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1 Not Exactly in Praise of Vagueness by Kees van Deemter University of Aberdeen (Draft. Please do not quote) 2 Preface Vagueness is the topic of quite a few scholarly books for professional academics. This book targets a much broader audience. For this reason, the exposition has been kept as informal as possible, focussing on the essence of an idea rather than its technical incarnation in formulas or computer programs. Most chapters of the book (with the exception of chapters 8 and 9, which rely on the material explained in earlier chapters) are self-explanatory. In a few areas of substantial controversy, I have used fictional dialogues – a tried and tested method since Plato’s days of course – to give readers a sense of the issues. Although some things have been too complex to yield willingly to this treatment, it has been a delight to discover how much complex material can be reduced to simple ideas. On a good day, it even seems to me that there are themes that are best explored in this informal way, free from the constraints of an academic straightjacket. The fact that the book is informal may not always make it an easy read. For one thing, it requires a somewhat philosophical spirit, in which one asks why certain well-known facts hold. Moreover, we will not be content when we, sort of, dimly understand why something happens: we will ask how this understanding can be given a place in a known theory. Essentially this means that we will insist on trying to understand vagueness in a way that is compatible with everything else we know about the world. Existing books on vagueness, written for philosophers, computer scientists or linguists for example, often focus on either the sorites paradox or Fuzzy Logic. The present book deals with both of these topics, and with much else besides. Part I discusses a variety of areas where vagueness is difficult to avoid, by putting the spotlight on some corners where it lurks unexpectedly. This part of the book, which draws on various walks of life, is intended as much to entertain as to instruct. Part II presents theories that aim to shed light on the meaning of vague expressions, often as a response to the sorites paradox. This middle part of the book is for the theoretically inclined: others may want to give it a miss and move on to the next part. Applications in Artificial Intelligence play a role throughout the book. Part III puts these applications centerstage and uses them to ask why and when it is a good idea to use vagueness “strategically” in communication. Although the subject matter of this book has seldom formed the core of my work – which has often been more practically oriented – it has engaged me intermittently over several decades, during which time I have worked in different research environments in various countries (England, Scotland, the Netherlands and the United States). Its long incubation time has caused the book to be influenced by a large and varied set of people, many of whom are a considerable number of handshakes removed from each other, as long as my own are excluded. I thank .... 3 Authors who thank their colleagues and friends often assure us (for reasons that escape me!) that all remaining errors are their own. But surely, so a famous argument goes, these authors would never commit anything to paper that they did not believe to be true. Why then are they apologising for errors that they do not believe to have committed? – Even though I have no doubt that the present book contains its share of errors as well, I am confident that it will allow its readership to see through the fallacy that underlies this argument, sometimes known as the Paradox of the Preface. 4 Contents Preface Prologue Part I: Vagueness, where one least expects it. Chapter 1 Introduction: False Clarity Chapter 2 Sex and similarity: on the Fiction of Species Chapter 3 Measurements that Matter Chapter 4 Identity and Gradual Change Chapter 5 Vagueness in Numbers and Maths Part II: Theories of vagueness Chapter 6 The Linguistics of Vagueness Chapter 7 Reasoning with Vague Information Chapter 8 Parrying a Paradox Chapter 9 Degrees of Truth Part III: Working Models of Vagueness Chapter 10 Artificial Intelligence Chapter 11 Computers as Authors: when to be vague? Chapter 12 Original sin and the expulsion from Boole’s paradise Epilogue: In the Antiques Shop. Notes Index [NOT THERE YET] 5 A barometric low hung over the Atlantic. It moved eastward towards a high-pressure area over Russia, without as yet showing any incli- nation to bypass it in a northerly direction. The air temperature was appropriate relative to the annual mean temperature (...) In a word that characterizes the facts fairly accurately, even if it is a bit old-fashioned: It was a fine day in August 1913. Robert Musil, The Man Without Qualities ‘Private investigator, huh?’, he said thoughtfully. ‘What kind of work do you do mostly?’ ‘Anything that’s reasonably honest,’ I said. He nodded. ‘Reasonably is a word you could stretch. So is honest.’ I gave him a shady leer. ‘You’re so right,’ I agreed. ‘Let’s get together some quiet afternoon and stretch them.’ Raymond Chandler, The Little Sister 6 Prologue The world may be best measured in terms of neatly quantifiable entities such as millimetres, grams, and millibars, but in ordinary conversation, we tend to speak more loosely. The weather, for example, can be assessed by measuring the temperature in Fahrenheit or Celcius, the atmospheric pressure in millibars, and so on. Yet, this morning’s weather report is likely to speak of a cold day and, if we’re un- lucky, another low-pressure zone. Categories like ‘cold’ and ‘low-pressure’ are not sharply delineated but vague around the edges. This book asks why vague concepts – concepts that allow borderline cases – play such an important role in our lives, and discusses various explanations for this fact. We shall see that vagueness is inherent in all our dealings with the world around us. Vagueness may be likened to original sin: a stain that can be diminished but never re- moved. We shall also argue, however, that vagueness is not always a sin but sometimes a virtue. There are often excellent reasons for avoiding precision. This view has practical consequences. It follows, for example, that the “intelli- gent agents” that are being built in Artificial Intelligence laboratories will gain in usefulness once they manage to use vague concepts judiciously. The mathematician Georg Kreisel is famous for having argued that informal ar- gumentation, instead of meticulous proof, can sometimes be a mathematician’s most powerful tool. The main thesis of the present book might be seen as a remote echo of Kreisel’s: sometimes, one just has to be sloppy. In defending this claim, we shall not only discuss colloquial conversation, where sloppiness is only to be expected; we shall also be concerned with the exchange of serious factual and scientific information. If we can come to understand why vagueness pervades even such fault-critical situations then we shall have achieved a lot. In the same spirit, the book will often focus on relatively simple things. Where complex notions are involved – like justice, beauty or happiness, to name but a few – the very idea of precision is difficult to imagine. This book is an attempt to account for the role of vagueness in our lives. This means that we shall ask such questions as: Why do people make such frequent use of words whose meaning is difficult to pin down?, and What do these words mean? Why is it that their meaning varies so much from one context to the next? Are all vague concepts basically alike in all these respects, or are there important differences between them? Finally, we shall ask If we were to build a robot that can communicate, how precise would we like it to be when it speaks to us? These questions will touch on many academic disciplines, from symbolic logic and game theory to computing science and biology, and from linguistics and legal theory to medicine and engineering. This book is full of examples. Describing people comes naturally to us, which is why quite a few of these examples will be about people. An American friend 7 (who, like me, is exceptionally tall) once pointed out that, in his social circle, height is just about the only aspect of the human body about which one can talk freely. To identify a person as ‘an old woman’, ‘the bald guy’, or ‘the skinny girl over there’ would be frowned upon. Height alone – according to my cautious friend – is safely neutral. I do not know whether his claim is correct in all particulars, but his point is well taken. Where I can, I shall avoid offence. Where I fail, I hope to be forgiven by readers of all descriptions. 8 Chapter 1 Introduction: False Clarity A few years ago, the BBC carried a news story entitled Students feel unsafe after dark. The story was essentially identical to reports in other news media. The Times Higher Education Supplement, who were close to the source of the story, wrote “The research (...) reveals that while students generally feel secure during the day, fewer than four in ten feel safe all of the time (...).” (THES, 16 April 2004) Worries about safety can be justified, yet one wonders what to make of a report of this kind. The research itself may be valid (see the footnotes at the end of the book) but by writing as they did, the journalists made it sound as if feel- ings of safety are an all-or-nothing affair.