. 66(1), 281–294 (2001)
Iemhoff R.: Intermediate logics and Visser’s rules. Notre. Dame. J. Formal. Log. 46(1), 65–81 (2005)
Iemhoff R.: On the rules of intermediate logics. Arch. Math. Log. 45(5), 581–599 (2006)
Kleene S.C.: Introduction to Metamathematics. D. Van Nostrand Co, New York (1952)
Mints, G.E.: Derivability of admissible rules. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 32, 85–89, 156 (1972) Investigations in constructive mathematics and mathematical logic, V
Roziére, P.: Régles admissibles en calcul propositionnel intuitionniste. PhD thesis, Université Paris VII (1992)
Rybakov V.V.: Admissibility of Logical Inference Rules, Studies in Logic and the Foundations of Mathematics. North-Holland Publishing, Amsterdam (1997)
Scott, D.: Completeness and axiomatizability in many-valued logic. In: Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., vol. XXV, Univ. California, Berkeley, Calif., 1971), Am. Math. Soc., Providence, R.I., pp. 411–435 (1974)
Shoesmith, D., Smiley, T.: Multiple-conclusion logic. Reprint of the 1978 hardback ed. Cambridge: Cambridge University Press. xiii, 396 p (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Citkin, A. A note on admissible rules and the disjunction property in intermediate logics. Arch. Math. Logic 51, 1–14 (2012). https://doi.org/10.1007/s00153-011-0250-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-011-0250-y