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 A Commentary on "Achilles and the Tortoise"

     Lewis Carroll’s purpose in this short paper is to demonstrate, by means of a reductio ad absurdum argument, that a rule of inference cannot be considered as a premiss of an argument. If a rule of inference is given as one of the premisses, then some other rule of inference must be accepted in order for the argument to be valid. Nevertheless, if this second rule is added to the premisses, then a third rule is needed and so on ad infinitum.

     Carroll establishes this point by means of the following 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.

(Z) The two sides of this triangle are equal to each other (Carroll, 118). In order for Z to follow validly from A and B, the reasoning process must be permitted by a rule of inference:

(C) If A and B are true, Z must be true (Carroll, 118). Moreover, in order for the argument that Z follows from A, B, and C to be valid, another rule of inference is necessary:

(D) If A and B and C are true, Z must be true (Carroll, 119).

Consequently, according to Carroll the argument can never be completed. If he is correct in this claim, then there is no compelling reason to accept any inference as legitimate. Carroll’s paper is therefore a strong argument for skepticism.

     I. M. Copi distinguishes between a logical relation and an argument’s premisses and conclusion. One might interpret this distinction to imply that the criterion of the correctness or incorrectness of arguments is not part of the specific argument. Since logic is a normative discipline, correct arguments must conform to rules, but this consideration is not a sufficient reason to presuppose that the rules are themselves premisses in specific arguments.

     While I cannot be certain that I. M. Copi would respond to Carroll’s argument in this way, this distinction (if correct) falsifies Carroll’s assumption that a rule of logic must be a premiss. In addition, I. M. Copi defines "rules of inference" as " rules that permit valid inferences from statements assumed as premisses" (Copi, 704). However, he does not explicitly write that elementary rules of inference are not part of the premisses. Indeed, if the rules were to be considered part of the premisses, I. M. Copi’s definition would fall prey to Carroll’s argument for logical skepticism.


1. Lewis Carroll, "Achilles and the Tortoise," in Readings on Logic, ed. I. M. Copi and J. A. Gould (New York: Macmillan, 1972), pp. 117-118.

2. I. M. Copi, Introduction to Logic (New York: Macmillan, 1994).

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