**"What the Tortoise Said to Achilles"** is a brief dialog by Lewis Carroll which playfully problematizes the foundations of logic. The dialog alludes to a Zeno paradox. 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: - (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."
- 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: - (1): ∀x,y: ∃c: (x=c and y=c) ⇒ x=y
- (2): ∃k: A=k and B=k
- (Z): A=B
The premise of the dialog is that the Tortoise wants Achilles to logically compel him to accept this as a valid argument. That is, if he grants (1) and (2), the Tortoise wishes Achilles to compel him logically to accept (Z). The Tortoise is obviously a troublemaker, since (Z) follows necessarily from (1) and (2) 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 (3): Achilles then asks the Tortoise to accept the expanded argument: - (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)
- 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 (4): - (4): (1) and (2) and (3) ⇒ (Z)
The list of premises thus 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"
And, to the great frustration of Achilles, the Tortoise refuses to accept every single one of them.
## What's wrong here Several philosophers have tried to resolve the Carroll paradox. Isashiki Takahiro (1999) summarizes past attempts and concludes they all fail before beginning yet another. See deduction theorem.
## 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 dialog, entitled "Two-Part Invention". - any number of websites, including [1] (
*http://www.lewiscarroll.org/achilles.html*), [2] (*http://www.ditext.com/carroll/tortoise.html*), and [3] (*http://home.earthlink.net/~lfdean/carroll/essays/achilles.html*) ## References - Isashiki Takahiro (1999). "What Can We Learn from Lewis Carroll's Paradox?". In
*Memoirs of the Faculty of Education, Miyazaki University: Humanities*, no. 86, pp. 79-98. The paper is in Japanese, although an extremely condensed summary by the author is available from [4] (*http://www.miyazaki-u.ac.jp/~e02702u/papers/eng_carroll.html*). Another author provides a more extended summary at [5] (*http://homepage2.nifty.com/Workshop-Alice/click/m-t.html*) |