Since the whole of the present facts are the infallible result of the whole of the past, so that if the prior state of the entire universe could recur it would be followed by the present, the process of ascertaining the relations of cause and effect is an analysis or resolution of this complex uniformity into the simpler uniformities which make it up. We must first mentally analyse the facts, not making this analysis minuter than is needed for our object at the time, but at the same time not regarding (as did the Greeks their verbal classifications) a mental decomposition of facts as ultimate. When we have thus succeeded in looking at any two successive chaotic masses (for such nature keeps at each instant presenting to us) as so many distinct antecedents and consequents, we must analyse the facts themselves, and try, by varying the circumstances, to discoverwhich of the antecedents and consequents (for many are always present together) are related to each other.
Experiment and observation are the two instruments for thus varying the circumstances. When the enquiry is, What are the effects of a given cause? experiment is far the superior, since it enables us not merely to produce many more and more opportune variations than nature, which is not arranged on the plan of facilitating our studies, offers spontaneously, but, what is a greater advantage, though one less attended to, also to insulate the phenomenon by placing it among known circumstances, which can be then infinitely varied by introducing a succession of well-defined new ones.
Observation cannot ascertain the effects of a given cause, because it cannot, except in the simplest cases, discover what are the concomitant circumstances; and therefore sciences in which experiment cannot be used, either at all, as in astronomy, or commonly, as in mental and social science, must be mainly deductive, not inductive. When, however, the object is to discover causes by means of their effects, observation alone is primarily available, since new effects could be artificially produced only through their causes, and these are, in the supposed case, unknown. But even then observation by itself cannot directly discover causes, as appears from the case of zoology, which yet contains many recognised uniformities. We have, indeed, ascertained a real uniformity when we observe some one antecedent to be invariably found along with the effects presented by nature. But it is only by reversing the process, andexperimentally producing the effects by means of that antecedent, that we can prove it to be unconditional, i.e. the cause.
Five canons may be laid down as the principles of experimental enquiry. The first is that of the Method of Agreement, viz.:If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the circumstances agree is the cause or the effect of the given phenomenon.The second canon is that of the Method of Difference, viz.:If an instance in which the phenomenon occurs and an instance in which it does not occur have every circumstance in common, save one, and that one occurs only in the former, that one circumstance is the effect, or the cause, or a necessary part of the cause, of the phenomenon.
These two are the simplest modes of singling out from the facts which precede or follow a phenomenon, those with which it is connected by an invariable law. Both are methods of elimination, their basis being, for the method of agreement, that whatever can be eliminatedis not, and for that of difference, that whatever cannot be eliminatedisconnected with the given phenomenon by a law. It is only, however,by the method of difference, which is a method of artificial experiment (and by experiment we can introduce into the pre-existing facts a change perfectly definite), that we can, at least by direct experience, arrive with certainty at causes. The method of agreement is chiefly useful as preliminary to and suggestive of applications of the method of difference, or as an inferior resource in its stead, when, as in the case of many spontaneous operations of nature, we have no power of producing the phenomenon.
When we have power to produce the phenomenon, but only by the agency, not of a single antecedent, but of a combination, the method of agreement can be improved (though it is even then inferior to the direct method of difference) by a double process being used, each proof being independent and corroborative of the other. This may be called the Indirect Method of Difference, or the Joint Method of Agreement and Difference, and its canon will be:If two or more instances in which the phenomenon occurs have only one circumstance in common, while two or more instances in which it does not occur have nothing in common save the absence of that circumstance, the circumstance in which alone the two sets of instances differ, is the effect, or the cause, or a necessary part of the cause, of the phenomenon.
The fourth canon is that of the Method of Residues, viz.:Subduct from any phenomenon such part as is known by previous inductions to be the effect of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents.This method is a modification of the method ofdifference, from which it differs in obtaining, of the two required instances, only the positive instance, by observation or experiment, but the negative, by deduction. Its certainty, therefore, in any given case, is conditional on the previous inductions having been obtained by the method of difference, and on there being in reality no remaining antecedentsbesidesthose given as such.
The fifth canon is that of the Method of Concomitant Variations, viz.:Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or(since they may be effects of a common cause)is connected with it through some fact of causation.Through this method alone can we find the laws of the permanent causes. For, though those of the permanent causes whose influence is local may be escaped from by changing the scene of the observation or experiment, many can neither be excluded nor even kept isolated from each other; and, therefore, in such cases, the method of difference, which requires a negative instance, and that of agreement, which requires the different instances to agree only in one circumstance, in order to prove causation, are (together with the methods which are merely forms of these) equally inapplicable. But, though many permanent antecedents insist on being always present, and never present alone, yet we have the resource of making or finding instances in which (the accompanying antecedents remaining unchanged) their influence isvariedandmodified. This method can be used most effectually when the variations of the cause arevariations of quantity; and then, if we know the absolute quantities of the cause and the effect, we may affirm generally that, at least within our limits of observation, the variations of the cause will be attended by similar variations of the effect; it being a corollary from the principle of the composition of causes, that more of the cause is followed by more of the effect. This method is employed usually when the method of difference is impossible; but it is also of use to determine according to what law the quantity or different relations of an effect ascertained by the method of difference follow those of the cause.
These four methods are the only possible modes of experimental enquiry. Dr. Whewell attacks them, first, on the ground (and the canon of ratiocination was attacked on the same) that they assume the reduction of an argument to formulæ, which (with the procuring the evidence) is itself the chief difficulty. And this is in truth the case: but, to reduce an argument to a particular form, we must first know what the form is; and in showing us this, Inductive Logic does a service the value of which is tested by the number of faulty inductions in vogue. Dr. Whewell next implies a complaint that no discoveries have ever been made by these four methods. But, as the analogous argument against the syllogism was invalidated by applying equally as against all reasoning, which must be reducible to syllogism, so this also falls by its own generality, since, if true against these methods, it must be true against all observation and experiment, since these must ever proceed by one of the four. And, moreover, even if the fourmethods were not methods of discovery, as they are, they would yet be subjects for logic, as being, at all events, the sole methods of Proof, which (unless Dr. Whewell be correct in his view that inductions are simply conceptions consistent with the facts they colligate) is the principal topic of logic.
FOOTNOTE:[1]Chap. IX. consists of 'Miscellaneous Examples of the Four Methods,' which cannot be well represented in an abridged form.
[1]Chap. IX. consists of 'Miscellaneous Examples of the Four Methods,' which cannot be well represented in an abridged form.
[1]Chap. IX. consists of 'Miscellaneous Examples of the Four Methods,' which cannot be well represented in an abridged form.
The difficulty in tracing the laws of nature arises chiefly from the Intermixture of Effects, and from the Plurality of Causes. The possibility of the latter in any given case—that is, the possibility that the same effect may have been produced by different causes—makes the Method of Agreement (when applied to positive instances) inconclusive, if the instances are few; for that Method involves a tacit supposition, that the same effect in different instances, which haveonecommon antecedent, must follow in all from the same cause, viz. from their common antecedent. When the instances are varied and very many (how many, it is for the Theory of Probability to consider), the supposition, that the presence in all of the common antecedent may be simply a coincidence, is rebutted; and this is the sole reason why merenumberof instances, differing only in immaterial points, is of any value. As applied, indeed, to negative instances, i.e. to those resembling each other in the absence of a certain circumstance, the Method of Agreementis not vitiated by Plurality of Causes. But the negative premiss cannot generally be worked unless an affirmative be joined with it: and then the Method is the Joint Method of Agreement and Difference. Thus, to find the cause of Transparency, we do not enquire in what circumstance the numberlessnon-transparent objects agree; but we enquire, first, in what the few transparent ones agree; and then, whether all the opaque do not agree in theabsenceof this circumstance.
Not only may there be Plurality of Causes, the whole of the effect being produced now by one, now by another antecedent; but there may also be Intermixture of Effects, through the interference of different causes with each other, so that part of the total effect is due to one, and part to another cause. This latter contingency, which, more than all else, complicates, the study of nature, does not affect the enquiry into those (the exceptional) cases, where, as in chemistry, the total effect is something quite different to the separate effects, and governed by different laws. There the great problem is to discover, not the properties, but the cause of the new phenomenon, i.e. the particular conjunction of agents whence it results; which could indeed never be ascertained by specific enquiry, were it not for the peculiarity, not of all these cases (e.g. not of mental phenomena), but of many, viz. that the heterogeneous effects of combined causes often reproduce, i.e. aretransformed intotheir causes (as, e.g. water into its components, hydrogen and oxygen). The great difficulty isnotthere to discover the properties of the new phenomenon itself, for these can be found by experiment like thesimpleeffects of any other cause; since, in this class of cases the effects of the separate causes give place to a new effect, and thereby cease to need consideration as separate effects. But in the far larger class of cases, viz. when the total effect is the exact sum of the separate effects of all the causes (the case of the Composition of Causes), at no point may it be overlooked that the effect is not simple but complex, the result of various separate causes, all of which are always tending to produce the whole of their several natural effects; having, it may be, theireffectsmodified, disturbed, or even prevented by each other, but always preserving theiraction, since laws of causation cannot have exceptions.
These complex effects must be investigated bydeducingthe law of the effect from the laws of the separate causes on the combination of which it depends. No inductive method is conclusive in such cases (e.g. in physiology, orà fortiori, in politics and history), whether it be the method of simple observation, which compares instances, whether positive or negative, to see if they agree in the presence or the absence of one common antecedent, or the empirical method, which proceeds by directly trying different combinations (either made or found) of causes, and watching what is the effect. Both are inconclusive; the former, because an effect may be due to the concurrence of many causes, and the latter, because we can rarely know what all the coexisting causes are; and still more rarely whether a certain portion (if not all) of the total effect is not due to these other causes, and not to the combination of causes which we are observing.
The deductive method is the main source of our knowledge of complex phenomena, and the sole source of all the theories through which vast and complicated facts have been embraced under a few simple laws. It consists of processes of Induction, Ratiocination, and Verification. First, by one of the four inductive methods, the simple laws (whence may bededucedthe complex) of each separate cause which shares in producing the effect, must be first ascertained. This is difficult, when the causes or rather tendencies cannot be observed singly. Such is the case in physiology, since the different agencies which make up an organized body cannot be separated without destroying the phenomenon; consequently there our sole resource is to produce experimentally, or find (as in the case of diseases), pathological instances in which only one organ at a time is affected. Secondly, when the laws of the causes have been found, we calculate the effect of any given combination of them by ratiocination, which may have (though not necessarily) among its premisses the theorems of the sciences of number and geometry. Lastly, as it might happen that some of the many concurring agencies have been unknown or overlooked, the conclusions of ratiocination must beverified; that is, it must be explained why they do not, or shown that they do, accord withobservedcases of at least equal complexity, and (which is the most effectual test) that the empirical laws and uniformities, if any, arrived at by direct observation,can be deduced from and so accounted for by them, as, e.g. Kepler's laws of the celestial motions by Newton's theory.
The aim, in the deductive method, is either to discover the law of the effect, or to account for it byexplainingit, that is, by pointing out some more general phenomenon (though often less familiar to us) of which this is a case and a partial exemplification, or some laws of causation which produce it by their joint or successive action. This explanation may be made, either—1. By resolving the laws of the complex effect into its elements, which consist as well of the separate laws of the causes which share in producing it, as also of their collocation, i.e. the fact that these separate laws have been so combined; or—2. By resolving the law which connects two links, not proximate, in a chain of causation, into the laws which connect each link with the intermediate links; or—3. By thesubsumptionor gathering up of several laws under one which amounts to the sum of them all, and which is the recognition of the same sequence in different sets of instances. In the first two of the processes, laws are resolved into others, which both extend to more cases, i.e. are moregeneral, and also, as being laws of nature, of which the complex laws are but results, are morecertain, i.e. moreunconditionaland moreuniversally true. In the third process, laws are resolved into others which are indeed moregeneral, but not morecertain, since they are in fact the same laws, and therefore, subject to the same exceptions.
Liebig's researches, e.g. into the Contagious Influence of Chemical Action, and his Theory of Respiration, are among the finest examples, since Newton's exposition of the law of gravitation, of the use of the deductive method forexplanation.[2]But the method is as available for explaining mental as physical facts. It is destined to predominate in philosophy. Before Bacon's time deductions were accepted as sufficient, when neither had the premisses been established by proper canons of experimental enquiry, nor the results tested by verification by specific experience. He therefore changed the method of the sciences from deductive to experimental. But, now that the principles of deduction are better understood, it is rapidly reverting from experimental to deductive. Only it must not be supposed that the inductive part of the process is yet complete. Probably, few of the great generalisations fitted to be the premisses for future deductions will be found among truths now known. Some, doubtless, are yet unthought of; others known only as laws of some limited class of facts, as electricity once was. They will probably appear first in the shape of hypotheses, needing to be tested by canons of legitimate induction.
FOOTNOTE:[2]These, and other illustrations in chap. xiii., cannot be usefully represented in an abridged form.
[2]These, and other illustrations in chap. xiii., cannot be usefully represented in an abridged form.
[2]These, and other illustrations in chap. xiii., cannot be usefully represented in an abridged form.
The constant tendency of science, operating by the Deductive Method, is to resolve all laws, even those which once seemed ultimate and not derivative, into others still more general. But no process ofresolvingwill ever reduce the number of ultimate laws below the number of those varieties of our feelings which are distinguishable in quality, and not merely in quantity or degree. Theideallimit of the explanation of natural phenomena is to show that each of these ultimate facts has (since the differences in the different cases of it affect our sensations as differences in degree only, and not in quality) only one sort of cause or mode of production; and that all the seemingly different modes of production or causes of it are resolvable into one. Butpracticallythis limit is never attained. Thus, though various laws of Causes of Motion have been resolved into others (e.g. the fall of bodies to the earth, and the motions of the planets, into the one law of mutual attraction), many causes of it remain still unresolved and distinct.
Hypotheses are made for the sake of this resolving and explaining of laws. When we do notknowof any more general laws into which to resolve an uniformity, we then (either on no or on insufficient evidence)supposesome, imagining either causes (as, e.g. Descartes did the Vortices), or the laws of their operation (as did Newton respecting the planetary central force); but we never feign both cause and law. The use ofa hypothesis is to enable us to apply the Deductive Method before the laws of the causes have been ascertained by Induction. In those cases where a false law could not have led to a true result (as was the case with Newton's hypothesis as to the law of the Attractive force) the third part of the process in the Deductive Method, viz. Verification, which shows that the results deduced are true, amounts to a complete induction, and one conforming to the canon of the Method of Difference. But this is the case only when either the cause is known to be one given agent (and only its law is unknown), or to be one of several given agents.
An assumed cause, on the other hand, cannot be accepted as true simplybecauseit explains the phenomena (since two conflicting hypotheses often do this even originally, or, as Dr. Whewell himself allows, may at any rate by modifications be made to do it); norbecauseit moreover leads to the prediction of other results which turn out true (since this shows only what was indeed apparent already from its agreement with the old facts, viz. that the phenomena are governed by laws partially identical with the laws of other causes); norbecausewe cannot imagine any other hypothesis which will account for the facts (since there may be causes unknown to our present experience which will equally account for them). The utility of such assumptionsof causesdepends on their being, in their own nature,capable(as Descartes' Vortices were not, though possibly the Luminiferous Ether may be) of being, at some time or other, proved directly by independent evidence to be the causes. And this was, perhaps, all that Newton meantby hisveræ causæ, which alone, he said, may be assigned as causes of phenomena. Assumptions of causes, which fulfil this condition, are, in science, even indispensable, with a view both to experimental inquiry, and still more to the application of the Deductive Method. They may be accepted, not indeed, as Dr. Whewell thinks they may be, as proof, but as suggesting a line of experiment and observation which may result in proof. And this is actually the method used by practical men for eliciting the truth from involved statements. They first extemporise, from a few of the particulars, a rude theory of the mode in which the event happened; and then keep altering it to square with the rest of the facts, which they review one by one.
The attempting, as in Geology, to conjecture, in conformity with known laws, in what former collocations of known agents (thoughnotknown to have been formerly present) individual existing facts may have originated, is not Hypothesis but Induction; for then we do notsupposecauses, but legitimately infer from known effects to unknown causes. Of this nature was Laplace's theory, whether weak or not, as to the origin of the earth and planets.
Sometimes a complex effect results, not (as has been supposed in the last four chapters) from several, but fromonelaw. The following is the way.
Some effects are instantaneous (e.g. somesensations), and are prolonged only by the prolongation of the causes; others are in their own nature permanent. In some cases of the latter class, the original is also the proximate cause (e.g. Exposure to moist air is both the original and the proximate cause of iron rust). But in others of the same class, the permanency of the effect is only the permanency of a series of changes. Thus, e.g. in cases of Motion, the original force is only theremotecause of any link (after the very first) in the series; and the motion immediately preceding it, being itself a compound of the original force and any retarding agent, is itsproximatecause. When the original cause is permanent as well as the effect (e.g. Suppose a continuance of the iron's exposure to moist air), we get a progressive series of effects arising from the cause's accumulating influence; and the sum of these effects amounts exactly to what a number of successively introduced similar causes would have produced. Such cases fall under the head of Composition of Causes, with this peculiarity, that, as the causes (to regard them as plural) do not come into play all at once, the effect at each instant is the sum of the effects only of the then acting causes, and the result will appear as an ascending series. Each addition in such case takes place according to a fixed law (equal quantities in equal times); and therefore it can be computed deductively. Even when, as is sometimes the case, a cause is at once permanent and progressive (as, e.g. the sun, by its position becoming more vertical, increases the heat in summer) so that the quantities added are unequal, the effect is still progressive, resulting from its cause's continuance and progressiveness combined.
Inallcases whatever of progressive effects, the succession not merely between the cause and the effect, but also between the first and latter stages of the effect, is uniform. Hence, from the invariable sequence of two terms (e.g. Spring and Summer) in a series going through any continued and uniform process of variation, we do not presume that one is the cause and the others the effect, but rather that the whole series is an effect.
Empirical laws are derivative laws, of which the derivation is not known. They are observed uniformities, which we compare with the result of any deduction to verify it; but of which thewhy, and also the limits, are unrevealed, through their being, though resolvable, not yet resolved into the simpler laws. They depend usually, not solely on the ultimate laws into which they are resolvable; but on those, together with an ultimate fact, viz. the mode of coexistence of some of the component elements of the universe. Hence their untrustworthiness for scientific purposes; for, till they have been resolved (and then a derivative law ceases to be empirical), we cannot know whether they result from the different effects of one cause, or from effects of different causes; that is, whether they depend on laws, or on laws and a collocation. And if they thus depend on a collocation, they can be received as true only within the limits of time andspace, and also circumstance, in which they have been observed, since the mode of the collocation of the permanent causes is not reducible to a law, there being no principle known to us as governing the distribution and relative proportions of the primæval natural agents.
Uniformities cannot be proved by the Method of Agreement alone to be laws of causation; they must be tested by the Method of Difference, or explained deductively. But laws of causation themselves are either ultimate or derivative. Signs, previous to actual proof byresolutionof them, of their being derivative, are, either that we cansurmisethe existence of a link between the known antecedent and the consequent, as e.g. in the laws of chemical action; or, that the antecedent is some very complex fact, the effects of which are probably (since most complex cases fall under the Composition of Causes) compounded of the effects of its different elements. But the laws which, though laws of causation, are thus presumably derivative laws only, need, equally with the uniformities which are not known to be laws of causation at all, to be explained by deduction (which they then in turn verify), and are lesscertainthan when they have been resolved into the ultimate laws. Consequently they come under the definition of Empirical Laws, equally with uniformities not known to be laws of causation. However, the latter are far moreuncertain; for as, till they are resolved, we cannot tell on how many collocations, as well as laws, they may not depend, we must not rely on them beyond the exact limits in which the observations were made. Therefore, the nameEmpirical Lawswill generally be confined here to these.
Empirical laws are certain only in those limits within which they have beenobservedto be true. But, even within those limits, the connection of two phenomena may, as the same effect may be produced by several different causes, be due to Chance; that is, it may, though being, as all facts must be, the result of some law, be a coincidence whence, simply because we do not know all the circumstances,wehave no ground to infer an uniformity. When neither Deduction, nor the Method of Difference, can be applied, the only way of inferring that coincidences are not casual, is by observing the frequency of their occurrence, not their absolute frequency, but whether they occurmoreoften than chance would (that is, more often than the positive frequency of the phenomena would) account for. If, in such cases, we could ascend to the causes of the two phenomena, we should find at some stage some cause or causes common to both. Till we can do this, the fact of the connection between them is only an empirical law; but still it is a law.
Sometimes an effect is the result partly of chance, and partly of law: viz. when the total effect is the result partly of the effects of casual conjunctions of causes, and partly of the effects of some constant cause which they blend with and modify. This is a case of Composition of Causes. The object being to findhow muchof the result is attributable to a givenconstant cause, the only resource, when the variable causes cannot be wholly excluded from the experiment, is to ascertain what is the effect of all ofthemtaken together, and then to eliminate this, which is the casual part of the effect, in reckoning up the results. If the results of frequent experiments, in which the constant cause is kept invariable, oscillate round one point, that average or middle point is due to the constant cause, and the variable remainder to chance; that is, to causes the coexistence of which with the constant cause was merely casual. The test of the sufficiency of such an induction is, whether or not an increase in the number of experiments materially alters the average.
We can thus discover not merelyhow muchof the effect, but even whetheranypart of it whatever is due to a constant cause, when this latter is so uninfluential as otherwise to escape notice (e.g. the loading of dice). This case of the Elimination of Chance is calledThe discovery of a residual phenomenon by eliminating the effects of chance.
The mathematical doctrine of chances, or Theory of Probabilities, considers what deviation from the average chance by itself can possibly occasion in some number of instances smaller than is required for a fair average.
In order to calculate chances, we must know that of several events one, and no more, must happen, and also not know, or have any reason to suspect, which of them that one will be. Thus, with the simple knowledge that the issue must be one of a certain number of possibilities, wemayconclude that one supposition is most probableto us. For this purpose it is notnecessarythat specific experience or reason should have also proved the occurrence of each of the several events to be, as a fact, equally frequent. For, the probability of an event is not a quality of the event (since every event is in itself certain), but is merely a name for the degree of groundwehave, with our present evidence, for expecting it. Thus, if we know that a box contains red, white, and black balls, though we do not know in what proportions they are mingled, we have numerically appreciable grounds for considering the probability to be two to one against any one colour. Our judgment may indeed be said in this case to rest on the experience we have of the laws governing the frequency of occurrence of the different cases; but such experience is universal and axiomatic, and not specific experience about a particular event. Except, however, in games of chance, the purpose of which requires ignorance, such specific experience can generally be, and should be gained. And a slight improvement in the data profits more than the most elaborate application of the calculus of probabilities to the bare original data,e.g. to such data, when we are calculating the credibility of a witness, as the proportion, even if it could be verified, between the number of true and of erroneous statements a man,quâman, may be supposed to make during his life. Before applying the Doctrine of Chance, therefore, we should lay a foundation for an evaluation of the chances by gaining positive knowledge of the facts. Hence, though not anecessary, yet a most usual condition for calculating the probability of a fact is, that we should possess aspecificknowledge of the proportion which the cases in which facts of the particular sort occur bear to the cases in which they do not occur.
Inferences drawn correctly according to the Doctrine of Chances depend ultimately on causation. This is clearest, when, as sometimes, the probability of an event is deduced from the frequency of the occurrence of the causes. When its probability is calculated by merely counting and comparing the number of cases in which it has occurred with those in which it has not, the law, being arrived at by the Method of Agreement, is only empirical. But even when, as indeed generally, the numerical data are obtained in the latter way (since usually we can judge of the frequency of the causes only through the medium of the empirical law, which is based on the frequency of the effects), still then, too, the inference really depends on causation alone. Thus, an actuary infers from his tables that, of any hundred living persons under like conditions, five will reach a given age, not simply because that proportion have reached it in times past, but because that fact shows the existence there of a particular proportion betweenthecauseswhich shorten and the causes which prolong life to the given extent.
Derivative laws are inferior to ultimate laws, both in the extent of the propositions, and in their degree of certainty within that extent. In particular, the uniformities of coexistence and sequence which obtain between effects depending on different primæval causes, vary along with any variation in the collocation of these causes. Even when the derivative uniformity is between different effects of the same cause, it cannot be trusted to, since one or more of the effects may be producible by another cause also. The effects, even, of derivative laws ofcausation(resulting, i.e. the laws, from the combination of several causes) are not independent of collocations; for, though laws of causation, whether ultimate or derivative, are themselves universal, being fulfilled even when counteracted, the peculiar probability of the latter kind of laws of causation being counteracted (as compared with ultimate laws, which are liable to frustration only from one set of counteracting causes) is fatal to the universality of the derivative uniformities made up of the sequences or coexistences of their effects; and, therefore, such derivative uniformities as the latter are to be relied on only when the collocations are known not to have changed.
Derivative laws, not causative, may certainly be extended beyond the limits of observation, but only to casesadjacentin time. Thus, we may not predict that the sun will rise this day 20,000 years, but we can predict that it will rise to-morrow, on the ground that it has risen every day for the last 5,000 years. The latter prediction is lawful,because, while we know the causes on which its rising depends, we know, also, that there has existed hitherto no perceptible cause to counteract them; and that it is opposed to experience that a cause imperceptible for so long should start into immensity in a day. If the uniformity is empirical only, that is, if we do not know the causes, and if we infer that they remain uncounteracted from their effects alone, we still can extend the law to adjacent cases, but only to cases still more closely adjacent in time; since we can know neither whether changes in these unknown causes may not have occurred, nor whether there may not exist now an adverse cause capable after a time of counteracting them.
An empirical law cannot generally be extended, in reference toPlace, even to adjacent cases (since there is no uniformity in the collocations of primæval causes). Such an extension is lawful only if the new cases arepresumablywithin the influence of the same individual causes, even though unknown. When, however, the causes are known, and the conjunction of the effects is deducible from laws of the causes, the derivative uniformity may be extended over a wider space, and with less abatement for the chance of counteracting causes.
One of the many meanings ofAnalogyis, Resemblance of Relations. The value of an analogical argument in this sense depends on the showing that, on the common circumstance which is thefundamentum relationis, the rest of the circumstances of the case depend. But, generally,to argue from analogysignifies to infer from resemblance in some points (not necessarily inrelations) resemblance in others. Induction does the same: but analogy differs from induction in not requiring the previous proof, by comparison of instances, of the invariable conjunction between the known and the unknown properties; though it requires that the latter should not have been ascertained to beunconnectedwith the common properties.
If a fair proportion of the properties of the two cases are known, every resemblance affords ground for expecting an indefinite number of other resemblances, among which the property in question may perhaps be found. On the other hand, every dissimilarity will lead us to expect that the two cases differ in an indefinite number of properties, including, perhaps, the one in question. These dissimilarities may even be such as would, in regard to one of the two cases, imply the absence of that property; and then every resemblance, as showing that the two cases have a similar nature, is even a reason for presuming against the presence of that property. Hence, the value of an analogical argument dependson the extent of ascertained resemblance as compared, first, with the amount of ascertained difference, and next, with the extent of the unexplored region of unascertained properties.
The conclusions of analogy are not of direct use, unless when the case to which we reason is a caseadjacent, not, as before, in time or place, but incircumstances. Even then a complete induction should be sought after. But the great value of analogy, even when faint, in science, is that it may suggest observations and experiments, with a view to establishing positive scientific truths, for which, however, the hypotheses based on analogies must never be mistaken.
The validity of all the four inductive methods depends on our assuming that there is a cause for every event. The belief in this, i.e. in the law of universal causation, some affirm, is an instinct which needs no warrant other than all men's disposition to believe it; and they argue that to demand evidence of it is to appeal to the intellect from the intellect. But, though there is no appeal from the faculties all together, there may be from one to another: and, as belief is not proof (for it may be generated by association of ideas as well as by evidence), a case of belief does require to be proved by an appeal to something else, viz. to the faculties of sense and consciousness.
The law of universal causation is, in fact, a generalisation from many partial uniformities of sequence. Consequently, like these, which cannot have been arrived at by any strict inductive method (for all such methods presuppose the law of causation itself), it must itself be based on inductionsper simplicem enumerationem, that is, generalisations of observed facts, from the mere absence of any known instances to the contrary. This unscientific process is, it is true, usually delusive; but only because, and in proportion as, the subject-matter of the observation is limited in extent. Its results, whenever the number of coincidences is too large for chance to explain, are empirical laws. These are ordinarily true only within certain limits of time, place, and circumstance, since, beyond these, there may be different collocations or counteracting agencies. But the subject-matter of the law of universal causation is so diffused that there is no time, place, or set of circumstances, at least within the portion of the universe within our observation, and adjacent cases, but must prove the law to be either true or false. It has, in fact, never been found to be false, but in ever increasing multitudes of cases to be true; and phenomena, even when, from their rarity or inaccessibility, or the number of modifying causes, they are not reducibleuniversallyto any law, yetin some instancesdo conform to this. Thus, it may be regarded as coextensive with all human experience, at which point the distinction between empirical laws and laws of nature vanishes. Formerly, indeed, it was only a very high probability; but, with our modern experience, it is, practically, absolutely certain, andit confirms the particular laws of causation, whence itself was drawn, when there seem to be exceptions to them. All narrower inductions got by simple enumeration are unsafe, till, by the application to them of the four methods, the supposition of their falsity is shown to contradictthislaw, though it was itself arrived at by simple enumeration.
Besides uniformities of succession, which always depend on causation, there are uniformities of coexistence. These also, whenever the coexisting phenomena are effects of causes, whether of one common cause or of several different causes, depend on the laws of their cause or causes; and, till resolved into these laws, are mere empirical laws. But there are some uniformities of coexistence, viz. those between the ultimate properties ofkinds, which do not depend on causation, and therefore seem entitled to be classed as a peculiar sort of laws of nature. As, however, the presumption always is (except in the case of thosekindswhich are calledsimple substancesor elementary natural agents), that a thing's properties really depend on causes though not traced, and wenevercan be certain that they do not; we cannot safely claim (though itmaybe an ultimate truth) higher certainty than that of an empirical law for any generalisation about coexistence, that is to say(sincekindsare known to us only by their properties, and, consequently, all assertions about them are assertions about the coexistence of something with those properties), about the properties ofkinds.
Besides, no rigorous inductive system can be applied to the uniformities of coexistence, since there is no general axiom related to them, as is the law of causation to those of succession, to serve as a basis for such a system. Thus, Bacon's practical applications of his method failed, from his supposing that we can have previous certainty that a property must have an invariable coexistent (as it must have an invariable antecedent), which he called its form. He ought to have seen that his great logical instrument, elimination, is inapplicable to coexistences, since things, which agree in having certain apparently ultimate properties, often agree in nothing else; even the properties which (e.g. Hotness) are effects of causes, generally being not connected with the ultimate resemblances or diversities in the objects, but depending on some outward circumstance.
Our only substitute for an universal law of coexistence is the ancients' inductionper enumerationem simplicem ubi non reperitur instantia contradictoria, that is, the improbability that an exception, if any existed, could have hitherto remained unobserved. But the certainty thus arrived at can be only that of an empirical law, true within the limits of the observations. For the coexistent property must be either a property of thekind, or an accident, that is, something due to an extrinsic cause, and not to thekind(whose own indigenous properties are always the same). And the ancients'class of induction can only prove thatwithin given limits, either (in the latter case) one common, though unknown, cause has always been operating, or (in the former case) that no newkindof the object hasas yetorby usbeen discovered.
The evidence is, of course (with respect both to the derivative and the ultimate uniformities of coexistence), stronger in proportion as the law is more general; for the greater the amount of experience from which it is derived, the more probable is it that counteracting causes, or that exceptions, if any, would have presented themselves. Consequently, it needs more evidence to establish an exception to a very general, than to a special, empirical law. And common usage agrees with this principle. Still, even the greater generalisations, when not based on connection by causation, are delusive, unless grounded on a separate examination ofeachof the includedinfimæ species, though certainly there is a probability (no more) that a sort of parallelism will be found in the properties of different kinds; and that their degree of unlikeness in one respect bears some proportion to their unlikeness in others.
The inferences calledprobablerest on approximate generalisations. Such generalisations, besides theinferior assurance with which they can be applied to individual cases, aregenerallyalmost useless as premisses in a deduction; and therefore inSciencethey are valuable chiefly as steps towards universal truths, the discovery of which is its proper end. But inpracticewe are forced to use them—1, when we have no others, in consequence of not knowing what general property distinguishes the portion of the class which have the attribute predicated, from the portion which have it not (though it is true that we can, in such a case, usually obtain a collection of exactly true propositions by subdividing the class into smaller classes); and, 2, when wedoknow this, but cannot examine whether that general property is present or not in the individual case; that is, when (as usually inmoralinquiries) we could get universal majors, but not minors to correspond to them. In any case an approximate generalisation can never be more than an empirical law. Its authority, however, is less when it composes the whole of our knowledge of the subject, than when it is merely the most available form of our knowledge for practical guidance, and the causes, or some certain mark of the attribute predicated, being known to us as well as the effects, the proposition can be tested by our trying to deduce it from the causes or mark. Thus, our belief that most Scotchmen can read, rests on our knowledge, not merely that most Scotchmen that we have known about could read, but also that most have been at efficient schools.
Either a single approximate generalisation may be applied to an individual instance, or several to the same instance. In the former case, theproposition, as stating a general average, must be applied only to average cases; it is, therefore, generally useless for guidance in affairs which do not concern large numbers, and simply supplies, as it were, the first term in a series of approximations. In the latter case, when two or more approximations (not connected with each other) areseparatelyapplicable to the instance, it is said that two (or more)probabilities are joined by addition, or, that there is aself-corroborative chainof evidence. Its type is: Most A are B; most C are B; this is both an A and a C; therefore it is probably a B. On the other hand, when the subsequent approximation or approximations is or are applicable only by virtue of the application of the first, this is joining two (or more) probabilities,by way of Deduction, which produces aself-infirmative chain; and the type is: Most A are B; most C are A; this is a C; therefore it is probably an A; therefore it is probably a B. As, in the former case, the probability increases at each step, so, in the latter, it progressively dwindles. It is measured by the probability arising from the first of the propositions, abated in the ratio of that arising from the subsequent; and the error of the conclusion amounts to the aggregate of the errors of all the premisses.
In two classes of cases (exceptions which prove the rule) approximate can be employed in deduction as usefully as complete generalisations. Thus, first, we stop at them sometimes, from the inconvenience, not the impossibility, of going further; and, by adding provisos, we might change the approximate into an universal proposition; the sum of the provisos beingthen the sum of the errors liable to affect the conclusion. Secondly, they are used in Social Science with reference to masses withabsolutecertainty, even without the addition of such provisos. Although the premisses in the Moral and Social Sciences are only probable, these sciences differ from the exact only in that we cannot decipher so many of the laws, and not in the conclusions that we do arrive at being less scientific or trustworthy.
There are, we have seen, five facts, one of which every proposition must assert, viz. Existence, Order in Place, Order in Time, Causation, and Resemblance. Causation is not fundamentally different from Coexistence and Sequence, which are the two modes of Order in Time. They have been already discussed. Of the rest, Existence, if of things in themselves, is a topic for Metaphysics, Logic regarding the existence ofphenomenaonly; and as this, when it is not perceived directly, is proved by proving that the unknown phenomenon is connected bysuccession or coexistencewith some known phenomenon, the fact of Existence is not amenable to anypeculiarinductive principles. There remain Resemblance and Order in Place.
As for Resemblance, Locke indeed, and, in a more unqualified way, his school, asserted that all reasoning is simply a comparison of two ideas by means of athird, and that knowledge is only the perception of the agreement or disagreement, that is, the resemblance or dissimilarity, of two ideas: they did not perceive, besides erring in supposing ideas, and not the phenomena themselves, to be the subjects of reasoning, that it is only sometimes (as, particularly, in the sciences of Quantity and Extension) that the agreement or disagreement of two things is the one thing to be established. Reasonings, however, aboutResemblances, whenever the two things cannot be directly compared by the virtually simultaneous application of our faculties to each, do agree with Locke's account of reasoning; being, in fact, simply such a comparison of two things through the medium of a third. There are laws or formulæ for guiding the comparison; but the only ones which do not come under the principles of Induction already discussed, are the mathematical axioms of Equality, Inequality, and Proportionality, and the theorems based on them. For these, which are true of all phenomena, or, at least, without distinction of origin, have no connection with laws of Causation, whereas all other theorems asserting resemblance have, being true only of special phenomena originating in a certain way, and the resemblances between which phenomena must be derived from, or be identical with, the laws of their causes.
In respect to Order in Place, as well as in respect to Resemblance, some mathematical truths are the only general propositions which, as being independent of Causation, require separate consideration. Such are certain geometrical laws, through which, from the position of certain points, lines, or spaces, we inferthe position of others, without any reference to their physical causes, or to their special nature, except as regards position or magnitude. There is no other peculiarity as respects Order in Place. For, the Order in Place of effects is of course a mere consequence of the laws of their causes; and, as for primæval causes, intheirOrder in Place, called theircollocation, no uniformities are traceable.
Hence, only the methods of Mathematics remain to be investigated; and they are partly discussed in the Second Book. The directly inductive truths of Mathematics are few: being, first, certain propositions about existence, tacitly involved in the so-called definitions; and secondly, the axioms, to which latter, though resting only on induction,per simplicem enumerationem, there could never have been even any apparent exceptions. Thus, every arithmetical calculation rests (and this is what makes Arithmetic the type of a deductive science) on the evidence of the axiom: The sums of equals are equals (which is coextensive with nature itself)—combined with the definitions of the numbers, which are severally made up of the explanation of the name, which connotes the way in which the particular agglomeration is composed, and of the assertion of a fact, viz. the physical property so connoted.
The propositions of Arithmetic affirm the modes of formation of given numbers, and are true of all things under the condition of being divided in a particular way. Algebraical propositions, on the other hand, affirm the equivalence of different modes of formation of numbers generally, and are true of all things under condition of being divided inanyway.
Though the laws of Extension are not, like those of Number, remote from visual and tactual imagination, Geometry has not commonly been recognised as a strictly physical science. The reason is, first, the possibility of collecting its facts as effectually from the ideas as from the objects; and secondly, the illusion that its ideal data are not mere hypotheses, like those in now deductive physical sciences, but a peculiar class of realities, and that therefore its conclusions areexceptionallydemonstrative. Really, all geometrical theorems are laws of external nature. They might have been got by generalising from actual comparison and measurement; only, that it was found practicable to deduce them from a few obviously true general laws, viz. The sums of equals are equals; things equal to the same thing are equal to one another (which two belong to the Science of Number also); and, thirdly (what is no merely verbal definition, though it has been so called): Lines, surfaces, solid spaces, which can be so applied to one another as to coincide, are equal. The rest of the premisses of Geometry consist of the so-called definitions, which assert, together with one or more properties, the real existence of objects corresponding to the names to be defined. The reason why the premisses are so few, and why Geometry is thus almost entirely deductive, is, that all questions of position and figure, that is, of quality, may be resolved into questions of quantity or magnitude, and so Geometry may be reduced to the one problem of the measurement of magnitudes; that is, to the finding the equalities between them.
Mathematical principles can be applied to othersciences. All causes operate according to mathematical laws; an effect being ever dependent on the quantity or a function of the agent, and generally on its position too. Mathematical principles cannot, indeed, as M. Comte has well explained, be usefully applied to physical questions, whenever the causes are either too inaccessible for their numerical laws to be ascertained, or are too complex forusto compute the effect, or are ever fluctuating. And, in proportion as physical questions cease to be abstract and hypothetical, mathematical solutions of them become imperfect. But the great value of mathematical training is, that we learn to use itsmethod(which is the most perfect type of the Deductive Method), that is, we learn to employ the laws of simpler phenomena to explain and predict those of the more complex.
The result of examining evidence is not always belief, or even suspension of judgment, but is sometimes positive disbelief. This can ensue only when the affirmative evidence does not amount to full proof, but is based on some approximate generalisation. In such cases, if the negative evidence consist of a stronger, though still only an approximate, generalisation, we think the fact improbable, and disbelieve it provisionally; but if of a complete generalisation based on a rigorous induction, it is disbelieved by us totally, and thought impossible. Hence, Hume declared miracles incredible, as being, he considered, contraryto a complete induction. Now, it is true thatin the absence of any adequate counteracting cause, a fact contrary to a complete induction is incredible, whatever evidence it may be grounded on; unless, indeed, the evidence go to prove the supposed law inconsistent with some better established one. But when a miracle is asserted, the presence of an adequate counteracting causeisasserted, viz. a direct interposition of an act of the will of a Being having power over nature. Therefore, all that Hume proved is, that we cannot believe in a miracle unless we believe in the power, andthe will, of the Deity to interfere with existing causes by introducing new ones; and that, in default of such belief, not the most satisfactory evidence of our senses or of testimony can hinder us from holding a seeming miracle to be merely the result of some unknown natural cause. The argument of Dr. Campbell and others against Hume, however, is untenable, viz. that, as we do not disbelieve an alleged fact (which may be something conforming to the uniform course of experience) merely because the chances are against it, therefore we need never disbelieve any fact supported by credible testimony (even if contrary to the uniform course of experience). But this is to confoundimprobability before the fact, which isnotalways a ground for disbelief, withimprobability after the fact, which always is.
Facts which conflict with special laws of causation are only improbable before the fact; that is, our disbelief depends on the improbability that there could have been present, without our knowledge, at the time and place of the event, an adequate counteracting cause. So, too, with facts which conflict withthe properties ofkinds(which are uniformities of mere coexistence not proved to be dependent on causation), that is, facts which assert the existence of a newkind; such facts we disbelieve only if, the generalisation being sufficiently comprehensive, some properties are said to have been found in the supposed newkinddisjoined from others which always have been known to accompany them. When the assertion would amount, if admitted, only to the existence of an unknown cause or an anomalouskind,unconformable, but, as Hume puts it,not contraryto experience, in circumstances so little explored, that it is credible hitherto unknown things may there be found, and when prejudice cannot have tempted to the assertion, one ought neither to admit nor to reject the testimony, but to suspend judgment till it be confirmed or disproved from other sources. Only facts, then, which are contradictory to the laws of Number, Extension, and Universal Causation (since these know no counteraction or anomaly), or to laws nearly as general, are improbable after, as well as before the fact, and only these we should termabsolutely impossible, calling other factsimprobableonly, or, at most,impossible in the circumstances of the case.
Between these two species of improbabilities liecoincidences; that is, combinations of chances presenting some unexpected regularity assimilating them in so far to the results of law. It was thought by d'Alembert that, though regular combinations are as probable as others according to the mathematical theory, some physical law prevents them from occurring so often. Now, stronger testimonymay indeed be needed to support the assertion of such a combination as, e.g. ten successive throws of sixes at dice, because such a regular series is more likely than an irregular series to be the result of design; and because even the desire to excite wonder is likely to tempt men to assert the occurrence falsely, though this probability must be estimated afresh in every instance. But though such a seriesseemspeculiarly improbable, it is only because the comparison is tacitly made, not between it and any one particular previously fixed series of throws, but between all regular and all irregular successions taken together. The fact is not in itself more improbable; and no stronger evidence is needed to give it credibility, apart from the reasons above mentioned, than in the case of ordinary events.