A,B ⊢ Z?
What the Tortoise Said to Achilles
Lewis Carroll · 1895
The puzzle
Achilles offers the inference: if A and B are true, then Z is true; A and B are true; therefore Z. The Tortoise refuses Z without first being granted, as a premise, the rule itself.
Note
Achilles obliges and adds C: if A, B, and C are true, then Z. But now the Tortoise demands rule D, the rule that lets you go from A, B, C to Z. And so on, forever. Carroll’s point is that you cannot prove an inference by adding the inference to the premises — at some point you have to act on the rule rather than describe it. The paper, three pages long in Mind, quietly demolishes the idea that logic is purely a matter of formal premises. There has to be a place where the rule is just followed. Wittgenstein, half a century later, made this the central problem of his later philosophy: no rule can determine its own application without already presupposing the practice of applying it.