Lewis Carroll Wrote a Three-Page Logic Paper That Has No Way Out

Lewis Carroll was, beyond the Alice books, a working Oxford logician. In 1895 he published a three-page dialogue in Mind called "What the Tortoise Said to Achilles." Premise A: things equal to the same thing are equal to each other. Premise B: the two sides of this triangle are equal to the same thing. Conclusion Z: the two sides are equal to each other. Achilles writes A, B, Z in his notebook and waits for assent.

The Tortoise refuses. He accepts A. He accepts B. He says he doesn't yet see why Z follows. So Achilles adds a premise C: "If A and B are true, then Z is true." The Tortoise nods: C, written down, accepted. Does Z follow now? No, says the Tortoise. You'd need a further premise D: "if A, B, and C are true, then Z is true." Achilles writes D. The Tortoise asks for E. The narrator gives up and goes to the bank.

The trick is that Achilles is treating the rule of inference as if it were a premise. Every time he writes down the rule that lets you get from premises to conclusion, the Tortoise insists that rule must itself be linked to the next step by yet another rule. This isn't a quirk of geometry. It happens for any deductive step. Modus ponens, the most basic rule in logic, has nothing underneath it that isn't another version of itself.

A hundred and thirty years of logicians have argued about the moral. One reading: rules of inference aren't premises, and trying to make them premises breaks logic. Another: at some point you have to act on a rule rather than describe it. Either way, the small triangle in Achilles's notebook never gets proved. The Tortoise is still asking for the next premise.