Logic, Inductive and Deductive - William Minto

The Sophism of Achilles and the Tortoise is the most triumphant of examples of Ignoratio Elenchi.

The point that the Sophism undertakes to prove is that Achilles can never overtake a Tortoise once it has a certain start: what it really proves, and proves indisputably, is that he cannot overtake the Tortoise within a certain space or time.

For simplicity of exposition, let us assume that the Tortoise has 100 yards start and that Achilles runs ten times as fast. Then, clearly, Achilles will not come up with it at the end of 100 yards, for while he has run 100, the Tortoise has run 10; nor at the end of 110, for then the Tortoise has run 1 more; nor at the end of 111, for then the Tortoise has run ¹⁄₁₀ more; nor at the end of 111¹⁄₁₀, for then the Tortoise has gained ¹⁄₁₀₀ more. So while Achilles runs this ¹⁄₁₀₀, the Tortoise runs ¹⁄₁₀₀₀; while he runs the ¹⁄₁₀₀₀, it runs ¹⁄₁₀₀₀₀. Thus it would seem that the Tortoise must always keep ahead: he can never overtake it.

But the conclusion is only a confusion of ideas: all that is really proved is that Achilles will not overtake the Tortoise while running

100 + 10 + 1 + ¹⁄₁₀ + ¹⁄₁₀₀ + ¹⁄₁₀₀₀ + ¹⁄₁₀₀₀₀, etc.

That is, that he will not overtake it till he has completed the sum of this series, 111¹⁄₉ yards. To prove this is an ignoratio elenchi; what the Sophist undertakes to prove is that Achilles will never overtake it, and he really proves that Achilles passes it between the 111th and 112th yards.

The exposure of this sophism is an example also of the value of a technical term. All attempts to expose it without using the term Ignoratio Elenchi or something equivalent to it, succeed only in bewildering the student. It is customary to say that the root of the fallacy lies in assuming that the sum of an infinite series is equal to infinity. This profound error may be implied: but if any assumption so hard to understand were really required, the fallacy would have little force with the generality.

It has often been argued that the Syllogism involves a petitio principii, because the Major Premiss contains the Conclusion, and would not be true unless the Conclusion were true. But this is really an Ignoratio Elenchi. The fact adduced, that the Major Premiss contains the Conclusion, is indisputable; but this does not prove the Syllogism guilty of Petitio. Petitio principii is an argumentative trick, a conscious or unconscious act of deception, a covert assumption, and the Syllogism, so far from favouring this, is an expositio principii, an explicit statement of premisses such that, if they are true, the conclusion is true. The Syllogism merely shows the interdependence of premisses and conclusion; its only tacit assumption is the Dictum de Omni.

If, indeed, an opponent challenges the truth of the conclusion, and you adduce premisses necessarily containing it as a refutation, that is an ignoratio elenchi unless your opponent admits those premisses. If he admits them and denies the conclusion, you convict him of inconsistency, but you do not prove the truth of the conclusion. Suppose a man to take up the position: "I am not mortal, for I have procured the elixir vitæ". You do not disprove this by saying, "All men are mortal, and you are a man". In denying that he is mortal, he denies that all men are mortal. Whatever is sufficient evidence that he is not mortal, is sufficient evidence that all men are not mortal. Perhaps it might be said that in arguing, "All men are mortal, and you are a man," it is not so much ignoratio elenchi as petitio principii that you commit. But be it always remembered that you may commit both fallacies at once. You may both argue beside the point and beg the question in the course of one and the same argument.

[Footnote 1:] Cp. Mr. Sidgwick's instructive treatise on Fallacies, International Scientific Series, p. 199.