On the Fusion of Coalgebraic Logics

Abstract

Fusion is arguably the simplest way to combine modal logics. For normal modal logics with Kripke semantics, many properties such as completeness and decidability are known to transfer from the component logics to their fusion. In this paper we investigate to what extent these results can be generalised to the case of arbitrary coalgebraic logics. Our main result generalises a construction of Kracht and Wolter and confirms that completeness transfers to fusion for a large class of logics over coalgebraic semantics. This result is independent of the rank of the logics and relies on generalising the notions of distance and box operator to coalgebraic models.

Coalgebraic completeness-via-canonicity for distributive substructural logics

“…First, it is modular in the choice of 'reasoning kernel' since the same formal functor can be overloaded to be used on several different choices of base categories. Second, and most importantly, it allows for a very concise definition and construction of the fusion of logics ( [DP11]); that is, the free combination of two logics defined on the same base categories. If…”

“…Second, it provides a generic framework in which to prove completeness-viacanonicity for a vast range of logics ([DP13]). Third, it is intrinsically modular; that is, it provides theorems about complicated logics by combining results for simpler ones ( [CP07,DP11]). We will return to the advantages of working coalgebraically throughout the paper.…”

Coalgebraic Completeness-via-Canonicity

“…Coalgebraic logics defined on a common minimal reasoning structure can be freely combined to form new logics combining the modalities of their constituents in a process called the fusion of modal logics (see [CP07,DP11]). Formally, if L 1 , L 2 : C → C are two syntax constructors, then the fusion of F L1 FV and F L2 FV is the language defined by the (point-wise) coproduct of these functors, i.e.…”

“…In fact, since the traditional boolean setting is a special case of the more general setting of positive coalgebraic logics, we will formulate most results in terms positive coalgebraic logics. A final advantage of the technique its modularity: we can combine strongly complete logics to create new strongly complete logics in a completely mechanical way (in the spirit of [CP07,DP11]). This work is in many ways a continuation and generalisation of the author's previous work with his PhD supervisor Dirk Pattinson ([DP13]).…”

Completeness via Canonicity for Distributive Substructural Logics: A Coalgebraic Perspective

“…Second, it provides a generic framework in which to prove completeness-via-canonicity for a vast range of logics ( [11]). Third, it is intrinsically modular; that is, it provides theorems about complicated logics by combining results for simpler ones ( [9,10]). …”