Systematizing Definability Theory

Publisher Summary This chapter explains that it is possible to systematize the questions of the different kinds of definability in first-order logic. The main syntactical tool used in this enterprise is the theory of distributive normal forms. For a given first-order theory, one can construct certain expressions that have a form of constituent but that can be inconsistent. In spite of their inconsistency, these expressions can be model-theoretically interpreted so that one can obtain model-theoretic arguments for results concerning certain familiar kinds of identifiability. A fully explicit treatment of the model-theoretical import of such inconsistent expressions would require recourse to ideas somewhat different from those of the traditional model theory. A set of tools appropriate to this task is in fact found suitably extended in the surface semantics of Hintikka. Model-theoretic significance of the ideas developed remains partly implicit. Although the focus of the concepts seems to be syntactic, it is hoped that the main outlines of their model-theoretical applications are apparent enough.