Lewis Carroll wrote a dialogue called “What the Tortoise Said to Achilles.” I first saw mention of this when I read Godel, Escher, Bach: An Eternal Golden Braid. It’s also been brought up a few times by James F. Harris in Against Relativism. Interestingly enough, the section I just finished reading, Harris critiques two relativists for using that very dialogue as a way of establishing that the laws of logic cannot be proven. After doing so, he falls into the very thing he criticized.
To demonstrate this, first a quick overview of the dialogue is necessary. Basically put, Achilles is attempting to convince Tortoise of the following logic:
A) Things that are equal to the same thing are equal to each other (or, mathematically put,:Â a = c; b = c; Therefore, a =Â b).
B) The two sides of a specific triangle are things that are equal to the same thing.
Z) Therefore the two sides of the triangle are equal to each other.
The last clause is put as Z) for a reason. This is because Tortoise responded to Achilles by saying that this modus ponens requires a reason for why he must accept Z), so Achilles gave the only option he could:
C) If A) and B) are true, then Z) must be true.
Tortoise then asks Achilles to prove that, so Achilles says:
D) If A) and B) and C) are true, then Z) must be true.
As you can see, this continues forever. When Harris first examed this dialogue, he concluded by saying:
Carroll’s point is clear: Unless one is willing to accept some logical rule or law and treat it on a different logical level from the infrences it generates, no justification is possible; and no actual inference can ever take place.
Harris, James F. (1992).  Against Relativism: A Philosophical Defense of Method, LaSalle: Open Court (p. 39)
It is ironic then that we see Harris make the following statements in disagreeing with David Bloor and Barry Barnes and their use of the dialogue:
Consider any person S who holds any belief B as the result of whatever method or upon whatever grounds one might choose–science, oracle, witchcraft, or voodoo. Let us call the method which produces B, M. Now suppose we ask how it is that S comes to believe that B is the result of M. If B is to be a belief upon which this person predicates any behavior whatsoever, then, as I have argued, both B and not-B cannot both follow from M (at the same time, under the same conditions, and so forth). Consequently, there must be some mechanism, some warrant, some criterion c by which S determines that B follows from M. Perhaps c is simply the recognition and implicit acknowledgment of an authority, or perhaps it is a certain ritual or procedure, or perhaps it is the application of a particular method to certain data. But whatever c is, S must be able to reason, ‘If c, then B follows from M and if not-c, then B does not follow from M’. This is the force of Wittgenstein’s famed call for the necessity of a criterion or rules.
(ibid, 105)
Does this not fall right back into the same infinite regress that the Tortoise moved Achilles into? After all, would there not need to be some meta-c to demonstrate why c is nessary in the first place?
In other words, there must be some criterion meta-c by which S determines that B follows from M due to c. And there must be some criterion meta-meta-c by which S determines that B follows from M due to c due to meta-c….etc.
In this instance, Harris falls into the very trap he was seeking to defuse in Bloor & Barnes. Ultimately, this is because the entire example is self-referential. That is, a person’s belief (B) in a method (M) is validated by that very method itself especially on properly basic philosophical levels. In other words, one asks: how do we know what criteria establishes a claim in the first place? Answer: by the method we employ. In other words, c presupposes M to be valid.
This is the only way to avoid falling into the Tortoise error.
