Implicit Logical Forces in Carroll's "What the Tortoise Said to Achilles"

by Corry Shores
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Achilles, having overtaken the Toroise, sat upon his shell and refers to the Zeno’s paradox that claims his victory is impossible, on account of an infinite series of distances.

The tortoise then begins to describe a race-course in which distances increase infinitely, and has Achilles take up pencil and paper.

The tortoise describes Euclid’s first proposition:

(A) Things that are equal to the same are equal to each other.

(B) The two sides of the Triangle are things that are equal to the same.

(Z) The two sides of this Triangle are equal to each other.

The Tortoise says, “Z follows logically from A and B, so that anyone who accepts A and B is true, must accept Z as true?”

and then claims that a first reader of the argument may doubt the truth of the premises, yet still concede that the argument used valid reasoning. Someone might accept the hypothetical proposition: “if A and B be true, Z must be true; but I don't accept A and B as true”

Yet furthermore, a second reader might accept A and B as true, while doubting the truth of the hypothetical proposition.

The Tortoise then challenges Achilles “to consider me as a reader of the second kind, and to force me, logically, to accept Z as true.”

In other words, Achilles must force the Tortoise to accept the above hypothetical, called ‘C:’

(C) If A and B are true, Z must be true."

and Achilles writes this new proposition in his notebook, hence together they make:

(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.

(Z) The two sides of this Triangle are equal to each other."

But then, implied in this argument is another hypothetical, call it ‘D:’

(D) If A and B and C are true, Z must be true.

and Achilles writes it down. The Tortoise moves one step further, and refuses again to accept D. But Achilles says, “Then Logic would take you by the throat, and force you to do it!”

But just like the hypotheticals, these forces of logic, however, are implicit in every explication of the argument. There is always an implied premise that grounds the inference. But Logic purports to explicate and clarify all its inner workings, or as the Tortoise claims: “Whatever Logic is good enough to tell me is worth writing down.”

Thus Achilles wrote down premise ‘E’ and so on indefinitely. Months later, Achilles was still entering hypotheticals into his notebook, with Tortoise wondering; “Have you got that last step written down? Unless I've lost count, that makes a thousand and one. There are several millions more to come.”

Carroll, Lewis. "What the Tortoise Said to Achilles." Available online at:

http://www.lewiscarroll.org/achilles.html