The Connectives

Cover

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Title Page, Copyright, Dedication

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Contents

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pp. vii-xii

Preface and Navigation Guide

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pp. xiii-xviii

This book is about the semantics and pragmatics of natural language sentence connectives and about the properties of and relations between analogous devices in the formal languages of numerous systems of propositional (or sentential) logic. And the intended readership is: all who find connectives, and the conceptual issues ...

0. Preliminaries

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pp. 1-46

Given n non-empty sets S₁, . . . , S_n and a relation R ⊆ S₁ × · · · × S_n, we call the structure (R, S₁, . . . , S_n an n-ary relational connection. When s_i ∈ S_i (for i = 1, . . . , n) we often write Rs₁, . . . , s_n for s₁, . . . , s_n ∈ R, using the familiar ‘infix’ notation s₁Rs₂ for the case of n = 2. ...

1. Elements of Sentential Logic

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pp. 47-194

This chapter and the next present some basic material in sentential logic. The intention is to stress those topics of particular relevance to what follows later, as well as to highlight the way individual connectives fare in the general development. We begin our discussion of logical matters by introducing the idea of a formal language. ...

2. A Survey of Sentential Logic

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pp. 195-374

There are two related respects in which our treatment of semantic matters up to this point might be generalized. As has been remarked, our boolean valuations are homomorphisms from the language on which we concentrated in §1.1 (with connectives ∧, ∨, ¬, ⊤, ⊥) to the two-element boolean algebra. ...

3. Connectives: Truth-Functional, Extensional, Congruential

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pp. 375-510

This chapter introduces and compares the three properties of connectives listed in its title. As they appear in the list, these properties go from the strongest, truth-functionality—the topic of the present section—to the weakest, congruentiality, to be reviewed in §3.3, via an interestingly intermediate property called here extensionality (§3.2). ...

4. Existence and Uniqueness of Connectives

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pp. 511-630

In this chapter, we will discuss the existence (§4.2) and uniqueness (§4.3) of sentential connectives in the sense of connective in which connectives are not simply syntactic operations, but operations building formulas with certain inferential properties and relationships (as in 1.19). ...

5. “And”

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pp. 631-766

Before passing to the three aspects of the study of natural language alluded to in the title of this subsection, a word by way of reminder should be said on the subject of logic. Here syntactic and semantic considerations also arise, in the shape of proof-theoretic or formal conditions on (generalized) consequence relations and of conditions on valuations. ...

6. “Or”

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pp. 767-924

Before beginning with an introduction proper, giving a brief overview of this section and of the rest of the chapter, we offer, by way of an hors d’oeuvres – an “Or” d’oeuvres, we might say – a glimpse at a problem besetting studies of this most intriguing of the sentence connectives, the word or. ...

7. “If”

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pp. 925-1162

This chapter addresses not only some of logical and semantic issues raised by ‘if__ then. . . ’, but also the behaviour of related connectives which have been proposed to formalize various special relations of implication and entailment. We associate the label conditionals with the former project ...

8. “Not”

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pp. 1163-1286

For a direct parallel with the preceding three chapters, we should open with an extended discussion of the points at which the treatment of negation in formal logic has been held inadequately to represent the behaviour of not (or it is not the case that) in English or corresponding locutions in other natural languages. ...

9. Universally Representative and Special Connectives

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pp. 1287-1336

This chapter introduces and compares two properties a connective may have or lack in a logic: the property of being ‘universally representative’ and the property of being what we shall call ‘special’. It will be sufficient to treat logics as individuated by the set of sequents they contain. ...

References

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pp. 1337-1439

Index

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pp. 1440-1492