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On the independent axiomatizability of modal and intermediate logics. (English) Zbl 0856.03017
The paper solves in the negative the problem of the existence of independent axiomatizations for modal and intermediate propositional logics (formulated by A. Tsytkin in 1986). The main idea is to use the following necessary condition: if a logic \(L\) is independently axiomatizable then every interval of logics \([L_1, L]\) with \(L_1\) finitely axiomatizable, contains an immediate predecessor of \(L\). A related property studied by W. J. Blok [Algebra Univers. 11, 285-294 (1980; Zbl 0457.08003)] is the strong coatomicity of a lattice of logics; it is the same as above, but \(L_1\) can be arbitrary. Blok proved that the lattice of normal modal logics is not strongly coatomic, but, as the authors observe, it seems unlikely that in his counterexample \(L_1\) is finitely axiomatizable.
The reviewed paper disproves independent axiomatizability for modal logics above K4 and for intermediate logics via differentiated (general) Kripke frames. For the intermediate case “Fine’s ladder” is used. The same construction yields also counterexamples above Grz and in the intervals [S4,S5] and [S4,Grz].

MSC:
03B45 Modal logic (including the logic of norms)
03B55 Intermediate logics
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