A System of Logic, Ratiocinative and Inductive

by John Stuart Mill

In one remarkable instance the scientific world was divided into two furiously hostile parties by an ambiguity of language affecting a branch of science which, more completely than most others, enjoys the advantage of a precise and well-defined terminology. I refer to the famous dispute respecting the vis viva, the history of which is given at large in Professor Playfair’s Dissertation. The question was, whether the force of a moving body was proportional (its mass being given) to its velocity simply, or to the square of its velocity: and the ambiguity was in the word Force. “One of the effects,” says Playfair, “produced by a moving body is proportional to the square of the velocity, while another is proportional to the velocity simply:” from whence clearer thinkers were subsequently led to establish a double measure of the efficiency of a moving power, one being called vis viva, and the other momentum. About the facts, both parties were from the first agreed: the only question was, with which of the two effects the term force should be, or could most conveniently be, associated. But the disputants were by no means aware that this was all; they thought that force was one thing, the production of effects another; and the question, by which set of effects the force which produced both the one and the other should be measured, was supposed to be a question not of terminology, but of fact.

The ambiguity of the word Infinite is the real fallacy in the amusing logical puzzle of Achilles and the Tortoise, a puzzle which has been too hard for the ingenuity or patience of many philosophers, and which no less a thinker than Sir William Hamilton considered as insoluble; as a sound argument, though leading to a palpable falsehood. The fallacy, as Hobbes hinted, lies in the tacit assumption that whatever is infinitely divisible is infinite; but the following solution (to the invention of which I have no claim) is more precise and satisfactory.

The argument is, let Achilles run ten times as fast as the tortoise, yet if the tortoise has the start, Achilles will never overtake him. For suppose them to be at first separated by an interval of a thousand feet: when Achilles has run these thousand feet, the tortoise will have got on a hundred; when Achilles has run those hundred, the tortoise will have run ten, and so on forever: therefore Achilles may run forever without overtaking the tortoise.

Now the “forever,” in the conclusion, means, for any length of time that can be supposed; but in the premises, “ever” does not mean any length of time; it means any number of subdivisions of time. It means that we may divide a thousand feet by ten, and that quotient again by ten, and so on as often as we please; that there never needs be an end to the subdivisions of the distance, nor consequently to those of the time in which it is performed. But an unlimited number of subdivisions may be made of that which is itself limited. The argument proves no other infinity of duration than may be embraced within five minutes. As long as the five minutes are not expired, what remains of them may be divided by ten, and again by ten, as often as we like, which is perfectly compatible with their being only five minutes altogether. It proves, in short, that to pass through this finite space requires a time which is infinitely divisible, but not an infinite time; the confounding of which distinction Hobbes had already seen to be the gist of the fallacy.

The following ambiguities of the word right (in addition to the obvious and familiar one of a right and the adjective right) are extracted from a forgotten paper of my own, in a periodical:

“Speaking morally, you are said to have a right to do a thing, if all persons are morally bound not to hinder you from doing it. But, in another sense, to have a right to do a thing is the opposite of having no right to do it, i.e., of being under a moral obligation to forbear doing it. In this sense, to say that you have a right to do a thing, means that you may do it without any breach of duty on your part; that other persons not only ought not to hinder you, but have no cause to think worse of you for doing it. This is a perfectly distinct proposition from the preceding. The right which you have by virtue of a duty incumbent upon other persons, is obviously quite a different thing from a right consisting in the absence of any duty incumbent upon yourself. Yet the two things are perpetually confounded. Thus, a man will say he has a right to publish his opinions; which may be true in this sense, that it would be a breach of duty in any other person to interfere and prevent the publication: but he assumes thereupon that, in publishing his opinions, he himself violates no duty; which may either be true or false, depending, as it does, on his having taken due pains to satisfy himself, first, that the opinions are true, and next, that their publication in this manner, and at this particular juncture, will probably be beneficial to the interests of truth on the whole.

“The second ambiguity is that of confounding a right of any kind, with a right to enforce that right by resisting or punishing a violation of it. People will say, for example, that they have a right to good government, which is undeniably true, it being the moral duty of their governors to govern them well. But in granting this, you are supposed to have admitted their right or liberty to turn out their governors, and perhaps to punish them, for having failed in the performance of this duty; which, far from being the same thing, is by no means universally true, but depends on an immense number of varying circumstances,” requiring to be conscientiously weighed before adopting or acting on such a resolution. This last example is (like others which have been cited) a case of fallacy within fallacy; it involves not only the second of the two ambiguities pointed out, but the first likewise.

One not unusual form of the Fallacy of Ambiguous Terms is known technically as the Fallacy of Composition and Division; when the same term is collective in the premises, distributive in the conclusion, or vicè [pg 570] versa; or when the middle term is collective in one premise, distributive in the other. As if one were to say (I quote from Archbishop Whately), “All the angles of a triangle are equal to two right angles: A B C is an angle of a triangle; therefore A B C is equal to two right angles.... There is no fallacy more common, or more likely to deceive, than the one now before us. The form in which it is most usually employed is to establish some truth, separately, concerning each single member of a certain class, and thence to infer the same of the whole collectively.” As in the argument one sometimes hears, to prove that the world could do without great men. If Columbus (it is said) had never lived, America would still have been discovered, at most only a few years later; if Newton had never lived, some other person would have discovered the law of gravitation; and so forth. Most true: these things would have been done, but in all probability not till some one had again been found with the qualities of Columbus or Newton. Because any one great man might have had his place supplied by other great men, the argument concludes that all great men could have been dispensed with. The term “great men” is distributive in the premises and collective in the conclusion.

“Such also is the fallacy which probably operates on most adventurers in lotteries; e.g., ‘the gaining of a high prize is no uncommon occurrence; and what is no uncommon occurrence may reasonably be expected; therefore the gaining of a high prize may reasonably be expected;’ the conclusion, when applied to the individual (as in practice it is), must be understood in the sense of ‘reasonably expected by a certain individual;’ therefore for the major premise to be true, the middle term must be understood to mean, ‘no uncommon occurrence to some one particular person;’ whereas for the minor (which has been placed first) to be true, you must understand it of ‘no uncommon occurrence to some one or other;’ and thus you will have the Fallacy of Composition.

“This is a Fallacy with which men are extremely apt to deceive themselves; for when a multitude of particulars are presented to the mind, many are too weak or too indolent to take a comprehensive view of them, but confine their attention to each single point, by turns; and then decide, infer, and act accordingly; e.g., the imprudent spendthrift, finding that he is able to afford this, or that, or the other expense, forgets that all of them together will ruin him.” The debauchee destroys his health by successive acts of intemperance, because no one of those acts would be of itself sufficient to do him any serious harm. A sick person reasons with himself, “one, and another, and another, of my symptoms do not prove that I have a fatal disease;” and practically concludes that all taken together do not prove it.