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.

Table of contents

1 Summary of the dialogue 2 What's wrong here 3 Where to find the article 4 References

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):

• (C): (A) and (B) ⇒ (Z)

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.

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], [1], and [1]

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 [1]. Another author provides a more extended summary at [1]