Not Exactly: In Praise of Vagueness

  • Kees van Deemter
Oxford University Press: 2010. 368 pp. £16.99 9780199545902 | ISBN: 978-0-1995-4590-2

Although scientists strive for increasing clarity in their measurements and concepts, it is often uncertainty that spurs new thinking. The haziness of the species notion set the young Charles Darwin pondering evolution. Francis Crick observed that if he and James Watson had worried about how to define the gene in the 1950s, progress in molecular biology would have stalled. “In research the front line is almost always in a fog,” Crick wrote in his autobiography. Even today there is no consensus definition of the gene.

In Not Exactly, a wide-ranging study of vagueness, computer scientist Kees van Deemter argues that precise definitions may not be meaningful or logical. Through his research background in artificial intelligence — he worked on the TENDUM question-answering machine developed at Philips Electronics in the 1980s — he knows how difficult it is to program computers to speak and write like humans. In the book, he brings a mix of logical, linguistic and philosophical perspectives to the topic of vagueness.

Natural languages — as opposed to the formal languages that are used in logic and computing — are full of imprecision and ambiguity. In English, the adjective 'large' is equally applicable to a spider, an elephant or a planet. Speakers infer the meaning of the word from the context of its usage. Thus 'large' is a vague concept by van Deemter's definition because “it allows borderline cases”. Although the term 'obese' would seem to be better defined, it is also vague: the borderlines between underweight, healthy weight, overweight and obese are, to a large extent, arbitrarily drawn.

Borderlines are essential for precision but their definition can defy reason. Much of the book explores the ramifications of the sorites paradox, an ancient Greek conundrum about the size of a heap (soros in Greek). Adding one grain of sand to another clearly does not make it a heap. But if you follow the reasoning of Aristotelian logic and Boolean algebra, which allows a statement to be either true or false, no matter how many grains you add, at no point does it become a heap. The threshold cannot be defined through classical logic.

Similarly, an object can retain its identity even though it has undergone many changes. In a 1990 London high court case, a seller of a vintage racing car sued a buyer who had withdrawn from the deal after claiming the car was not authentic because of the successive replacement of its parts. The judge ruled in favour of the seller: “Any new parts were assimilated into the whole at such a rate and over such a period of time that they never caused the car to lose its identity.”

There is no satisfactory resolution of the sorites paradox by modifying classical logic, van Deemter argues. Rather than statements being either true or false, what is needed is a logic based on degrees of truth, ranging from zero to 100% certainty.

Allowing for such gradations in boundary definitions can help in decision-making. The author tells the story of the stealing of a diamond from the emperor of China by one of a thousand eunuchs. The single witness exclaims on his death bed only that “The thief is tall.” How is the emperor to catch him? A classical logician — who might categorize suspects as either tall or not tall — would define an average height and advise searching everyone who is taller than the average. A logician who allows for degrees of truth might find the culprit more quickly: the taller the thief is, the more likely the witness is to have described him as tall. Therefore, the search should begin with the tallest.

For multiple constraints, the degrees of truth are combined using further logical operations. Such 'fuzzy logic' systems are widely used in computing, for example in providing automated decision-support systems for physicians. But, van Deemter maintains, because these combinations still rely on assumptions of truth or falsity, fuzzy logic cannot address all the ambiguities of natural language, including the sorites paradox.

Not Exactly is often a tough read for those without training in formal logic, although van Deemter intersperses it with lively fictitious dialogues. The book's argument that in public discourse we need more use of vagueness and less of the 'false clarity' of formal logic is convincingly made. In science, vagueness is sometimes a virtue and must be better understood if computers are ever to pass the Turing test for demonstrating human intelligence.