A System of Logic: Ratiocinative and Inductive Volume I Part 18

[6] Supra, p. 128.

[7] _Logic_, p. 239 (9th ed.).

[8] It is hardly necessary to say, that I am not contending for any such absurdity as that we _actually_ "ought to have known" and considered the case of every individual man, past, present, and future, before affirming that all men are mortal: although this interpretation has been, strangely enough, put upon the preceding observations. There is no difference between me and Archbishop Whately, or any other defender of the syllogism, on the practical part of the matter; I am only pointing out an inconsistency in the logical theory of it, as conceived by almost all writers. I do not say that a person who affirmed, before the Duke of Wellington was born, that all men are mortal, _knew_ that the Duke of Wellington was mortal; but I do say that he _a.s.serted_ it; and I ask for an explanation of the apparent logical fallacy, of adducing in proof of the Duke of Wellington's mortality, a general statement which presupposes it. Finding no sufficient resolution of this difficulty in any of the writers on Logic, I have attempted to supply one.

[9] The language of ratiocination would, I think, be brought into closer agreement with the real nature of the process, if the general propositions employed in reasoning, instead of being in the form All men are mortal, or Every man is mortal, were expressed in the form Any man is mortal. This mode of expression, exhibiting as the type of all reasoning from experience "The men A, B, C, &c. are so and so, therefore _any_ man is so and so," would much better manifest the true idea--that inductive reasoning is always, at bottom, inference from particulars to particulars, and that the whole function of general propositions in reasoning, is to vouch for the legitimacy of such inferences.

[10] Review of Quetelet on Probabilities, _Essays_, p. 367.

[11] _Philosophy of Discovery_, p. 289.

[12] _Theory of Reasoning_, ch. iv. to which I may refer for an able statement and enforcement of the grounds of the doctrine.

[13] It is very probable that the doctrine is not new, and that it was, as Sir John Herschel thinks, substantially antic.i.p.ated by Berkeley. But I certainly am not aware that it is (as has been affirmed by one of my ablest and most candid critics) "among the standing marks of what is called the empirical philosophy."

[14] _Logic_, book iv. ch. i. sect. 1.

[15] See the important chapter on Belief, in Professor Bain's great treatise, _The Emotions and the Will_, pp. 581-4.

[16] A writer in the "British Quarterly Review" (August 1846), in a review of this treatise, endeavours to show that there is no _pet.i.tio principii_ in the syllogism, by denying that the proposition, All men are mortal, a.s.serts or a.s.sumes that Socrates is mortal. In support of this denial, he argues that we may, and in fact do, admit the general proposition that all men are mortal, without having particularly examined the case of Socrates, and even without knowing whether the individual so named is a man or something else. But this of course was never denied. That we can and do draw conclusions concerning cases specifically unknown to us, is the datum from which all who discuss this subject must set out. The question is, in what terms the evidence, or ground, on which we draw these conclusions, may best be designated--whether it is most correct to say, that the unknown case is proved by known cases, or that it is proved by a general proposition including both sets of cases, the unknown and the known? I contend for the former mode of expression. I hold it an abuse of language to say, that the proof that Socrates is mortal, is that all men are mortal. Turn it in what way we will, this seems to me to be a.s.serting that a thing is the proof of itself. Whoever p.r.o.nounces the words, All men are mortal, has affirmed that Socrates is mortal, though he may never have heard of Socrates; for since Socrates, whether known to be so or not, really is a man, he is included in the words, All men, and in every a.s.sertion of which they are the subject. If the reviewer does not see that there is a difficulty here, I can only advise him to reconsider the subject until he does: after which he will be a better judge of the success or failure of an attempt to remove the difficulty. That he had reflected very little on the point when he wrote his remarks, is shown by his oversight respecting the _dictum de omni et nullo_. He acknowledges that this maxim as commonly expressed,--"Whatever is true of a cla.s.s, is true of everything included in the cla.s.s," is a mere identical proposition, since the cla.s.s _is_ nothing but the things included in it. But he thinks this defect would be cured by wording the maxim thus,--"Whatever is true of a cla.s.s, is true of everything which _can be shown_ to be a member of the cla.s.s:" as if a thing could "be shown" to be a member of the cla.s.s without being one. If a cla.s.s means the sum of all the things included in the cla.s.s, the things which can "be shown" to be included in it are part of the sum, and the _dictum_ is as much an identical proposition with respect to them as to the rest. One would almost imagine that, in the reviewer's opinion, things are not members of a cla.s.s until they are called up publicly to take their place in it--that so long, in fact, as Socrates is not known to be a man, he _is not_ a man, and any a.s.sertion which can be made concerning men does not at all regard him, nor is affected as to its truth or falsity by anything in which he is concerned.

The difference between the reviewer's theory and mine may be thus stated. Both admit that when we say, All men are mortal, we make an a.s.sertion reaching beyond the sphere of our knowledge of individual cases; and that when a new individual, Socrates, is brought within the field of our knowledge by means of the minor premise, we learn that we have already made an a.s.sertion respecting Socrates without knowing it: our own general formula being, to that extent, for the first time _interpreted_ to us. But according to the reviewer's theory, the smaller a.s.sertion is proved by the larger: while I contend, that both a.s.sertions are proved together, by the same evidence, namely, the grounds of experience on which the general a.s.sertion was made, and by which it must be justified.

The reviewer says, that if the major premise included the conclusion, "we should be able to affirm the conclusion without the intervention of the minor premise; but every one sees that that is impossible." A similar argument is urged by Mr. De Morgan (_Formal Logic_, p. 259): "The whole objection tacitly a.s.sumes the superfluity of the minor; that is, tacitly a.s.sumes we know Socrates[46] to be a man as soon as we know him to be Socrates." The objection would be well grounded if the a.s.sertion that the major premise includes the conclusion, meant that it individually specifies all it includes. As however the only indication it gives is a description by marks, we have still to compare any new individual with the marks; and to show that this comparison has been made, is the office of the minor. But since, by supposition, the new individual has the marks, whether we have ascertained him to have them or not; if we have affirmed the major premise, we have a.s.serted him to be mortal. Now my position is that this a.s.sertion cannot be a necessary part of the argument. It cannot be a necessary condition of reasoning that we should begin by making an a.s.sertion, which is afterwards to be employed in proving a part of itself. I can conceive only one way out of this difficulty, viz. that what really forms the proof is _the other_ part of the a.s.sertion; the portion of it, the truth of which has been ascertained previously: and that the unproved part is bound up in one formula with the proved part in mere antic.i.p.ation, and as a memorandum of the nature of the conclusions which we are prepared to prove.

With respect to the minor premise in its formal shape, the minor as it stands in the syllogism, predicating of Socrates a definite cla.s.s name, I readily admit that it is no more a necessary part of reasoning than the major. When there is a major, doing its work by means of a cla.s.s name, minors are needed to interpret it: but reasoning can be carried on without either the one or the other. They are not the conditions of reasoning, but a precaution against erroneous reasoning. The only minor premise necessary to reasoning in the example under consideration, is, Socrates is _like_ A, B, C, and the other individuals who are known to have died. And this is the only universal type of that step in the reasoning process which is represented by the minor. Experience, however, of the uncertainty of this loose mode of inference, teaches the expediency of determining beforehand what _kind_ of likeness to the cases observed, is necessary to bring an un.o.bserved case within the same predicate; and the answer to this question is the major. Thus the syllogistic major and the syllogistic minor start into existence together, and are called forth by the same exigency. When we conclude from personal experience without referring to any record--to any general theorems, either written, or traditional, or mentally registered by ourselves as conclusions of our own drawing, we do not use, in our thoughts, either a major or a minor, such as the syllogism puts into words. When, however, we revise this rough inference from particulars to particulars, and subst.i.tute a careful one, the revision consists in selecting two syllogistic premises. But this neither alters nor adds to the evidence we had before; it only puts us in a better position for judging whether our inference from particulars to particulars is well grounded.

[17] Infra, book iii. ch. ii.

[18] Infra, book iii. ch. iv. 3, and elsewhere.

[19] _Mechanical Euclid_, pp. 149 _et seqq._

[20] We might, it is true, insert this property into the definition of parallel lines, framing the definition so as to require, both that when produced indefinitely they shall never meet, and also that any straight line which intersects one of them shall, if prolonged, meet the other.

But by doing this we by no means get rid of the a.s.sumption; we are still obliged to take for granted the geometrical truth, that all straight lines in the same plane, which have the former of these properties, have also the latter. For if it were possible that they should not, that is, if any straight lines other than those which are parallel according to the definition, had the property of never meeting although indefinitely produced, the demonstrations of the subsequent portions of the theory of parallels could not be maintained.

[21] Some persons find themselves prevented from believing that the axiom, Two straight lines cannot inclose a s.p.a.ce, could ever become known to us through experience, by a difficulty which may be stated as follows. If the straight lines spoken of are those contemplated in the definition--lines absolutely without breadth and absolutely straight;--that such are incapable of inclosing a s.p.a.ce is not proved by experience, for lines such as these do not present themselves in our experience. If, on the other hand, the lines meant are such straight lines as we do meet with in experience, lines straight enough for practical purposes, but in reality slightly zigzag, and with some, however trifling, breadth; as applied to these lines the axiom is not true, for two of them may, and sometimes do, inclose a small portion of s.p.a.ce. In neither case, therefore, does experience prove the axiom.

Those who employ this argument to show that geometrical axioms cannot be proved by induction, show themselves unfamiliar with a common and perfectly valid mode of inductive proof; proof by approximation. Though experience furnishes us with no lines so unimpeachably straight that two of them are incapable of inclosing the smallest s.p.a.ce, it presents us with gradations of lines possessing less and less either of breadth or of flexure, of which series the straight line of the definition is the ideal limit. And observation shows that just as much, and as nearly, as the straight lines of experience approximate to having no breadth or flexure, so much and so nearly does the s.p.a.ce-inclosing power of any two of them approach to zero. The inference that if they had no breadth or flexure at all, they would inclose no s.p.a.ce at all, is a correct inductive inference from these facts, conformable to one of the four Inductive Methods hereinafter characterized, the Method of Concomitant Variations; of which the mathematical Doctrine of Limits presents the extreme case.

[22] Whewell's _History of Scientific Ideas_, i. 140.

[23] Dr. Whewell (_Philosophy of Discovery_, p. 289) thinks it unreasonable to contend that we know by experience, that our idea of a line exactly resembles a real line. "It does not appear," he says, "how we can compare our ideas with the realities, since we know the realities only by our ideas." We know the realities (I conceive) by our senses.

Dr. Whewell surely does not hold the "doctrine of perception by means of ideas," which Reid gave himself so much trouble to refute.

If Dr. Whewell doubts whether we compare our ideas with the corresponding sensations, and a.s.sume that they resemble, let me ask on what evidence do we judge that a portrait of a person not present is like the original. Surely because it is like our idea, or mental image of the person, and because our idea is like the man himself.

Dr. Whewell also says, that it does not appear why this resemblance of ideas to the sensations of which they are copies, should be spoken of as if it were a peculiarity of one cla.s.s of ideas, those of s.p.a.ce. My reply is, that I do not so speak of it. The peculiarity I contend for is only one of degree. All our ideas of sensation of course resemble the corresponding sensations, but they do so with very different degrees of exactness and of reliability. No one, I presume, can recal in imagination a colour or an odour with the same distinctness and accuracy with which almost every one can mentally reproduce an image of a straight line or a triangle. To the extent, however, of their capabilities of accuracy, our recollections of colours or of odours may serve as subjects of experimentation, as well as those of lines and s.p.a.ces, and may yield conclusions which will be true of their external prototypes. A person in whom, either from natural gift or from cultivation, the impressions of colour were peculiarly vivid and distinct, if asked which of two blue flowers was of the darkest tinge, though he might never have compared the two, or even looked at them together, might be able to give a confident answer on the faith of his distinct recollection of the colours; that is, he might examine his mental pictures, and find there a property of the outward objects. But in hardly any case except that of simple geometrical forms, could this be done by mankind generally, with a degree of a.s.surance equal to that which is given by a contemplation of the objects themselves. Persons differ most widely in the precision of their recollection, even of forms: one person, when he has looked any one in the face for half a minute, can draw an accurate likeness of him from memory; another may have seen him every day for six months, and hardly know whether his nose is long or short. But everybody has a perfectly distinct mental image of a straight line, a circle, or a rectangle. And every one concludes confidently from these mental images to the corresponding outward things. The truth is, that we may, and continually do, study nature in our recollections, when the objects themselves are absent; and in the case of geometrical forms we can perfectly, but in most other cases only imperfectly, trust our recollections.

[24] _History of Scientific Ideas_, i. 65-67.

[25] Ibid. 60.

[26] _History of Scientific Ideas_, i. 58, 59.

[27] "If all mankind had spoken one language, we cannot doubt that there would have been a powerful, perhaps a universal, school of philosophers, who would have believed in the inherent connexion between names and things, who would have taken the sound _man_ to be the mode of agitating the air which is essentially communicative of the ideas of reason, cookery, bipedality, &c."--De Morgan, _Formal Logic_, p. 246.

[28] It would be difficult to name a man more remarkable at once for the greatness and the wide range of his mental accomplishments, than Leibnitz. Yet this eminent man gave as a reason for rejecting Newton's scheme of the solar system, that G.o.d _could not_ make a body revolve round a distant centre, unless either by some impelling mechanism, or by miracle:--"Tout ce qui n'est pas explicable" says he in a letter to the Abb Conti, "par la nature des cratures, est miraculeux. Il ne suffit pas de dire: Dieu a fait une telle loi de nature; donc la chose est naturelle. Il faut que la loi soit excutable par les natures des cratures. Si Dieu donnait cette loi, par exemple, un corps libre, de tourner l'entour d'un certain centre, _il faudrait ou qu'il y joignt d'autres corps qui par leur impulsion l'obligea.s.sent de rester toujours dans son orbite circulaire, ou qu'il mt un ange ses trousses, ou enfin il faudrait qu'il y concourt extraordinairement_; car naturellement il s'cartera par la tangente."--_Works of Leibnitz_, ed.

Dutens, iii. 446.

[29] _Novum Organum Renovatum_, pp. 32, 33.

[30] _History of Scientific Ideas_, i. 264.

[31] _Hist. Sc. Id._, i. 263.

[32] Ibid. 240.

[33] _Hist. Sc. Id._, ii. 25, 26.

[34] _Phil. of Disc._, p. 339.

[35] _Phil. of Disc._, p. 338.

[36] Ib. p. 463.

[37] _Phil. of Disc._, pp. 472, 473.

[38] The _Quarterly Review_ for June 1841, contained an article of great ability on Dr. Whewell's two great works (since acknowledged and reprinted in Sir John Herschel's Essays) which maintains, on the subject of axioms, the doctrine advanced in the text, that they are generalizations from experience, and supports that opinion by a line of argument strikingly coinciding with mine. When I state that the whole of the present chapter (except the last four pages, added in the fifth edition) was written before I had seen the article, (the greater part, indeed, before it was published,) it is not my object to occupy the reader's attention with a matter so unimportant as the degree of originality which may or may not belong to any portion of my own speculations, but to obtain for an opinion which is opposed to reigning doctrines, the recommendation derived from a striking concurrence of sentiment between two inquirers entirely independent of one another. I embrace the opportunity of citing from a writer of the extensive acquirements in physical and metaphysical knowledge and the capacity of systematic thought which the article evinces, pa.s.sages so remarkably in unison with my own views as the following:--

"The truths of geometry are summed up and embodied in its definitions and axioms.... Let us turn to the axioms, and what do we find? A string of propositions concerning magnitude in the abstract, which are equally true of s.p.a.ce, time, force, number, and every other magnitude susceptible of aggregation and subdivision. Such propositions, where they are not mere definitions, as some of them are, carry their inductive origin on the face of their enunciation.... Those which declare that two straight lines cannot inclose a s.p.a.ce, and that two straight lines which cut one another cannot both be parallel to a third, are in reality the only ones which express characteristic properties of s.p.a.ce, and these it will be worth while to consider more nearly. Now the only clear notion we can form of straightness is uniformity of direction, for s.p.a.ce in its ultimate a.n.a.lysis is nothing but an a.s.semblage of distances and directions. And (not to dwell on the notion of continued contemplation, _i.e._, mental experience, as included in the very idea of uniformity; nor on that of transfer of the contemplating being from point to point, and of experience, during such transfer, of the h.o.m.ogeneity of the interval pa.s.sed over) we cannot even propose the proposition in an intelligible form to any one whose experience ever since he was born has not a.s.sured him of the fact. The unity of direction, or that we cannot march from a given point by more than one path direct to the same object, is matter of practical experience long before it can by possibility become matter of abstract thought. _We cannot attempt mentally to exemplify the conditions of the a.s.sertion in an imaginary case opposed to it, without violating our habitual recollection of this experience, and defacing our mental picture of s.p.a.ce as grounded on it._ What but experience, we may ask, can possibly a.s.sure us of the h.o.m.ogeneity of the parts of distance, time, force, and measurable aggregates in general, on which the truth of the other axioms depends? As regards the latter axiom, after what has been said it must be clear that the very same course of remarks equally applies to its case, and that its truth is quite as much forced on the mind as that of the former by daily and hourly experience, ...

_including always, be it observed, in our notion of experience, that which is gained by contemplation of the inward picture which the mind forms to itself in any proposed case, or which it arbitrarily selects as an example--such picture, in virtue of the extreme simplicity of these primary relations, being called up by the imagination with as much vividness and clearness as could be done by any external impression, which is the only meaning we can attach to the word intuition, as applied to such relations_."

And again, of the axioms of mechanics:--"As we admit no such propositions, other than as truths inductively collected from observation, even in geometry itself, it can hardly be expected that, in a science of obviously contingent relations, we should acquiesce in a contrary view. Let us take one of these axioms and examine its evidence: for instance, that equal forces perpendicularly applied at the opposite ends of equal arms of a straight lever will balance each other. What but experience, we may ask, in the first place, can possibly inform us that a force so applied will have any tendency to turn the lever on its centre at all? or that force can be so transmitted along a rigid line perpendicular to its direction, as to act elsewhere in s.p.a.ce than along its own line of action? Surely this is so far from being self-evident that it has even a paradoxical appearance, which is only to be removed by giving our lever thickness, material composition, and molecular powers. Again, we conclude, that the two forces, being equal and applied under precisely similar circ.u.mstances, must, if they exert any effort at all to turn the lever, exert equal and opposite efforts: but what _ priori_ reasoning can possibly a.s.sure us that they _do_ act under precisely similar circ.u.mstances? that points which differ in place _are_ similarly circ.u.mstanced as regards the exertion of force? that universal s.p.a.ce may not have relations to universal force--or, at all events, that the organization of the material universe may not be such as to place that portion of s.p.a.ce occupied by it in such relations to the forces exerted in it, as may invalidate the absolute similarity of circ.u.mstances a.s.sumed? Or we may argue, what have we to do with the notion of angular movement in the lever at all? The case is one of rest, and of quiescent destruction of force by force. Now how is this destruction effected? a.s.suredly by the counter-pressure which supports the fulcrum. But would not this destruction equally arise, and by the same amount of counter-acting force, if each force simply pressed its own half of the lever against the fulcrum? And what can a.s.sure us that it is not so, except removal of one or other force, and consequent tilting of the lever? The other fundamental axiom of statics, that the pressure on the point of support is the sum of the weights ... is merely a scientific transformation and more refined mode of stating a coa.r.s.e and obvious result of universal experience, viz. that the weight of a rigid body is the same, handle it or suspend it in what position or by what point we will, and that whatever sustains it sustains its total weight. a.s.suredly, as Mr. Whewell justly remarks, 'No one probably ever made a trial for the purpose of showing that the pressure on the support is equal to the sum of the weights.' ... But it is precisely because in every action of his life from earliest infancy he has been continually making the trial, and seeing it made by every other living being about him, that he never dreams of staking its result on one additional attempt made with scientific accuracy. This would be as if a man should resolve to decide by experiment whether his eyes were useful for the purpose of seeing, by hermetically sealing himself up for half an hour in a metal case."

On the "paradox of universal propositions obtained by experience," the same writer says: "If there be necessary and universal truths expressible in propositions of axiomatic simplicity and obviousness, and having for their subject-matter the elements of all our experience and all our knowledge, surely these are the truths which, if experience suggest to us any truths at all, it ought to suggest most readily, clearly, and unceasingly. If it were a truth, universal and necessary, that a net is spread over the whole surface of every planetary globe, we should not travel far on our own without getting entangled in its meshes, and making the necessity of some means of extrication an axiom of locomotion.... There is, therefore, nothing paradoxical, but the reverse, in our being led by observation to a recognition of such truths, as _general_ propositions, coextensive at least with all human experience. That they pervade all the objects of experience, must ensure their continual suggestion _by_ experience; that they are true, must ensure that consistency of suggestion, that iteration of uncontradicted a.s.sertion, which commands implicit a.s.sent, and removes all occasion of exception; that they are simple, and admit of no misunderstanding, must secure their admission by every mind."

"A truth, necessary and universal, relative to any object of our knowledge, must verify itself in every instance where that object is before our contemplation, and if at the same time it be simple and intelligible, its verification must be obvious. _The sentiment of such a truth cannot, therefore, but be present to our minds whenever that object is contemplated, and must therefore make a part of the mental picture or idea of that object which we may on any occasion summon before our imagination.... All propositions, therefore, become not only untrue but inconceivable_, if ... axioms be violated in their enunciation."

Another eminent mathematician had previously sanctioned by his authority the doctrine of the origin of geometrical axioms in experience.

"Geometry is thus founded likewise on observation; but of a kind so familiar and obvious, that the primary notions which it furnishes might seem intuitive."--_Sir John Leslie_, quoted by Sir William Hamilton, _Discourses_, &c. p. 272.

[39] _Principles of Psychology._

[40] Mr. Spencer is mistaken in supposing me to claim any peculiar "necessity" for this axiom as compared with others. I have corrected the expressions which led him into that misapprehension of my meaning.

[41] Mr. Spencer makes a distinction between conceiving myself looking into darkness, and conceiving _that I am_ then and there looking into darkness. To me it seems that this change of the expression to the form _I am_, just marks the transition from conception to belief, and that the phrase "to conceive that _I am_," or "that anything _is_," is not consistent with using the word conceive in its rigorous sense.