Introduction

1.1.1 The first purpose of this chapter is to review classical propositional logic, including semantic tableaux. The chapter also sets out some basic terminology and notational conventions for the rest of the book.

1.1.2 In the second half of the chapter we also look at the notion of the conditional that classical propositional logic gives, and, specifically, at some of its shortcomings.

1.1.3 The point of logic is to give an account of the notion of validity: what follows from what. Standardly, validity is defined for inferences couched in a formal language, a language with a well-defined vocabulary and grammar, the object language. The relationship of the symbols of the formal language to the words of the vernacular, English in this case, is always an important issue.

1.1.4 Accounts of validity themselves are in a language that is normally distinct from the object language. This is called the metalanguage. In our case, this is simply mathematical English. Note that ‘iff’ means ‘if and only if’.

1.1.5 It is also standard to define two notions of validity. The first is semantic. A valid inference is one that preserves truth, in a certain sense. Specifically, every interpretation (that is, crudely, a way of assigning truth values) that makes all the premises true makes the conclusion true. We use the metalinguistic symbol ‘⊧’ for this. What distinguishes different logics is the different notions of interpretation they employ.

Introduction

17.1.1 In this chapter we will look at the behaviour of contingent identity in modal logic. We assume that the logic to which identity is being added is any quantified normal modal logic, constant or variable domain, without the Negativity Constraint. Recall that if L is any logic L(CI) is L augmented by contingent identity.

17.1.2 First, we will take all constants to be rigid designators. We then look at the addition of descriptors.

17.1.3 Finally, we will take up briefly two important philosophical issues concerning identity, tense and modality.

Contingent Identity

17.2.1 In 16.4.1 we looked at a problem concerning SI. Later in 16.4 we saw how the distinction between rigid and non-rigid designators solves the problem. But there would appear to be a more virulent form of it. The Morning Star is the planet Venus, as is the Evening Star, m = v = e. But arguably, these noun phrases are rigid. So it follows by NI that □m = e; and this would seem not to be true. It would seem to be a contingent matter that the heavenly object that appears in the sky around dawn, and christened by the Ancients ‘the Morning Star’, turned out to be identical with the heavenly body that appears in the sky around dusk, and christened by the Ancients ‘the Evening Star’. The latter, for example, could have turned out to be Mercury.

Introduction

6.1.1 In this chapter, we look at another logic that has a natural possibleworld semantics: intuitionist logic, a logic that arose originally out of certain views in the philosophy of mathematics called intuitionism.

6.1.2 We will also look briefly at the philosophical foundations of intuitionism, and at the distinctive account of the conditional that intuitionist logic provides.

Intuitionism: The Rationale

6.2.1 Let us start with a look at the original rationale for intuitionism. Consider the sentence ‘Granny had led a sedate life until she decided to start pushing crack on a small tropical island just south of the Equator.’ You can understand this, and indefinitely many other sentences that you have never (I presume) heard before. How is this possible?

6.2.2 We can understand a sentence of this kind because we understand its individual parts and the way they are put together; the meaning of a sentence is determined by the meanings of its parts, and of the grammatical construction which composes these. This fact is called compositionality.

6.2.3 An orthodox view, usually attributed to Frege, is that the meaning of a statement is given by the conditions under which it is true, its truth conditions. Thus, by compositionality, the truth conditions of a statement must be given in terms of the truth conditions of its parts. Thus, for example, ¬A is true iff A is not true; A ∧ B is true iff A is true and B is true; and so on.

Introduction

11.1.1 In this chapter we look at fuzzy logic, that is, logic in which sentences can take as a truth value any real number between 0 and 1.

11.1.2 We look at one of the major motivations for such a logic: vagueness. We also show some of the connections between fuzzy logic and relevant logics.

11.1.3 Finally, fuzzy logic gives a very distinctive account of the conditional, since modus ponens may fail. The chapter examines what fuzzy conditionals are like.

Sorites Paradoxes

11.2.1 Suppose that Mary is aged five, and hence is a child. If someone is a child, they are a child one second later: there is no second at which a person turns from a child to an adult. (We are talking about biological childhood here, not legal childhood. The latter does terminate at the instant someone turns eighteen, in many jurisdictions.) So in one second's time, Mary will still be a child. Hence, one second after that, she will still be a child; and one second after that; and one second after that … Hence, Mary will be a child after any number of seconds have elapsed. But this is, of course, absurd. After an appropriate number of seconds have elapsed, so have thirty years, by which time Mary is thirty-five, and so certainly not a child.

11.2.2 The argument of 11.2.1 is known as a sorites paradox. It arises because the predicate ‘is a child’ is vague in a certain sense.

Introduction

10.1.1 In this chapter we look at logics in the family of main stream relevant logics. These are obtained by employing a ternary relation to formulate the truth conditions of →. In the most basic logic, there are no constraints on the relation. Stronger logics are obtained by adding constraints.

10.1.2 We also see how these semantics can be combined with the semantics of conditional logics of chapter 5 to give an account of ceteris paribus enthymemes.

The Logic B

10.2.1 N4 and N* are relevant logics, but, as relevant logics go, they are relatively weak. Many proponents of relevant logic have thought that the relevant logics of the last chapter are too weak, on the ground that there are intuitively correct principles concerning the conditional that they do not validate. A way to accommodate such principles within a possible-world semantics is to use a relation on worlds to give the truth conditions of conditionals at non-normal worlds. Unlike the binary relation of modal logic, xRy, though, this relation is a ternary, that is, three-place, relation, Rxyz.

10.2.2 Intuitively, the ternary relation Rxyz means something like: for all A and B, if A → B is true at x, and A is true at y, then B is true at z. What philosophical sense to make of this, we will come back to later.

Introduction

8.1.1 In this chapter we look at a logic called first degree entailment (FDE). This is formulated, first, as a logic where interpretations are relations between formulas and standard truth values, rather than as the more usual functions. Connections between FDE and the many-valued logics of the last chapter will emerge.

8.1.2 We also look at an alternative possible-world semantics for FDE, which will introduce us to a new kind of semantics for negation.

8.1.3 Finally, we look at the relation of all this to the explosion of contradictions, and to the disjunctive syllogism.

The Semantics of FDE

8.2.1 The language of FDE contains just the connectives ∧, ∨ and ¬. A ⊃ B is defined, as usual, as ¬A ∨ B.

8.2.2 In the classical propositional calculus, an interpretation is a function from formulas to the truth values 0 and 1, written thus: ν(A) = 1 (or 0). Packed into this formalism is the assumption (usually made without comment in elementary logic texts) that every formula is either true or false; never neither, and never both.

8.2.3 As we saw in the last chapter, there are reasons to doubt this assumption. If one does, it is natural to formulate an interpretation, not as a function, but as a relation between formulas and truth values. Thus, a formula may relate to 1; it may relate to 0; it may relate to both; or it may relate to neither.

In expositions of modern logic, the use of some mathematics is unavoidable. The amount of mathematics used in this text is rather minimal, but it may yet throw a reader who is unfamiliar with it. In this section I will explain briefly three bits of mathematics that will help a reader through the text. The first is some simple set-theoretic notation and its meaning. The second is the notion of proof by induction. The third concerns the notion of equivalence relations and equivalence classes. It is not necessary to master the following before starting the book; the material can be consulted if and when required.

Set-theoretic Notation

0.1.1 The text makes use of standard set-theoretic notation from time to time (though never in a very essential way). Here is a brief explanation of it.

0.1.2 A set, X, is a collection of objects. If the set comprises the objects a1, …, an, this may be written as {a1, …, an}. If it is the set of objects satisfying some condition, A(x), then it may be written as {x :A(x)}. a ∈ X means that a is a member of the set X, that is, a is one of the objects in X. a ∉ X means that a is not a member of X.

Introduction

21.1.1 In this chapter we leave world-semantics for the time being, and turn to many-valued logics.

21.1.2 We will start with a brief look at the general situation concerning many-valued logics, before turning to the special cases of the 3-valued logics of chapter 7 for more detailed consideration.

21.1.3 Free versions of these logics are next on the agenda – in particular, now that we have the machinery of truth value gaps at our finger tips, the neutral free logics mentioned in 13.4.7. This will occasion a discussion of the behaviour of the existence predicate in a many-valued logic, and the question of whether it might make good philosophical sense for a statement of existence to have a non-classical value.

21.1.4 Next, we turn to the behaviour of identity in many-valued logics, and particularly the 3-valued logics of chapter 7. This will occasion a discussion of whether identity statements may plausibly be taken to have non-classical values.

21.1.5 We will finish with a few comments on supervaluations and subvaluations in the context of quantificational logic.

Quantified Many-valued Logics

21.2.1 As we saw in 7.2.2, a propositional many-valued logic is characterised by a structure 〈V,D,{fc : c ∈ C}〉, where V is the set of truth values, D ⊆ V is the set of designated values, and for each connective, c, fc is the truth function it denotes.

Around the turn of the twentieth century, a major revolution occurred in logic. Mathematical techniques of a quite novel kind were applied to the subject, and a new theory of what is logically correct was developed by Gottlob Frege, Bertrand Russell and others. This theory has now come to be called ‘classical logic’. The name is rather inappropriate, since the logic has only a somewhat tenuous connection with logic as it was taught and understood in Ancient Greece or the Roman Empire. But it is classical in another sense of that term, namely standard. It is now the logic that people normally learn when they take a first course in formal logic. They do not learn it in the form that Frege and Russell gave it, of course. Several generations of logicians have polished it up since then; but the logic is the logic of Frege and Russell none the less.

Despite this, many of the most interesting developments in logic in the last forty years, especially in philosophy, have occurred in quite different areas: intuitionism, conditional logics, relevant logics, paraconsistent logics, free logics, quantum logics, fuzzy logics, and so on. These are all logics which are intended either to supplement classical logic, or else to replace it where it goes wrong. The logics are now usually grouped under the title ‘non-classical logics’; and this book is an introduction to them.

Introduction

13.1.1 The family of free logics is a family of systems of logic that dispense with a number of the existential assumptions of classical logic.

13.1.2 In this chapter, we will look at the semantics of, and tableau systems for, various free logics.

13.1.3 We will then discuss how these logics handle some issues concerning existence.

13.1.4 Until further notice, we assume that the language does not contain the identity predicate. In the final part of the chapter, we will see how its addition affects matters.

Syntax and Semantics

13.2.1 The vocabulary of free logic is the same as that of classical first-order logic, except that we single out one of the one-place predicates for special treatment. Let this be. We will write this as, and think of it as an existence predicate. Thus, a can be thought of as ‘a exists’.

13.2.2 An interpretation for the language is a triple 〈D, E, ν〉, where D is a non-empty set, and E (the ‘inner domain’) is a (possibly empty) subset of D. One can think of D as the set of all objects, and E as the set of all existent objects. Thus, one might think of D as containing objects such as Sherlock Holmes, Pegasus and Julius Caesar. Only the last of these would be in E.

13.2.3 As in classical logic, ν assigns every constant in the language a member of D, and every n-place predicate a subset of Dn.

Introduction

11a.1.1 In standard modal logics, the worlds are two-valued, in the following sense: there are two values (true and false) that a sentence may take at a world. Technically, however, there is no reason why this has to be the case: the worlds could be many-valued. This chapter looks at many-valued modal logics.

11a.1.2 We will start with the general structure of a many-valued modal logic. To illustrate the general structure, we will look briefly at modal logic based on Łukasiewicz continuum-valued logic.

11a.1.3 We will then look at one particular many-valued modal logic in more detail, modal First Degree Entailment (FDE), and its special cases, modal K3 and modal LP. In particular, tableau systems for these logics will be given.

11a.1.4 Modal many-valued logics engage with a number of philosophical issues. The final part of the chapter will illustrate by returning to the issue of future contingents.

General Structure

11a.2.1 As we observed in 7.2, semantically, a propositional many-valued logic is characterised by a structure 〈V, D, {fc : c ∈ C}〉, where V is the set of semantic values, D ⊇ V is the set of designated values, and for each connective, c, fc is the truth function it denotes. An interpretation, ν, assigns values in V to propositional parameters; the values of all formulas can then be computed using the fcs; and a valid inference is one that preserves designated values in every interpretation.

Introduction

3.1.1 In this chapter we look at some well-known extensions of K, the system of modal logic that we considered in the last chapter.

3.1.2 We then look at the question of which systems of modal logic are appropriate for which notions of necessity.

3.1.3 We will end the chapter with a brief look at logics with more than one pair of modal operators, in the shape of tense logic. (This can be skipped without loss of continuity for Part I of the book.)

Semantics for Normal Modal Logics

3.2.1 There are many systems of modal logic. If there is any doubt as to which one is being considered in what follows, we subscript the turnstile (⊧ or ⊢) used. Thus, the consequence relation of K is written as ⊧K.

3.2.2 The most important class of modal logics is the class of normal logics. The basic normal logic is the logic K.

3.2.3 Other normal modal logics are obtained by defining validity in terms of truth preservation in some special class of interpretations. Typically, the special class of interpretations is one containing all and only those interpretations whose accessibility relation, R, satisfies some constraint or other.

Introduction

2.1.1 In this chapter, we look at the basic technique – possible-world semantics – variations on which will occupy us for most of the following chapters. (We will return to the subject of the conditional in chapter 4.)

2.1.2 This will take us into an area called modal logic. This chapter concerns the most basic modal logic, K (after Kripke).

Necessity and Possibility

2.2.1 Modal logic concerns itself with the modes in which things may be true/false, particularly their possibility, necessity and impossibility. These notions are highly ambiguous, a subject to which we will return in the next chapter.

2.2.2 The modal semantics that we will examine employ the notion of a possible world. Exactly what possible worlds are, we will return to later in this chapter. For the present, the following will suffice. We can all imagine that things might have been different. For example, you can imagine that things are exactly the same, except that you are a centimetre taller. What you are imagining here is a different situation, or possible world. Of course, the actual world is a possible world too, and there are indefinitely many others as well, where you are two centimetres taller, three centimetres taller, where you have a different colour hair, where you were born in another country, and so on.

Introduction

25.1.1 In this chapter, we will look at the addition of quantifiers to the Łukasiewicz continuum-valued logic.

25.1.2 We will then look at the behaviour of identity in this logic.

25.1.3 This will occasion a discussion of some philosophical issues concerning fuzzy identity, connected, in particular, with the sorites paradox and with vague objects.

25.1.4 A technical appendix describes the addition of quantifiers and identity to the general class of t-norm logics.

Quantified Łukasiewicz Logic

25.2.1 In the language we are concerned with, the set of connectives, C, is {∧, ∨, ¬, →}, and the set of quantifiers, Q, is {∀, ∃}. (A ↔ B can be taken as defined as (A → B) ∧ (B → A).)

25.2.2 As we saw in 21.2, an interpretation for a quantified many-valued logic is a structure 〈D, V, D, {fc : c ∈ C}, {fq : q ∈ Q}, ν〉. D is a non-empty domain of quantification. For every constant, c, ν(c) ∈ D, and for every n-place predicate, P, ν(P) is an n-place function that maps members of D into the truth values, V. In Łukasiewicz continuum-valued logic, V = [0, 1], the set of real numbers between 0 and 1, ordered in the usual way. f∨, f∧, f¬ and f→ are as in the propositional case (11.4.2).

Introduction

7.1.1 In this chapter, we leave possible-world semantics for a time, and turn to the subject of propositional many-valued logics. These are logics in which there are more than two truth values.

7.1.2 We have a look at the general structure of a many-valued logic, and some simple but important examples of many-valued logics. The treatment will be purely semantic: we do not look at tableaux for the logics, nor at any other form of proof procedure. Tableaux for some many-valued logics will emerge in the next chapter.

7.1.3 We also look at some of the philosophical issues that have motivated many-valued logics, how many-valuedness affects the issue of the conditional, and a few other noteworthy issues.

Many-valued Logic: The General Structure

7.2.1 Let us start with the general structure of a many-valued logic. To simplify things, we take, henceforth, A ≡ B to be defined as (A ⊃ B) ∧ (B ⊃ A).

7.2.2 Let C be the class of connectives of classical propositional logic {∧, ∨, ¬, ⊃}. The classical propositional calculus can be thought of as defined by the structure 〈V, D, {fc; c ∈ C}〉. V is the set of truth values {1, 0}. D is the set of designated values {1}; these are the values that are preserved in valid inferences. For every connective, c, fc is the truth function it denotes.