## Fusions of modal logics revisited (1998)

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Venue: | In Advances in modal logic |

Citations: | 44 - 7 self |

### BibTeX

@INPROCEEDINGS{Wolter98fusionsof,

author = {Frank Wolter},

title = {Fusions of modal logics revisited},

booktitle = {In Advances in modal logic},

year = {1998},

pages = {361--379},

publisher = {CSLI}

}

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### Abstract

The fusion Ll Lr of two normal modal logics formulated in languages with disjoint sets of modal operators is the smallest normal modal logic containing Ll [ Lr. This paper proves that decidability, interpolation, uniform interpolation, and Halldencompleteness are preserved under forming fusions of normal polyadic polymodal logics. Those problems remained open in [Fine & Schurz [3]] and [Kracht &Wolter [10]]. The paper de nes the fusion `l `r of two classical modal consequence relations and proves that decidability transfers also in this case. Finally, these results are used to prove a general decidability result for modal logics based on superintuitionistic logics. Given two logical system L1 and L2 it is natural to ask whether the fusion (or join) L1 L2 of them inherits the common properties of both L1 and L2. Let us consider some examples: (i) It is known that the rst order theory of one equivalence relation has the nite model property and is decidable. However, the rst order theory of two equivalence relations does not have the nite model property and is in fact undecidable (see Janiczak [7]). This result shows that even if we know the rst order properties of the individual relations of a theory, there may be no algorithm to determine the purely logical consequences of these properties. (ii) Various positive and negative results are known for joins of term rewriting systems (TRSs) whose vocabularies are disjoint. For example, the join of two TRSs is con uent i the two TRSs are con uent but there are complete TRSs whose join is not complete (see e.g. Klop [8]). In fact, the literature on TRSs shows how useful the study of joins of systems can be. (iii) In contrast to rst order theories the join of two decidable equational theories in disjoint languages is decidable as well. This was proved by Pigozzi in [12]. So we observe interesting di erences between logical systems by investigating the behavior of joins. To form the join of two modal logics (in languages with disjoint sets of modal operators) is { in a sense { a generalization of forming the join of two equational theories in disjoint languages. Namely, it is well-known that each modal logic corresponds to an equational theory of boolean algebras with operators. So the join of two modal logics corresponds to