What the Tortoise Said to Achilles

"What the Tortoise Said to Achilles", written by Lewis Carroll in 1895 for the philosophical journal Mind, is a brief allegorical dialogue on the foundations of logic. The title alludes to one of Zeno's paradoxes of motion, in which Achilles could never overtake the tortoise in a race. In Carroll's dialogue, the tortoise challenges Achilles to use the force of logic to make him accept the conclusion of a simple deductive argument. Ultimately, Achilles fails, because the clever tortoise leads him into an infinite regression. The discussion begins by considering the following logical argument: A: "Things that are equal to the same are equal to each other" (a Euclidean relation) B: "The two sides of this triangle are things that are equal to the same" Therefore, Z: "The two sides of this triangle are equal to each other" The Tortoise asks Achilles whether the conclusion logically follows from the premises, and Achilles grants that it obviously does. The Tortoise then asks Achilles whether there might be a reader of Euclid who grants that the argument is logically valid, as a sequence, while denying that A and B are true. Achilles accepts that such a reader might exist (that is, a reader who denies the premises), and that he would hold that if A and B are true, then Z must be true, while not yet accepting that A and B are true. The Tortoise then asks Achilles whether a second kind of reader might exist, who accepts that A and B are true, but who does not yet accept the principle that if A and B are both true, then Z must be true. Achilles grants the Tortoise that this second kind of reader might also exist. The Tortoise, then, asks Achilles to treat the Tortoise as a reader of this second kind. Achilles must now logically compel the Tortoise to accept that Z must be true. (The tortoise is a reader who denies the argument form itself; the syllogism's conclusion, structure, or validity.

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Related concepts (2)

Zeno's paradoxes

Zeno's paradoxes are a set of philosophical problems devised by the Eleatic Greek philosopher Zeno of Elea (c. 490–430 BC). The origins of the paradoxes are somewhat unclear, but they are generally thought to have been developed to support Parmenides' doctrine of monism, that all of reality is one, and that all change is impossible. Diogenes Laërtius, citing Favorinus, says that Zeno's teacher Parmenides was the first to introduce the paradox of Achilles and the tortoise.