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1 Not Exactly in Praise of Vagueness 1 by Kees van Deemter University of Aberdeen (Draft. Please do not quote)

2 2 Preface Vagueness is the topic of quite a few scholarly books for professional linguists, philosophers and computer scientists. This book targets a much broader audience. For this reason, I have kept the exposition as informal as possible, focussing on the essence of an idea rather than its technical incarnation in formulas or computer programs. In a few cases where there is substantial controversy about the right approach to a problem, 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 informal 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. The fact that the book is informal does not necessarily make it an easy read. For one thing, it requires a certain 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: in many cases we will ask how this understanding can be given a place in a known model or theory (such as Symbolic Logic or Game Theory). Essentially this means that we will insist on understanding vagueness in a way that is compatible with everything else we know about the world. Existing books on vagueness often focus on one of two topics known as the sorites paradox and Fuzzy Logic. The present book will deal with both of these, and with much else besides. The structure of the book is as follows: Part I demonstrates how vagueness can be unavoidable, by discussing a variety of areas where vagueness plays a role, even where it is least expected. This part of the book, which draws on examples from biology, medicine, law and engineering, is intended as much to entertain as to instruct. Part II presents logical and linguistic theories that aim to shed light on the meaning of vague expressions, often as a response to the sorites paradox. This 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 focusses on the question why and when it is a good idea to use vagueness strategically, when it would have been possible to be more precise. Because the subject matter of this book has engaged me intermittently over several decades, and because it touches on a number of different disciplines, my thanks go to a varied set of people, many of whom are a substantial number of handshakes removed from each other, as long as my own are excluded. I thank... Authors frequently thank their colleagues, then continue humbly that all remaining errors are their own. But surely, (so a famous argument called the

3 Paradox of the Preface goes) these authors would never commit anything to paper that they did not believe to be true. Why 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, I am confident that it will allow its readership to understand the fallacy that underlies the Paradox of the Preface. 3

4 4 Contents Preface Prologue Part I: Vagueness, where one least expects it. Chapter 1 Introduction: Vagueness and Crispness Chapter 2 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 Chapter 12 Something like a Conclusion Epilogue: In the Antiques Shop.

5 5 A barometric low hung over the Atlantic. It moved eastward towards a high-pressure area over Russia, without as yet showing any inclination 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 Why are we satisfied to speak vaguely of a red nose, without specifying what shade of red even though degrees of red can be stated precisely to the micromillimeter of a wavelength (...)? Robert Musil, The Man Without Qualities

6 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 unlucky, 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 tendency. 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 removed. 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 intelligent 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 argumentation 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. The primary purpose of this book is 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 to me 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. 7

8 8

9 Chapter 1 Introduction: Vagueness and Crispness 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 To get the subject of this book squarely in our sights, let us start with some examples from daily life, where information is presented as if everything was clear cut, but where clarity disappears under scrutiny. 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 (...). Only a quarter of students in the capital feel safe travelling to and from the university campus at night. (THES, 16 April 2004) Sadly, worries about safety can be justified, yet one wonders what to make of a report of this kind, when it does not clarify the terms that it uses. By writing as they did, the journalists made it sound as if feelings of safety or unsafety are an all-or-nothing affair. But feelings come in degrees, and it is doubtful that there exists a generally-agreed point where a feeling of safety suddenly turns into one of unsafety. In the absence of such a point, the percentages in the news report 9

10 10 CHAPTER 1. INTRODUCTION: VAGUENESS AND CRISPNESS are meaningless. Feelings may be particularly difficult to quantify, but that s not the point. The point is that numerical information is often taken for granted when inquisitive questions should have been asked. We lament high incidences of failing schools, violent crime, and obesity, without questioning the norms that underlie these assessments. Yet, before you can count something, you have to know what you are counting. Consider obesity: we are told that 40% of British children are obese these days, and we swallow the information. Yet no matter how slim they are, one could always put the threshold for obesity at such a point that the percentage of obese children is 40%. If someone told you that 40% of people are tall, you would not take him seriously. Obesity may be different, but why exactly? It has been observed that people have a tendency to paint reality in black and white, rather than richly varied grey tones. We seem to like clarity so much that we see it where it does not exist. The terms black and white themselves are an interesting example, particularly when applied to skin colour. We are all familar with predictions saying that this or that city in Western Europe will, in this and this future year, have a majority of black inhabitants. Such figures are seldom called into question by inquiring just how dark one s skin has to be to count as black, a question that becomes harder and harder to answer as the number of children from diverse parentage grows. The biologist Richard Dawkins has a catchy phrase for the sleight of hand that allows us to think in black and white both literally and figuratively calling it the tyranny of the discontinuous mind. We shall soon see what drove him to this characterisation. These are not isolated examples: even the cornerstones of our mental vocabulary are affected. The important notion of causation is a case in point: court cases have been fought over the question whether smoking causes cancer, for example, even though the likelihood of cancer is affected by many different factors, so the notion of a cause is problematic. It is true that smoking is a very strong factor, and this could arguably justify the simplification. Other cases, however, suggest a genuine lack of awareness that causation comes in degrees. Blastland and Dilnot, in their recent book The Tiger that Isn t discuss cases where scientists are reported to have found the gene for multiple sclerosis or asthma. In fact, however, the gene is present in a percentage of the people suffering from the disease that is only slightly higher than in the general population. Simplifications of this kind can be misleading, because they overstate the importance of a discovery. Other authors have made similar observations. John Allen Paulos, for instance, targetted the notion of food safety. He reports on a clause of the American FDA Act of 1958, which requires that no food additive shall be deemed safe if it is found... to induce cancer in man or animal, but without specifying a minimum allowed level of each of the relevant substances. We like to pretend that there is a sharp division between substances that are safe and ones that are poisonous

11 11 but, in reality, there is only a continuuum, with water at one extreme perhaps (because it takes many litres, swallowed in quick succession, to kill a person), chemical weapons at the other, and things like salt and alcohol somewhere in between. Once again, we think in black and white, whereas reality is subtly shaded. Examples of what Blastland and Dilnot call false clarity might be easiest to find in the social and medical domains, but examples in history and geography abound as well. It is often thought, for example, that expressions like the Great Wall of China and the Silk Road denote well-defined entities. Yet the former only denotes a loosely delineated group of spatially separated walls which are extremely different from each other in terms of their structure, height and age; one person s Great Wall is not necessarily someone else s, because it is by no means obvious which walls are part of the great one. Similarly, the Silk Road denotes a diffuse network of paths all of which were important before ships replaced camels as the vehicles of choice for travel between East and West. In both cases it would be easy to point at boundary cases: a path, for example, that was only used during severe winters. The idea that there is a definite thing to which words like the Silk Road refer is largely illusory. It is sometimes useful to simplify a bit by thinking in black and white. In some areas, extremes are the norm: by and large, water is either frozen or fluid, and a person is either dead or alive (but see our chapter on Identity and Gradual Change). Even if you hear that someone has the flu or is in love, you may not ask How badly?, since you know how uncompromising these afflictions tend to be. In such cases the world around us might be argued to support our liking for clarity. But we bring the same attitudes to situations where nuance is vital. In the middle part of the book, for example, I shall argue that people s tendency to think in terms of all or nothing may be responsible for the attitude that students of language and communication have brought to vagueness. For now, let me note another, more dangerous consequence of this tendency of ours, relating to public life rather than academic thought. The point is that false clarity leaves us open to manipulation, as when politicians redefine words for political gain. In the nineteen eighties, for example, Margaret Thatcher s successive Tory governments are reputed to have re-defined the crucial notion of unemployment dozens of times, narrowing it further and further, thereby allowing the figures to be polished as more and more people lost their jobs. If we were more aware that concepts like unemployment are not cast in stone then leaders might be less tempted to lead us up the garden path. Politicians are not on their own in the deception game of course: sales people must surely count as the champions of the genre, and even academics join the game when reporting about their research and begging for funding. We are all used to hearing consumer products being recommended for being powerful (when it s a hoover or a car engine), healthy (when it s food), fast (when it s a car or a phone), or excellent value (when it s pretty much anything),

12 12 CHAPTER 1. INTRODUCTION: VAGUENESS AND CRISPNESS even though it would be very difficult to test such claims, because the words in question are essentially undefined. Research, these days, is invariably excellent, innovative and world leading. My personal favourites are the assurances that some new product is simply better, leaving it to our own imagination to decide what it is better than. It is easy to see why such claims are left vague: if you claimed that your product is better than some particular product, in some well-specified respect, you might be proven wrong, with potentially unpleasant consequences. What is remarkable is that meaningless claims are nevertheless thought to be persuasive. Why else do companies pay good money for them?

13 13 (TO BE DISPLAYED IN SEPARATE BOX) BBC News, Wednesday 26 November 2008 Apple made to drop iphone advert An Apple iphone advert has been banned by the advertising standards watchdog for exaggerating the phone s speed. The advert boasted the new 3G model was really fast and showed it loading internet pages in under a second. The Advertising Standards Authority (ASA) upheld complaints by 17 people who said the TV advert had misled them as to its speed. (...) After upholding the viewers complaints, the ASA said the advert must not appear again in the same form. Apple said its claims were relative rather than absolute in nature - implying the 3G iphone was really fast in comparison to the previous generation - and therefore the advert was not misleading. The company also said the average consumer would realise the phone s performance would vary - a point they said was made clear by the text stating network performance will vary by location. The following chapter will flesh out one case of false clarity: the clarity associated with the biological notion of a species. But before we go there, let us define our theme more precisely and look ahead towards the rest of the book. Vagueness Having found that what looks like an all-or-nothing affair is often, on closer inspection, a matter of degrees, let us put the spotlight on words that wear their vagueness on their sleaves, so to speak, making it very evident that they are vague. I use the word vague in a specific sense: a sense of the word that is common in academic books but less so in ordinary conversation. Let me explain. A concept or word is called vague in this book if it allows borderline cases. Grey is a vague concept, for instance: some birds are grey and some bird are not, but others are borderline grey, for example because they are so dark that you might be tempted to call them black. Good people may differ on whether to call them grey or not. The fact that such grey-ish birds can exist makes grey a vague concept. Words like large and small, many and few, are all vague for the same reason. Fewer than five, by contrast, is not vague in the sense of this book, because it creates a sharp division between two classes of numbers. There is no difference of opinion on whether four counts as fewer than five. Until further notice, we shall use the word vagueness to denote expressions such as grey, because they allow borderline cases. (Expressions that are not vague will be called crisp, for lack of a better word.) It is only in the chapter Measurements that Matter that this standard use of the word vagueness will be challenged.

14 14 CHAPTER 1. INTRODUCTION: VAGUENESS AND CRISPNESS In the course of this book, it will become clear that vagueness is everywhere: if you believe a concept to be completely crisp, then examine it more closely and it will often prove to be vague. Size-denoting terms like small and large are obviously vague, for example but, to a lesser extent, so are colour terms, at least in ordinary language, where sharp boundaries are not artificially imposed on them. Crucially, this is no accident of English. Travel anywhere, and you will find that colour terms are vague. Even when artificial languages like Esperanto are created and taught, little is done to explain the meaning of vague expressions, other than by translating them to words in existing languages. The relevant sense of granda, in Esperanto, translates to English tall,for example, but whether this makes a man of 175cm granda is left to the imagination of the learner. The aim of this book is to explore how vagueness works, and why it pervades communication. It is part and parcel of this enterprise to ask why vagueness is not always a bad thing: we shall see that sometimes vagueness is simply unavoidable (for example, when our observations do not allow us to be precise) while, on other occasions, vagueness is actually preferable to precision. We shall also devote considerable space discussing the implications of our findings for the construction of Artificially Intelligent systems such as speaking robots which are slowly starting to be endowed with a human-like capacity to produce and understand vague words. But vagueness does not only pose practical problems: it also poses difficult theoretical challenges. Paradox In October 2007, The List Universe, a web site devoted to top ten lists, voted the sorites paradox, also known as the Paradox of the Heap, one of the 10 greatest unsolved problems of science. (Other problems included the existence of black holes, the cause of the Great Depression, and the chemical origin of life.) The mechanism through which the list was composed is not known to me, but I do not contest the weight of the problems posed by the sorites paradox and its relatives, which will play a central role in this book. I will introduces these paradoxes informally here; a more detailed discussion follows in later chapters. The sorites paradox. In the sixth century BC Euboulides of Milete gave us the following puzzle, centering around the vague word stoneheap (Greek: sorites). One stone does not make a stoneheap, Euboulides observed. But if something is too small to be a stoneheap, you cannot turn it into a stoneheap by adding just one stone. Clearly then, two stones do not make a stoneheap either. But by the same reasoning, nor do three stones, and so on. Consequently, no finite number of stones can ever make a stoneheap. I take it that no sane person has ever been swayed by the argument, just like no-one has ever been swayed by the famous proof that Achilles can never beat a turtoise in a run-

15 15 ning contest. The challenge is to say what exactly is wrong with the argument, and to offer a principled approach to reasoning with vague concepts that does not have strange consequences like sorites. Because it lies at the heart of the problem of vagueness, we shall look at various forms of the sorites argument in detail. And because it is a challenge to a mathematical approach to reasoning, known as Classical Logic, our story will require a certain (very modest) amount of mathematical symbolism. I was educated in a hard-nosed research tradition in which language is analysed as a means for expressing statements about the world. Human beings, in this view, are a somewhat inconvenient species who have unfairly manoeuvred themselves in between language and the world. But in the case of vague language, it is hard to say much about language without involving our perception of the world. To see what I mean, consider a variant of Euboulides paradox, located in the laboratory of a manufacturer of Hi-Fi music equipment. The human ear has limited sensitivity: some sounds are too soft to be audible. Likewise, the ear has limited resolution: a difference of 0.5 db is almost certainly too small to be perceived by any person. Now a modern version of Euboulides paradox is obtained as follows: A sound with a loudness of -30 db is too weak to be audible; if it is amplified to db, it must still be inaudible (since 0.5 db cannot make the difference); but then a further amplification to -29 db must also be inaudible, and so on, ultimately implying that even a sound of 150dB well above the average person s pain threshold must also be inaudible. Euboulides has struck again! This version is harnassed by science: we actually know that a difference of 0.5 db is undetectable. A curious difference between scientific disciplines is worth mentioning here: acousticians are not too perturbed by this paradox, because they do not care too much about the meaning of a word like audible : they are happy if they can model a person s hearing, and their models do not require them to define this concept precisely. For students of language, cognition and communication, however, the paradox is harder to push aside, since they earn their bread building models of the meaning of words, sentences, and so on. This difference in attitude between academic disciplines will be further explored in the chapter on Degree Theories. The sorites paradox plays tricks on us every day. A nice collection of examples can be found in a recent bestseller by Penn and Zalesne, entitled Microtrends. Chapter after chapter, the book offers up changes in society that have gone so slowly and gradually that they have managed to stay under the radar for a long time, even though the changes are so substantial that everyone would have noticed had they occurred overnight. One example is the gradually increasing percentage of women in America who stay unmarried; another is the gradually decreasing amount of time that people sleep per night. It is said that a frog that falls into hot water will jump out immediately; but if it is sitting in water that heats up gradually, then reportedly the poor animal will fail to notice

16 16 CHAPTER 1. INTRODUCTION: VAGUENESS AND CRISPNESS the change, and allow itself to be boiled. (Don t try this at home.) In many respects, people are just like frogs. Although we shall pay a fair amount of attention to the sorites paradox, it will not be studied in isolation, but connected with everything else we know about perception and communication. What, after all, is the use of solving one riddle by creating a new one? Synopsis of the book The core of the book can be divided into three parts, each of which can be read on its own. Part I (comprising chapters 2-5) offers a Survey of vagueness in human endeavours, demonstrating that vagueness pervades all of them, and introducing informally some of the main issues connected with vagueness. Part II (comprising chapters 6-9) focusses on linguistic and logical Theories of Vagueness, many of which can be seen as responses to the sorites paradox. Part III (comprising chapters 10-12) focusses on Working Models of Vagueness and the question when vaguenes plays a useful role in communication. As befits a book on vagueness, these parts are not crisply separated from each other, particularly because because applications are well represented in large sections of Part II. Part I: Vagueness, where one least expects it. This part of the book has an exploratory and introductory character. Chapter 1 (Introduction: Vagueness and Crispness) is where we are now. Various cases of false clarity have been described, and their relevance for society has been discussed. The main themes of the book have been introduced (see also the Preface), and a synopsis of the book is being offered. Chapter 2 (The Fiction of Species) examines the biological concept of a species, which is often thought to be crisp (i.e., not vague) but which, on closer inspection, constitutes another case of false clarity. The idea of defining species in terms of inter-breeding will prove to be incoherent. In Chapter 3 (Measurements that Matter) we explore the concepts of intelligence, poverty, obesity, and statistical significance. We follow the development of these concepts over time, and to examine what factors have shaped them. Science will emerge as an important determinant of the way in which vague concepts are made artificially crisp. Chapter 4 (Identity and Gradual Change) shows how vagueness complicates the concept of identity. If a car is repaired very often, is it still the same car? Some practical instantiations of this questions will be discussed, including a famous court case involving a vintage racing car. Chapter 5 (Vagueness in Numbers and Maths) shows that numerical precision is often an illusion. The chapter concludes with discussions of two

17 17 areas of mathematics where subtler perspectives on quantity are necessary: statistical significance and computational complexity. Part II: Theories of vagueness This part of the book focusses on logical and linguistic theory. Chapter 6 (The Linguistics of Vagueness) introduces some basic tenets of Linguistics, explaining how the form and meaning of sentences can be modelled, and introducing the notions of ambiguity, vagueness, and nonspecificity, which are of particular relevance to later chapters. Special attention will be devoted to the various ways in which linguistic expressions are affected by the context in which they are uttered. Chapter 7 (Reasoning with Vague Information) is the first of two chapters where Symbolic Logic will be shown informally at work, showing that vagueness is not always the mortal enemy of logic. We describe the sorites paradox, discuss its ancient roots in Aristotelian logic, and explain why the paradox is a challenge for Symbolic Logic. Chapter 8 (Parrying a Paradox) asks what it would mean to solve the sorites paradox. Some of the more cautious appraoches to the paradox are discussed, all of which stay close to the oldest and most venerable variant of Symbolic Logic, known as Classical Logic. We also discuss a new way of looking at the paradox, inspired by the idea of a speaking robot. Variation in human judgment will emerge as the main difficulty for the approaches discussed in this chapter. Chapter 9 (Degrees of Truth) This chapter turns to a set of more revolutionary responses to the paradox, which add other truth values to the usual two (i.e., true and false). We discuss the best known member of this family, named Fuzzy Logic, and argue that it has brought undeserved disrepute to the family as a whole. We argue that a probabilistic degree theory offers the best prospects, but that its sophistication comes at a price because, like all degree theories, it complicates our view of the word. Part III: Working Models of Vagueness This part of the book deals with Artificial Intelligence and Natural Language Generation. Chapter 10 (Artificial Intelligence) offers a quick introduction into Artificial Intelligence (AI). After a brief history of the subject, we exemplify some of the ways in which AI has sought to deal with vagueness and degrees. The main areas covered here are fuzzy expert systems and Qualitative Reasoning. The Turing test is presented as an important perspective on assessing the quality of computer programs that interact with people. Chapter 11 (Computers as Authors) zooms in on Natural Language Generation, the part of AI which involves the construction of computer programs that write and speak. We show how vagueness makes it possible to avoid information overload, and to add useful bias to an otherwise

18 18 CHAPTER 1. INTRODUCTION: VAGUENESS AND CRISPNESS neutral piece of information. Ideas from Game Theory will shed further light on these issues. Referring expressions will be used as a case study. Chapter 12 (Something like a Conclusion) summarises the main themes of the book and highlights a number of issues that had so far allen by the wayside, such as the phenomenon of hedging, the psychological theory of Prototypes, and the legal theory of Open Texture. We shall also ask how vague concepts differ, using colour perception as an illustration. More systematically than before we ask When does it pay to be vague? Perspectives on vagueness Let us say a few things about the academic subjects that will dominate this book: Symbolic Logic, Linguistics, Artificial Intelligence and, to a smaller extent, Game Theory. Symbolic Logic is the academic discipline whose goal it is to explain the idea of a valid (i.e., correct) argument, and to do so in a mathematically rigourous way. Euboulides paradox is a case of logical argumentation gone mad, so if we want to understand exactly what is wrong with sorites, Symbolic Logic is a natural place to look. Symbolic Logic is most often associated with classical logic, which is essentially the oldest and most well-established variant of Symbolic Logic. The basic principles of classical logic will be sketched in chapter 8. The main thing worth knowing about it is that it uses a simple dichotomy between true and false. No statement can be both true and false, and no statement can be neither, which is what logicians call the Law of Excluded Middle. Euboulides paradox appears to challenge this Law of Excluded Middle, which makes it hard to deal with subtleties like something being almost or very true. Classical logic has a hard time making sense of statements like this collection of stones is a rather marginal stoneheap, or this sound is barely audible. Some logicians have argued that this is a flaw in classical logic, and constructed alternative logical systems, in some of which true and false are no longer the only options. The middle part of the book will be devoted in large part to these developments. The second main ingredient of this book is Linguistics: the science of language. Linguists have become a modest tribe these days. Long gone are the schoolmasterly days when they told the rest of us of how to speak or write: linguistics has become an emphatically empirical (i.e., observation-driven) enterprise. Because observation of human behaviour lies at the heart of this method, it is ordinary speakers and writers who determine collectively whether linguists are right not the other way around. The present book does contain a certain amount of work that fits into this type of linguistics, but much else as well. This is partly because I will be asking why we speak or write the way we do, a question that will steer us towards psychology and game theory. It is also because I feel a certain reluctant affinity with philosophers like Bertrand Russell, who did not

19 19 shy away from occasionally criticising the way language works. Artificial Intelligence (AI) is that part of Computing Science where programs are built that mimic such abilities, including reasoning and communicating, many of which are infested by vagueness. AI involves the construction of working models (e.g., computer programs) of a human ability, which is a good way to learn what that ability amounts to. The same spirit has moved some archeologists, for example: to understand what it must have meant to build a particular ancient building, they do not stop at passively analysing its structure: they proceed to construct a very similar building, using original materials. (Reputedly, this approach led to the discovery that rice was used in the construction of the Great Wall of China.) This constructive spirit is one of the attractions of AI. One of the important challenges for AI is to build programs that communicate effectively with people. We will have ample opportunity to investigate this challenge, culminating in the question when and why it is helpful to communicate vaguely. This challenge brings us to the last of the disciplines from which we will borrow substantially, namely Game Theory. Symbolic Logic is well equipped to answer questions about meaning and inference, but it is less obvious that it has something to say about social interaction and the ways in which linguistic utterances form a part of it. For this purpose, we shall now and then look at Game Theory the mathematics of rational social interaction for inspiration. Game Theory will come to our aid when we try to reason about the utility of an utterance and give us a vocabulary to talk about the usefulness of vagueness. Things to remember This book is concerned with the role of vague words (and other expressions) in communication. By definition, vague expressions admit borderline cases. Expressions that are not vague will be called crisp. The main questions posed by this book are How the meaning of vague expressions is best understood, Why vagueness is so prevalent in human communication, and Under what circumstances vagueness is preferable over crispness. We shall be alert to any differences that may exist between different types of vagueness. Anecdotal evidence suggests that vague concepts are often treated as if they were crisp, without clarification of the thresholds that were employed to make them crisp. The result is a quasi-precision that can end up misinforming. Many examples of this phenomenon can be found in recent books by Blastland and Dilnot, by Paulos, and by Dawkins. The book consist of three parts, each of which may be read independently from the others. The main academic protagonists of this book are the disciplines of Symbolic Logic, Linguistics and Artificial Intelligence.

20 20 CHAPTER 1. INTRODUCTION: VAGUENESS AND CRISPNESS Subsidiary roles are played by other areas of Computing Science and by Philosophy, Game Theory and Psychology.

21 21 Part III Vagueness, where one least expects it

22 22 CHAPTER 1. INTRODUCTION: VAGUENESS AND CRISPNESS

23 Chapter 2 The fiction of species We have seen that vagueness affects many of the qualities that we ascribe to people and things. But surely, this is not true for the central concepts that we use to classify the things around us? It is one thing to say that words like tall, or even cause, lack crisp boundaries, but it would be something else to claim that words like man, chimpanzee and tiger are also vague. Roughly speaking, the distinction appears to coincide with the distinction between English adjectives and nouns: adjectives (such as tall ) denote subtly varying qualities and may therefore be vague, but nouns (such as tiger ) denote natural classes of things and are therefore crisp, one might hope. To show that this hope is illfounded, we will now focus on a central building block of our thinking at a level of common sense, but in biological taxonomy too which is the concept of a species. Species, of course, is a somewhat abstract word: Fido may be a dog but that does not make Fido a species. The concept dog itself is a species. Likewise, common chimpanzee is a species, and so is Homo sapiens (i.e., man). Species are the bedrocks of biology, far stabler than more inclusive biological groupings such as genus, class and order, and also stabler than less inclusive groupings, which subdivide the species. All of these other groupings are more difficult to justify scientifically; equally, none are as entrenched in everyday conversation as the names of (at least some) species. One might therefore expect that species-denoting terms have well-defined, crisp borderlines. Historically, biologists only started thinking systematically about these matters fairly recently. Even the famous Linnaeus who put biologists thinking about species on a solid footing (around the year 1750), gave every appearance of believing that there is not much of a problem here, at least in principle. All species are different from each other, aren t they, so it is just a matter of working hard to discover what the differences are. But in later years, particularly when it became plausible that species had developed gradually over time (rather than being created collectively in one mighty gesture), biologists realised that it would be rather nice to have a firm principle for deciding whether two animals belonged to the same species. Around 1940 this realisation had started to 23

24 24 CHAPTER 2. THE FICTION OF SPECIES culminate in something approaching consensus. In what follows, let us sketch what this near-consensus amounts to. Simply put, a species was defined to be a group of organisms whose members interbreed with each other. The idea is essentially the following: when looking for boundaries between species, not just any boundary will do: we want a species to consist of organisms that are reasonably similar to each other in important respects. (What else is the point of grouping them together?) But how similar exactly? How do we prevent a situation in which each individual biologist has his own idiosyncratic understanding of what it takes to be a lion? Biologists came up with an elegant idea, namely to invoke interbreeding as a criterion: if two animals of different sexes can interbreed then they belong to the same species, otherwise they are not. The beauty of the idea lies in the fact that it uses the crisp concept of interbreeding to give sharpness to what would otherwise threaten to be a fuzzy boundary. This is done on the plausible assumption that if two animals are similar enough to interbreed then their offspring must once again be quite similar to the parents. (Nature could conceivably have worked differently, for example by making offspring as different form their parents as possible, but this is clearly not what we see around us.) But is interbreeding a well defined concept, and is it as crisp as one would like it to be? On reflection, some uncomfortable questions may be asked. For example, The notion of inter-breeding only applies to organisms that reproduce sexually, so the standard definition does not apply to other species. For our purposes, I propose not to worry too much over this objection. Let s leave single-celled organisms and other celibate life forms aside and concentrate on the rest of us. Horses and donkeys (and reputedly even lions and tigers) can produce offspring together, but none that is fertile, so their mating does not have any long-term effects. Presumably, fertility of offspring should be taken into account in the definition of a species. Some types of animals that do not interbreed under normal circumstances can be induced to interbreed. Not much encouragement is needed, in some cases. Should these animals be counted as interbreeding with each other or not? If we really believed that two animals have to be able to interbreed to belong to the same species, then two men could not both be human, because they cannot interbreed. And, in all likelihood, your grandmother (assuming she is too old to have children) would belong to a species with just one member. Clearly, the notion of interbreeding has to be taken with a pinch of salt, disregarding such trifles as age and gender. Chihuahuas and Great Danes do not produce puppies together, but this is arguably for no deeper reason than their difference in size (although a lack of inclination might play a subsidiary role). Does this justify regarding

25 25 them as different species? The standard view would answer this in the negative. Some groups of animals fail to interbreed solely because they are geographically apart from each other. If only that waterfall, mountain range, or stretch of desert didn t exist, they would happily interbreed. (I m reminded of school trips during which our teachers tried to keep us in our own sleeping quarters, separate from the girls.) It seems reasonable to disregard geographical separation, and focus on whether two animals could interbreed, if given a reasonable chance. Temporal separation can have the same effect as geographical separation: the fact that you are unlikely to have children with any of your great grandparents does not mean you belong to a different species. Taking complications of this kind into account, and disregarding organisms that do not reproduce sexually, a species is usually thought to be something like the following: Species: a maximally large group of animals, such that healthy young specimens of the right age and sex are able in principle to produce fertile offspring under favourable circumstances such as occur naturally. One might think that all the obvious wrinkles in the notion of a species have now been ironed out. Enter the Ensatina salamander. Ensatina salamanders live along the hilly edge of California s Central Valley. They tend to avoid the centre of the Valley, presumably because of the heat there. Ensatina comes in six-or-so forms, which are usually viewed as its subspecies. Two of these, Ensatina eschscholtzii and Ensatina klauberi (both of which live in the south of the valley, with eschscoltzii dominating one side and klauberi the other) do not interbreed with each other, but a reasonable case can nevertheless be made that they belong to the same species: eschscholtzii does mate with a third subspecies living just north of it; these mate with a fourth subspecies, these mate with a fifth; and these, in turn, mate with our old friends Ensatina klauberi. The reasons why two salamanders can or cannot get fertile offspring are buried somewhere in their biology. For us, however, these reasons are not what matters: we are only interested in the facts on the ground, so to speak. FIG (perhaps plate 21 from Dawkins, or the original from Stebbins 2003, or some abstraction) Let us now schematise the story a bit, focussing on those aspects that matter most to us. In doing so we will use a broad brush. A more precise account would talk about individual animals. It will be convenient, however, to simplify a little, by grouping all the salamanders in a particular sub-species (e.g., all the members of Ensatina eschscholtzii) together, pretending that they are

26 26 CHAPTER 2. THE FICTION OF SPECIES all alike. (Having read the next page, you probably won t have difficulty reconstructing the story in a more precise style, focussing on individuals rather than subspecies.) We arrange the six kinds of Ensatina in a sequence from E 1 (this is eschscholtzii) to E 6 (this is klauberi), in such a way that each member of the sequence interbreeds with the previous one and the next one in the sequence: E 1 E 2 E 3 E 4 E 5 E 6 For concreteness and in order not to make the creatures look more promiscuous than necessary let us assume that this shows all the interbreeding that goes on: E 1 does not interbreed with E 3,.., E 6, for example; similarly, E 2 does not interbreed with E 4,.., E 6, and so on. If these are the facts about breeding, what does this mean for the definition of Ensatina? Do all six types of salamanders count as Ensatina? The answer, based on the standard definition of species, is No! Sure enough, E 1 and E 2 form a species together, and so do E 2 and E 3. But E 1 and E 3 do not, hence they do not belong to the same species. Note, however, that the two species that we have just found ({E 1, E 2 } and {E 2, E 3 }) overlap, because E 2 occurs in each of the two. Instead of being nicely separate chunks of reality, these two species end up all intertwined. This, surely, is not how biologists (let alone us regular folk) like to think about animals. A director of a large multinational company once remarked that he wanted to think of his company as a plate of asparagus, with all divisions and components neatly defined and separate, not as a plate of spaghetti, on which everything was all mixed up and intertwined. I suppose we re essentially all a bit like him when we try to make sense of the world around us. PICTURE How have biologists responded to this problem? Their dominant reaction appears to be to let common sense prevail over formal definitions. The standard view of the situation goes roughly as follows: E 2 belongs to the same species as E 1. But E 3 belongs to the same species as E 2, therefore E 3 must belong to the same species as E 1. The argument can be repeated for E 4 : it belongs to the same species as E 3, therefore it too must belong to the same species as E 1. Similary, the argument can be repeated for E 4, for E 5, and for E 6, showing that all six types of salamanders belong to one big species. This standard view, however, lumps together all the different types of salamanders into what is sometimes called a ring species. This includes E 1 and E 6, which do not interbreed together and which would therefore not belong to the same species, if the standard definition of species had anything to do with it. Fortunately, E 1 and E 6 are still a bit similar they are all ordinary salamanders. If there happened to be lots of other animals in the world, then perhaps there would exist animals E 7, E 8, and so on, each of which a little different from the previous one, includ-

27 27 ing some E n (where n is some seriously large number) which is entirely unlike any salamander... and a biological version of the sorites paradox would arise. How should we view the story of the Ensatina salamander? Is it indicative of an important flaw in the concept of a species, or just a little anomaly about which we should not worry? We shall use the remainder of this chapter to answer this question. Admittedly, ring species are unusual. Before concluding that there is no reason for worry, let us follow the biologist Richard Dawkins, who wrote engagingly about Ensatina, by thinking about more ordinary situations, when a species develops over time. Our own species will provide as good an example as any. Suppose you were to draw up a huge list of your ancestors, always choosing a parent whose sex is opposite your own. So if you are a woman, you list your father, parental grandfather, parental great grandfather, and so on, in a huge sequence. Suppose you had super-human amounts of time and patience, going back to the person who lived generations or so ago (something like 6 million years BC). Now we need to ask some rather personal questions. Would you be able to interbreed with the first ancestor in the sequence (your mother or father, that is)? Unpalatable though the thought is to most of us, I take it that the answer is probably yes. How about the second ancestor (one of your grand parents)? Well, if we gloss over some minor problems posed by time and emotion, then probably, the answer must be yes. But at some time t in the past, so many biological differences will have accumulated between you and the ancestor in the list who lived at that time t that it would be impossible for the two of you to have fertile offspring even if you had been the fondest of contemporaries. I do not know precisely how far back in time we need to go to find a suitable t, but 6 million years ago is almost certainly long enough ago, since this is estimated to be the time when your ancestor was also an ancestor of today s chimpanzees: probably an ape who walked on four legs. For future reference, let s call him Richard. Recall the story of the Ensatina salamaner: Ensatina is often treated as just one species of salamander. This seems sensible. What do all these tiny differences between six types of animals matter, given that each interbreeds with the next one? But suppose we applied the same logic to the human species, based on the story involving Richard? We would start reasoning that you are the same species as your parents, who are the same species as their parents, and so on. After or so reasoning steps, we would be concluding that your ancestors 6 million years ago walking on four legs, having no language and hardly using any tools were human too. If this is not bad enough for your taste, then let s reverse direction: starting from Richard, we keep choosing a child instead of a parent: first we choose one of Richard s children, then one of its children, and so on. By making the right choices often enough, we arrive at a chimpanzee now living in the London zoo. Using the same reasoning as before, we conclude that this chimp is of the same species as Richard, who is of the same species as you. In other words: chimps are human. The conclusion is that people and chimps belong to the same species.

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