A System of Logic, Ratiocinative and Inductive · John Stuart Mill
Chapter VII.
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 __ 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.
The employment of a proposition to prove that on which it is itself dependent for proof, by no means implies the degree of mental imbecility which might at first be supposed. The difficulty of comprehending how this fallacy could possibly be committed, disappears when we reflect that all persons, even the instructed, hold a great number of opinions without exactly recollecting how they came by them. Believing that they have at some former time verified them by sufficient evidence, but having forgotten what the evidence was, they may easily be betrayed into deducing from them the very propositions which are alone capable of serving as premises for their establishment. “As if,” says Archbishop Whately, “one should attempt to prove the being of a God from the authority of Holy Writ;” which might easily happen to one with whom both doctrines, as fundamental tenets of his religious creed, stand on the same ground of familiar and traditional belief.
Arguing in a circle, however, is a stronger case of the fallacy, and implies more than the mere passive reception of a premise by one who does not remember how it is to be proved. It implies an actual attempt to prove two propositions reciprocally from one another; and is seldom resorted to, at least in express terms, by any person in his own speculations, but is committed by those who, being hard pressed by an adversary, are forced into giving reasons for an opinion of which, when they began to argue, they had not sufficiently considered the grounds. As in the following example from Archbishop Whately: “Some mechanicians attempt to prove (what they ought to lay down as a probable but doubtful hypothesis)(265) that every particle of matter gravitates equally: ‘why?’ ‘because those bodies which contain more particles ever gravitate more strongly, _i.e._, are heavier:’ ‘but (it may be urged) those which are heaviest are not always more bulky;’ ‘no, but they contain more particles, though more closely condensed:’ ‘how do you know that?’ ‘because they are heavier:’ ‘how does that prove it?’ ‘because all particles of matter gravitating equally, that mass which is specifically the heavier must needs have the more of them in the same space.’” It appears to me that the fallacious reasoner, in his private thoughts, would not be likely to proceed beyond the first step. He would acquiesce in the sufficiency of the reason first given, “bodies which contain more particles are heavier.” It is when he finds this questioned, and is called upon to prove it, without knowing how, that he tries to establish his premise by supposing proved what he is attempting to prove by it. The most effectual way, in fact, of exposing a petitio principii, when circumstances allow of it, is by challenging the reasoner to prove his premises; which if he attempts to do, he is necessarily driven into arguing in a circle.
Mihi a docto doctore, Demandatur causam et rationem quare Opium facit dormire. A quoi respondeo, Quia est in eo Virtus dormitiva, Cujus est natura Sensus assoupire. The words Nature and Essence are grand instruments of this mode of begging the question, as in the well-known argument of the scholastic theologians, that the mind thinks always, because the _essence_ of the mind is to think. Locke had to point out, that if by essence is here meant some property which must manifest itself by actual exercise at all times, the premise is a direct assumption of the conclusion; while if it only means that to think is the distinctive property of a mind, there is no connection between the premise and the conclusion, since it is not necessary that a distinctive property should be perpetually in action.
The following is one of the modes in which these abstract terms, Nature and Essence, are used as instruments of this fallacy. Some particular properties of a thing are selected, more or less arbitrarily, to be termed its nature or essence; and when this has been done, these properties are supposed to be invested with a kind of indefeasibleness; to have become paramount to all the other properties of the thing, and incapable of being prevailed over or counteracted by them. As when Aristotle, in a passage already cited, “decides that there is no void on such arguments as this: in a void there could be no difference of up and down; for as in nothing there are no differences, so there are none in a privation or negation; but a void is merely a privation or negation of matter; therefore, in a void, bodies could not move up and down, which it is in their _nature_ to do.”(266) In other words, it is in the _nature_ of bodies to move up and down, _ergo_ any physical fact which supposes them not so to move, can not be authentic. This mode of reasoning, by which a bad generalization is made to overrule all facts which contradict it, is Petitio Principii in one of its most palpable forms.
None of the modes of assuming what should be proved are in more frequent use than what are termed by Bentham “question-begging appellatives;” names which beg the question under the disguise of stating it. The most potent of these are such as have a laudatory or vituperative character. For instance, in politics, the word Innovation. The dictionary meaning of this term being merely “a change to something new,” it is difficult for the defenders even of the most salutary improvement to deny that it is an innovation; yet the word having acquired in common usage a vituperative connotation in addition to its dictionary meaning, the admission is always construed as a large concession to the disadvantage of the thing proposed.
The following passage from the argument in refutation of the Epicureans, in the second book of Cicero, “De Finibus,” affords a fine example of this sort of fallacy: “Et quidem illud ipsum non nimium probo (et tantum patior) philosophum loqui de cupiditatibus finiendis. An potest cupiditas finiri? tollenda est, atque extrahenda radicitus. Quis est enim, in quo sit cupiditas, quin recte cupidus dici possit? Ergo et avarus erit, sed finite: adulter, verum habebit modum: et luxuriosus eodem modo. Qualis ista philosophia est, quæ non interitum afferat pravitatis, sed sit contenta mediocritate vitiorum?” The question was, whether certain desires, when kept within bounds, are vices or not; and the argument decides the point by applying to them a word (_cupiditas_) which _implies_ vice. It is shown, however, in the remarks which follow, that Cicero did not intend this as a serious argument, but as a criticism on what he deemed an inappropriate expression. “Rem ipsam prorsus probo: elegantiam desidero. Appellet hæc _desideria naturæ_; cupiditatis nomen servet alio,” etc. But many persons, both ancient and modern, have employed this, or something equivalent to it, as a real and conclusive argument. We may remark that the passage respecting _cupiditas_ and _cupidus_ is also an example of another fallacy already noticed, that of Paronymous Terms.
Many more of the arguments of the ancient moralists, and especially of the Stoics, fall within the definition of Petitio Principii. In the “De Finibus,” for example, which I continue to quote as being probably the best extant exemplification at once of the doctrines and the methods of the schools of philosophy existing at that time; of what value as arguments are such pleas as those of Cato in the third book: That if virtue were not happiness, it could not be a thing to _boast_ of: That if death or pain were evils, it would be impossible not to fear them, and it could not, therefore, be laudable to despise them, etc. In one way of viewing these arguments, they may be regarded as appeals to the authority of the general sentiment of mankind which had stamped its approval upon certain actions and characters by the phrases referred to; but that such could have been the meaning intended is very unlikely, considering the contempt of the ancient philosophers for vulgar opinion. In any other sense they are clear cases of Petitio Principii, since the word laudable, and the idea of boasting, imply principles of conduct; and practical maxims can only be proved from speculative truths, namely, from the properties of the subject-matter, and can not, therefore, be employed to prove those properties. As well might it be argued that a government is good because we ought to support it, or that there is a God because it is our duty to pray to him.
It is assumed by all the disputants in the “De Finibus” as the foundation of the inquiry into the _summum bonum_, that “sapiens semper beatus est.” Not simply that wisdom gives the best chance of happiness, or that wisdom consists in knowing what happiness is, and by what things it is promoted; these propositions would not have been enough for them; but that the sage always is, and must of necessity be, happy. The idea that wisdom could be consistent with unhappiness, was always rejected as inadmissible: the reason assigned by one of the interlocutors, near the beginning of the third book, being, that if the wise could be unhappy, there was little use in pursuing wisdom. But by unhappiness they did not mean pain or suffering; to that it was granted that the wisest person was liable in common with others: he was happy, because in possessing wisdom he had the most valuable of all possessions, the most to be sought and prized of all things, and to possess the most valuable thing was to be the most happy. By laying it down, therefore, at the commencement of the inquiry, that the sage must be happy, the disputed question respecting the _summum bonum_ was in fact begged; with the further assumption, that pain and suffering, so far as they can co-exist with wisdom, are not unhappiness, and are no evil.
The following are additional instances of Petitio Principii, under more or less of disguise. Plato, in the _Sophistes_, attempts to prove that things may exist which are incorporeal, by the argument that justice and wisdom are incorporeal, and justice and wisdom must be something. Here, if by _something_ be meant, as Plato did in fact mean, a thing capable of existing in and by itself, and not as a quality of some other thing, he begs the question in asserting that justice and wisdom must be something; if he means any thing else, his conclusion is not proved. This fallacy might also be classed under ambiguous middle-term; _something_, in the one premise, meaning some substance, in the other merely some object of thought, whether substance or attribute.
It was formerly an argument employed in proof of what is now no longer a popular doctrine, the infinite divisibility of matter, that every portion of matter however small, must at least have an upper and an under surface. Those who used this argument did not see that it assumed the very point in dispute, the impossibility of arriving at a minimum of thickness; for if there be a minimum, its upper and under surface will of course be one; it will be itself a surface, and no more. The argument owes its very considerable plausibility to this, that the premise does actually seem more obvious than the conclusion, though really identical with it. As expressed in the premise, the proposition appeals directly and in concrete language to the incapacity of the human imagination for conceiving a minimum. Viewed in this light, it becomes a case of the _a priori_ fallacy or natural prejudice, that whatever can not be conceived can not exist. Every fallacy of Confusion (it is almost unnecessary to repeat) will, if cleared up, become a fallacy of some other sort; and it will be found of deductive or ratiocinative fallacies generally, that when they mislead, there is mostly, as in this case, a fallacy of some other description lurking under them, by virtue of which chiefly it is that the verbal juggle, which is the outside or body of this kind of fallacy, passes undetected.
Euler’s Algebra, a book otherwise of great merit, but full, to overflowing, of logical errors in respect to the foundation of the science, contains the following argument to prove that _minus_ multiplied by _minus_ gives _plus_, a doctrine the opprobrium of all mere mathematicians, and which Euler had not a glimpse of the true method of proving. He says _minus_ multiplied by _minus_ can not give _minus_; for _minus_ multiplied by _plus_ gives _minus_, and _minus_ multiplied by _minus_ can not give the same product as _minus_ multiplied by _plus_. Now one is obliged to ask, why minus multiplied by minus must give any product at all? and if it does, why its product can not be the same as that of minus multiplied by plus? for this would seem, at the first glance, not more absurd than that minus by minus should give the same as plus by plus, the proposition which Euler prefers to it. The premise requires proof, as much as the conclusion; nor can it be proved, except by that more comprehensive view of the nature of multiplication, and of algebraic processes in general, which would also supply a far better proof of the mysterious doctrine which Euler is here endeavoring to demonstrate.
A striking instance of reasoning in a circle is that of some ethical writers, who first take for their standard of moral truth what, being the general, they deem to be the natural or instinctive sentiments and perceptions of mankind, and then explain away the numerous instances of divergence from their assumed standard, by representing them as cases in which the perceptions are unhealthy. Some particular mode of conduct or feeling is affirmed to be _unnatural_; why? because it is abhorrent to the universal and natural sentiments of mankind. Finding no such sentiment in yourself, you question the fact; and the answer is (if your antagonist is polite), that you are an exception, a peculiar case. But neither (say you) do I find in the people of some other country, or of some former age, any such feeling of abhorrence; “ay, but their feelings were sophisticated and unhealthy.”
One of the most notable specimens of reasoning in a circle is the doctrine of Hobbes, Rousseau, and others, which rests the obligations by which human beings are bound as members of society, on a supposed social compact. I waive the consideration of the fictitious nature of the compact itself; but when Hobbes, through the whole Leviathan, elaborately deduces the obligation of obeying the sovereign, not from the necessity or utility of doing so, but from a promise supposed to have been made by our ancestors, on renouncing savage life and agreeing to establish political society, it is impossible not to retort by the question, Why are we bound to keep a promise made for us by others? or why bound to keep a promise at all? No satisfactory ground can be assigned for the obligation, except the mischievous consequences of the absence of faith and mutual confidence among mankind. We are, therefore, brought round to the interests of society, as the ultimate ground of the obligation of a promise; and yet those interests are not admitted to be a sufficient justification for the existence of government and law. Without a promise it is thought that we should not be bound to that which is implied in all modes of living in society, namely, to yield a general obedience to the laws therein established; and so necessary is the promise deemed, that if none has actually been made, some additional safety is supposed to be given to the foundations of society by feigning one.
So also if any one has pointed out the extenuating circumstances in some particular case of offense, so as to show that it differs widely from the generality of the same class, the sophist, if he finds himself unable to disprove these circumstances, may do away the force of them, by simply _referring the action to that very class_, which no one can deny that it belongs to, and the very name of which will excite a feeling of disgust sufficient to counteract the extenuation; _e.g._, let it be a case of peculation, and that many _mitigating_ circumstances have been brought forward which can not be denied; the sophistical opponent will reply, ‘Well, but after all, the man is a _rogue_, and there is an end of it;’ now in reality this was (by hypothesis) never the question; and the mere assertion of what was never denied _ought_ not, in fairness, to be regarded as decisive; but, practically, the odiousness of the word, arising in great measure from the association of those very circumstances which belong to most of the class, but which we have supposed to be _absent_ in _this particular_ instance, excites precisely that feeling of disgust which, in effect, destroys the force of the defense. In like manner we may refer to this head all cases of improper appeal to the passions, and every thing else which is mentioned by Aristotle as extraneous to the matter in hand (ἔξω τοῦ πράγματος).”
Again, “instead of proving that ‘this prisoner has committed an atrocious fraud,’ you prove that the fraud he is accused of is atrocious; instead of proving (as in the well-known tale of Cyrus and the two coats) that the taller boy had a right to force the other boy to exchange coats with him, you prove that the exchange would have been advantageous to both; instead of proving that the poor ought to be relieved in this way rather than in that, you prove that the poor ought to be relieved; instead of proving that the irrational agent—whether a brute or a madman—can never be deterred from any act by apprehension of punishment (as, for instance, a dog from sheep-biting, by fear of being beaten), you prove that the beating of one dog does not operate as an _example_ to _other_ dogs, etc.
“It is evident that Ignoratio Elenchi may be employed as well for the apparent refutation of your opponent’s proposition, as for the apparent establishment of your own; for it is substantially the same thing, to prove what was not denied or to disprove what was not asserted. The latter practice is not less common, and it is more offensive, because it frequently amounts to a personal affront, in attributing to a person opinions, etc., which he perhaps holds in abhorrence. Thus, when in a discussion one party vindicates, on the ground of general expediency, a particular instance of resistance to government in a case of intolerable oppression, the opponent may gravely maintain, ‘that we ought not to do evil that good may come;’ a proposition which of course had never been denied, the point in dispute being, ‘whether resistance in this particular case _were_ doing evil or not.’ Or again, by way of disproving the assertion of the right of private judgment in religion, one may hear a grave argument to prove that ‘it is impossible every one can be _right in his judgment_.’”
The works of controversial writers are seldom free from this fallacy. The attempts, for instance, to disprove the population doctrines of Malthus, have been mostly cases of _ignoratio elenchi_. Malthus has been supposed to be refuted if it could be shown that in some countries or ages population has been nearly stationary; as if he had asserted that population always increases in a given ratio, or had not expressly declared that it increases only in so far as it is not restrained by prudence, or kept down by poverty and disease. Or, perhaps, a collection of facts is produced to prove that in some one country the people are better off with a dense population than they are in another country with a thin one; or that the people have become more numerous and better off at the same time. As if the assertion were that a dense population could not possibly be well off; as if it were not part of the very doctrine, and essential to it, that where there is a more abundant production there may be a greater population without any increase of poverty, or even with a diminution of it.
The favorite argument against Berkeley’s theory of the non-existence of matter, and the most popularly effective, next to a “grin”(267)—an argument, moreover, which is not confined to “coxcombs,” nor to men like Samuel Johnson, whose greatly overrated ability certainly did not lie in the direction of metaphysical speculation, but is the stock argument of the Scotch school of metaphysicians—is a palpable Ignoratio Elenchi. The argument is perhaps as frequently expressed by gesture as by words, and one of its commonest forms consists in knocking a stick against the ground. This short and easy confutation overlooks the fact, that in denying matter, Berkeley did not deny any thing to which our senses bear witness, and therefore can not be answered by any appeal to them. His skepticism related to the supposed substratum, or hidden cause of the appearances perceived by our senses; the evidence of which, whatever may be thought of its conclusiveness, is certainly not the evidence of sense. And it will always remain a signal proof of the want of metaphysical profundity of Reid, Stewart, and, I am sorry to add, of Brown, that they should have persisted in asserting that Berkeley, if he believed his own doctrine, was bound to walk into the kennel, or run his head against a post. As if persons who do not recognize an occult cause of their sensations could not possibly believe that a fixed order subsists among the sensations themselves. Such a want of comprehension of the distinction between a thing and its sensible manifestation, or, in metaphysical language, between the noumenon and the phenomenon, would be impossible to even the dullest disciple of Kant or Coleridge.
Ideas discussed in this chapter
Text from Project Gutenberg, public domain in the United States.