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.

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.

From 1912, C. I. Lewis attempted to construct a logic of entailment. In doing so, he created his modal logics, S1–S5, of which his chosen logic of entailment was S2. Although his logics avoid the so-called paradoxes of material implication, they still fall prey to the problem of explosion (that every proposition follows from any contradiction) and the problem of implosion (that every tautology follows from every proposition). These problems, and the inadequate treatment of nested entailments, make Lewis’s logics of limited use as logics of entailment. The chapter also discusses the systems devised by Lewis’s students Everett Nelson and William Parry. Nelson’s connexive logic avoids many of the problems with Lewis’s system but is found to have severe difficulties of its own, and Parry’s analytic implication, although it introduces an interesting version of the notion of meaning containment, does not adequately avoid the problems with Lewis’s logics.

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”.

One difficulty with Lewis’s logics favourite systems, S1–S3, is that they have no intuitive semantics or proof theory. Another approach to constructing a logic of entailment is to begin with a semantic intuition and then adopt the logic characterized by the semantics. This is the approach of Carnap’s Meaning and Necessity. On Carnap’s view, entailment is just strict implication in the sense of the logic S5. The chapter examines Carnap’s semantics and its successor developed by Nino Cocchiarella, and finds that whereas they may give a good representation of the notion of logical truth, they do not provide an adequate analysis of entailment. Once again, the problem of implosion and nested entailments are problematic. This chapter also looks at attempts to solve these difficulties using worlds at which the logical truths differ, and raises philosophical worries about them.