The 2,500-Year Hunt for the Glue of Thought

The Glue Words: A 2,000-Year-Old Question

You’re listening to a friend tell a story. She says, “I wanted ice cream, and the shop was open, but it started raining.” The words “and” and “but” don’t point to anything you can see or touch—they aren’t things like “ice cream” or “shop.” Yet without them, the story would fall apart. Words like these act as invisible glue.

For more than two thousand years, philosophers have wondered: what makes these “glue words” different from ordinary ones? And why do we trust arguments that rely on them so deeply? The question is more than just a word game. If you can sort the glue words from the rest, you start to see what makes an argument valid no matter what you’re talking about—whether it’s ice cream, galaxies, or numbers.

Medieval logicians called the picture-like words categorematic terms: words that name a thing or a property, like “cat” or “runs.” The glue words they called syncategorematic terms: words that mean nothing by themselves, but show how the meaningful terms are combined. The 14th-century philosopher Jean Buridan said that syncategorematic words give a sentence its form, while categorematic words provide its matter—like the steel bars that shape a concrete wall. “Every,” “only,” “not,” “or,” “if”—all mark the structure of a thought without adding any extra “stuff.”

For centuries, this division seemed natural and complete. But then a quiet revolution in logic shattered it.

When Frege Broke the Glue

Gottlob Frege (1848–1925) changed the way we think about the inner shape of sentences. Before him, everyone assumed a sentence was just a subject stuck to a predicate by a glue word. “Every cat runs” was “cat” (subject) plus “runs” (predicate), joined by “every.” But Frege saw something deeper.

He noticed that you can break a sentence into functions and arguments, like in mathematics. A function is an incomplete pattern with a blank spot. When you fill the spot with an argument, you get a complete thought. For example, “__ runs” is a function with one blank; fill it with “Whiskers,” and you get “Whiskers runs.” Even the glue word “every” is itself a second-level function: it takes two simpler functions and says the first is included in the second.

Suddenly, the old syncategorematic/categorematic split stopped working. If a predicate like “runs” is an incomplete function, it too is “unsaturated” and cannot stand on its own—just like the glue words. Should “runs” count as syncategorematic? Or should we call it categorematic and limit the syncategorematic to variables and parentheses? Neither option gave a sharp dividing line. The medieval distinction, so clear inside the subject-predicate view, collapsed.

Frege’s insight opened the door to modern logic, but it also left a problem: if you can’t tell logical words from ordinary ones by whether they are “glue,” how do you tell them apart?

Too Neutral to Be About Anything?

A different path came from a simple idea: logic is supposed to apply everywhere, to every subject. So perhaps logical constants are the words that are completely topic‑neutral—they don’t care what you’re talking about. The philosopher Gilbert Ryle (1900–1976) suggested that a foreigner who understood only topic‑neutral words could get no clue from a paragraph what it was about.

That sounds promising, but it’s frustratingly vague. Take “because.” Does it give a clue about subject matter? Yes: it tells you the paragraph involves causes or explanations. Is “because” topic‑neutral enough to be logical? Ryle’s test doesn’t give a clear yes or no. Even words like “every” reveal that the topic involves countable things. Topic neutrality might come in degrees, not a clean boundary.

To make this idea precise, several philosophers turned to mathematics. The Polish logician Alfred Tarski (1901–1983) proposed that a logical expression is one that stays the same no matter how you swap the objects in the world. This is called permutation invariance.

Imagine you have a bag of marbles, and you replace each marble with a different one while keeping the total number fixed. If the truth of a sentence using a certain word stays unchanged under every possible swap, that word is permutation‑invariant. For instance, the word “every” is invariant because “Every marble is red” might become false when you swap colors, but the operation “every” itself doesn’t depend on which particular marbles are involved. The identity relation “is identical to” is invariant because swapping objects doesn’t change whether something is itself. On the other hand, “is a cat” is not invariant: swap all cats with dogs, and that predicate behaves differently.

Tarski’s test is mathematically rigorous. It counts the standard quantifiers (“all,” “some”), truth‑functional connectives (“and,” “or,” “not”), and identity as logical. It also includes higher‑order quantifiers and cardinality words like “infinitely many.” Set membership, however, fails the test, which pleased many philosophers who thought mathematics and logic should be kept separate.

But a tricky problem emerged. What if a word is permutation‑invariant only because of a strange accident in the real world? Suppose you invent a connector “%” that behaves exactly like “not” in all sentences, but its definition also mentions “there are no male widows” (which is necessarily true). Under permutation invariance, “%” gets counted as logical, because its truth conditions match “not.” Yet it seems absurd to say that a definition that drags in the idea of widows and maleness is purely logical. The problem, many argued, is that permutation invariance looks only at the reference of a word—what it picks out—rather than its meaning or sense.

Games with Rules: Can We Define Logic by Its Moves?

Another approach starts not from what words are about, but from how we use them in reasoning. Suppose you could learn the complete meaning of a word just by mastering a few simple inference rules—introduction and elimination rules. For example, to grasp “and,” you only need to know:

• If you have A and B, you can conclude “A and B.”
• If you have “A and B,” you can conclude A. And you can conclude B.

No worldly knowledge is needed. The rules are purely formal. The German logician Gerhard Gentzen (1909–1945) suggested that the introduction rules for a logical constant act as a “definition” of its meaning, and the elimination rules are just consequences of that definition. If a word can be fully characterized by such inferential rules, perhaps it deserves to be called logical.

This idea gives a clean way to filter out imposters. Consider a made‑up word “tonk,” introduced by the philosopher Arthur Prior (1914–1969) as a joke. Its rules say: from A you can infer “A tonk B,” and from “A tonk B” you can infer B. Put them together, and you can prove anything at all—the system explodes. Prior’s “tonk” shows that not just any set of rules determines a coherent meaning. For a logical constant, the introduction and elimination rules must be in harmony: the elimination rules shouldn’t let you infer more than the introduction rules allow.

But even this harmony requirement isn’t a complete fix. Critics point out that to grasp the universal quantifier “every,” you may need more than just formal rules; you need a feel for what kind of evidence supports “all” claims. And there are tricky cases where two different connectives, like “or” and a made‑up “‡” (which means “not both not-A and not-B”), share the same truth table but feel different in meaning. If logic is about meaning, shouldn’t the two be distinguished? Those who focus on inference rules say yes: they are grasped through different sets of primitively compelling rules, so they are different logical constants. That keeps the boundary sharp, but it forces you to commit to a specific view of what it means to understand a word.

So Who Wins the Fight? (And Does It Even Matter?)

After all these attempts, you might expect a clear winner. Instead, philosophers have divided into four main camps.

The Demarcaters think there is a real, important boundary between logical and non‑logical words, and we just need to find the right criterion—permuation invariance, inferential rules, or something else. They say that logic is about formal validity: an argument is valid because of its logical structure, not because of facts about cats and dogs. Without a sharp list of logical constants, you can’t nail down what formal validity means.

The Debunkers reply that the whole problem is a pseudoproblem. The job of a logician is to study validity in general, not just a special “formal” kind. The words that appear in logical formulas—the “glue words”—are just tools that logicians happen to find useful at a given moment. Asking which words are really logical is like asking which mountains are really the ones worth climbing. The answer depends on the climber.

The Relativists take a middle path. They agree that logical consequence depends on a choice of logical constants, but they say there is no single correct choice. For any set of words you pick as logical, you get a corresponding notion of “C-consequence.” None is the one true consequence; different choices suit different purposes. This avoids fighting over the One True List, but it also means that the boundary of logic is, in the end, a matter of decision.

The Deflaters also find a middle ground. They think “logical constant” is a family resemblance term: there’s no one thread running through every example, only overlapping similarities. You can tell a logical word from a non‑logical one, but you can’t give a perfectly sharp, exception‑free definition—just like you can’t define “game” perfectly, yet you know a game when you see one.

Why does this centuries‑old puzzle matter today? Because every time you evaluate an argument—whether a politician’s speech, a friend’s excuse, or a mathematical proof—you rely on an implicit idea of which words carry the logical structure. If you get that idea wrong, you can be tricked by a sentence that sounds persuasive but has no solid logical backbone. The hunt for the glue of thought isn’t just a museum piece; it’s a reminder that clear thinking starts with seeing the invisible architecture of the sentences we use every day.

Think about it

1. If all the “glue words” suddenly vanished from your language, could you still think clearly? What would be lost first?
2. Suppose someone invents a new word “plog” with the rule: from “A plog B” you can infer both A and B. Would that word be logical? Does it feel like it belongs with “and” or “or”?
3. Imagine two detectives solving the same crime using different lists of logical words. Could they end up believing different arguments are valid—even if they agree on all the facts?