In this chapter, the semantical framework of Chapter 7 is fine-tuned in order to provide a semantics for Anderson and Belnap’s logic E of relevant entailment. It is shown how a form of logical necessity can be represented in the semantics, and this is used to motivate the postulates that are needed to characterize that logic. A labelled deduction system (introduced in Chapter 7) is also used to show how to derive the axioms of E easily.

The model theory for quantified relevant logic developed by Robert Goldblatt and Mares is adapted to the present semantical framework. A universally quantified statement ‘For all x A(x)’ is taken to mean that there is some proposition in the present theory that entails every instance of A(x). An axiomatization of the logic is given, and completeness is proven in the Appendix to the book. Identity, the nature of domains, and higher-order quantification are also discussed.

In this chapter, negation and disjunction are integrated into the semantics developed in Chapters 7 and 8. Here, the semantics of negation is given in terms of an incompatibility relation between theories. A corresponding incompatibility relation is added to the formal language, and a more intuitive and conceptually satisfying set of rules for negation are added to the natural deduction system.

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.

This chapter introduces Anderson and Belnap’s natural deduction treatment of entailment and the idea that hypotheses in deductions should really be used in those deductions. The idea of real use motivates relevant logic and is a key idea in the chapters that follow. The chapter outlines the development of Fitch-style natural deduction systems and introduces the reader to them.

This chapter generalizes the ideas given in the previous chapter. It sets out the notion of a model based on a set of theories. One of these theories is the logic itself. It is, so to speak, the correct theory of theories. It correctly states the principles under which all the theories (including itself) are closed. But each theory has associated with it a closure operator. Some of these operators get the principles of theory closure quite wrong in the sense that they do not apply correctly to every theory in the model. The interaction between these closure operators can be altered in various ways, giving rise to different logical systems. The resulting formal semantics can be represented in the manner of Kit Fine’s “Models for Entailment”.

This chapter discusses the application of the logic to actual theories. In particular, it discusses the use of this relevant logic to make inferences about classical theories. It also examines the internal structure of theories and the nature of the conditionals in those theories. At the end of the chapter, some suggestions are made about generalizing the semantical theory of the book to treat the application of background scientific theories to other theories and some consequences for scientific confirmation.