WHAT THE TORTOISE SAID TO ACHILLES

"What the Tortoise Said to Achilles" is a brief dialog by Lewis Carroll which playfully problematizes the foundations of logic. The tortoise challenges Achilles to use the force of logic to make him accept a particular 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"
(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"

If we take A and B as the two indicated sides, we can formalize these statements in mathematical symbols as:

(A): ∀x,y,c: (x=c and y=c) ⇒ x=y
(B): ∃k: A=k and B=k
(Z): A=B

The premise of the dialog is that the Tortoise wants Achilles to logically compell him to accept this as a valid argument. That is, if he grants (A) and (B), the Tortoise wishes Achilles to logically compell him to accept (Z).

The Tortoise is obviously a troublemaker, since (Z) follows necessarily from (A) and (B) given the standard laws of logic. Again using mathematical symbols, we can rigorously show this as follows:

Let s be the "same" to which A and B are equal. (The second premise guarantees that there is such an s)
A=s and B=s.
(A=s and B=s) ⇒ A=B. (Specialization of (A))
A=B. (Modus ponens)

The Tortoise will not let Achilles off so easily, however. He refuses to accept the argument, although he soon grants Achilles an additional premise (C):