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 inductions _per 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 circ.u.mstance, 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 circ.u.mstances, 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 mult.i.tudes of cases to be true; and phenomena, even when, from their rarity or inaccessibility, or the number of modifying causes, they are not reducible _universally_ to any law, yet _in some instances_ do 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, and it 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 contradict _this_ law, though it was itself arrived at by simple enumeration.
CHAPTER XXII.
UNIFORMITIES OF COEXISTENCE NOT DEPENDENT ON CAUSATION.
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 of _kinds_, which do not depend on causation, and therefore seem ent.i.tled to be cla.s.sed as a peculiar sort of laws of nature. As, however, the presumption always is (except in the case of those _kinds_ which are called _simple substances_ or elementary natural agents), that a thing"s properties really depend on causes though not traced, and we _never_ can be certain that they do not; we cannot safely claim (though it _may_ be an ultimate truth) higher certainty than that of an empirical law for any generalisation about coexistence, that is to say (since _kinds_ are known to us only by their properties, and, consequently, all a.s.sertions about them are a.s.sertions about the coexistence of something with those properties), about the properties of _kinds_.
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 circ.u.mstance.
Our only subst.i.tute for an universal law of coexistence is the ancients"
induction _per enumerationem simplicem ubi non reperitur instantia contradictoria_, that is, the improbability that an exception, if any existed, could have hitherto remained un.o.bserved. 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 the _kind_, or an accident, that is, something due to an extrinsic cause, and not to the _kind_ (whose own indigenous properties are always the same). And the ancients" cla.s.s of induction can only prove that _within given limits_, either (in the latter case) one common, though unknown, cause has always been operating, or (in the former case) that no new _kind_ of the object has _as yet_ or _by us_ been 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 of _each_ of the included _infimae 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.
CHAPTER XXIII.
APPROXIMATE GENERALISATIONS, AND PROBABLE EVIDENCE.
The inferences called _probable_ rest on approximate generalisations.
Such generalisations, besides the inferior a.s.surance with which they can be applied to individual cases, are _generally_ almost useless as premisses in a deduction; and therefore in _Science_ they are valuable chiefly as steps towards universal truths, the discovery of which is its proper end. But in _practice_ we are forced to use them--1, when we have no others, in consequence of not knowing what general property distinguishes the portion of the cla.s.s 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 cla.s.s into smaller cla.s.ses); and, 2, when we _do_ know this, but cannot examine whether that general property is present or not in the individual case; that is, when (as usually in _moral_ inquiries) 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, the proposition, 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) are _separately_ applicable to the instance, it is said that two (or more) _probabilities are joined by addition_, or, that there is a _self-corroborative chain_ of 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 a _self-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 cla.s.ses 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 being then the sum of the errors liable to affect the conclusion. Secondly, they are used in Social Science with reference to ma.s.ses with _absolute_ certainty, 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.
CHAPTER XXIV.
THE REMAINING LAWS OF NATURE.
There are, we have seen, five facts, one of which every proposition must a.s.sert, 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 of _phenomena_ only; and as this, when it is not perceived directly, is proved by proving that the unknown phenomenon is connected by _succession or coexistence_ with some known phenomenon, the fact of Existence is not amenable to any _peculiar_ inductive principles. There remain Resemblance and Order in Place.
As for Resemblance, Locke indeed, and, in a more unqualified way, his school, a.s.serted that all reasoning is simply a comparison of two ideas by means of a third, 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 Quant.i.ty and Extension) that the agreement or disagreement of two things is the one thing to be established. Reasonings, however, about _Resemblances_, 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 formulae 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 a.s.serting 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 s.p.a.ces, we infer the 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 primaeval causes, in _their_ Order in Place, called their _collocation_, 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 a.s.sertion 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 in _any_ way.
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 cla.s.s of realities, and that therefore its conclusions are _exceptionally_ demonstrative. 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 s.p.a.ces, 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 a.s.sert, 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 quant.i.ty 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 other sciences. All causes operate according to mathematical laws; an effect being ever dependent on the quant.i.ty 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 for _us_ to 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 its _method_ (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.
CHAPTER XXV.
THE GROUNDS OF DISBELIEF.
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, contrary to a complete induction. Now, it is true that _in 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 a.s.serted, the presence of an adequate counteracting cause _is_ a.s.serted, 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, and _the 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 confound _improbability before the fact_, which is _not_ always a ground for disbelief, with _improbability 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 with the properties of _kinds_ (which are uniformities of mere coexistence not proved to be dependent on causation), that is, facts which a.s.sert the existence of a new _kind_; such facts we disbelieve only if, the generalisation being sufficiently comprehensive, some properties are said to have been found in the supposed new _kind_ disjoined from others which always have been known to accompany them. When the a.s.sertion would amount, if admitted, only to the existence of an unknown cause or an anomalous _kind_, _unconformable_, but, as Hume puts it, _not contrary_ to experience, in circ.u.mstances so little explored, that it is credible hitherto unknown things may there be found, and when prejudice cannot have tempted to the a.s.sertion, 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 term _absolutely impossible_, calling other facts _improbable_ only, or, at most, _impossible in the circ.u.mstances of the case_.
Between these two species of improbabilities lie _coincidences_; that is, combinations of chances presenting some unexpected regularity a.s.similating 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 testimony may indeed be needed to support the a.s.sertion 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 a.s.sert the occurrence falsely, though this probability must be estimated afresh in every instance. But though such a series _seems_ peculiarly 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.
BOOK IV.
OPERATIONS SUBSIDIARY TO INDUCTION.
CHAPTER I.
OBSERVATION AND DESCRIPTION.
The mental process which Logic deals with, viz. the investigation of truth by means of evidence, is always a process of Induction. Since Induction is simply the extension to a cla.s.s of something observed to be true of certain members of it, Observation is the first preliminary to it. It is, therefore, right to consider, not indeed how or what to observe (for this belongs to the art of Education), but under what conditions observation is to be relied on. The sole condition is, that the supposed observation should really be an observation, and not an inference, whereas it is usually a compound of both, there being, in our propositions, besides observation which relates only to the sensations, an inference from the sensations to the objects themselves. Thus so-called errors of sense are only erroneous inferences from sense. The sensations themselves must be genuine; but, as they generally arise on a certain arrangement of outward objects being present to the organs, we, as though by instinct, infer this arrangement even when not existing.
The sole object, then, of the logic of observation, is to separate the inferences from observation from the observations themselves, the only thing really observed by the mind (to waive the metaphysical problem as to the _perception_ of objects) being its own feelings or states of consciousness, outward, viz. Sensations, and inward, viz. Thoughts, Emotions, and Volitions.
As in the simplest observation much is inference, so, in describing an observed fact, we not merely describe the fact, but are always forced to cla.s.s it, affirming the resemblance, in regard of whatever is the ground of the name being given, between it and all other things denoted by the name. The resemblance is sometimes perceived by direct comparison of the objects together; sometimes (as, e.g. in the description of the earth"s figure as globular and so forth) it is inferred through intermediate marks, i.e. deductively. When a hypothesis is made (e.g. by Kepler, as to the figure of the earth"s...o...b..t), and then verified by comparison with actual observations, Dr. Whewell calls the process Colligation of Facts by appropriate Conceptions, and affirms it to be the whole of Induction. But this also is only description, being really the ordinary process of ascertaining resemblance by a comparison of phenomena; and, though subsidiary to Induction, it is not itself Induction at all.
CHAPTER II.
ABSTRACTION, OR THE FORMATION OF CONCEPTIONS.