This chapter examines Dana Scott’s project of treating a logic of entailment as one that captures its own deducibility relation in the sense that it represents (and vindicates) the way in which the theorems of the logic themselves are derived. For example, a reflexive logic that is axiomatized using the rule of modus ponens also contains the entailment ‘(A and A entails B) entails B’. It is argued in this chapter that the reflexivity constraints get in the way of the logic’s being used as a general theory of theory closure. A logic should be closed under its own principles of inference, but the logic should be able to be used with theories that are radically different from itself.

The semantical framework for the positive view of this book is one in which entailment is understood primarily in terms of theory closure. This chapter outlines both the history of the notion, beginning with Alfred Tarski’s theory of closure operators, and the relationship between closure operators and the entailment connective. At the end of the chapter, it is shown how closure operators can be used to model a simple logic, Graham Priest’s logic N4.