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# 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:

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.