On the interpolation property of some intuitionistic modal logics. (English) Zbl 0844.03008
Summary: Let \({\mathbf L}\) be one of the intuitionistic modal logics considered by H. Ono [Publ. Res. Inst. Math. Sci., Kyoto Univ. 13, 687-722 (1977; Zbl 0373.02026)] (or one of its extensions) and let \(M_{{\mathbf L}}\) be the “algebraic semantics” of \({\mathbf L}\). In this paper, we extend to \({\mathbf L}\) the equivalence, proved in the classical case [see L. L. Maksimova, Algebra Logika 18, 556-586 (1979; Zbl 0436.03011)], among the weak Craig interpolation theorem, the Robinson theorem and the amalgamation property of variety \(M_{{\mathbf L}}\). We also prove the equivalence between the Craig interpolation theorem and the super-amalgamation property of variety \(M_{{\mathbf L}}\). Then we obtain the Craig interpolation theorem and Robinson theorem for two intuitionistic modal logics, one of \(S_4\)-type and the other one of \(S_5\)-type, showing the super-amalgamation property of the corresponding algebraic semantics.
MSC:
| 03B45 | Modal logic (including the logic of norms) |
| 03G25 | Other algebras related to logic |
| 03C40 | Interpolation, preservation, definability |
Keywords:
topological pseudoboolean algebras; intuitionistic modal logics; weak Craig interpolation theorem; Robinson theorem; amalgamation property; variety; super-amalgamation property; algebraic semanticsReferences:
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