It is merely an accident when general names are names of classes of real objects: e.g. The unity of God, in the Christian sense, and the non-existence of the things called dragons, do not prevent those names being general names. The using a name to connote attributes, turns the things, whether real or imaginary, into a class. But, in predicating the name, we predicate only the attributes; and evenwhen a name (as, e.g. those in Cuvier's system) is introduced as a means of grouping certain objects together, and not, as usually, as a means of predication, it still signifies nothing but the possession of certain attributes.
Classification (as resulting from the use of general language) is the subject of the Aristotelians' Five Predicables, viz.Genus,Species,Differentia,Proprium,Accidens. These are a division of general names, not based on a distinction in their meaning, i.e. in the attributes connoted, but on a distinction in the class denoted. They express, not the meaning of the predicate itself, but its relation (a varying one) to the subject. Commonly, the names of any two classes (or, popularly, the classes themselves), one of which includes all the other and more, are called respectivelygenusandspecies. But the Aristotelians, i.e. the schoolmen, meant bydifferences in kind(genereorspecie) something which was in its nature (and not merely with reference to the connotation of the name) distinct fromdifferencesin theaccidents. Now, it is the fact that, though a fresh class may be founded on the smallest distinction in attributes, yet that some classes have, to separate them from other classes, no common attributes except those connoted by the name, while others have innumerable common qualities (from which we have to select a few samples for connotation) not referrible to a common source. The ends of language and of classification would be subverted if the latter (not if the former) sorts ofdifferencewere disregarded. Now, it was these only that the Aristotelians calledkinds(generaorspecies), holdingdifferencesmade up ofcertainanddefiniteproperties to bedifferencesin theaccidentsof things. In conformity with this distinction—and it is a true one—any class, e.g. negro as opposed to white man, may, according as physiology shall show thedifferencesto be infinite or finite, be discovered to be a distinctkindorspecies(though not according to the naturalist's construction ofspecies, as including all descended from the same stock), or merely a subdivision of thekindorspecies, Man. Amongkinds, agenusis a class divisible into otherkinds, though it may be itself a species in reference to highergenera; that which is not so divisible, is an individual'sproximate kindorinfima species(species prædicabilisand alsosubjicibilis), whose common properties must include all the common properties of every other realkindto which the individual can be referred.
The Aristotelians said that thedifferentiamust be of theessenceof the subject. They vaguely understood, indeed, by theessenceof a thing, that which makes it thekindof thing that it is. But, as akindis such from innumerable qualities not flowing from a common source, logicians selected the qualities which make the thing be what it is called, and termed these the essence, not merely of thespecies, but, in the case of theinfima species, of the individual also. Hence, the distinction between the predicables, Differentia, Proprium, and Accidens, is founded, not on the nature of things, but on the connotation of names. Thespecific differenceis that which must be added to the connotation of thegenusto complete the connotation of thespecies. Aspeciesmay have variousdifferences, according to the principle of the particular classification. Akind, and notmerely a class, may be founded on any one of these, if there be a host of properties behind, of which this one is the index, and not the source. Sometimes a name has a technical as well as an ordinary connotation (e.g. the name Man, in the Linnæan system, connotes a certain number of incisor and canine teeth, instead of its usual connotation of rationality and a certain general form); and then the word is in fact ambiguous, i.e. two names.GenusandDifferentiaare said to be of the essence; that is, the properties signified by them are connoted by the name denoting thespecies. But bothpropriumandaccidensare said to be predicated of the speciesaccidentally. A proprium of the species, however, is predicated of the species necessarily being an attribute, not indeed connoted by the name, but following from an attribute connoted by it. It follows, either by way of demonstration as a conclusion from premisses, or by way of causation as effect from cause; but, in either case,necessarily. Inseparable accidents, on the other hand, are attributes universal, so far as we know, to the species (e.g. blackness to crows), but notnecessary; i.e. neither involved in the meaning of the name of the species, nor following from attributes which are. Separable accidents do not belong to all, or if to all, not at all times (e.g. the fact of being born, to man), and sometimes are not constant even in the same individual (e.g. to be hot or cold).
A definition is a proposition declaring either the special or the ordinary meaning, i.e. in the case of connotative names, the connotation, of a word. This may be effected by stating directly the attributes connoted; but it is more usual to predicate of the subject of definition one name of synonymous, or several which, when combined, are of equivalent, connotation. So that, a definition of a name being thus generally the sum total of the essential propositions which could be framed with that name for subject, is really, as Condillac says, ananalysis. Even when a name connotes only a single attribute, it (and also the corresponding abstract name itself) can yet be defined (in this sense of being analysed or resolved into its elements) by declaring the connotation of that attribute, whether, if it be a union of several attributes (e.g. Humanity), by enumerating them, or, if only one (e.g. Eloquence), by dissecting the fact which is its foundation. Even when the fact which is the foundation of the attribute is a simple feeling, and therefore incapable of analysis, still, if the simple feeling have a name, the attribute and the object possessing it may be defined by reference to the fact: e.g. a white object is definable as one exciting the sensation of white; and whiteness, as the power of exciting that sensation. The only names, abstract or concrete, incapable of analysis, and therefore of definition, are proper names, as having no meaning, and also the names of the simple feelings themselves,since these can be explained only by the resemblance of the feelings to former feelings called by the same or by an exactly synonymous name, which consequently equally needs definition.
Though the only accurate definition is one declaring all the facts involved in the name, i.e. its connotation, men are usually satisfied with anything which will serve as an index to its denotation, so as to guard them from applying it inconsistently. This was the object of logicians when they laid down that a species must be definedper genus et differentiam, meaning by thedifferentia oneattribute included in the essence, i.e. in the connotation. And, in fact, one attribute, e.g. in defining man, Rationality (Swift's Houyhnhms having not been as yet discovered) often does sufficiently mark out the objects denoted. But, besides that a definition of this kind ought, in order to be complete, to beper genus et differentias, i.e. byallthe connoted attributes not implied in the name of thegenus, still, even if all were given, asummum genuscould not be so defined, since it has no superior genus. And for merely marking out the objects denoted, Description, in which none of the connoted attributes are given, answers as well as logicians' so-calledessentialdefinition. In Description, any one or a combination of attributes may be given, the object being to make it exactly coextensive with the name, so as to be predicable of the same things. Such a description may be turned into an essential definition by a change of the connotation (not the denotation) of the name; and, in fact, thus are manufactured almost all scientific definitions, which, being landmarks ofclassification, and not meant to declare the meaning of the name (though, in fact, they do declare it in its new use), are ever being modified (as is the definition of a science itself) with the advance of knowledge. Thus, a technical definition helps to expound the artificial classification from which it grows; but ordinary definition cannot expound, as the Aristotelians fancied it could, the natural classification of things, i.e. explain their division intokinds, and the relations among thekinds: for the properties of everykindare innumerable, and all that definition can do is to state the connotation of the name.
Both these two modes, viz. the essential but incomplete Definition, and the accidental, or Description, are imperfect; but the Realists' distinction between definition of names and of things is quite erroneous. Their doctrine is now exploded; but many propositions consistent with it alone (e.g. that the science of geometry is deduced from definitions) have been retained by Nominalists, such as Hobbes. Really a definition, as such, cannot explain a thing's nature, being merely an identical proposition explaining the meaning of a word. But definitions of namesknown to be names of really existing objects, as in geometry, include two propositions, one a definition and another a postulate. The latter affirms the existence of a thing answering to the name. The science is based on the postulates (whether they rest on intuition or proof), for the demonstration appeals to them alone, and not on the definitions, which indeed might, though at some cost of brevity, be dispensed with entirely. It has been argued that, atany rate, definitions are premisses of science,providedthey give such meanings to terms as suit existing things: but even so, the inference would obviously be from the existence, not of the name which means, but of the thing which has the properties.
One reason for the belief that demonstrative truths follow from the definitions, not from the postulates, was because the postulates are never quite true (though in reality so much of them is true as is true of the conclusions). Philosophers, therefore, searching for something more accurately true, surmised that definitions must be statements and analyses, neither of words nor of things, as such, but of ideas; and they supposed the subject-matter of all demonstrative sciences to be abstractions of the mind. But even allowing this (though, in fact, the mind cannot so abstract one property, e.g. length, from all others; it onlyattendsto the one exclusively), yet the conclusions would still follow, not from the mere definitions, but from the postulates of the real existence of the ideas.
Definitions, in short, are of names, not things: yet they are not therefore arbitrary; and to determine whatshould bethe meaning of a term, it is often necessary to look at the objects. The obscurity as to the connotation arises through the objects being named before the attributes (though it is from the latter that the concrete general terms get their meaning), and through the same name being popularly applied to different objects on the ground of general resemblance, without any distinct perception of their common qualities, especially when these are complex. The philosopher, indeed, uses generalnames with a definite connotation; but philosophers do not make language—it grows: so that, by degrees, the same name often ceases to connote even general resemblance. The object in remodelling language is to discover if the things denoted have common qualities, i.e. if they form a class; and, if they do not, to form one artificially for them. A language's rude classifications often serve, when retouched, for philosophy. The transitions in signification, which often go on till the different members of the group seem to connote nought in common, indicate, at any rate, a striking resemblance among the objects denoted, and are frequently an index to a real connection; so that arguments turning apparently on the double meaning of a term, may perhaps depend on the connection of two ideas. To ascertain the link of connection, and to procure for the name a distinct connotation, the resemblances of things must be considered. Till the name has got a distinct connotation, it cannot be defined. The philosopher chooses for his connotation of the name the attributes most important, either directly, or as the differentiæ leading to the most interesting propria. The enquiry into the more hidden agreement on which these obvious agreements depend, often itself arises under the guise of enquiries into the definition of a name.
The preceding book treated, not of the proper subject of logic, viz. the nature of proof, but of assertion. Assertions (as, e.g. definitions) which relate to the meaning of words, are, sincethatis arbitrary, incapable of truth or falsehood, and therefore of proof or disproof. But there are assertions which are subjects for proof or disproof, viz. the propositions (the real, and not the verbal) whose subject is some fact of consciousness, or its hidden cause, about which is predicated, in the affirmative or negative, one of five things, viz. existence, order in place, order in time, causation, resemblance: in which, in short, it is asserted, that some given subject does or does not possess some attribute, or that two attributes, or sets of attributes, do or do not (constantly or occasionally) coexist.
A proposition not believed on its own evidence, but inferred from another, is said to beproved; and this process of inferring, whether syllogistically or not, isreasoning. But whenever, as in the deduction of a particular from a universal, or, in Conversion, the assertion in the new proposition is the same as thewhole or part of the assertion in the original proposition, the inference is only apparent; and such processes, however useful for cultivating a habit of detecting quickly the concealed identity of assertions, are not reasoning.
Reasoning, or Inference, properly so called, is, 1, Induction, when a proposition is inferred from another, which, whether particular or general, is less general than itself; 2, Ratiocination, or Syllogism, when a proposition is inferred from others equally or more general; 3, a kind which falls under neither of these descriptions, yet is the basis of both.
The syllogistic figures are determined by the position of the middle term. There are four, or, if the fourth be classed under the first, three. But syllogisms in the other figures can be reduced to the first by conversion. Such reduction may not indeed be necessary, for different arguments are suited to different figures; the first figure, says Lambert, being best adapted to the discovery or proof of the properties of things; the second, of the distinctions between things; the third, of instances and exceptions; the fourth, to the discovery or exclusion of the different species of a genus. Still, as the premisses of the first figure, got by reduction, are really the same as the original ones, and as the only arguments of great scientific importance, viz. those in which the conclusion is a universal affirmative, can be proved in the first figurealone, it is best to hold that the two elementary forms of the first figure are the universal types, the one in affirmatives, the other in negatives, of all correct ratiocination.
Thedictum de omni et nullo, viz. that whatever can be affirmed or denied of a class can be affirmed or denied of everything included in the class, which is a true account generalised of the constituent parts of the syllogism in the first figure, was thought the basis of the syllogistic theory. The fact is, that when universals were supposed to have an independent objective existence, this dictum stated a supposed law, viz. that thesubstantia secundaformed part of the properties of each individual substance bearing the name. But, now that we know that a class or universal is nothing but the individuals in the class, the dictum is nothing but the identical proposition, that whatever is true of certain objects is true of each of them, and, to mean anything, must be considered, not as an axiom, but as a circuitous definition of the wordclass.
It was the attempt to combine the nominalist view of the signification of general terms with the retention of the dictum as the basis of all reasoning, that led to the self-contradictory theories disguised under the ultra-nominalism of Hobbes and Condillac, the ontology of the later Kantians, and (in a less degree) the abstract ideas of Locke. It was fancied that the process of inferring new truths was only the substitution of one arbitrary sign for another; and Condillac even described science asune langue bien faite. But language merely enables us to remember and impart our thoughts; it strengthens, like anartificial memory, our power of thought, and is thought's powerful instrument, but not its exclusive subject. If, indeed, propositions in a syllogism did nothing but refer something to or exclude it from a class, then certainly syllogisms might have the dictum for their basis, and import only that the classification is consistent with itself. But such is not the primary object of propositions (and it is on this account, as well as because men will never be persuaded in common discourse toquantifythe predicate, that Mr. De Morgan's or Sir William Hamilton'squantification of the predicateis a device of little value). What is asserted in every proposition which conveys real knowledge, is a fact dependent, not on artificial classification, but on the laws of nature; and as ratiocination is a mode of gaining real knowledge, the principle or law of all syllogisms, with propositions not purely verbal, must be, for affirmative syllogisms, that; Things coexisting with the same thing coexist with one another; and for negative, that; A thing coexisting with another, with which a third thing does not coexist, does not coexist with that third thing. But if (seesuprà, p. 26) propositions (and, of course, all combinations of them) be regarded, not speculatively, as portions of our knowledge of nature, but as memoranda for practical guidance, to enable us, when we know that a thing has one of two attributes, to infer it has the other, these two axioms may be translated into one, viz. Whatever has any mark has that which it is a mark of; or, if both premisses are universal, Whatever is a mark of any mark, is a mark of that of which this last is a mark.
The question is, whether the syllogistic process is one of inference, i.e. a process from the known to the unknown. Its assailants say, and truly, that in every syllogism, considered as an argument to prove the conclusion, there is apetitio principii; and Dr. Whately's defence of it, that its object is to unfold assertions wrapped up and implied (i.e. in fact,asserted unconsciously) in those with which we set out, represents it as a sort of trap. Yet, though no reasoning from generals to particulars can, as such, prove anything, the conclusionisabonâ fideinference, though not an inference from the general proposition. The general proposition (i.e. in the first figure, the major premiss) contains not only a record of many particular facts which we have observed or inferred, but also instructions for making inferences in unforeseen cases. Thus the inference is completed in the major premiss; and the rest of the syllogism serves only to decipher, as it were, our own notes.
Dr. Whately fails to make out that syllogising, i.e. reasoning from generals to particulars, is theonlymode of reasoning. No additional evidence is gained by interpolating a general proposition, and therefore we may, if we please, reason directly from the individual cases, since it is on these alone that the general proposition, if made, would rest. Indeed, thus are in fact drawn, as well the inferences ofchildren and savages, and of animals (which latter having no signs, can frame no general propositions), as even those drawn by grown men generally, from personal experience, and particularly the inferences of men of high practical genius, who, not having been trained to generalise, can apply, but not state, their principles of action. Even when we have general propositions we need not use them. Thus Dugald Stewart showed that the axioms need not be expressly adverted to in order to make good the demonstrations in Euclid; though he held, inconsistently, that the definitions must be. All general propositions, whether called axioms, or definitions, or laws of nature, are merely abridged statements of the particular facts, which, as occasion arises, we either think we may proceed on as proved, or intend to assume.
In short, all inference is from particulars to particulars; and general propositions are both registers or memoranda of such former inferences, and also short formulæ for making more. The major premiss is such a formula; and the conclusion is an inference drawn, not from, but according to that formula. Theactualpremisses are the particular facts whence the general proposition was collected inductively; and the syllogistic rules are to guide us in reading the register, so as to ascertain what it was that we formerly thought might be inferred from those facts. Even where ratiocination is independent of induction, as, when we accept from a man of science the doctrine that allAisB; or from a legislator, the law that all men shall do this or that, the operation of drawing thence any particular conclusion is a process, not ofinference, but of interpretation. In fact, whether the premisses are given by authority, or derived from our own (or predecessors') observation, the object is always simply to interpret, by reference to certain marks, an intention, whether that of the propounder of the principle or enactment, or that which we or our predecessors had when we framed the general proposition, so that we may draw no inferences that were notintendedto be drawn. We assent to the conclusion in a syllogism on account of its consistency with what we interpret to have been the intention of the framer of the major premiss, and not, as Dr. Whately held, because the supposition of a false conclusion from the premisses involves a contradiction, since, in fact, the denial, e.g. that an individual now living will die, is notin termscontradictory to the assertion that his ancestors and their contemporaries (to which the general proposition, as a record of facts, really amounts) have all died.
But the syllogistic form, though the process of inference, which there always is when a syllogism is used, lies not in this form, but in the act of generalisation, is yet a great collateral security for the correctness of that generalisation. When all possible inferences from a given set of particulars are thrown into one general expression (and, if the particulars support one inference, they always will support an indefinite number), we are more likely both to feel the need of weighing carefully the sufficiency of the experience, and also, through seeing that the general proposition would equally support some conclusion which weknowto be false, to detect any defect in the evidence, which, from bias or negligence, wemight otherwise have overlooked. But the syllogistic form, besides being useful (and, when the validity of the reasoning is doubtful, even indispensable) for verifying arguments, has the acknowledged merit of all general language, that it enables us to make an induction once for all. Wecan, indeed, and in simple cases habituallydo, reason straight from particulars; but in cases at all complicated, all but the most sagacious of men, and they also, unless their experience readily supplied them with parallel instances, would be as helpless as the brutes. The only counterbalancing danger is, that general inferences from insufficient premisses may become hardened into general maxims, and escape being confronted with the particulars.
The major premiss is not really part of the argument. Brown saw that there would be apetitio principiiif it were. He, therefore, contended that the conclusion in reasoning follows from the minor premiss alone, thus suppressing the appeal to experience. He argued, that to reason is merely to analyse our general notions or abstract ideas, and that,providedthat the relation between the two ideas, e.g. ofmanand ofmortal, has been first perceived, we can evolve the one directly from the other. But (to waive the error that a proposition relates to ideas instead of things), besides that thisprovisois itself a surrender of the doctrine that an argument consists simply of the minor and the conclusion, the perception of the relation between two ideas, one of which is not implied in the name of the other, must obviously be the result, not of analysis, but of experience. In fact, both the minor premiss, and alsothe expression of our former experience, mustbothbe present in our reasonings, or the conclusion will not follow. Thus, it appears that the universal type of the reasoning process is: Certain individuals possess (as I or others have observed) a given attribute; An individual resembles the former in certain other attributes: Therefore (the conclusion, however, not being conclusive from its form, as is the conclusion in a syllogism, but requiring to be sanctioned by the canons of induction) he resembles them also in the given attribute. But, though this, and not the syllogistic, is the universal type of reasoning, yet the syllogistic process is a useful test of inferences. It is expedient,first, to ascertain generally what attributes are marks of a certain other attribute, so as, subsequently, to have to consider,secondly, only whether any given individuals have those former marks. Every process, then, by which anything is inferred respecting an unobserved case, we will consider to consist of both these last-mentioned processes. Both are equally induction; but the name may be conveniently confined to the process of establishing the general formula, while the interpretation of this will be called 'Deduction.'
The minor premiss always asserts a resemblance between a new case and cases previously known. When this resemblance is not obvious to the senses,or ascertainable at once by direct observation, but is itself matter of inference, the conclusion is the result of a train of reasoning. However, even then the conclusion is really the result of induction, the only difference being that there are two or more inductions instead of one. The inference is still from particulars to particulars, though drawn in conformity, not to one, but to several formulæ. This need of several formulæ arises merely from the fact that the marks by which we perceive that an inference can be drawn (and of which marks the formulæ are records) happen to be recognisable, not directly, but only through the medium of other marks, which were, by a previous induction, collected to be marks of them.
All reasoning, then, is induction: but the difficulties in sciences often lie (as, e.g. in geometry, where the inductions are the simple ones of which the axioms and a few definitions are the formulæ) not at all in the inductions, but only in the formation of trains of reasoning to prove the minors; that is, in so combining a few simple inductions as to bring a new case, by means of one induction within which it evidently falls, within others in which it cannot be directly seen to be included. In proportion as this is more or less completely effected (that is, in proportion as we are able to discover marks of marks), a science, though always remaining inductive, tends to become alsodeductive, and, to the same extent, to cease to be one of theexperimentalsciences, in which, as still in chemistry, though no longer in mechanics, optics, hydrostatics, acoustics, thermology, and astronomy, each generalisation rests on a special induction, and the reasonings consist but of one step each.
An experimental science may become deductive by the mere progress of experiment. The mere connecting together of a few detached generalisations, or even the discovery of a great generalisation working only in a limited sphere, as, e.g. the doctrine of chemical equivalents, does not make a science deductive as a whole; but a science is thus transformed when some comprehensive induction is discovered connecting hosts of formerly isolated inductions, as, e.g. when Newton showed that the motions of all the bodies in the solar system (though each motion had been separately inferred and from separate marks) are all marks of one like movement. Sciences have become deductive usually through its being shown, either by deduction or by direct experiment, that the varieties of some phenomenon in them uniformly attend upon those of a better known phenomenon, e.g. every variety of sound, on a distinct variety of oscillatory motion. The science of number has been the grand agent in thus making sciences deductive. The truths of numbers are, indeed, affirmable of all things only in respect of their quantity; but since the variations ofqualityin various classes of phenomena have (e.g. in mechanics and in astronomy) been found to correspond regularly to variations ofquantityin the same or some other phenomena, every mathematical formula applicable to quantities so varying becomes a mark of a corresponding general truth respecting the accompanying variations in quality; and as the science of quantity is, so far as a science can be, quite deductive, the theory of that special kind of qualities becomes so likewise. It was thus that Descartes and Clairaut made geometry,which was already partially deductive, still more so, by pointing out the correspondence between geometrical and algebraical properties.
All sciences are based on induction; yet some, e.g. mathematics, and commonly also those branches of natural philosophy which have been made deductive through mathematics, are called Exact Sciences, and systems of Necessary Truth. Now, their necessity, and even their alleged certainty, are illusions. For the conclusions, e.g. of geometry, flow only seemingly from the definitions (since from definitions, as such, only propositions about the meaning of words can be deduced): really, they flow from an implied assumption of the existence of real things corresponding to the definitions. But, besides that the existence of such things is not actual or possible consistently with the constitution of the earth, neither can they even beconceivedas existing. In fact, geometrical points, lines, circles, and squares, are simply copies of those in nature, to a part alone of which we choose toattend; and the definitions are merely some of our first generalisations about these natural objects, which being, though equally true of all, not exactly true of any one, must, actually, when extended to cases where the error would be appreciable (e.g. to lines of perceptible breadth), be corrected by the joining to them of new propositions about the aberration. The exact correspondence, then, between the factsand those first principles of geometry which are involved in the so-called definitions, is a fiction, and is merelysupposed. Geometry has, indeed (what Dugald Stewart did not perceive), some first principles which are true without any mixture of hypothesis, viz. the axioms, as well those which are indemonstrable (e.g. Two straight lines cannot enclose a space) as also the demonstrable ones; and so have all sciences some exactly true general propositions: e.g. Mechanics has the first law of motion. But, generally, the necessity of the conclusions in geometry consists only in their following necessarily from certainhypotheses, for which same reason the ancients styled the conclusions of all deductive sciencesnecessary. That the hypotheses, which form part of the premisses of geometry, must, as Dr. Whewell says, not be arbitrary—that is, that in their positive part they are observed facts, and only in their negative part hypothetical—happens simply because our aim in geometry is to deduce conclusions which may be true of real objects: for, when our object in reasoning is not to investigate, but to illustrate truths, arbitrary hypotheses (e.g. the operation of British political principles in Utopia) are quite legitimate.
The ground of our belief in axioms is a disputed point, and one which, through the belief arising too early to be traced by the believer's own recollection, or by other persons' observation, cannot be settled by reference to actual dates. The axioms are really only generalisations from experience. Dr. Whewell, however, and others think that, though suggested, they are not proved by experience, and that their truth is recognisedà prioriby theconstitution of the mind as soon as the meaning of the proposition is understood. But this assumption of anà priorirecognition is gratuitous. It has never been shown that there is anything in the facts inconsistent with the view that the recognition of the truth of the axioms, however exceptionally complete and instant, originates simply in experience, equally with the recognition of ordinary physical generalisations. Thus, that we see a property of geometrical forms to be true, without inspection of the material forms, is fully explained by the capacity of geometrical forms of being painted in the imagination with a distinctness equal to reality, and by the fact that experience has informed us of that capacity; so that a conclusion on the faith of the imaginary forms is really an induction from observation. Then, again, there is nothing inconsistent with the theory that we learn by experience the truth of the axioms, in the fact that they are conceived by the mind as universally and necessarily true, that is, that we cannot figure them to ourselves as being false. Our capacity or incapacity of conceiving depends on our associations. Educated minds can break up their associations more easily than the uneducated; but even the former not entirely at will, even when, as is proved later, they are erroneous. The Greeks, from ignorance of foreign languages, believed in an inherent connection between names and things. Even Newton imagined the existence of a subtle ether between the sun and bodies on which it acts, because, like his rivals the Cartesians, he could not conceive a body acting where it is not. Indeed, inconceivableness depends so completely onthe accident of our mental habits, that it is the essence of scientific triumphs to make the contraries of once inconceivable views themselves appear inconceivable. For instance, suppositions opposed even to laws so recently discovered as those of chemical composition appear to Dr. Whewell himself to be inconceivable. What wonder, then, that an acquired incapacity should be mistaken for a natural one, when not merely (as in the attempt to conceive space or time as finite) does experience afford no model on which to shape an opposed conception, but when, as in geometry, we are unable even to call up the geometrical ideas (which, being impressions of form, exactly resemble, as has been already remarked, their prototypes), e.g. of two straight lines, in order to try to conceive them inclosing a space, without, by the very act, repeating the scientific experiment which establishes the contrary.
Since, then, the axioms and the misnamed definitions are but inductions from experience, and since the definitions are only hypothetically true, the deductive or demonstrative sciences—of which these axioms and definitions form together the first principles—must really be themselves inductive and hypothetical. Indeed, it is to the fact that the results are thus only conditionally true, that the necessity and certainty ascribed to demonstration are due.
It is so even with the Science of Number, i.e. arithmetic and algebra. But here the truth has been hidden through the errors of two opposite schools; for while many held the truths in this science to beà priori, others paradoxically considered them tobe merely verbal, and every process to be simply a succession of changes in terminology, by which equivalent expressions are substituted one for another. The excuse for such a theory as this latter was, that in arithmetic and algebra we carry no ideas with us (not even, as in a geometrical demonstration, a mental diagram) from the beginning, when the premisses are translated into signs, till the end, when the conclusion is translated back into things. But, though this is so, yet in every step of the calculation, there is a real inference of facts from facts: but it is disguised by the comprehensive nature of the induction, and the consequent generality of the language. For numbers, though they must be numbers of something, may be numbers of anything; and therefore, as we need not, when using an algebraical symbol (which represents all numbers without distinction), or an arithmetical number, picture to ourselves all that it stands for, we may picture to ourselves (and this not as a sign of things, but as being itself a thing) the number or symbol itself as conveniently as any other single thing. That we are conscious of the numbers or symbols, in their character of things, and not of mere signs, is shown by the fact that our whole process of reasoning is carried on by predicating of them the properties of things.
Another reason why the propositions in arithmetic and algebra have been thought merely verbal, is that they seem to beidenticalpropositions. But in 'Two pebbles and one pebble are equal to three pebbles,' equality but not identity is affirmed; the subject and predicate, though names of the sameobjects, being names of them in different states, that is, as producing different impressions on the senses. It is on such inductive truths, resting on the evidence of sense, that the Science of Number is based; and it is, therefore, like the other deductive sciences, an inductive science. It is also, like them, hypothetical. Its inductions are the definitions (which, as in geometry, assert a fact as well as explain a name) of the numbers, and two axioms, viz. The sums of equals are equal; the differences of equals are equal. These axioms, and so-called definitions are themselves exactly, and not merely hypothetically, true. Yet the conclusions are true only on the assumption that, 1 = 1, i.e. that all the numbers are numbers of the same or equal units. Otherwise, the certainty in arithmetical processes, as in those of geometry or mechanics, is notmathematical, i.e. unconditional certainty, but only certainty of inference. It is the enquiry (which can be gone through once for all) into the inferences which can be drawn from assumptions, which properly constitutes all demonstrative science.
New conclusions may be got as well from fictitious as from real inductions; and this is even consciously done, viz. in thereductio ad absurdum, in order to show the falsity of an assumption. It has even been argued that all ratiocination rests, in the last resort, on this process. But as this is itself syllogistic, it is useless, as a proof of a syllogism, against a man who denies the validity of this kind of reasoning process itself. Such a man cannot in fact be forced to a contradiction in terms, but only to a contradiction, or rather an infringement, of the fundamental maxim of ratiocination, viz. 'Whatever has a mark, haswhat it is a mark of;' and, since it is only by admitting premisses, and yet rejecting a conclusion from them, that this axiom is infringed, consequently nothing isnecessaryexcept the connection between a conclusion and premisses.
As all knowledge not intuitive comes exclusively from inductions, induction is the main topic of Logic; and yet neither have metaphysicians analysed this operation with a view to practice, nor, on the other hand, have discoverers in physics cared to generalise the methods they employed.
Inferences are equallyinductive, whether, as in science, which needs its conclusions for record, not for instant use, they pass through the intermediate stage of a general proposition (to which class Dr. Whewell, without sanction from facts, or from the usage of Reid and Stewart, the founders of modern English metaphysical terminology, limits the term induction), or are drawn direct from particulars to a supposed parallel case. Neither does it make any difference in thecharacterof the induction, whether the process be experiment or ratiocination, and whether the object be to infer a general proposition or an individual fact. That, in the latter case, the difficulty of the practical enquiries, e.g. of a judge or an advocate, lies chiefly in selecting from among all approved general propositions those inductions which suit hiscase (just as, even in deductive sciences, the ascertaining of the inductions is easy, their combination to solve a problem hard) is not to the point: the legitimacy of the inductions so selected must at all events be tried by the same test as a new general truth in science. Induction, then, may be treated here as though it were the operation of discovering and proving general propositions; but this is so only because the evidence which justifies an inference respecting one unknown case, would justify a like inference about a whole class, and is really only another form of the same process: because, in short, the logic of science is the universal logic applicable to all human enquiries.
Induction is the process by which what is true at certain times, or of certain individuals, is inferred to be true in like circumstances at all times, or of a whole class. There must be an inference from the known to the unknown, and not merely from a less to a more general expression. Consequently, there is no valid induction, 1, in those cases laid down in the common works on Logic as the only perfect instances of induction, viz. where what we affirm of the class has already been ascertained to be true of each individual in it, and in which the seemingly general proposition in the conclusion is simply a number of singular propositions written in an abridged form;or, 2, when, as often in mathematics, the conclusion, though really general, is a mere summing up of the different propositions from which it is drawn (whether actually ascertained, or, as in the case of the uncalculated terms of an arithmetical series, when once its law is known, readily to be understood); or, 3, when the several parts of a complex phenomenon, which are only capable of being observed separately, have been pieced together by one conception, and made, as it were, one fact represented in a single proposition.
Dr. Whewell sets out this last operation, which he terms thecolligation of facts, as induction, and even as the type of induction generally. But, though induction is always colligation, or (as we may, with equal accuracy, characterise such a general expression obtained by abstraction simply connecting observed facts by means of common characters)description, colligation, or description, as such, though a necessary preparation for induction, is not induction. Induction explains and predicts (and, as an incident of these powers, describes). Different explanations collected by real induction from supposed parallel cases (e.g. the Newtonian and theImpactdoctrines as to the motions of the heavenly bodies), or different predictions, i.e. different determinations of the conditions under which similar facts may be expected again to occur (e.g. the stating that the position of one planet or satellite so as to overshadow another, and, on the other hand, that the impending over mankind of some great calamity, is the condition of an eclipse), cannot be true together. But, for a colligation to be correct, it is enough that it enables the mind to represent to itself as a whole all the separate facts ascertained ata given time, so that successive tentative descriptions of a phenomenon, got by guessing till a guess is found which tallies with the facts, may, though conflicting (e.g. the theories respecting the motions of the heavenly bodies), beallcorrectso far as they go. Induction is proof, the inferring something unobserved from something observed; and to provide a proper test of proof is the special purpose of inductive logic. But colligation simply sums up the facts observed, as seen under a new point of view. Dr. Whewell contends that, besides the sum of the facts, colligation introduces, as a principle of connection, a conception of the mind not existing in the facts. But, in fact, it is only because this conception is a copy of something in the facts, although our senses are too weak to recognise it directly, that the facts are rightly classed under the conception. The conception is often even got by abstraction from the facts which it colligates; but also when it is a hypothesis, borrowed from strange phenomena, it still is accepted as true only because found actually, and as a fact, whatever the origin of the knowledge of the fact, to fit and to describe as a whole the separate observations. Thus, though Kepler's consequent inference that,becausethe orbit of a planet is an ellipse, the planet wouldcontinueto revolve in that same ellipse, was an induction, his previous application of the conception of an ellipse, abstracted from other phenomena, to sum up his direct observations of the successive positions occupied by the different planets, and thus to describe their orbits, was no induction. It altered only thepredicate, changing—The successive places of, e.g. Mars, are A, B, C, and so forth, into—Thesuccessive places of, e.g. Mars, are points in an ellipse: whereas induction always widens thesubject.
Induction is generalisation from experience. It assumes, that whatever is true in any one case, is true in all cases of a certain description, whether past, present, or future (and not merely in future cases, as is wrongly implied in the statement by Reid's and Stewart's school, that the principle of induction is 'our intuitive conviction that the future will resemble the past'). It assumes, in short, that the course of nature is uniform, that is, that all things take place according to general laws. But this general axiom of induction, though by it were discovered the obscure laws of nature, is no explanation of the inductive process, but is itself an induction (not, as some think, an intuitive principle which experienceverifiesonly), and is arrived at after many separate phenomena have been first observed to take place according to general laws. It does not, then,proveall other inductions. But it is aconditionof their proof. For any induction can be turned into a syllogism by supplying a major premiss, viz. What is true of this, that, &c. is true of the whole class; and the process by which we arrive at this immediate major may be itself represented by another syllogism or train of syllogisms, the major of the ultimate syllogism, and which therefore is the warrant for the immediatemajor, being this axiom, viz. that there is uniformity, at all events, in the class of phenomena to which the induction relates, and a uniformity which, if not foreknown, may now be known.
But though the course of nature is uniform, it is also infinitely various. Hence there is no certainty in the induction in use with the ancients, and all non-scientific men, and which Bacon attacked, viz. 'Inductio per enumerationem simplicem, ubi non reperitur instantia contradictoria'—unless, as in a few cases, we must have known of the contradictory instances if existing. The scientific theory of induction alone can show why a general law of nature may sometimes, as when the chemist first discovers the existence and properties of a before unknown substance, be inferred from a single instance, and sometimes (e.g. the blackness of all crows) not from a million.
The uniformity of the course of nature is a complex fact made up of all the separate uniformities in respect to single phenomena. Each of these separate uniformities, if it be not a mere case of and result from others, is a law of nature; for, thoughlawis used for any general proposition expressing a uniformity,law of natureis restricted to cases where it has been thought that a separate act of creative will is necessary to account for the uniformity. Laws of nature, in the aggregate, are the fewest generalpropositions from which all the uniformities in the universe might be deducted. Science is ever tending to resolve one law into a higher. Thus, Kepler's three propositions, since having been resolved by Newton into, and shown to be cases of the three laws of motion, may be indeed called laws, but not laws of nature.
Since every correct inductive generalisation is either a law of nature, or a result from one, the problem of inductive logic is to unravel the web of nature, tracing each thread separately, with the view, 1, of ascertaining what are theseverallaws of nature, and, 2, of following them into their results. But it is impossible to frame a scientific method of induction, or test of inductions, unless, unlike Descartes, we start with the hypothesis that some trustworthy inductions have been already ascertained by man's involuntary observation. These spontaneous generalisations must be revised; and the same principle which common sense has employed to revise them, correcting the narrower by the wider (for, in the end, experience must be its own test), serves also, only made more precise, as the real type of scientific induction. As preliminary to the employment of this test, nature must be surveyed, that we may discover which are respectively the invariable and the variable inductions at which man has already arrived unscientifically. Then, by connecting these different ascertained inductions with one another through ratiocination, they become mutually confirmative, the strongest being made still stronger when bound up with the weaker, and the weakest at least as strong as the weakest of those from which they are deduced (as in the case of the Torricellian experiment) while those leading deductively toincompatible consequences become each other's test, showing that one must be given up (e.g. the old farmers' bad induction that seed never throve if not sown during the increase of the moon). It is because a survey of the uniformities ascertained to exist in nature makes it clear that there are certain and universal uniformities serving as premisses whence crowds of lower inductions may be deduced, and so be raised to the same degree of certainty, that a logic of induction is possible.
Phenomena in nature stand to each other in two relations, that of simultaneity, and that of succession. On a knowledge of the truths respecting the succession of facts depends our power of predicting and influencing the future. The object, therefore, must be to find some law of succession not liable to be defeated or suspended by any change of circumstances, by being tested by, and deduced from which law, all other uniformities of succession may be raised to equal certainty. Such a law is not to be found in the class of laws of number or of space; for though these are certain and universal, no laws except those of space and number can be deduced from them by themselves (however importantelementsthey may be in the ascertainment of uniformities of succession). But causation is such a law; and of this, moreover, all cases of succession whatever are examples.
ThisLaw of Causationimplies no particular theory as to the ultimate production of effects byefficientcauses, but simply implies the existence of an invariable order of succession (on our assurance of which the validity of the canons of inductive logic depends) found by observation, or, when not yet observed, believed, to obtain between an invariable antecedent, i.e. thephysicalcause, and an invariable consequent, the effect. This sequence is generally between a consequent and thesumof several antecedents. The cause is really the sum total of the conditions, positive and negative; the negative being stated as one condition, the same always, viz. the absence of counteracting causes (since one cause generally counteracts another by the same law whereby it produces its own effects, and, therefore, the particular mode in which it counteracts another may be classed under the positive causes). But it is usual, even with men of science, to reserve the namecausefor an antecedenteventwhich completes the assemblage of conditions, and begins to exist immediately before the effect (e.g. in the case of death from a fall, the slipping of the foot, and not the weight of the body), and to style the permanent facts orstates, which, though existing immediately before, have also existed long previously, theconditions. But indeed, popularly, any condition which the hearer is least likely to be aware of, or which needs to be dwelt upon with reference to the particular occasion, will be selected as the cause, even a negative condition (e.g. the sentinel's absence from his post, as the cause of a surprise), though from a mere negation no consequence can really proceed.On the other hand, the object which is popularly regarded as standing in the relation ofpatient, and as being the mere theatre of the effect, is never styledcause, being included in the phrase describing the effect, viz. as the object, of which the effect isa state. But really these so-calledpatientsare themselves agents, and their properties are positive conditions of the effect. Thus, the death of a man who has taken prussic acid is as directly the effect of the organic properties of the man, i.e. thepatient, as of the poison, i.e. theagent.
To be a cause, it is not enough that the sequencehas beeninvariable. Otherwise, night might be called the cause of day; whereas it is not even a condition of it. Such relations of succession or coexistence, as the succession of day and night (which Dr. Whewell contrasts aslaws of phenomenawithcauses, though, indeed, the latter also are laws of phenomena, only more universal ones), result from the coexistence of real causes. The causes themselves are followed by their effects, not only invariably, but alsonecessarily, i.e.unconditionally, or subject to none but negative conditions.Thisis material to the notion of a cause. But another question is not material, viz. whether causesmustprecede, or may, at times, be simultaneous with (they certainly are never preceded by) their effects. In some, though not in all cases, the causes do invariably continuetogether withtheir effects, in accordance with the schools' dogma,Cessante causâ, cessat et effectus; and the hypothesis that, in such cases, the effects are producedafreshat each instant by their cause, is only a verbal explanation. But the questiondoes not affect the theory of causation, which remains intact, even if (in order to take in cases of simultaneity of cause and effect) we have to define a cause, as the assemblage of phenomena, which occurring, some other phenomenon invariably and unconditionally commences, or has its origin.
There exist certain original natural agents, called permanent causes (some being objects, e.g. the earth, air, and sun; others, cycles of events, e.g. the rotation of the earth), which together make up nature. All other phenomena are immediate or remote effects of these causes. Consequently, as the state of the universe at one instant is the consequence of its state at the previous instant, a person (but only if of more than human powers of calculation, and subject also to the possibility of the order being changed by a new volition of a supreme power) might predict the whole future order of the universe, if he knew the original distribution of all the permanent causes, with the laws of the succession between each of them and its different mutually independent effects. But, in fact, the distribution of these permanent causes, with the reason for the proportions in which they coexist, has not been reduced to a law; and this is why the sequences or coexistences among the effects of several of them together cannot rank as laws of nature, though they are invariable while the causes coexist. For this same reason (since the proximate causes are traceable ultimately to permanent causes) there are no original and independent uniformities of coexistence between effects of different (proximate) causes, though there may be such between different effects of the same cause.
Some, and particularly Reid, have regarded man's voluntary agency as the true type of causation and the exclusive source of the idea. The facts of inanimate nature, they argue, exhibit only antecedence and sequence, while in volition (and this would distinguish it from physical causes) we are conscious, prior to experience, of power to produce effects: volition, therefore, whether of men or of God, must be, they contend, an efficient cause, and the only one, of all phenomena. But, in fact, they bring no positive evidence to show that we could have known, apart from experience, that the effect, e.g. the motion of the limbs, would follow from the volition, or that a volition is more than a physical cause. In lieu of positive evidence, they appeal to the supposed conceivableness of the direct action of will on matter, and inconceivableness of the direct action of matter on matter. But there is no inherent law, to this effect, of the conceptive faculty: it is only because our voluntary acts are, from the first, the most direct and familiar to us of all cases of causation, that men, as is seen from the structure of languages (e.g. their active and passive voices, and impersonations of inanimate objects), get thehabitof borrowing them to explain other phenomena by a sort of original Fetichism. Even Reid allows that there is a tendency to assume volition where it does not exist, and that the belief in it has its sphere gradually limited, in proportion as fixed laws of succession among external objects are discovered.
This proneness to require the appearance of some necessary and natural connection between the cause and its effect, i.e. some reasonper sewhy the oneshould produce the other, has infected most theories of causation. But the selection of the particular agency which is to make the connection between the physical antecedent and its consequent seemconceivable, has perpetually varied, since it depends on a person's special habits of thought. Thus, the Greeks, Thales, Anaximenes, and Pythagoras, thought respectively that water, air, or number is such an agency explaining the production of physical effects. Many moderns, again, have been unable toconceivethe production of effects by volition itself, without some intervening agency to connect it with them. This medium, Leibnitz thought, was someper seefficient physical antecedent; while the Cartesians imagined for the purpose the theory of Occasional Causes, that is, supposed that God, notquâmind, orquâvolition, butquâomnipotent, intervenes to connect the volition and the motion: so far is the mind from being forced to think the action of mind on matter morenaturalthan that of matter on matter. Those who believe volition to be an efficient cause are guilty of exactly the same error as the Greeks, or Leibnitz or Descartes; that is, of requiring anexplanationof physical sequences by something ἁνευ οὑ τὁ αἱτιον οὑκ ἁν ποτ εἱη αἱτιον [Greek: aneu hou to aition ouk an pot' eiê aition]. But they are guilty of another error also, in inferring that volition, even if it is anefficientcause of so peculiar a phenomenon as nervous action, must therefore be the efficient cause of all other phenomena, though having scarcely a single circumstance in common with them.
An effect is almost always the result of the concurrence of several causes. When all have their full effect, precisely as if they had operatedsuccessively, the joint effect (and it is not inconsistent to give the name ofjoint effecteven to the mutual obliteration of the separate ones) may bededucedfrom the laws which govern the causes when acting separately. Sciences in which, as in mechanics, this principle, viz. thecomposition of causes, prevails, are deductive and demonstrative. Phenomena, in effect, do generally follow this principle. But in some classes, e.g. chemical, vital, and mental phenomena, the laws of the elements when called on to work together, cease and give place to others, so that the joint effect is not the sum of the separate effects. Yet even here the more general principle is exemplified. For the newheteropathiclaws, besides that they never supersedeallthe old laws (thus, The weight of a chemical compound is equal to the sum of the weight of the elements), have been often found, especially in the case of vital and mental phenomena, to enterunalteredinto composition with one another, so that complex facts may thus bededuciblefrom comparatively simple laws. It is even possible that, as has been already partly effected by Dalton's law of definite proportions, and the law of isomorphism, chemistry itself, which is now the least deductive of sciences, may be made deductive, through the laws of the combinations being ascertained to be, thoughnot compounded of the laws of the separate agencies, yet derived from them according to a fixed principle.
The proposition, that effects are proportional to their causes, is sometimes laid down as an independent axiom of causation: it is really only a particular case of the composition of causes; and it fails at the same point as the latter principle, viz. when an addition does not become compounded with the original cause, but the two together generate a new phenomenon.