What the Tortoise Said to Achilles
"What the Tortoise Said to Achilles",[1] written by Lewis Carroll in 1895 for the philosophical journal Mind,[1] is a brief allegorical dialogue on the foundations of logic.[1] The title alludes to one of Zeno's paradoxes of motion,[2] 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.[1]
The discussion begins by considering the following logical argument:[1][3]
The tortoise accepts premises A and B as true but not the hypothetical:
The Tortoise claims that it is not "under any logical necessity to accept Z as true". The tortoise then challenges Achilles to force it logically to accept Z as true. Instead of searching the tortoise’s reasons for not accepting C, Achilles asks it to accept C, which it does. After which, Achilles says:
The tortoise responds, "That's another Hypothetical, isn't it? And, if I failed to see its truth, I might accept A and B and C, and still not accept Z, mightn't I?"[1][3]
Again, instead of requesting reasons for not accepting D, he asks the tortoise to accept D. And again, it is "quite willing to grant it",[1][3] but it still refuses to accept Z. It then tells Achilles to write into his book,
Following this, the Tortoise says: "until I’ve granted that [i.e., E], of course I needn’t grant Z. So it's quite a necessary step".[1] With a touch of sadness, Achilles sees the point.[1][3]
The story ends by suggesting that the list of premises continues to grow without end, but without explaining the point of the regress.[1][3]
Discussion[edit]
Several philosophers have tried to resolve Carroll's 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."
Peter Winch, a Wittgensteinian philosopher, 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).
Carroll's dialogue is apparently the first description of an obstacle to conventionalism about logical truth,[4] later reworked in more sober philosophical terms by W.V.O. Quine.[5]
Lewis Carroll (April 1895). "What the Tortoise Said to Achilles". Mind. IV (14): 278–280. doi:10.1093/mind/IV.14.278.
Reprinted:
As audio: