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.

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.

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”.

A logic of entailment is one in which it represents its own derivability relation as a connective. This allows it to express nested claims about what is derivable from what. It can say, for example, that if B is derivable from A and if C is derivable from B, then C is derivable from A. Mathematicians and logicians think this way when making proof plans. This chapter sets out the problem of constructing proof plans and looks at the other uses to which logics of entailment have been put. One key use, especially in the context of this book, is as a theory of the closure of scientific theories.

Is logic intrinsically normative? Given that we often make errors in reasoning, one might hold that logic is not about how we do think but about how we ought to think. However, logical laws say nothing about thinking. How does logic gain normative traction on psychological processes such as belief formation or reasoning? To answer this question, the chapter begins by examining the debate between descriptivists, who hold that logic simply describes truth-preserving entailment relations, and normativists, who hold that logic is intrinsically normative. Husserl was a descriptivist, and Heidegger’s early work follows him. But Heidegger’s phenomenological account of the truth-predicate as grounded in the comportment of assertion, and his analysis of the “hermeneutic as” that grounds such comportment, affords a different perspective on the debate. Logic provides constitutive norms for the practice of reason-giving, a kind of rationality – being answerable to others – that is not an essential property of, but is nevertheless demanded by, Dasein’s being as “care.” The chapter concludes by showing how Heidegger’s phenomenological approach can affirm the main claims of descriptivism while insisting that “the reign of logic” in philosophy “disintegrates into the turbulence of a more original questioning.”

Nicholas Allott considers how relevance theory can be seen as responding to doubts about the possibility of any kind of systematic pragmatic theory. He considers three sceptical positions: Fodor’s argument that pragmatic processes are not amenable to scientific study because they are unencapsulated (highly context-sensitive), Chomsky’s claim that human intentional action is a mystery rather than a scientifically tractable problem, and a third view which maintains that intentional communication is too complex for systematic study. Allott argues that work in relevance theory can be seen as successfully challenging these sceptical views and he gives concrete examples of its achievements.

In this chapter, Jacques Moeschler addresses some complex issues about the function of negation and its interaction with metarepresentation.He identifies three distinct uses of negation, namely, descriptive negation and two kinds of metarepresentational negation (one metalinguistic, the other presupposition-cancelling), which differ in their semantic entailments. His key claim is that all three of them have what he calls ‘representational’ (or propositional) effects on the context, specifically the elimination and/or the strengthening of existing assumptions, albeit different for each of the uses.