Fuzzy Logic

Mathematical Creatures and Where to Find Them, 1

The Anatomy of Thought

Everyone has encountered logic at some point. It might have come through a watertight argument that persuaded you to change your mind, a pithy Vulcan saying from a Star Trek episode, or a dusty high school textbook that tried to teach you how to navigate truth tables. Logic is a two-headed creature. It mostly grows out of the trunk of philosophy, but its branches are tightly entangled with those of mathematics.

At its most basic, logic is a kind of grammar. Just as grammar teaches you how to structure sentences from words and phrases, logic shows you how to structure reasoning from its simplest parts. It deals in formal systems: languages made of symbols, governed by precise rules, that let us build complex ideas step by step. And it asks questions like: What does it mean for something to be true? How do we know a proof is valid? Can we construct a universal language for reasoning itself?

The Lego blocks of logic are propositions — statements that are either true or false. “It is raining” or “The cat is on the mat” are typical examples. We can combine these propositions into more complex ones using connectives such as AND, OR, and NOT, or flip their truth values entirely. With the help of quantifiers such as “for all” (∀) and “there exists” (∃), we can express broader claims, like “Every number has a successor” or “There exists a smallest prime.” And at a higher level, we can assemble entire logical systems by choosing a set of assumed rules, called axioms, and then deriving new statements, called theorems, which must be true if the axioms are.

Many different logical systems can be formulated and explored, but most of the time, when we talk about logic, we are referring to classical logic. One of its most famous principles is the Law of the Excluded Middle: every proposition is either true or false. There is no third option. You can be dead or alive, but not both at once. This principle, and others like it, feel like common sense. But logic, like geometry, allows us to play with different rules and see what happens. The geometry most people learn in school is a modern descendant of what Euclid taught over two thousand years ago. Yet in the nineteenth century, mathematicians discovered that by changing just one of Euclid’s assumptions, they could create new kinds of geometry. These systems are no less valid than the classical one, and far from being just a curiosity, they now play a vital role, among other things, in understanding the shape of the universe.

When Logic Learns to Hesitate

The first of the creatures we shall encounter is called fuzzy logic, and it fittingly dwells within 03-XX (Mathematical Logic and Foundations), more specifically under 03Bxx (General Logic). Its precise designation is 03B52: Fuzzy logic; logic of vagueness.

Unlike classical logic, its defining feature is, well, its fuzziness. Fuzzy logic rejects the Law of the Excluded Middle, which in classical logic insists that every proposition must be either true or false. Instead, fuzzy logic allows for degrees of truth. Rather than just two truth values — 0 (false) and 1 (true) — it permits any value between 0 and 1, so a statement like “This soup is hot” might be 0.8 true if it’s fairly hot but not boiling. These intermediate values reflect partial truth or membership, and fuzzy logic defines versions of AND, OR, and NOT that operate smoothly across this continuum:

• Fuzzy AND is often defined as the minimum of two truth values.

• Fuzzy OR as the maximum.

• Fuzzy NOT as one minus the truth value.

So, for example:
“Soup is hot” = 0.8
“Soup is spicy” = 0.6
→ “Soup is hot AND spicy” = min(0.8, 0.6) = 0.6

If we think of fuzzy logic as a creature, a good mental image might be that of a chameleon, or any animal skilled in mimicry. When you glance at a patch of forest where such a creature is hiding, you may think you’ve spotted it — camouflaged among leaves — but how sure are you? You might find yourself assigning different degrees of confidence to the edges of its shape. Is that a leaf, or is it its tail? Where does the creature end and the background begin?

The Rise and Fall of Fuzzy Logic

While fuzzy logic as a field is relatively recent (it was introduced by Lotfi A. Zadeh in 1965 as an extension of classical Boolean logic) it gained notable popularity during the 1980s and 1990s. This was especially true in Japan, where it was embraced for consumer electronics and industrial automation. Companies like Matsushita and Subaru developed fuzzy-controlled air conditioners, rice cookers, and even applied it to automatic transmissions. There was talk of "fuzzy thinking" as a new cognitive mode, a style of reasoning closer to how humans think and more adaptable to natural language. It also appeared to be computationally cheap and interpretable, especially in contrast to the more opaque statistical methods that all this talk of fuzziness may already have brought to your mind.

Yet the hype faded almost as quickly as it had arisen. Among the reasons were the rise of more robust methods in probabilistic modeling, such as Bayesian approaches, which proved better for quantifying risk, making predictions, and updating beliefs. Fuzzy logic also lacked a single, unified theoretical foundation, which limited its coherence as a field. The rapid growth of machine learning further sidelined it. Fuzzy systems are largely rule-based, interpretable, and require hand-crafted design, whereas techniques like neural networks, support vector machines, and deep learning prioritize performance and can learn automatically from large data sets. Finally, fuzzy logic struggled with problems of scale, which became increasingly central in the era of big data.

Fuzzy Logic: A Fuzzy Legacy

Fuzzy logic occupies a curious position. While it is neither obsolete nor in any way mistaken, it has not lived up to the expectations once placed on it. It is still used in many practical applications (aerospace engineering, automotive traffic control, business decision-making, industrial processes) and can be useful, elegant, and effective in specific contexts (especially where interpretability and human-like reasoning are important). But as we have seen, it does not scale well to the kinds of big data and predictive tasks that define much of modern artificial intelligence.

Perhaps its deeper value lies not in replacing other approaches but in reminding us that truth can be a matter of degree, and that logic need not always be razor-sharp. This might sound trivial, but it does not feel that way to me. Much of what draws me to mathematics is its inflexibility and certainty (I got enough of the opposite in my humanities education to last me a lifetime), but fuzzy logic offers a salutary reminder that the choice of our axioms is arbitrary or shaped by preferences that are themselves open to debate.