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A note on admissible rules and the disjunction property in intermediate logics. (English) Zbl 1248.03047
Summary: 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.

##### MSC:
 03B55 Intermediate logics 06D20 Heyting algebras (lattice-theoretic aspects) 08C15 Quasivarieties
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##### References:
 [1] 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 · Zbl 0964.06008 [2] Chagrov A., Zakharyaschev M.: Modal Logic, Oxford Logic Guides. The Clarendon Press Oxford University Press, Oxford Science Publications, New York (1997) · Zbl 0871.03007 [3] Citkin A.: On admissible rules of intuitionistic propositional logic. Math. USSR, Sb 31, 279–288 (1977) (A. Tsitkin) · Zbl 0386.03011 [4] Citkin A.: On structurally complete superintuitionistic logics. Sov. Math. Dokl. 19, 816–819 (1978) (A. Tsitkin) · Zbl 0412.03009 [5] Iemhoff R.: On the admissible rules of intuitionistic propositional logic. J. Symbol. Log. 66(1), 281–294 (2001) · Zbl 0986.03013 [6] Iemhoff R.: Intermediate logics and Visser’s rules. Notre. Dame. J. Formal. Log. 46(1), 65–81 (2005) · Zbl 1102.03032 [7] Iemhoff R.: On the rules of intermediate logics. Arch. Math. Log. 45(5), 581–599 (2006) · Zbl 1096.03025 [8] Kleene S.C.: Introduction to Metamathematics. D. Van Nostrand Co, New York (1952) · Zbl 0047.00703 [9] 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 · Zbl 0358.02031 [10] Roziére, P.: Régles admissibles en calcul propositionnel intuitionniste. PhD thesis, Université Paris VII (1992) [11] Rybakov V.V.: Admissibility of Logical Inference Rules, Studies in Logic and the Foundations of Mathematics. North-Holland Publishing, Amsterdam (1997) · Zbl 0872.03002 [12] 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) [13] Shoesmith, D., Smiley, T.: Multiple-conclusion logic. Reprint of the 1978 hardback ed. Cambridge: Cambridge University Press. xiii, 396 p (2009) · Zbl 0381.03001
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