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Encyclopedia > 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 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.

 Contents

## Summary of the dialogue GA_googleFillSlot("encyclopedia_square");

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):

• (3): (1) and (2) ⇒ (Z)

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.

## 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)

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