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.


03B45 Modal logic (including the logic of norms)
03B25 Decidability of theories and sets of sentences
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