Volume 10, No. 4 - May 2000
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Baldness is a fuzzy concept. Just consider: A man with a full head of hair is not bald. And if you remove a single hair from his head, he still is not bald. But if you repeat this inferential step, plucking a new hair at every iteration, eventually you arrive at the conclusion that a man with no hairs left is not bald — which, though perhaps reassuring to some, is absurd.

This baldness puzzle can be found in the writings of Diogenes Laertius. It is an example of a sorites (from soros, Greek for "heap"), a series of logical propositions designed so that each collapses neatly into the next and in the conclusion they all fold up like an accordion. Sorites paradoxes allegedly go back to Eubulides of Miletus in the fourth century b.c. They continue to irritate contemporary philosophers of language, who have responded by writing at length about vagueness. For engineers and computer scientists, these paradoxes have occasioned a more joyous creation: fuzzy logic.

Logically, there are three possible responses to the baldness paradox. First, you can assert that a single hair can be the difference between non-bald and bald. This move commits you to a cutoff point — a critical number of hairs — between baldness and non-baldness.

Second, you can admit that the adjective "bald" is irredeemably vague and try to purge it — and similar adjectives, like "poor" and "drunk," and nouns like "life" and "pornography" — from the language. This has been the dream of philosophers from Leibniz to Frege to Carnap, all of whom urged the construction of a new language in which each term would have a precise meaning.

Third, you can deny that a sentence like "Jones is bald" need be either true or false. That is, you can jettison the classical logic of Aristotle. During the 1920s, the Polish logician Jan Lukasiewicz experimented with a logic whose propositions could have fractional truth-values between 0 and 1. In 1937, the Cornell philosopher Max Black proposed using such a "many-valued logic" to analyze vague concepts. And in the 1960s, a Berkeley engineering professor named Lotfi Zadeh speculated that a many-valued logic capable of handling vague concepts might allow machines to reason as flexibly and intuitively as humans do. Zadeh gave the enterprise a catchy name: "fuzzy logic."

A mechanism controlled by Zadeh's fuzzy logic would use rules of thumb couched in the vague terms of everyday language. Take an air conditioner. It might have two rules: "If the room is cool, blow soft," and "If the room is warm, blow hard." At sixty-nine degrees Fahrenheit, the air conditioner's minicomputer might decide that the first proposition was 70 percent true, and the second proposition 30 percent true. So the air conditioner would blow at a rate that was 70 percent soft plus 30 percent hard. (As a practical matter, fuzzy-logic answers tend to come out as weighted averages.)

Zadeh spawned something of a cult. His disciples invoked the omnivalent Buddha against the bivalent Aristotle. And although those outside the cult have denounced fuzzy logic as "the cocaine of science," its practical success has been undeniable. In the early 1970s, engineers in London used it to control the operation of a small steam engine. By the 1980s, fuzzy rules were directing a driverless subway system in Sendai, Japan. The early 1990s saw a "fuzzy boom" in consumer products, with the introduction of hundreds of smart gadgets employing fuzzy logic — air conditioners, camcorders, washing machines. Today fuzzy logic is propelling "soft" computing, in which neural-net computers learn fuzzy rules that offer "good enough" solutions — all of which is nicely described by Arturo Sangalli in his recent book The Importance of Being Fuzzy (Princeton, 1998).

Not a bad legacy, one might think, for an antique paradox. And to champions of the fuzzy worldview, it's just the beginning. In The Fuzzy Future (Harmony, 1999), Bart Kosko — a self-described polymath with a fondness for meditating in hot tubs — argues that fuzzy logic is on the verge of ousting its binary rival in every aspect of life, from politics to warfare to art. (Some of this, it should be said, descends from fuzzy into woolly. Kosko's approach to the abortion issue, for instance, involves polling people to see at what point during pregnancy they believe personhood begins, then aggregating the data to draw a "fuzzy life curve.")

But practical success has done nothing to establish fuzzy logic's philosophical bona fides with its critics. In Deviant Logic, Fuzzy Logic (Chicago, 1996), the philosopher Susan Haack voices her conviction that "truth does not come in degrees" and argues that fuzzy logic sacrifices all the virtues of its classical counterpart, like formal rules of inference and provable consistency.

The deepest disagreement between the fuzzies and the anti-fuzzies, though, may be over where the vagueness really lies. Is it in us or in the world? It has been argued, for example, that Mount Everest is a vague object, like a cloud, because it is impossible to say whether certain molecules are inside it or outside it. Others contend that the world is precise, and vagueness is subjective — a kind of ignorance. Which brings us back to the first and most straightforward answer to the sorites paradox: Although our perception may be too gross to detect it, there may be an exact point at which a man moves from non-bald to bald. After all, to a supremely perceptive eye, the very hairs on your head are numbered.

In addition to "Hypotheses," Jim Holt writes a regular column for the Wall Street Journal covering books on science and philosophy. His most recent book is Worlds Within Worlds: How the Infitesimal Revolutionized Thought (Four Walls/Eight Windows).