Dov Gabbay: Fibring Logics. a Review

Studying combined logics is a growing trend in applied logic. The general question is this: Suppose language L 3 has been formed by fusion of two languages L 1 and L 2. Let two logics L 1 and L 2 be defined over L 1 and L 2 , respectively. What can be said about conservative extensions L 3 in L 3 ? This is a fairly broad problem. Conservative extensions are not unique, and they may have different properties. In modal logics, several extensions have been studied in detail. For example [1] studied bimodal modal logics which are the minimal conservative extensions of their monomodal fragments (called fusions). Another example are the products of modal logics, to which the author of the present book has made important contributions. Numerous questions arise: what are the properties of the combination and can they be predicted from the properties of the fragments? Is there a general recipe by which we can obtain combined systems? The present book promotes fibring as a general model construction for combined systems. The general idea is this. Let the languages L 1 and L 2 and two classes of models be given. Fibred models can be thought of as models for one of the languages enriched with an oracle that answers about the truth of formulae of the other language. This oracle takes the form of a function F , which, given a model M (which typically contains a distinguished world or state), returns a model F (M) of the other language in which ϕ is evaluated. F is called the fibring function. If ϕ actually contains operators of the first language , this construction thus toggles between the two model classes until ϕ is totally decomposed. The possibilities for combination are galore. We can try to combine logics whose signature is more or less similar (like two predicate logics with different but possibly nondisjoint signature), modal logics with different sets of operators, but we may also try something more radical by combining logics that are basically dissimilar. Two paradigm examples are intuitionistic modal logic (where the modal operator with its modal semantics is added on top of the intuitionistic connectives) and the temporalisation of systems. The latter is interesting insofar as it shows that it might be useful to study systems that do not use the