A Note on Relativised Products of Modal Logics

One may think of many ways of combining modal logics representing various aspects of an application domain. Two ‘canonical’ constructions, supported by a well-developed mathematical theory, are fusions [17, 6, 7] and products [8, 7]. The fusion L1⊗ · · ·⊗Ln of n ≥ 2 normal propositional unimodal logics Li with the boxes 2i is the smallest multimodal logic in the language with n boxes 21, . . . ,2n (and their duals 31, . . . ,3n) that contains all the Li. This means that if the Li are axiomatised by sets Axi of axioms, then L1⊗ · · · ⊗Ln is axiomatised by the union Ax1 ∪ · · · ∪Axn. Thus, fusions are useful when the modal operators of the combined logics are not supposed to interact (see e.g. [1] which provides numerous examples of applications of fusions in description logic). It is this absence of interaction axioms that ensures the transfer of good algorithmic properties from the components to their fusion [17, 6, 32]. In particular, it is possible to reduce reasoning in the fusion to reasoning in the components. On the semantic level, the fusion of Kripke complete modal logics L1, . . . , Ln can be characterised by the class of all n-frames 〈W,R1, . . . , Rn〉 such that each 〈W,Ri〉 is a frame for Li [17, 6]. Note that although frames for fusions have n different accessibility relations, they cannot be regarded as ‘genuinely many-dimensional’ in the geometric sense. Products of modal logics do have real many-dimensional frames. Given n Kripke frames F1 = 〈U1, R1〉, . . . , Fn = 〈Un, Rn〉, their product F1 × · · · × Fn is the n-frame 〈 U1 × · · · × Un, R1, . . . , Rn 〉 ,