Abstract
There has been a significant interest in modal logics with intersection, prominent examples including epistemic and doxastic logics with distributed knowledge, propositional dynamic logic with intersection, and description logics with concept intersection. Completeness proofs for such logics tend to be complicated, in particular on model classes such as S5 used, e.g., in standard epistemic logic, mainly due to the undefinability of intersection of modalities in standard modal logic. A standard proof method for the S5 case uses an “unraveling-folding” technique to achieve a treelike model to deal with the problem of undefinability. This method, however, is not easily adapted to other logics, due to its reliance on S5 in a number of steps. In this paper we demonstrate a simpler and more general proof technique by building a treelike canonical model directly, which avoids the complications in the processes of unraveling and folding. We illustrate the technique by showing completeness of the normal modal logics K, D, T, B, S4 and S5 extended with intersection modalities. Furthermore, these treelike canonical models are compatible with Fischer-Ladner-style closures, and we combine the methods to show the completeness of the mentioned logics further extended with transitive closure of union modalities known from PDL or epistemic logic. Some of these completeness results are new.
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Notes
- 1.
Although the symbol \(\biguplus \) is sometimes used for disjoint union, we repurpose it here for transitive closure of the union.
- 2.
The subscript i of a unary modal operator \(\Box _i\) typically stands for an agent in epistemic logic or a role in description logic. In epistemic logic, a finite number of agents is assumed, and the intersection modality (i.e., a distributed knowledge operator) is an arbitrary intersection over a finite domain. In description logic, the number of roles are typically unbounded, but the intersection is binary, which is in effect equivalent to finite intersection.
- 3.
There are two major differences however. First, the Kleene star in both logics are the reflexive-transitive closure, and we consider the transitive closure which is denoted by a “\(+\)” in the symbol \(\uplus \). Second, \(\uplus _{I}\) is a compound modality (union and then take the transitive closure), while in those logics the Kleene star is separated from the union, and as a result, the Kleene star applies to the intersection as well, which we do not consider here.
- 4.
D\(\cap \), 4\(\cap \), B\(\cap \) and N\(\cap \) are not needed in the sense that they are derivable.
- 5.
D\(\uplus \), T\(\uplus \), 4\(\uplus \), B\(\uplus \), 5\(\uplus \) and N\(\uplus \) are not needed in the sense that they are derivable.
- 6.
We refer to a modal logic textbook, say [8], for a definition of a (maximal) consistent set of formulas.
- 7.
This is the place where the use of \(\iota \) is essential to make sure that a closure is finite.
- 8.
In an extension of [2], to appear.
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Acknowledgments
We thank the anonymous reviewers for very detailed and useful comments and suggestions. Yì N. Wáng acknowledges funding support by the National Social Science Foundation of China (Grant No. 16CZX048, 18ZDA290), and the Fundamental Research Funds for the Central Universities of China.
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Wáng, Y.N., Ågotnes, T. (2020). Simpler Completeness Proofs for Modal Logics with Intersection. In: Martins, M.A., Sedlár, I. (eds) Dynamic Logic. New Trends and Applications. DaLi 2020. Lecture Notes in Computer Science(), vol 12569. Springer, Cham. https://doi.org/10.1007/978-3-030-65840-3_16
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