# 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.

The discussion begins by considering the following logical argument:

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):

Achilles then asks the Tortoise to accept the expanded 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): (A) and (B) ⇒ (Z)
- therefore (Z): "The two sides of this triangle are equal to each other"

The Tortoise refuses to accept this new argument, although he soon grants Achilles an additional premise (D):

- (D): (A) and (B) and (C) ⇒ (Z)

The list of premises thus continues to grow without end, leaving the argument always in the form:

- (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): (A) and (B) ⇒ (Z)
- (D): (A) and (B) and (C) ⇒ (Z)
- ...
- (n): (A) and (B) and (C) and (D) and ... and (n) ⇒ (Z)
- therefore (Z): "The two sides of this triangle are equal to each other"

And, to the great frustration of Achilles, the Tortoise refuses to accept every single one of them.

Several philosophers have tried to resolve the Carroll paradox. Isashiki Takahiro (1999) summarizes past attempts and concludes they all fail before beginning yet another.

- 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 dialog, entitled "Two-Part Invention".
- any number of websites, including [1], [1], and [1]

## References