Introduction

In this chapter we present the ‘relational’ semantics for relevant logic. We begin by introducing the motivations for truth-theoretic semantics. We also look at the possible world semantics for modal logics. The semantics for relevant logic is a modification of the semantics for modal logics. Looking at the semantics for modal logics will allow us to introduce in an easier form certain ideas that are repeated in the relevant semantics. In addition, as we shall see, the modal semantics has certain important virtues. We shall show, in this and the next chapter, that the relevant semantics has these virtues as well.

Why truth-theoretic semantics?

In the last chapter we looked at a fragment of the natural deduction system for a relevant logic. We saw that it uses similar rules to those of the natural deduction system for classical logic, but restricts them by forcing the premises of an argument really to be used in that argument. Why not stop there and say that this is all there is to relevant logic? There is something still missing. First, we need to understand the rest of the logic – its treatment of conjunction, disjunction and negation. As we shall see, it is difficult to interpret them entirely in terms of the natural deduction system. Second, what we have said so far is inadequate as an interpretation of implication. We also need some way of understanding, not only how to prove that an implication is true, but what it means for an implication to be true.

This book is a philosophical interpretation of relevant logic. Relevant logic, also called ‘relevance logic’, has been around for at least half a century. It has been extensively developed and studied in terms of its mathematical properties. So relevant logic is a highly developed and mathematically well-understood branch of non-classical logic. But what is it good for and why should we adopt it? I think that it is a good tool for understanding ordinary deductive reasoning and that it provides us with the tools to understand conditionals. And that is what this book is all about.

Unlike intuitionist logic, relevant logic does not come packaged with its own philosophy. There are intuitionists, and they all share a large number of important philosophical views that non-intuitionists reject. Although some relevant logicians have talked about ‘the relevantist’, relevantism is not a well-developed view, nor one that is widely held even by relevant logicians. By and large, we are free to adopt their own philosophical interpretation of relevant logic.

Historically, my own view developed out of my acquaintance with the possible worlds approach to semantics. When I was a graduate student, I studied modal logic and Montague grammar and found the framework of possible worlds to be a very intuitive, elegant and powerful framework in which to do semantics. Later, after I had become immersed in relevant logic, I wanted to give others the same sort of feeling of being at home in relevant logic that I had felt when I was first exposed to possible world semantics. This book is the latest product of that attempt.

I begin with the possible worlds framework.

This book is an interpretation and defence of relevant logic. These two projects of interpreting and defending relevant logic are not distinct. For it has been the chief complaint about relevant logic that it has no reasonable interpretation.

My interpretation has used as its central notion the real use of premises in a deductive inference. This, I think, makes sense. For it makes relevance the key notion in understanding relevant logic. We saw this notion at work in the understanding of relevant deduction, situated inference, the theory of implication, and, in a somewhat weaker sense, in the theory of conditionals. In other words, relevance as the real use of premises (or the real use of antecedents, in the case of implications and conditionals) permeates my whole interpretation of relevant logic.

I also have tried to short-circuit metaphysical criticisms of relevant logic by constructing a model for the logic from an ontology that is very close to that standardly used by modal logicians. The one difference is that my ontology uses non-well-founded set theory rather than standard set theory. But this is a minor difference, for the theory can be reconstructed using standard set theory, albeit in a less elegant fashion, and, besides, there are other very good reasons for accepting non-well-founded set theory. The upshot of all of this is that we can use relevant logic without any pangs of philosophical guilt. It has a reasonable interpretation and it does not commit us to an extravagant ontology any more than modal logic does.

Nor should we feel guilty about the alleged logical weakness of relevant logic.

Semantics and metaphysics

In this book I am urging people to accept relevant logic and its semantics. In this chapter we discuss what metaphysical consequences are entailed by accepting the semantics. But before we can get into this issue properly, we should discuss what it means to accept a theory.

Theories can be treated either realistically or anti-realistically. A realist about a subject like semantics, will ask of the theory whether or not it is literally true. If it is found not to be true, then the realist will reject the theory. For a realist, to accept a theory is to believe that it is literally true. Anti-realists, on the other hand, do not use the literal truth of a theory as the criterion of acceptance.

There are various brands of anti-realism. Those that are most relevant to the subject of semantic theories are instrumentalists and fictionalists. Instrumentalists about semantic theories hold that talk about things like possible worlds is useful for understanding our semantic intuitions but are not to be taken literally (see e.g., (Kripke 1972)).

Fictionalists also hold that we should not treat the claims made in a semantic theory as being literally true. Suppose, for example, that we adopt a fictionalist attitude towards the semantics for relevant logic. Then, when we say, for example, ‘There are situations that make contradictions true’ (as we shall say in chapter 5 below), what we mean is that according to this semantics there are situations that make contradictions true. In addition, the fictionalist about this semantics will also hold that the semantics is an appropriate fiction to use to analyse the meaning and truth value of our statements about implication.

The problem of implication

We introduced the problem of implication in the previous chapter. The problem is to give an intuitive semantics for relevant implication. In the previous chapter, we looked at the Routley–Meyer semantics for relevant logic, but we did not give it an interpretation. We cannot claim that relevant logic has an intuitive semantics until we do so. That is the task of the present chapter.

We begin with our basic ontology – the list of things that we presuppose in our semantics. The elements of our ontology are situations and possible worlds, as well as individuals and sets. We have met situations and worlds already in this book, but here and in the next chapter we will look at them in much more depth.

After we introduce our ontology, we give an intuitive semantics for implication using neighbourhoods, which were briefly introduced in the previous chapter. The bulk of the chapter shows how, given a few reasonable assumptions, we can view the Routley–Meyer semantics to be just the intuitive semantics in disguise. In addition, given the resulting interpretation of the Routley–Meyer semantics, we can justify the postulates (given in section 2.13) which yield a semantics for the relevant logic R.

Situations and worlds

We have already been introduced in chapter 2 to the notion of a possible world. Situations are somewhat less familiar objects to philosophers and semanticists and we will begin with them. They were introduced into logic in the early 1960s by Saul Kripke, who used ‘evidential situations’ in his model theory for intuitionist logic (Kripke 1965a).