Analysis of Mr. Mill's System of Logic Part 2

Definitions, in short, are of names, not things: yet they are not therefore arbitrary; and to determine what _should be_ the 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 general names 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 cla.s.s; and, if they do not, to form one artificially for them. A language's rude cla.s.sifications 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 differentiae 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.

BOOK II.

REASONING.

CHAPTER I.

INFERENCE, OR REASONING IN GENERAL.

The preceding book treated, not of the proper subject of logic, viz. the nature of proof, but of a.s.sertion. a.s.sertions (as, e.g. definitions) which relate to the meaning of words, are, since _that_ is arbitrary, incapable of truth or falsehood, and therefore of proof or disproof. But there are a.s.sertions 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 a.s.serted, 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 be _proved_; and this process of inferring, whether syllogistically or not, is _reasoning_. But whenever, as in the deduction of a particular from a universal, or, in Conversion, the a.s.sertion in the new proposition is the same as the whole or part of the a.s.sertion in the original proposition, the inference is only apparent; and such processes, however useful for cultivating a habit of detecting quickly the concealed ident.i.ty of a.s.sertions, 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.

CHAPTER II.

RATIOCINATION, OR SYLLOGISM.

The syllogistic figures are determined by the position of the middle term. There are four, or, if the fourth be cla.s.sed 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 figure alone, 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.

The _dictum de omni et nullo_, viz. that whatever can be affirmed or denied of a cla.s.s can be affirmed or denied of everything included in the cla.s.s, which is a true account generalised of the const.i.tuent 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 the _substantia secunda_ formed part of the properties of each individual substance bearing the name. But, now that we know that a cla.s.s or universal is nothing but the individuals in the cla.s.s, 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 word _cla.s.s_.

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 subst.i.tution of one arbitrary sign for another; and Condillac even described science as _une langue bien faite_. But language merely enables us to remember and impart our thoughts; it strengthens, like an artificial 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 cla.s.s, then certainly syllogisms might have the dictum for their basis, and import only that the cla.s.sification 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 to _quantify_ the predicate, that Mr. De Morgan's or Sir William Hamilton's _quantification of the predicate_ is a device of little value). What is a.s.serted in every proposition which conveys real knowledge, is a fact dependent, not on artificial cla.s.sification, 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 (see _supra_, 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.

CHAPTER III.

THE FUNCTIONS AND LOGICAL VALUE OF THE SYLLOGISM.

The question is, whether the syllogistic process is one of inference, i.e. a process from the known to the unknown. Its a.s.sailants say, and truly, that in every syllogism, considered as an argument to prove the conclusion, there is a _pet.i.tio principii_; and Dr. Whately's defence of it, that its object is to unfold a.s.sertions wrapped up and implied (i.e.

in fact, _a.s.serted 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 conclusion _is_ a _bona fide_ inference, 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 the _only_ mode 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 of children 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 a.s.sume.

In short, all inference is from particulars to particulars; and general propositions are both registers or memoranda of such former inferences, and also short formulae for making more. The major premiss is such a formula; and the conclusion is an inference drawn, not from, but according to that formula. The _actual_ premisses 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 all A is B; 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 of inference, 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 not _intended_ to be drawn. We a.s.sent 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 not _in terms_ contradictory to the a.s.sertion 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 we _know_ to be false, to detect any defect in the evidence, which, from bias or negligence, we might 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. We _can_, indeed, and in simple cases habitually _do_, 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 a _pet.i.tio principii_ if 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 a.n.a.lyse our general notions or abstract ideas, and that, _provided_ that the relation between the two ideas, e.g. of _man_ and of _mortal_, 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 this _proviso_ is 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 a.n.a.lysis, but of experience. In fact, both the minor premiss, and also the expression of our former experience, must _both_ be 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 un.o.bserved 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 establis.h.i.+ng the general formula, while the interpretation of this will be called 'Deduction.'

CHAPTER IV.

TRAINS OF REASONING, AND DEDUCTIVE SCIENCES.

The minor premiss always a.s.serts 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 formulae. This need of several formulae arises merely from the fact that the marks by which we perceive that an inference can be drawn (and of which marks the formulae 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 formulae) 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 also _deductive_, and, to the same extent, to cease to be one of the _experimental_ sciences, 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 quant.i.ty; but since the variations of _quality_ in various cla.s.ses of phenomena have (e.g. in mechanics and in astronomy) been found to correspond regularly to variations of _quant.i.ty_ in the same or some other phenomena, every mathematical formula applicable to quant.i.ties so varying becomes a mark of a corresponding general truth respecting the accompanying variations in quality; and as the science of quant.i.ty 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.

CHAPTERS V. AND VI.

DEMONSTRATION AND NECESSARY TRUTHS.

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 a.s.sumption 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 const.i.tution of the earth, neither can they even be _conceived_ as existing. In fact, geometrical points, lines, circles, and squares, are simply copies of those in nature, to a part alone of which we choose to _attend_; 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 facts and those first principles of geometry which are involved in the so-called definitions, is a fiction, and is merely _supposed_. 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 s.p.a.ce) 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 certain _hypotheses_, for which same reason the ancients styled the conclusions of all deductive sciences _necessary_. 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 ill.u.s.trate 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 recognised _a priori_ by the const.i.tution of the mind as soon as the meaning of the proposition is understood. But this a.s.sumption of an _a priori_ recognition 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 a.s.sociations. Educated minds can break up their a.s.sociations 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 on the 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 s.p.a.ce 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 s.p.a.ce, 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 be _a priori_, others paradoxically considered them to be merely verbal, and every process to be simply a succession of changes in terminology, by which equivalent expressions are subst.i.tuted 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 be _identical_ propositions.

But in 'Two pebbles and one pebble are equal to three pebbles,' equality but not ident.i.ty is affirmed; the subject and predicate, though names of the same objects, 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, a.s.sert 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 a.s.sumption 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 not _mathematical_, 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 a.s.sumptions, which properly const.i.tutes all demonstrative science.

New conclusions may be got as well from fict.i.tious as from real inductions; and this is even consciously done, viz. in the _reductio ad absurdum_, in order to show the falsity of an a.s.sumption. 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, has what 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 is _necessary_ except the connection between a conclusion and premisses.