Definition
Carroll's regress is a paradox in Lewis Carroll's 1895 essay What the Tortoise Said to Achilles. The Tortoise asks Achilles to use modus ponens: from "A" and "if A then B" conclude "B." Achilles agrees. The Tortoise then asks for the meta-premise "if A and (if A then B) then B" to be written down. Achilles writes it. The Tortoise asks for the next meta-meta-premise. The act of applying the rule is never captured by any finite list of premises.
Why it matters
How it works
To apply modus ponens, you have premises A and A ⊃ B. The conclusion B follows. The Tortoise's trick: ask Achilles to make explicit the principle "if (A and A ⊃ B) then B." Once written, the principle is itself a premise. Now to apply modus ponens to A, A ⊃ B, and the new principle, you need a higher-order principle: "if (A and A ⊃ B and ((A and A ⊃ B) ⊃ B)) then B." And so on. The act of crossing from premises to conclusion is a meta-level operation that no premise contains.