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:

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, 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 (that is, a reader who denies the premises).

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.)

After writing down A, B, and Z in his notebook, Achilles asks the Tortoise to accept the hypothetical:

The Tortoise agrees to accept C, if Achilles will write down what it has to accept in his notebook, making the new argument:

But now that the Tortoise accepts premise C, it still refuses to accept the expanded argument. When Achilles demands that "If you accept A and B and C, you must accept Z," the Tortoise remarks that that's another hypothetical proposition, and suggests even if it accepts C, it could still fail to conclude Z if it did not see the truth of:

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