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Decidable modal logic with undecidable admissibility problem. (English. Russian original) Zbl 0782.03005
Algebra Logic 31, No. 1, 53-61 (1992); translation from Algebra Logika 31, No. 1, 83-93 (1992).
The admissibility problem for a given logic \(L\) is to determine whether an arbitrary given inference rule \(A_ 1(p_ 1,\dots,p_ n),\dots,A_ m(p_ 1,\dots,p_ n)/B(p_ 1,\dots,p_ n)\) is admissible in \(L\), i.e., for all formulas \(C_ 1,\dots,C_ n\), \(B(C_ 1,\dots,C_ n)\in L\) whenever \(A_ 1(C_ 1,\dots,C_ n)\in L,\dots,A_ m(C_ 1,\dots,C_ n)\in L\).
As is known, V. Rybakov proved the decidability of the admissibility problem for a number of intermediate and modal logics.
In this paper, the author constructs a decidable normal modal logic for which the admissibility problem is undecidable. The logic is an extension of K4 of width 3 and has infinitely many axioms.

MSC:
03B45 Modal logic (including the logic of norms)
03B25 Decidability of theories and sets of sentences
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References:
[1] V. V. Rybakov, ”Problems of admissibility and substitution, logical equations and restricted theories of free algebras,” in: Logic, Methodology and Philosophy of Science VIII, Elsevier (1989), pp. 121–139. · Zbl 0691.03012
[2] V. V. Rybakov, ”On admissibility of inference rules in modal logicG,” Tr. Inst. Mat. Sib. Otd. Akad. Nauk SSSR,12, 120–138, (1989).
[3] V. V. Rybakov, ”Equations in free closure algebras and the substitution problem,” Dokl. Akad. Nauk SSSR,287, No. 3, 554–557 (1986).
[4] G. D. Birkhoff, Lattice Theory, Amer. Math. Soc. (1979).
[5] K. Fine, ”Logics containing K4, Part I,” J. Symb. Logic,39, No. 1, 31–42 (1974). · Zbl 0287.02010
[6] A. I. Mal’tsev, ”Identical relations on quasigroup varieties,” Mat. Sb.,69, No. 1, 3–12 (1966).
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