Monotone and Antitone Semantic and Syntactic Interpolation
Chapters 4 (page 113) and 5 (page 153) are probably the core of the present book.We treat very general interpolation problems for monotone and antitone, two-valued and many-valued logics in the present chapter, splitting the question into two parts, “semantical interpolation” and “syntactic interpolation”, show that the first problem, existence of semantical interpolation, has a simple and general answer, and reduce the second question, existence of syntactic interpolation, to a definability problem. The first problem leads us to consider particular, universal interpolants, which form the boundaries of all interpolants, and depend only on one formula and the set of common variables, but not on the details of the second formula. We further investigate these universal interpolants, and define new semantical, and then also syntactic, operators from them. These results are summarized briefly in Section 188.8.131.52 (page 115). For the latter, i.e., semantical interpolation, we also examine the concrete example of finite Goedel logics.