What the Tortoise Said to Achilles

"What the Tortoise Said to Achilles" is a brief dialogue by Lewis Carroll which playfully problematises 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.

Summary of the dialogue
The discussion begins by considering the following logical argument:
 * A: "Things that are equal to the same are equal to each other" (transitive property)
 * 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, 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 him as a reader of this second kind, and then to logically compel him to accept that Z must be true.

After writing down A, B and Z in his notebook, Achilles asks the Tortoise to accept the hypothetical:
 * C: "If A and B are true, Z must be true"

The Tortoise agrees to accept C, if Achilles will write down what he has to accept in his note-book, making the new argument:
 * A: "Things that are equal to the same are equal to each other"
 * B: "The two sides of this triangle are things that are equal to the same"
 * C: "If A and B are true, Z must be true"
 * Therefore Z: "The two sides of this triangle are equal to each other"

But now that the Tortoise accepts premise C, he 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 he accepts C, he could still fail to conclude Z if he did not see the truth of:
 * D: "If A and B and C are true, Z must be true"

The Tortoise continues to accept each hypothetical premise once Achilles writes it down, but denies that the conclusion necessarily follows, since each time he denies the hypothetical that if all the premises written down so far are true, Z must be true:
 * "And at last we've got to the end of this ideal race-course! Now that you accept A and B and C and D, of course you accept Z."
 * "Do I?" said the Tortoise innocently. "Let's make that quite clear. I accept A and B and C and D. Suppose I still refused to accept Z?"
 * "Then Logic would take you by the throat, and force you to do it!" Achilles triumphantly replied. "Logic would tell you, 'You can't help yourself. Now that you've accepted A and B and C and D, you must accept Z!' So you've no choice, you see."
 * "Whatever Logic is good enough to tell me is worth writing down," said the Tortoise. "So enter it in your note-book, please. We will call it
 * (E) If A and B and C and D are true, Z must be true. Until I've granted that, of course I needn't grant Z. So it's quite a necessary step, you see?"
 * "I see," said Achilles; and there was a touch of sadness in his tone.
 * "Whatever Logic is good enough to tell me is worth writing down," said the Tortoise. "So enter it in your note-book, please. We will call it
 * (E) If A and B and C and D are true, Z must be true. Until I've granted that, of course I needn't grant Z. So it's quite a necessary step, you see?"
 * "I see," said Achilles; and there was a touch of sadness in his tone.
 * "I see," said Achilles; and there was a touch of sadness in his tone.

Thus, the list of premises continues to grow without end, leaving the argument always in the form:
 * (1): "Things that are equal to the same are equal to each other"
 * (2): "The two sides of this triangle are things that are equal to the same"
 * (3): (1) and (2) ⇒ (Z)
 * (4): (1) and (2) and (3) ⇒ (Z)
 * (n): (1) and (2) and (3) and (4) and ... and (n − 1) ⇒ (Z)
 * Therefore (Z): "The two sides of this triangle are equal to each other"
 * Therefore (Z): "The two sides of this triangle are equal to each other"

At each step, the Tortoise argues that even though he accepts all the premises that have been written down, there is some further premise (that if all of (1)–(n) are true, then (Z) must be true) that he still needs to accept before he is compelled to accept that (Z) is true.

Discussion
Several philosophers have tried to resolve the Carroll paradox. Bertrand Russell discussed the paradox briefly in § 38 of The Principles of Mathematics (1903), distinguishing between implication (associated with the form "if p, then q"), which he held to be a relation between unasserted propositions, and inference (associated with the form "p, therefore q"), which he held to be a relation between asserted propositions; having made this distinction, Russell could deny that the Tortoise's attempt to treat inferring Z from A and B as equivalent to, or dependent on, agreeing to the hypothetical "If A and B are true, then Z is true".

The Wittgensteinian philosopher Peter Winch discussed the paradox in The Idea of a Social Science and its Relation to Philosophy (1958), where he argued that the paradox showed that "the actual process of drawing an inference, which is after all at the heart of logic, is something which cannot be represented as a logical formula … Learning to infer is not just a matter of being taught about explicit logical relations between propositions; it is learning to do something" (p.57). Winch goes on to suggest that the moral of the dialogue is a particular case of a general lesson, to the effect that the proper application of rules governing a form of human activity cannot itself be summed up with a set of further rules, and so that "a form of human activity can never be summed up in a set of explicit precepts" (p.53).

Isashiki Takahiro (1999) summarizes past attempts and concludes they all fail before beginning yet another.

Where to find the article

 * Carroll, Lewis. "What the Tortoise Said to Achilles". Mind, n.s., 4 (1895), pp. 278–80.
 * Hofstadter, Douglas. Gödel, Escher, Bach: an Eternal Golden Braid. See the second dialogue, entitled "Two-Part Invention." Dr. Hofstadter appropriated the characters of Achilles and the Tortoise for other, original, dialogues in the book which alternate contrapuntally with prose chapters. In doing so, Hofstadter unconsciously (and to his later distaste) ascribes the male sex to the Tortoise, where Carroll makes no such attribution.
 * A number of websites, including, , , and