The Day Someone Invented a Word That Destroyed Logic
The Word That Could Prove Anything
Imagine you’re building a chain of reasoning. You start with a simple fact: “It’s raining.” Using logical words like and, or, and if…then, you combine facts to reach a conclusion. These words are like the glue of reason.
But what if you could invent a new logical word, one so powerful that it could turn any statement into any other statement? You could start with “The sky is blue” and end with “The moon is made of green cheese.” And you could do it in just two steps.
In 1960, philosopher A.N. Prior proposed exactly such a word. He called it Tonk. It was meant as a joke, but it shook the foundations of logic. It showed that not just any set of rules can give meaning to a word. Something deeper has to be right.
Meet Tonk: The Monster Connective
To understand why Tonk was a monster, you need to know how logical connectives usually work. Think of conjunction — the word and. If you know “It’s raining” and you also know “I have an umbrella,” you can put them together: “It’s raining and I have an umbrella.” That’s an introduction rule: it tells you how to build a compound sentence using and. There’s also an elimination rule: from “It’s raining and I have an umbrella,” you can pull out either half: “It’s raining.” These two kinds of rules — introduction and elimination — define what and means in logic. They set the boundaries of the game.
Now, Tonk had its own introduction and elimination rules. The introduction rule said: from any statement A, you may infer A Tonk B (for any B). For example, from “The sky is blue,” you get “The sky is blue Tonk the moon is cheese.” The elimination rule said: from A Tonk B, you may infer B. So from that Tonk-sentence, you get “The moon is cheese.”
Put these together, and you can go from absolutely any true statement to any other statement — true or false. Logic becomes a game where every move wins. That’s not reasoning; it’s chaos.
Why Logical Words Have Rules
Prior’s Tonk was a joke, but it contained a serious lesson. When we add a new logical word to a language, we have to check that it doesn’t wreck everything else. Philosopher Nuel Belnap argued that a reasonable new connective must pass at least two tests.
The first test is conservativity. Adding Tonk to a system with only plain statements like “It’s raining” and “It’s sunny” should not let you prove new things with those old words alone — things you couldn’t prove before. But with Tonk, you could prove “It’s sunny” from “It’s raining” without using Tonk again. That’s non-conservative. It’s like bringing a chess piece that, once on the board, makes your pawn able to checkmate instantly, even after you remove the new piece.
The second test is about uniqueness. Imagine you duplicated a logical word — say you had two versions of and, written and₁ and and₂ — with exactly the same introduction and elimination rules. You should be able to prove that “A and₁ B” and “A and₂ B” mean the same thing, using only those rules. Connective rules that pass this test uniquely characterize the word. Tonk passed uniqueness easily (its rules were so strong, anything provable with one Tonk was provable with the other), but it failed conservativity.
So a healthy connective needs both: it must not smuggle in new powers for old words, and its rules must pick out exactly one logical job for it to do.
The Clash Over ‘Not’: Classicists vs Intuitionists
These tests matter because they reveal deep disagreements. Take the word not — logical negation. In classical logic, you’re allowed to cancel double negations: from “It’s not true that it’s not raining” you can conclude “It’s raining.” This is the rule of double negation elimination.
But in intuitionistic logic, developed by thinkers like L.E.J. Brouwer, that rule is rejected. An intuitionist accepts that you can go from “It’s raining” to “It’s not not raining,” but not the reverse. The intuitionist rules for not uniquely characterize their negation within their own logical system, and the classical rules do the same for theirs.
Now, what if you tried to be friendly and said: “Let’s put both classical-not and intuitionistic-not in the same language, each with its own rules, and let people use whichever they like”? Here’s the problem. Because the rules for intuitionistic negation already uniquely characterize it, adding the stronger classical rules would make the two negations equivalent — you could prove that classical-not and intuitionistic-not behave identically. The classical rules would infect the intuitionistic one, forcing it to accept double negation elimination. So you can’t have both peacefully. This shows that what a logical word means is bound tightly to the whole system of reasoning you accept.
Why This Matters to You
You use logical words every day: “You can have cake if you eat your vegetables,” “We can go to the park or the beach.” These words seem simple, but behind them is a network of rules that keep reasoning straight.
The Tonk disaster shows that meaning isn’t just about spelling or dictionary definitions. It’s about what a word lets you do in an argument. A word with broken rules can make nonsense look like proof.
And the clash over not shows that logical rules aren’t set in stone. Reasonable people can disagree about them, and those disagreements have consequences: they change what counts as a valid argument. So next time you say “if” or “or,” remember: you’re playing a game with ancient rules, and what you can say depends on keeping those rules consistent.
Think about it
- Imagine a new connective “blip” with the rule: from “A blip B” you can always infer “A,” and from “A” alone you can infer “A blip B.” What weird results would follow? How does blip compare to Tonk?
- In everyday talk, we sometimes use “or” to mean “one or the other but not both.” Would it make sense to have two different logical ors? What rules would you need to keep them from colliding?
- If two people follow different rules for not, can they really be disagreeing, or are they just using the same word for different ideas? How could you tell?
