The authors’ work focuses mainly on propositional logic. It consists of seven chapters, where the first chapter is an introduction that presents a useful summary of the topics dealt with in the other chapters, and main results for each of the topics. In general, the authors’ attention is directed to the following subjects: interpolation (in classical, nonmonotonic and some other nonclassical logics), conditionals and neighbourhood semantics. These themes are related in the book to the concepts of modularity and/or independence. We believe this work is suitable for a research seminar in logic which would focus on some of those subjects or related topics. In what follows, we present in more detail the main contents of the book.
One of the authors’ main motivations is the issue of interpolation for monotonic, antitonic, and nonmonotonic logics. They distinguish between semantic and syntactic interpolation and show that existence of semantic interpolation has a simple and general answer for monotonic and antitonic logics, be they two-valued or many-valued, that is, there is always semantic interpolation for (two-valued or many-valued) monotonic and antitonic logics. In the case of nonmonotonic logics this is not always the case. For this sort of logics they consider three variants of semantic interpolation and show that existence of interpolation for two of the variants is closely related to multiplication laws defining the nonmonotonic logic.
The authors also pay attention to the relationship between semantic and syntactic interpolation. They show that semantic interpolation will not necessarily result in syntactic interpolation. They do this by focusing on so-called finite Gödel logics. In general, for both monotonic and nonmonotonic logics, semantic interpolation results in syntactic interpolation if the language and the operators are sufficiently rich to express the semantical interpolants.
The idea of constructing a general theory of conditionals is another motivation for the authors. They present different possible classification criteria for conditionals and point out that it is an impossible, hopeless task to provide an exhaustive list of ordering principles for conditionals as well as an exhaustive enumeration of all possible conditionals. However, they propose general criteria for ordering conditionals based, for example, on the properties of their model choice functions. When conditionals are based on binary relations, the authors propose to classify conditionals by looking at the properties of those relations and the way the relations are used. They discuss some sorts of conditionals in more detail, such as revision, update and counterfactual conditionals.
A topic the authors also pay considerable attention to is that of neighbourhood semantics. They consider different ways neighbourhood models can be defined and explore some of the possible connections between these different definitions. Also, they take into account different uses of neighbourhoods.