How Can You Search for a Unicorn That Doesn’t Exist?
The Unicorn Puzzle
Imagine you’re daydreaming about a unicorn with a silver horn. Your friend asks what you’re doing. “I’m searching for a unicorn,” you say. That sentence feels perfectly ordinary — but it contains a puzzle that baffled thinkers for hundreds of years. Does your searching prove that a unicorn exists? If not, how can you search for something that isn’t anywhere?
Now change the verb: “I’m finding a unicorn.” Something shifts. If you say I found a unicorn, that forces the idea that the unicorn is real. Yet when you seek a unicorn, you don’t have to believe one exists. This split — between de dicto (the “notional” reading, which doesn’t require existence) and de re (the “objectual” reading, which does) — was a classic headache in the philosophy of language. It seemed that English, our everyday code, was playing tricks.
Then a mathematician named Richard Montague (1930–1971) stepped in. He argued that the difference between seek and find wasn’t random magic. It came from hidden logical structure — the same kind of structure that makes “2 + 2 = 4” feel inevitable. Montague believed you could treat English itself as a formal system, with precise rules that build the meaning of every sentence from its pieces. The unicorn puzzle was his favorite test case. And his solution opened a door into a whole new way of understanding what words do.
Math Meets Language
Montague was a logician, not a linguist. He studied set theory and modal logic — tools that mathematicians use to talk about necessity, possibility, and infinity. When he looked at a normal sentence like “John walks,” he saw something his colleagues might have expected to find only in an artificial language like algebra.
His revolutionary claim was this: there is no deep theoretical difference between natural languages like English and the made-up languages of logicians. Both can be understood with one “mathematically precise theory.” That didn’t mean he thought people speak in equations. It meant he believed the meaning of a sentence could be defined using a model — a mathematical universe built from sets, functions, and truth-values — in the same way you define the truth of “∃x (x is a unicorn).”
A model, in this sense, isn’t a miniature copy of reality. It gives more than reality (it can include imaginary worlds and impossible situations) and also less (it only cares about the structure that language itself assumes). By treating language this way, Montague opened a new discipline: model-theoretic semantics. The goal was to pin down what makes a sentence true — and when one sentence forces another to be true, a relationship called entailment. “John walks and sings” entails “John walks,” because there’s no model where the first is true and the second isn’t.
Build It Like a Recipe
Montague’s system had one cardinal rule: the meaning of a whole expression is a function of the meanings of its parts and the way they are combined. This is the Principle of Compositionality. If you know what walk means and what sing means, and you know the rule that joins them with and, you automatically get the meaning of walk and sing. It works like a cooking recipe: the finished dish depends on the ingredients and the steps, not on magic.
To make this recipe visible, Montague borrowed an instrument from logic called the lambda operator (λ). Think of λ as a placeholder that waits for a property to plug in. If we say that the name John stands for a set of properties — the properties that hold for John — we can write that as λP[P(John)]. The λ says: “I am a function looking for a property P. I’ll give you the value True if John has that property.” That sounds fancy, but it’s just a precise way of describing what we do every time we say “John is singing.”
The beauty of this notation is that it lets you handle both “John” and “every man” with the same basic machinery. When you say Every man is singing, the word every acts as a recipe that takes the property of being a man and the property of singing, and then says: “For all x, if x is a man, x sings.” This idea, that noun phrases denote chunks of logical functions, became the seed of generalized quantifier theory — one of Montague’s most lasting gifts to semantics.
When One Sentence Has Two Stories
Take the sentence Every man loves a woman. It seems to have exactly one arrangement of words. Yet you can hear it two ways: either there is a single special woman (say, a famous queen) whom all men love, or each man loves some woman (maybe his mother or partner). No word is ambiguous; the difference lies entirely in the order in which the meaning pieces are combined.
Montague gave an account of this scope ambiguity by keeping track of the sentence’s derivational history — the steps the grammar took to assemble it. Even if the final sentence looks the same, its “construction manual” matters. If the quantifier “every man” is added first, it takes wide scope, and you get the reading where each man’s love is independent. If “a woman” slips in via a special later rule, it takes narrow scope, suggesting one shared woman.
This move felt clunky to later researchers, so they invented more elegant ways to keep the recipe straight — like storing the meanings of nouns in a “Cooper store” and retrieving them when needed — but Montague’s insight stuck: meaning doesn’t just float on top of words; it’s carried by the invisible blueprint of how the sentence is built. That same blueprint also explains why a phrase like No man ever walks makes sense while Some man ever walks sounds terrible. The word “ever” is a negative polarity item, licensed only by downward-entailing expressions like “no.” Generalized quantifier theory, born from Montague’s analysis, gave us a sharp explanation for these subtle patterns.
Imagine a Thermometer That Rises
Consider another riddle: The temperature is ninety, and The temperature is rising. If the first sentence is true, the temperature equals 90. But 90 can’t “rise” — it’s just a fixed number. So the meaning of “the temperature” must shift: sometimes it picks out a concrete measurement (the extension), and sometimes it picks out a pattern across possible worlds and times — its intension.
Montague borrowed the idea of possible worlds from Saul Kripke and welded it to his type logic. The meaning of a declarative sentence became an intension: a function that tells you its truth value at any given world and moment. So The temperature is rising is true at a world-time if the extension of the temperature is going up there. This handling of intensionality allowed Montague to solve puzzles where we talk about non-existent fish (John wishes to catch a fish and eat it, which doesn’t commit you to any particular fish) or about beliefs that don’t reduce to simple truth. The tools of logic suddenly reached the fuzzy, creative corners of ordinary speech.
One price of this approach: all tautologies — sentences that are always true by their form — get the same intension. “Green grass is green” and “White snow is white” become logically equivalent, which bothers anyone who feels they mean different things. Philosophers have since tinkered with the system, sometimes adding “impossible worlds” or more structured propositions, to respect those differences. Montague pointed in that direction himself, but the core lesson endured: meaning isn’t just about what’s actually true in the real world; it’s about the web of possibilities that language lets us spin.
The Code Still Running
Richard Montague died young, but the revolution he lit never stopped. His demand that semantics be done with mathematical precision — using model theory, compositionality, and a typed logical language — reshaped an entire field. Before Montague, many philosophers thought that the logic of ordinary English was too messy to formalize. After Montague, every researcher who wants to explain how language works has to ask: what is the function that each piece contributes, and how do the pieces combine?
Today, the lambda calculus isn’t just a chapter in a dusty textbook. It’s the engine behind the grammars that linguists like Irene Heim and Angelika Kratzer used to build a science of meaning. And while your digital assistant doesn’t literally “think,” the algorithms that let it parse your question — find the weather in Tokyo — rely on compositional strategies that Montague championed: break the request into chunks, give each chunk a precise logical job, and compute the whole.
When you pause over a sentence that seems to say two things at once, or puzzle over how a joke plays with existence and nothingness, you’re walking the same path Montague walked in the 1970s. The hidden logic of everyday talk isn’t a flaw — it’s a vast, beautiful structure. And the key to that structure is a mathematician’s conviction that words aren’t just sounds or feelings. They’re a formal system. They’re a code. And you already know it.
Think about it
- Suppose a friend says, “I believe the tallest mountain is in Nepal,” but she’s never heard of Mount Everest. Does she believe Mount Everest is in Nepal? What does her sentence mean about the mountain itself?
- If you had to write a recipe — step by step — that a computer could use to combine words into the meaning of a new sentence, what would be the hardest part to capture? Would “sarcasm” fit in the recipe?
- Imagine a language spoken only by a community of deep-sea divers. Could its logical rules for words like “here” or “now” be completely different from English? How would we prove it?
