Introduction

23.1.1 This chapter brings together the techniques of previous chapters, to look at a variety of logics that they may generate. The chapter also acts as a bridge between the basic system of relevant logic of the last chapter, First Degree Entailment, and the full relevant logics of the next.

23.1.2 By this stage of the book we have many independent techniques that may be employed in constructing the semantics of a logic: normal and non-normal worlds, constant and variable domains, different numbers of truth values, negation using many values and the * semantics, necessary and contingent identity. These techniques can be combined to produce a vast variety of logics, far too many to consider here. We will consider only some of the more notable ones.

23.1.3 We begin with the basic relevant logics N4 and N*, starting with the former. This will require an application of the matrix semantics employed for non-normal modal logics in chapter 18. The logics K4 and K* are then obtained as special cases. We will look only at the constant domain versions of the logics.

23.1.4 Identity for the logics in question is next on the agenda. We will concern ourselves only with necessary identity, though the behaviour of identity at non-normal worlds gives it something of the flavour of contingent identity.

23.1.5 There is then a philosophical interlude concerning one application of identity in relevant logic: relevant predication.

Part I of the book has explored, in various ways, a relevant account of conditionals. Such an account seems to me to be better than any of the other accounts that we have traversed in the course of Part I. This is, naturally, a contentious view. Logic is a contentious subject, and the conditional has been particularly so since the earliest years of the discipline. It was the Stoic logicians who first discussed conditionals explicitly, and they had at least four competing accounts. These accounts survived – one way or another – and others were added throughout the Middle Ages. Consensus might have been reached locally, but only locally.

The changes in logic at the beginning of the twentieth century were revolutionary. The power of the mathematical techniques employed by the founders of modern logic made anything before obsolete. (Which is not to say that there is not now a good deal to be learned from it – just that whatever is of value in it must be seen through radically new eyes.) It is perhaps not surprising, then, that their work established a very general consensus over the conditional. The view of the conditional as material became highly orthodox – though never universal, as C. I. Lewis bears witness.

Digesting the results of the revolution occupied logicians in the first half of the century. But the second half was quite different.

Introduction

14.1.1 In this chapter we will start to look at quantified normal modal logics. These come in two varieties: constant domain (where the domain of quantification is the same in all worlds), and variable domain (where the domain may vary from world to world).

14.1.2 Where it is necessary to distinguish between the two, I will use the following notation. If S is any system of propositional modal logic, CS will denote the constant domain quantified version, and VS will denote the variable domain quantified version.

14.1.3 In this chapter we will look at the semantics and tableaux for constant domain logics, saving variable domains for the next.

14.1.4 For these two chapters we will take it that identity is not part of the language. We will turn to the topic of identity in modal logic in chapter 16.

14.1.5 We will also take a quick look at one of the major philosophical issues to which quantified modal logic gives rise: the issue of essentialism.

14.1.6 The chapter ends by showing how the semantic and tableau techniques of normal modal logic extend to tense logic.

Constant Domain K

14.2.1 The syntax of quantified modal logic augments the language of first order classical logic (12.2) with the operators □ and ◇, as propositional modal logic extends classical propositional logic (2.3.1, 2.3.2).

Introduction

4.1.1 In this chapter we look at some systems of modal logic weaker than K (and so non-normal). These involve so-called non-normal worlds. Nonnormal worlds are worlds where the truth conditions of modal operators are different.

4.1.2 We are then in a position to return to the issue of the conditional, and have a look at an account of a modal conditional called the strict conditional.

Non-normal Worlds

4.2.1 Let us start by looking at the technicalities concerning non-normality. In due course we will be able to discuss what they mean.

4.2.2 A non-normal interpretation of a modal propositional language is a structure, 〈W, N, R, ν〉, where W, R and ν are as in previous chapters, and N ⊆ W. Worlds in N are called normal. Worlds in W – N (the worlds that are not normal) are called non-normal.

4.2.3 The truth conditions for the truth functions, ∧, ∨, ¬, etc. are the same as before (2.3.4). The truth conditions for □ and ◇ at normal worlds are also as before (2.3.5). But if w is non-normal:

1. νw(□A) = 0

2. νw(◇A) = 1

In a sense, at non-normal worlds, everything is possible, and nothing is necessary.

4.2.4 Note that at every world, w, ¬□A and ◇¬A still have the same truth value, as do ¬◇A and □¬A. We saw this to be the case for normal worlds in 2.3.9 and 2.3.10.

Introduction

16.1.1 In this chapter we will start to look at the behaviour of identity in modal logic. (Henceforth, I use ‘modal logic’ to include tense logic.) There are, in fact, two kinds of semantics for identity in modal logic: necessary and contingent.

16.1.2 Where it is necessary to distinguish between the two notions of identity, I will use the following notation. If S is any system of logic without identity, S(NI) will denote the system augmented by necessary identity, and S(CI) will denote the system of logic augmented by contingent identity. In this chapter we will deal with necessary identity, which is simpler; in the next chapter, we will turn to contingent identity.

16.1.3 We will assume, first, that the Negativity Constraint is not in operation. We will then see how its addition affects matters.

16.1.4 Next, we will look at the distinction between rigid and non-rigid designators, and see how non-rigid designators can be added to the logic.

16.1.5 Finally, there is a short philosophical discussion of how this distinction applies to names and descriptions in a natural language such as English.

Necessary Identity

16.2.1 Assume that we are dealing with any quantified (constant or variable domain) normal modal logic (without the Negativity Constraint). As in the classical case (12.5.1), we now distinguish one of the binary predicates as the identity predicate.

16.2.2 The denotation of the identity predicate is the same in every world, w, of an interpretation: νw(=) = {〈d, d〉 : d ∈ D}.

Introduction

15.1.1 In this chapter we will look at the other variety of semantics for quantified modal (and tense) logic: variable domain.

15.1.2 We will start with K and its normal extensions. Next we observe how matters can be extended to tense logic.

15.1.3 There are then some comments on other extensions of the logics involved.

15.1.4 The chapter ends with a brief discussion of two major philosophical issues that variable domain semantics throw into prominence: the question of existence across worlds, and the connection (or lack thereof) between existence and the particular quantifier.

Prolegomenon

15.2.1 Perhaps the most obvious objection to constant domain semantics is as follows. Just as the properties of objects may vary from world to world, what exists at a world, it is natural to suppose, may vary from world to world. Thus, I exist at this world, but in a world where my parents never met, I do not exist. Or, at this world, Sherlock Holmes does not exist, but in a world that realises the stories of Arthur Conan Doyle, he does.

15.2.2 Another way of making the point is as follows. Consider the following formulas:

These are usually called the Barcan Formula and the Converse Barcan Formula, respectively. Both of these are valid in CK (and a fortiori stronger constant domain logics), as may be checked. But intuitively they are invalid.

Introduction

18.1.1 The techniques concerning quantification and identity in normal modal logics carry over in a natural way to other logics which have possible world semantics. In this chapter we will look at one of these, non-normal modal logics.

18.1.2 We will ignore identity to start with, and look at the constant and variable domain versions of non-normal modal logics (without descriptors).

18.1.3 We will then look at the addition of identity to these logics.

18.1.4 Non-normal worlds are important since, being worlds where logical truths may fail (as we saw in 4.4.7), they are harbingers of the impossible worlds of relevant logics (9.7). But the addition of quantifiers and identity to non-normal worlds appears to raise no novel philosophical issues. There is therefore no philosophical discussion in this chapter.

Non-normal Modal Logics and Matrices

18.2.1 In a non-normal modal logic, formulas of the form □A and ◇A are assigned truth values at non-normal worlds in a way that does not depend on the value of A. When quantification is involved, employing this strategy in the simple-minded way may cause a problem. Most obviously, □Pa and □Pb may be assigned different values at a world, even though a = b is true there. (More generally, the Denotation Lemma, which is integral to the correct functioning of quantifiers, breaks down.)

18.2.2 To overcome this problem, we have to treat formulas of the form □A and ◇A, with n free variables, effectively as n-place predicates.

The first edition of Introduction to Non-Classical Logic deals with just propositional logics. In 2004, Cambridge University Press and I decided to produce a second volume dealing with quantification and identity in nonclassical logics. Late in the piece, it was decided to put the old and the new volumes together, and simply bring out one omnibus volume. The practical decision caused a theoretical problem. Was it the same book as the old Introduction or a different one? The answer – as befits a book on non-classical logic – was, of course, both. So the name of the book had to be the same and different. We decided to achieve this seeming impossibility by adding an appropriate sub-title to the book, ‘From If to Is’. Though there are many propositional operators and connectives, the conditional, ‘if’, is perhaps the most vexed. It is, at any rate, the focus around which the old Introduction moves. Whether or not ‘if’ is univocal is a contentious matter; but ‘is’ is certainly said in many ways. There is the ‘is’ of predication (‘Ponting is Australian’), the ‘is’ of existence (‘There is a spider in the bathtub’, ‘Socrates no longer is’), and the ‘is’ of identity (‘2 plus 2 is 4’). All of these are in play in first-order logic; they provide the focus around which the new part of the book moves.

I conclude with a few comments of a methodological nature concerning the investigations of this book.

Let us start by returning to the objection to the theory of vague objects voiced in 25.7.9. The point there, recall, was that the theory given was not really a theory about vague objects at all. In the semantics of the language, the identity relation is the standard crisp one. What this shows is that the objects we are dealing with are really crisp objects. The identity relation of the object language is not really identity, just some sort of similarity relation.

There is something wrong about this objection, and something right. It is certainly the case that the identity relation of the object language and the identity relation of the metalanguage (in which the semantics are expressed) are different. It does not follow that it is the relation of the object language that is not the real notion. It is open to someone who holds that there are genuine vague objects to maintain that it is the identity relation of the metalanguage that is not really identity. To claim otherwise in this context would be to beg the question.

It remains the case, however, that the identity relation of the object language and the metalanguage are out of kilter. There is therefore something prima facie awry in the situation.

Introduction

22.1.1 The present chapter is devoted to another many-valued logic, one that will lead us into a discussion of relevant logic: First Degree Entailment (FDE).

22.1.2 We start with the relational semantics for FDE, and see that this is equivalent to a many-valued semantics.

22.1.3 We will then look at tableaux for quantified FDE, in the process obtaining tableau systems for the 3-valued logics of the last chapter.

22.1.4 A quick look at free logics in the context of relational semantics is next on the agenda.

22.1.5 After that, we move on to the * semantics and tableaux for FDE, and note their equivalence with the relational semantics.

22.1.6 Finally, we will look at the behaviour of identity in both semantics for FDE.

22.1.7 The philosophical issues that tend to be raised by quantification and identity in FDE are much the same as those which we met in connection with the three-valued logics of the last chapter. There is therefore no new philosophical discussion in this chapter.

Relational and Many-valued Semantics

22.2.1 An interpretation for quantified FDE is a structure 〈D, ν〉, where D is the non-empty domain of quantification. For every constant in the language, c, ν(c) ∈ D, and for every n-place predicate, P, ν(P) is a pair 〈ε, A〉, where ε and A are subsets of Dn.

Introduction

9.1.1 In this chapter, we will see how the techniques of modal logic and many-valued logic can be combined. More specifically, we will look at logics that add some kind of strict conditional with world semantics on top of a many-valued base-logic, specifically, FDE.

9.1.2 The non-normal worlds of chapter 4 will also make a reappearance, giving us some basic relevant logics. This will allow us to discuss further what, exactly, non-normal worlds are.

9.1.3 We will end the chapter with a brief look at so called logics of constructible negation, which have close connections with intuitionist logic; and an even briefer look at connexive logics.

Adding →

9.2.1 FDE has no conditional operator. The material conditional, A ⊃ B, does not even satisfy modus ponens, as we saw in 8.6.5. In any case, as we have seen, using possible-world semantics provides a much more promising approach to the logic of conditional operators. Thus, an obvious thing to do is to build a possible-world semantics on top of the relational semantics of FDE.

9.2.2 To effect this, let us add a new binary connective, →, to the language of FDE to represent the conditional. By analogy with Kν, a relational interpretation for such a language is a pair 〈W, ρ〉, where W is a set of worlds, and for every w ∈ W, ρw is a relation between propositional parameters and the values 1 and 0.

Introduction

20.1.1 In this chapter we will look at one more logic with possible-world semantics: intuitionist logic.

20.1.2 After a brief prolegomenon, we will look at the semantics for this. We will then look at two tableau systems. The first is close to the tableau system for variable domain modal logic of chapter 15. The second is slightly more complicated to formulate, but produces simpler tableaux.

20.1.3 All this without identity, which is thrown into play in the second half of the chapter.

20.1.4 En route, we will also look at some philosophical issues concerning existence, construction and identity.

Existence and Construction

20.2.1 Mathematical Platonists think of mathematical objects as existing in some objective realm, just like (we normally think that) stones and stars do; it is just a realm that is out of causal contact with us – or anything else. As we observed (6.2.5) mathematical intuitionists reject this view.

20.2.2 So what, according to them, does it mean to say that a mathematical object exists? It means that we are able to construct it; that is, that there is some recipe we can follow to produce it. Obviously, the entity constructed is not a physical entity; we may call it a mental (or maybe social) entity. Thus, mathematical objects have no cognition-independent existence.

20.2.3 As we also observed (6.2.6), an intuitionist needs to give the proof conditions for sentences (where a proof is something that can be recognised as such).