TABLE I.
PRIMARY CLASSIFICATION OF TERMS.
The Traditional doctrine of quantity is not only miserably defective, but is also in such inextricable confusion that it is difficult to give any intelligible account of it. The following statements are authorised by a leading text book; all of them are corroborated by other text books, and all but one by every text book that I have been able to consult.
In the examination and description of terms, quantity is not mentioned; nevertheless, terms are divided into Singular and General, and General terms into Collective and Distributive.
Quantity means, in Logic, the quantity of propositions, as Universal or Particular. The logical doctrine of quantity does not apply to terms, which cannot be either Universal or Particular; at any rate, neither the Universal term nor the Particular term is ever mentioned in Logic. Nevertheless, every term must be either distributed or undistributed; which means that it is viewed quantitatively, and must possess one of these two quantities.
The quantities known to Traditional Logic are two only, viz:—the Universal, the Particular, the Distributed, the Undistributed, the Singular, the General, the Distributive, the Collective, and the Indesignate.
The predicate of a proposition is never quantified: it is always, however, of either distributed or undistributed quantity. Universal propositions are those in which the predicate explicitly refers to the whole of the subject.
Particular propositions are those in which the predicate does not refer to the whole of the subject.
There is no proposition in which the predicate does not refer to the whole of the subject. This rule is not to be found in any book on Logic. It is, however, indisputably true.
How logicians reconcile in their own minds these several statements, is no business of mine. Whether they forget, when they come to one page, what they have said on a previous page; or whether the Universal quantity is the same as the Distributed quantity, and the Particular the same as the Undistributed; or whether they are respectively sometimes the same and sometimes different; or whether the Singular and General quantities are, is the estimation of Logic, not quantities; or whether Logic does not know what quantity is; or whether Distributive quantity is or is not the same as Distributed quantity; or whether logicians are unable to count up to more than two; I do not know, and it is no business of mine to determine. These subtle questions must be left to the acumen of logicians themselves; but there is one matter on which I am clear, and that is that in no circumstances whatever does the predicate of a proposition refer to part only of the subject of that proposition. The subject of a proposition may, indeed, be but a small part of an individual or of a class, but however small a part it may be, the predicate refers to the whole of that part. If I say that a ten-millionth of a grain of radium is a recognisable quantity, or that one of the innumerable host of stars is brighter than the rest, or that one of the countless grains of sand on the sea-shore is blown into my eye, the predication refers not to part of the one-millionth, very small part though it is of a very small part; nor to part of the star or part of the grain of sand, one only though each is of an incalculable multitude. The predicate refers in each case to the whole of the subject, small part of a whole though that subject is. What logicians mean when they say that a Particular proposition is one in which the predicate does not refer to the whole of the subject, is, no doubt, that the subject of a Particular proposition is part of a class; and this, though not universally true, is at any rate sometimes true; but this is not what they say, or what most of them say, and what they do say is never true. According to their definition, there is no such thing as a Particular proposition.
Another distinction between Universal and Particular propositions is said to be that, while the subject of the Universal is understood in its whole denotation, the denotation of the subject of the Particular proposition is left indefinite. According to this distinction, ‘Some of the men were killed’ is a particular proposition, and to this logicians would, I suppose, agree. But according to this distinction, ‘Twenty-two of the men were killed’ is a Universal proposition; yet ‘twenty-two of them’ is certainly part of a class, so that the proposition is Particular according to one definition of Particular, and not Particular—in fact Universal—according to another definition. No doubt, if twenty-two of them were killed, all of the twenty-two were killed, and in this sense, the proposition is Universal; and no doubt if twenty-two of them were killed, it is not specifically asserted that more than twenty-two of them were not killed, and logicians may claim that this renders the numbers indefinite, and determines decisively that the proposition is Particular; but if logicians make this claim, they are confusing definiteness with exactness, which are very different things. For my own part, I am unable to decide, on the principles of Traditional Logic, whether ‘Twenty-two of them were killed’ is, in that Logic, a Universal or a Particular proposition; and I shall be very much surprised if logicians give a unanimous answer to the question. Fortunately for themselves, however, logicians are always ready with a short way to deal with difficulties. If any example, found in experience, of a Ratio, a term, a quantity, a mode of argument, or anything else, will not fit in with the scheme of Traditional Logic, out that example has to go. Such a Ratio, such a term, such a quantity, such a mode of argument, is ‘not logical.’ It doesn’t count. We are reminded of little Curran’s offer to his gigantic antagonist who complained of the unfairness of a duel between them. Curran offered to chalk out a figure of his own size on the body of his huge adversary, and to agree that every bullet that hit outside the chalk marks should not count. It is little consolation to a reasoner, whose arguments have been smashed, shattered and pulverised, to be told that the crushing retort of his adversary is outside the scheme of Traditional Logic. The Austrian generals who were defeated by Napoleon proved to their own satisfaction that Napoleon’s methods were not warranted by the rules of war. The demonstration was irrefragable, but it did not alter the results of the campaigns. What were altered were the rules of war.
The doctrine of Traditional Logic, that quantity inheres in the proposition, seems to me fundamentally erroneous. In my view, quantity resides in the terms alone, and the proposition as a whole can no more be Universal or Particular than it can be white or soft. A proposition expresses the formation in the mind of a relation; and though, with some straining of the meaning of words, a relation may be said to be Universal or Particular, the formation of a relation cannot possibly be either. If there is a relation that subsists between all things, that relation may properly be termed Universal; and in this sense the relations of space and time are approximately Universal. They are universal in the material universe. And relations, such as likeness and unlikeness, that exist between certain things only, may be termed Particular relations. But the formation of a relation, cannot be either Universal or Particular; and it is the formation of a relation that is expressed by a proposition.
Traditional Logic, as now taught, denies utterly that the Predicate is quantified; and abjures, abhors, and repudiates the Hamiltonian doctrine of the quantification of the Predicate, as utterly heretical and damnable. Nevertheless, Traditional Logic still teaches, by the mouths of those very logicians who pour contempt upon the Hamiltonian doctrine, that the predicate of every proposition must be either distributed or undistributed. Of the four forms of propositions that alone exist, E and O distribute their predicates, A and I leave the predicate undistributed. In no proposition is the predicate quantified, but in every proposition the predicate possesses either distributed or undistributed quantity. That is the doctrine taught by every logician at the present time. How the two portions of the doctrine are reconciled is no business of mine, I am thankful to say, for, not having been brought up in Traditional Logic, I find some difficulty in reconciling flat contradictories.
In my view, we may or may not quantify the predicate, or rather, the Object-term, of a proposition; and if we do choose to quantify the Object-term, we must apply to it a quantity that is applicable to it, and not one that is inapplicable. There is one mode of quantity—the Extensive—that is applicable to quantitative terms alone, and cannot rightfully be applied to qualitative terms. There is another mode of quantity—the Intensive—that is applicable to qualitative terms only, and cannot rightfully be applied to quantitative terms. Traditional Logic does not explicitly or formally deny these doctrines, but its practice is completely inconsistent with them, and we may therefore take it that if Traditional Logic did know of them, it would deny them.
As already shown, Traditional Logic, though it formally and verbally denies that the predicate is quantified, yet declares that the predicate always possesses quantity. It declares moreover that the predicate is always, or ought always, to be understood as a qualitative term; and that, being a qualitative term, it must be invested with extensive quantity, of which qualitative terms are not susceptible. I say that in ‘All gold is heavy,’ the predicate ‘heavy’ is to be understood qualitatively, that is to say, as a quality belonging to gold; and so far Logic agrees, But I say further, that ‘heavy’ cannot be every heavy, or a few heavy, or many heavy, or half heavy, or have any other numerical quantity attached to it. As a quality, it is susceptible of intensive quantity only. It may be very heavy, or moderately heavy, or rather heavy, or extremely heavy. This Logic does not admit, or at least does not declare; for no mention of intensive quantity is to be found in books on Logic. On the contrary, Logic declares that ‘heavy’ in the given proposition, does possess extensive quantity, for it is undistributed, and an undistributed term ‘refers to a portion of a class’ and ‘leaves the extent of the denotation absolutely indefinite.’ The sole aim, purpose, and meaning of conferring distributive quantity upon a term is to enable us, or disenable us, to include some other term within it. Logicians are compelled, by their doctrine that all reasoning is inclusion in classes or exclusion from classes, to make every term a class. They do not seem able to appreciate the difference between a quantitative term and a qualitative term, or to recognise that any change has taken place when the term ‘heavy’ is altered into ‘heavy things.’
But although logicians are forced by their erroneous doctrine of reasoning to pretend that every qualitative term is quantitative, and expresses a class or part of a class, it seems never to have occurred to any logician, from the days of Aristotle down to this present time of writing, that the predicate or Object-term of a proposition may explicitly express a class or part of a class. That All men are mortal is a proposition that every logician can understand, appreciate, and admit within the precincts of Logic; but that Mortality can be an attribute of all men, is beyond the utmost range of his imagination.
The teaching of every logician is that the subject is to be understood in Extension, the Predicate in Intension. The practice of every logician is to understand both in Extension. “‘All men are mortal,”’ say logicians, ‘means Every man has the attribute of mortality; and therefore we will always take it to mean Every man belongs to the class of mortal beings. “No man is perfect” means No man possesses the attribute of perfection; and therefore we will always take it to mean No man is included in the class of perfect beings.’
The sole justification—no, that is wrong; there can be no justification for the constant infraction of a rule that is laid down as universally applicable. The sole foundation for this strange doctrine of the quantification of the predicate, which logicians formally repudiate and universally adopt, is the fancied necessity of finding means to convert the proposition according to the conventional rule. Unless we understand ‘All men are mortal’ to mean ‘All men are included in the class of mortals,’ we cannot get the logical converse, ‘Some mortals are men,’ and without this converse, the Moods of the Syllogism must go by the board.
In my view, the signs of quantity, whether of Extensive or Intensive quantity, can as readily be affixed to the Object-term as to the Subject-term, and are, in the actual reasonings of daily life, as often affixed to the one as to the other. Either Extensive or Intensive quantity may be expressed in the Subject alone, in the Object alone, or in both, or in neither.
Extensive Quantity.
Subject alone quantified—All men are liars.
Object alone quantified—Lying is common to all men.
Neither term quantified—Cloven-footed mammals are ruminants.
Both terms quantified—One volunteer is worth twenty pressed men.
Intensive Quantity.
Subject alone quantified—Gross ignorance is deplorable.
Object alone quantified—Lying is very shameful.
Neither term quantified—Patience is virtuous.
Both terms quantified—Intense hunger is very demoralising.
All of these variations of quantity except the first, would, I suppose, be rejected by logicians as ‘not logical,’ but, as I owe no allegiance to Traditional Logic, I accept them all, and have no difficulty in arguing and drawing conclusions from them. If Lying is common to all men, then All men are liars. If Cloven-footed mammals are ruminants, then we may expect any cloven-footed mammal to ruminate. If One volunteer is worth twenty pressed men, he is worth more than ten pressed men. If Gross ignorance is deplorable, we should not encourage it. If Lying is very shameful, it is not to be commended. If Patience is virtuous, it is virtuous to be patient. If Intense hunger is very demoralising, starving them is not the best way to cultivate bravery in troops. These arguments are no more difficult to construct than they are easy to refute; but none of them can be reached by any method known to Traditional Logic as I understand it.
If the distribution or non-distribution of the predicate is as important, nay, as necessary, in our estimation of propositions, as Traditional Logic asserts, then it is not justified in rejecting Hamilton’s scheme of quantification of the predicate, which, however, all logicians do reject. Hamilton’s quantification merely carries to its logical conclusion that quantification of the predicate which all logicians admit and proclaim. It merely states explicitly that which they assert is implicit in the proposition. Hamilton says in effect, ‘It is admitted that in every proposition the predicate is either distributed or it is not. It is admitted that this quantification is not explicit in the proposition, but must be inferred from the quality of the proposition. It would be much more satisfactory to state the quantity of the predicate openly and explicitly; and this I do. When the quantities of the predicate are thus formally displayed, it is found that the quantification in vogue is imperfect. My scheme brings to light quantities that have lain unsuspected. Logic declares that ‘“All A is B” means, of necessity, “all A constitutes a part only of the class B.” But this does not exhaust the possibilities. All A may constitute the whole of the class B; and it is not justifiable to make an assumption that may be inaccurate. Moreover, not only is the assumption, that all A is some B, not necessarily true, but when it happens that All A is all B, we can obtain, by so stating it, a more precise and correct conclusion than we can by limiting our statement to All A is some B.’
It seems to me that this reasoning is irrefutable. If quantification, in the logical sense of ‘distribution’ does reside in the predicate; if the predicate may be either ‘distributed’ or ‘undistributed’; and if the ‘distribution’ or ‘non-distribution’ of the predicate is as important as Traditional Logic says it is; then it seems to me beyond the pale of discussion that it is desirable to state openly on the face of the proposition whether or not the predicate is distributed, and not to leave the distribution to be inferred from the quality of the proposition.
Hamilton’s scheme, as far as it relates to the affirmative propositions, A and I, of Traditional Logic, is as follows:—
1. The Toto-total—All A is all B.
2. The Toto-partial—All A is some B.
3. The Parti-total—Some A is all B.
4. The Parti-partial—Some A is some B,
and there is a corresponding series of negatives.
Now, it cannot be denied, that if the proposition does in fact state the inclusion of the Subject in a class, or its exclusion from a class, these alternatives are possible, and do give us more insight into the meaning of the proposition,—do state more clearly the relation of predicate to Subject—than the mere statement All A (or some A) is B. Logicians find, however, that when the negative series is examined, it leads to self-contradiction and absurdities; and for this reason they reject the Hamiltonian scheme of quantification of the predicate. They do not recognise, however, that that scheme is a strictly logical extension and application of their own doctrine; and that if it is rejected, their own doctrine must go with it. If an hypothesis, when strictly applied, leads to absurdity and self-contradiction, the logical conclusion is that the hypothesis is erroneous. No Logic but Traditional Logic would allow us, in such a case, to retain the hypothesis with respect to the cases it covers, and reject it with respect to the cases it does not cover. The Ptolemaic hypothesis of crystal spheres accounts perfectly for the movements of the fixed stars; but when applied to the movements of the planets and comets, it breaks down, and is found inefficient. Would astronomers, then, be justified in retaining it in the case of the stars, and rejecting it as far as the planets and comets are concerned? No practical reasoner would countenance such a proceeding for a moment; but this is what Logicians do with respect to the quantification of the predicate.
Consideration of other propositions also destroys the scheme of quantity enunciated by Traditional Logic. ‘Cæsar was killed by Brutus.’ Neither the Subject nor the Predicate is a class, and neither, therefore, can very well be considered ‘distributed.’ By some logicians, the proposition is regarded as a Universal, since the Predicate refers to the whole of the Subject; but in the first place, as already shown, in every particular proposition, the Predicate refers to the whole of the Subject; and in the second, it is incongruous and anomalous to speak of the whole of Cæsar being killed. We do not think of Cæsar, in this connection, as a whole composed of parts. We think of him as an individual; and if we think of Cæsar as a whole, we do not, unless we are logicians, think of killed-by-Brutus as a class or as an attribute. The proposition certainly does not express, as Traditional Logic asserts that it does, that Cæsar was one of the class of persons killed by Brutus. It expresses, not the inclusion of Cæsar in a class, but that Cæsar underwent an experience—that he, as patient, suffered an experience at the hands of the agent, Brutus. Some logicians, recognising these difficulties, relegate the singular proposition to a separate class, that is neither Universal nor Particular; but then the logical scheme of quantity falls to the ground.
My view is that quantity is not vital in reasoning. In Traditional Logic, in which all reasoning consists in inclusion in classes, or exclusion from classes, quantity is, of course, of paramount importance; but to a Logic which looks at reasoning by and large, and reasons of relations of all orders, of which the relations of class inclusion and exclusion form but a small minority, quantity is of less importance.
But if quantity is to be admitted into reasoning at all, it must be admitted freely, and in all its kinds, and degrees, and varieties. To admit the two quantities, All and Some, and to exclude all other quantities, every one of which is quite as important as these, and quite as frequently employed in argument, is monstrous, and cannot be allowed. Nothing but the blind infatuation with which logicians regard the syllogism, would render such a course possible; and it speaks loudly for the strength and depth of this infatuation, that the multitude of quantities other than All and Some should have been ostracised by Logic for so many generations. The time is come, however, to break down this absurd restriction.
CHAPTER VI
NEW DOCTRINE OF QUANTITY
QUANTITY
MY view of quantities is as follows:—
Terms, whether quantitative or qualitative, may be contemplated intrinsically, that is to say, without regard to any term of the other kind; and thus regarded, are susceptible of quantity; the quantity of quantitative terms being different in kind from the quantity of qualitative terms.
Or terms of either kind may be contemplated in association with a term of the other kind, as follows:—
Quantitative terms may be contemplated with respect to the qualities they possess; and, thus contemplated, become susceptible of classification. Qualitative terms, on the other hand, may be contemplated with respect to the concretes they qualify; and, thus contemplated, they form the means of classification.
It follows that our task is first to investigate the quantity of quantitative terms, then the quantity of qualitative terms, and then to investigate the means of classification, and the process of classification.
Quantitative terms are either individuals or classes, each of which has its characteristic intrinsic quantity, but both these kinds of quantity have the common character of extensity, and the two may be considered together as Extensive quantities.
Qualitative terms are either abstracts or attributes, each of which, again, has its characteristic intrinsic quantity, but both kinds of quantity have the common character of intensity or degree, and the two may be considered together as Intensive or Graduate quantities.
Extensive quantity is, then, of two kinds, one of which is applicable to individuals only, while they are regarded as individuals, and are not compared, or grouped, or contrasted, or associated, with other individuals. Thus regarded, an individual is susceptible of contemplation as a whole composed of parts, and we may concentrate our attention on the whole, or on a part, or on whole and part in association, or on parts associated together. When contemplated as a whole, without regard to its parts, a thing is an individual, or unit, and its quantity is Singular; when contemplated with respect to its parts it is a Composite individual, and is susceptible of a quantity that is no longer Singular, but that for want of a better name I call Massive quantity.
The other kind of Extensive quantity is applicable to individuals, contemplated, not singly, but with respect to other individuals, associated together in one or more classes. Such Extensive quantity is Numerical quantity.
Intensive quantity also is of two kinds, according as it is applicable to Abstract or to Attributive quality. Abstract quality is in a sense absolute—in the sense, that is, that abstract quality is absolved from its connection with any concrete but itself, and is contemplated alone. Attributive quality, on the other hand, is always related to the concrete by which it is manifested, and therefore may be termed relative. It must be admitted, however, that absolute and relative, in this connection, mean little, if anything more than abstract and attributive.
The complete scheme of quantity will therefore be as follows:—
TABLE II.
SCHEME OF QUANTITY.
This is the proper logical order, but Traditional Logic gives such enormous preponderance to the Distributive quantity, that in order to bring my scheme into comparison and contrast with that of Traditional Logic, it will be desirable to bow myself in the House of Rimmon, and rearrange the Extensive quantities so as to take the Distributive before the Singular and the Collective. The order in which these quantities will be taken is therefore as follows:—
TABLE III.
EXTENSIVE QUANTITIES.
Extensive quantities refer to individual things, either singly, or aggregated into classes by the possession of some common quality. The class is of course conceptual; that is to say, when we speak of things being grouped, or associated, or aggregated, together in classes, we do not mean that the things are grouped or associated together in physical propinquity; we mean that they are regarded or contemplated together by the mind, as alike in some respect. Similarly, all logical quantities are conceptual. They are aspects of things. They are ways of looking at things; and, as the same thing may be looked at from different aspects and contemplated with respect to different qualities, so it may have different quantities, according to the mode of contemplation.
When we contemplate individual things as grouped together in classes, we may contemplate them in two ways as thus grouped. We may contemplate them in the class discriminately, or one by one, preserving their separateness from one another, and saying what we have to say of them of each and every individual in the class. Thus contemplated, they are contemplated distributively, and the quantity is the Distributive Extensive. But we may contemplate them otherwise. We may contemplate them indiscriminately, without distinguishing one from another, or keeping them separate in thought; and then whatever predication is made about them is made of all taken together in bulk or in the mass, and is not true of each and every one. Thus contemplated, the individuals are contemplated collectively, and the quantity is the Collective Extensive. In ‘Every one of the books is a first edition’ the Subject term is Distributive. In ‘All the books together cost £300’ the Subject term is Collective.
THE INDESIGNATE QUANTITY.
An indesignate quantity is, strictly speaking, no quantity. In many propositions no quantity is designated, either in the Subject or in the Object term; and, as Traditional Logic cannot construct an argument with unquantified propositions, such propositions have always been to it a stumbling block. In such propositions as ‘The weather is fine,’ ‘Beauty is skin deep,’ ‘The falling out of faithful friends renewing is of love,’ there is no quantity in either term. This being so, what can Traditional Logic do, since it cannot construct an argument with an unquantified proposition? The obvious expedient, and one that is adopted by many cautious logicians, is to refuse to such propositions any place in Logic. Such logicians are wise in their own generation, for they evade an insuperable difficulty. That the price paid for this relief is such a narrowing and restriction of the province of Logic as reduces it to inefficiency in a large field of reasoning, is a consideration that has never troubled logicians. Indeed, there is no reason why it should trouble them in this case; for, when all Modals are excluded; when all propositions not expressed by the copula are excluded; and when all quantities but All and Some are excluded; the diminution, by the exclusion of the Indesignate out of the attenuated remnant that is left, is really of little consequence. When a man’s arms and legs have been amputated, it makes little difference to him to lose one of his ears; and the difficulty of bringing the Indesignate proposition into the scheme of Traditional Logic is so great, that those logicians are wisest who exclude this proposition altogether.
Other logicians, however, admit it; and the difficulty of deciding whether it is Universal or Particular is neatly surmounted, by some logicians making it Particular, and others making it Universal. A third class declare that it is Universal in some cases and Particular in others; while a fourth decide that the indesignate quantity results from carelessness on the part of the proposer, who might have investigated and declared the quantity of the indesignate Subject, but failed to do so.
In my view, the Indesignate term is neither Universal nor Particular, but Indesignate. It remains indesignate until its quantity is designated, and it is just as capable of entering into arguments and of forming the basis of sound and valid conclusions as are propositions that are specifically and definitely quantified. If I say ‘The poor are discontented’ or ‘Traditional Logic is unduly restricted,’ my statement must go for what it is worth; and any conclusion I may draw from it must be as indesignate in quantity as the premiss from which I started; but as long as it remains indesignate throughout the argument, and appears indesignate in the conclusion, the argument and the conclusion are, as far as quantity is concerned, perfectly sound; and it is just as easy, just as logical, just as permissible, to argue from an indesignate, as from a definitely quantified proposition.
These are perfectly valid arguments. In the second the Predicate, and in the first both the Subject and Predicate of the conclusion are indesignate; and in both arguments the middle term is indesignate in both premisses. Yet the arguments are valid, and why? Because the term which is indesignate in the premiss remains indesignate in the conclusion. The Indesignate term is neither Universal nor Particular. It is, and it remains throughout the argument, Indesignate.
THE DESIGNATE QUANTITY.
If a Class term is preceded by a sign of quantity, it becomes a Designate term, for the sign designates the quantity that is to be attached to the term. As already stated, Designate terms are of two kinds, Distributive and Collective, the Distributive referring discriminately and separately to every member of the class, and the Collective term referring indiscriminately to all the members of the class taken together in the lump. In either case, the class designated by the sign of quantity may be regarded as a whole, without reference to any larger class of which it is a part; or it may be regarded as part of a larger class. In the first case, the quantity is called universal; in the second, it is called particular. Thus far, the statements made are true of both the Distributive and the Collective terms, but from this point they must be considered separately.
THE DISTRIBUTIVE QUANTITY.
This is practically the only quantity treated of by Traditional Logic. It is true that most logicians mention the Collective term, and point out the difference between it and the Distributive. It is true that, in modern times, logicians have admitted the Singular quantity as a possible subject of predication and argument. It is true that, under the name of the Substantial term, a logician here and there has shown that he is aware of the Uniform quantity; but beyond a bare mention, or a very perfunctory description, these quantities receive no consideration or examination at the hands of logicians. No logician known to me recognises that the Collective quantity as well as the Distributive is divisible into the Universal and the Particular; no logician of recent times discusses the nature of the Individual, or is aware of more than two kinds of Individual. Every logician confuses the Aggregate individual with the Collective class, and both with the Uniform individual; and no logician has ever yet recognised that there is an Intensive quantity as well as a Numerical quantity; nor has any logician regarded the Comprehensive quantity as one of the quantities of terms, or relegated it to its proper place in Logic. To all intents and purposes, the logical scheme of quantity is limited to the Distributive quantity alone.
Nor does the irrational limitation of the logical scheme of quantity end here. Logic divides the Distributive quantity into the Universal and the Particular, and then sits down contentedly in the belief that the last word has been said, and that no further division is required. It does not appear to have dawned upon the minds of logicians that any further division is possible or practicable, or that there are, in fact, any other quantities of terms than the bare Universal and the bare Particular; or any other signs of quantity than All and Some. Of the three varieties of the Universal it knows but one; of the innumerable classes, sub-classes, varieties and sub-varieties of the Particular quantity, Traditional Logic knows not one.
The following is the skeleton Table of Distributive quantities:
TABLE IV.
DISTRIBUTIVE QUANTITIES.
THE UNIVERSAL DISTRIBUTIVE.
When all the Individuals in a class, regarded as a whole class, are referred to discriminately, or one by one, the quantity is the Universal Distributive, and the sign of this quantity is All. The Universal Distributive quantity is, however, itself a class, and includes three separate individuals, as follows:—
The Simultaneous Universal quantity is characterised by the sign Every. This is the Universal of Traditional Logic, and the only Universal known to that body of doctrine. When Traditional Logic uses the sign All, it means Every one, and has not, until lately, recognised that ‘All’ includes three different meanings. The Universal Distributive term refers to all the individuals in a class taken discriminately, or one by one; but things can be taken one by one in different ways. If two things are on the table, we may take one in each hand simultaneously, and then we take every one of them. If there were fifty things on the table, and we were Briareus, we could lift them all, simultaneously and separately, and then we should lift every one of them; and the universal lifting is the Universal of Traditional Logic. Or without lifting them, we may contemplate them simultaneously, and make the same predication simultaneously of each separate one, and still we contemplate and speak of every one.
But there is another way of taking them one by one, besides taking them simultaneously. We may lift them one by one in turn, taking them one after the other until all are lifted; or we may contemplate or speak of them one by one in turn, going on until all have been thus contemplated or predicated of; and in this case we deal with each one. By Each we mean one by one in turn until all have been taken. If we take each, we may not arrest the process until the tale is complete: we must go on to the end. Any predication made about each is not true unless it includes all.
But a predication made about every individual of a class is true of any one of them taken at random; and vice versâ, what is true of any one of them, is true, as far as the class likeness extends, of any other, and of every other. If we select one, as we must if we take them in turn, then we are not entitled, until we have selected every other one, to predicate of all what is predicable of that selected one; because the very fact that it was selected raises a presumption that it was in some respect different from the rest. But if we take one at random, the very fact that it is so taken indicates the absence of any difference between it and the rest, and what is predicable of one taken at random is predicable, as far as the class likeness extends, of every other. Hence Any one, though it refers ostensibly to an individual only, is in practice a sign of the Universal. This, which we may call the Alternative Universal, is very closely akin to the Representative Individual. ‘Any pen is useful’ has very nearly the same meaning as ‘A pen is a useful thing.’ It has not quite the same meaning, however, as will be shown when we examine the Representative Individual.
These, then, are the three varieties of the Universal quantity; but though they are here explained in detail in connection with the Distributive quantity, it must be clearly understood that the Universal is not restricted to the Distributive quantity, but is common to it with the Collective quantity and the Unified Individual. The Collective quantity, however, has but one form of Universal.
THE PARTICULAR DISTRIBUTIVE.
If Traditional Logic lumps together not merely the three varieties of the Distributive Universal, but the Collective and some forms of the Individual; if it makes no distinction between things so different as the Aggregate and the Corporate Individuals; if it is blind to Intensive quantity and deaf to Indesignate quantity, we need not be surprised that it fails to distinguish some of the varieties of the particular Distributive, for the characterisation of some of them is subtle and elusive; but that Logic should fail to distinguish or acknowledge any variety whatever of the Particular, except, in a halting way, one additional meaning of ‘Some,’ is really portentous, and calls to mind Dr. Johnson’s saying with respect to Thomas Sheridan. ‘Sherry is dull, Sir, naturally dull; but he must have taken great pains to become what we see him. Such an excess of stupidity is not in nature.’ Logicians are blind to the obvious, naturally blind; but they must have taken great pains not to see some at least of the varieties of the Particular. Such an excess of unobservation is not in nature.
A Particular Distributive term refers discriminately to the individuals in a class; but the class is regarded, not as a whole, but as part of a larger and including class. Now, in examining the Universal term, we have already found that the individuals composing a class may, when contemplated discriminately, or one by one, be contemplated simultaneously, or successively, or alternately; and this is equally true whether the class they compose is or is not part of a larger class. But when the individuals compose a class within a larger class, the double composition offers us more ways of contemplating them than when they are regarded as forming a primary class only. The individuals composing the subordinate class, or sub-class, may be grouped together in the mind, and contemplated either with respect to the larger or including class, or with respect to other sub-classes within that including class.
Every quantity has its appropriate sign, which, when prefixed to a term, determines the quantity of the term; and every such sign is the answer to a question that may be put with respect to the quantity of the term. The following Table gives a fairly complete list of the Particular Distributive quantities, together with a specimen sign of each.
TABLE V.
PARTICULAR DISTRIBUTIVE QUANTITIES.
Each of these quantities is a class, and includes several distinct sub-classes; and most of the sub-classes are again divisible into varieties and sub-varieties, of which the individual members are, in many cases, indefinitely numerous. The Enumerative, the Proportional, the Ordinal, the Comparative, and the Indefinite Selective, have each their Indefinite, Semi-definite, and Definite sub-classes. The Residual quantity is divisible into the Universal Residual, and the Particular Residual; and has as many varieties of these as has the whole class. The Demonstrative quantity is divisible into Appropriative and Repudiative, and each of these into Singular and Plural; and the Purposive quantity is divisible into the Suitable and the Unsuitable, the latter being again subdivided. The total number of Particular quantities is therefore indefinitely multitudinous.
The weary student will be disquieted to learn that the varieties enumerated above by no means exhaust all the recognisable and distinguishable varieties of the Particular Distributive quantity. Every particular quantity is further susceptible of three Forms, in addition to the unqualified, vague or Indesignate form that alone has hitherto been referred to. The three additional Forms are the Minimal, the Maximal, and the Exact.
The Minimal form of a quantity fixes that quantity as a minimum, below which the quantity does not extend, but fixes no maximum, and leaves it uncertain whether the part referred to in the term, does or does not extend beyond the minimum, and even to the whole class. The signs of the minimum are ‘not less than’ and ‘at least,’ and the indefinite ‘Some at least’ fixes a minimum, in this case an indefinite minimum, to the quantity to which the term refers, but leaves the maximum in doubt, and lets it be understood that the quantity may, for aught it expresses, extend to the whole class.
The Maximal form of a quantity has the opposite effect. It fixes as a maximum the quantity referred to by the term, but leaves the minimum uncertain. It gives us the assurance that the quantity is not more than is expressed by the term, but gives us no assurance that it is as many, or even that there are any at all. The sign of the maximum is ‘Only,’ or ‘Not more than,’ to which it may be necessary, in some cases, for the sake of greater precision, to add ‘If any.’ The indefinite ‘Some only if any’ positively fixes the maximum at some, and assures us that it is not all; but it fixes no minimum, and the ‘Some’ may be an inappreciably small number, or even none at all.
The Exact form of a quantity fixes both the maximum and the minimum, and assures us, first, that the quantity does not extend to all, nor to any more than the quantity expressed, and second, that the whole of the quantity expressed is certainly within the reference of the term, which, therefore, does not, as the maximal form does, or may, express a vanishing quantity. The indefinite quantity cannot be expressed in its exact form by any sign less cumbrous than ‘Some certainly, but some only,’ or ‘Some only, but certainly some’; in other quantities, however, it admits of much neater expression, and there are several forms of quantity in which the quantity itself and its form can be expressed in a single word.
Out of the three primary kinds of quantity enumerated in Table II., Traditional Logic selects the Extensive, and explicitly or implicitly declares that we can reason of no other. Out of the two kinds of Extensive quantity, Logic selects one, the Class-quantity, and, though Logic does not now declare that we can reason of no other, there was a time when it did make this declaration, and it now gives no more attention to the Singular quantity than is barely decent. Out of the two kinds of Class-quantity, Logic selects one, the Designate, and proclaims its inability to reason of the other. Out of two kinds of Designate quantity, Logic selects one, the Distributive, and though it goes so far as to admit the existence of the other, Logic does not allow that this other, the Collective quantity, is susceptible of subordinate quantities. The two primary kinds of Distributive quantity, the Universal and the Particular, are, indeed, distinguished by Logic, but here its discrimination ends, or almost ends. Of the three kinds of Universals, Logic uses but one, though it is aware of the existence of the other two; and of the eight classes of Particular Distributive quantities, each with its sub-classes and varieties, Logic knows nothing at all.
Logic selects one form of one variety of one kind of the Particular quantity, and explicitly excludes another form of this variety; while of all other kinds, with their varieties and forms, incalculably numerous though they are, Logic is profoundly ignorant. It never recognises their existence, and apparently has never realised or imagined that they do or can exist. Out of them all, Logic selects the Minimal Indefinite Enumerative, and declares that this is the only Particular quantity of which we do or can reason. It would be nearer the mark to say that it is the quantity of which we reason least often. It is clear that if we are to restrict our statements and reasonings to All and Some, we must divest these statements and reasonings of all precision and definiteness, and can reach none but vague conclusions. That the reasonings of Logic are but vague must be apparent to everyone who has ever opened a book on the subject. The first thing with respect to these reasonings that strikes a new comer is that they seem to have so little practical application; and when he seeks the reason for this detachment and aloofness from practical affairs, he finds it in the indefiniteness of the Particular quantity that alone is used. In the practical statements and reasonings of daily life, we do not restrict our dealings to ‘Some.’ We never do and never ought to employ this quantity if it is possible to use one more definite. In our household and family affairs, in business and. professions, in Parliament, in the pulpit, in the shop, the factory, the mine, and the railway, in work and in play, we never, if we can help it, use the indefinite ‘Some.’ What sort of a world would it be, how much business could be transacted, how many things could be done, if assertion and reasoning were limited to All and Some? ‘A house to let, with some rooms in it, some distance from some town in some county, rent some pounds per annum, fare from London some money, distance some miles. The trains run sometimes, and the journey takes some time.’ ‘Please sell me some shares in some stocks and buy me some shares in some other stocks. My address is some number in some street in some town.’ ‘He is sure to be elected, for though some voters are pledged to vote for his adversary, some have promised to vote for him.’ ‘I am not in your debt, for though it is true that I owed you some money last week, I paid you yesterday all I had.’ ‘Since some people live to fifty, and some live to a hundred, it is evident that the same number of people live to a hundred as live to fifty.’ No? Do you doubt? Well, you may perhaps have reason, but Logic has no terms to express differences of quantity more precise than that between All and Some, and therefore in matters of such extreme exactitude it would be unfair to expect guidance from Logic. ‘An excursion train will start for some place next week, at some time in the day, and the return fare will be some shillings.’ ‘Ample provision has been made for the party, for some people are coming and some food has been provided.’ Are these the modes of statement and reasoning that are in use?
Do we understand Some slimy things, or All slimy things? and did they crawl with Some legs, or with All legs? and if with Some, did they necessarily crawl with some-at-least-and-perhaps-all legs?
‘From that hour he never put pen to paper.’ Some pen or all pens? Some paper, or some at least, it may be all paper?
Let us try how the General Confession looks in logical terms:—
‘We have erred and strayed from some, it may be all, of thy
‘ways, like some; it may be all, lost sheep. We have followed
‘of too much some or all the devices and some or all the
‘desires our own hearts. We have offended against some, it
‘may be all, of may be all, of thy holy laws. We have left
‘undone some, it those things that we ought to have done, and
‘we have done some, it may be all, of those things that we
‘ought not to have done. And there is no health in us. But
‘thou, O Lord, have mercy upon some and perhaps all of us,’
&c. These are the terms in which we ought to express ourselves if we follow strict logical form.
But Logic is not content with excluding from its purview all Particular distributives except the indefinite Some. We have seen that there are three forms of Some—the minimal form, Some at least; the maximal form, Some only; and the exact form Some certainly, but not all, or Some at least, but some only. To these may possibly be added a fourth, viz.:—‘Some, but I do not say whether some at least or some only.’ Of these three and possibly four, Logic selects the first, drives the second out of its precincts, and does not recognise the existence of the third. Why Logic should take to its bosom ‘Some at least,’ pitch ‘Some only’ neck and crop out of the house, and wilfully shut its eyes to the exact ‘Some,’ it is difficult to conjecture. Logicians make it a grave charge against Sir William Hamilton, that he has polluted the virgin purity of the logical Some, by befouling it with the hateful meaning Some only; but for the exclusion of the maximal Some only, Logic gives neither rhyme nor reason. And there is no reason for it. There can be no reason. ‘Some only’ is a quantity as useful, as accurate, as frequently employed, as capable of entering into statement and reasoning, as ‘Some at least,’ and its exclusion from Logic is utterly unwarrantable.
If Some were drowned, Logic declares that all may have been drowned, and will not admit the possibility that some must have been saved; nor will it allow us to count on the probability that some may have been saved, for this is a modal, and modals are excluded from Logic. I aver, on the contrary, that if some only were drowned—a possibility that Traditional Logic will not admit—then it is certain that some were not drowned; and the anxiety of us, who are not logicians, as to the fate of Some at least of the party, is relieved; but logicians must still remain in anxious uncertainty. We, who are not logicians, know that, if logicians recognise some modes of argument only, and some Particular quantities only, it is certain that there are other modes of argument and other Particular quantities that logicians are not aware of; but logicians themselves are precluded by their own rules from drawing this inference, or making this admission. They claim that as they are acquainted with Some modes of argument and Some particular quantities, and as Some has, in Logic, but the one meaning of ‘Some, it may be all,’ the modes of argument and the quantities with which they are acquainted may be all there are. We know better, but we cannot convince a logician of his error, for to him every Some is potentially All.
*I am ashamed to say that not until too late did I recognise that this is not a logical argument at all. The predicate of the major is explicitly quantified, and though the predicate of every proposition must be quantified, It must be quantified sub rosâ, and must not brazenly avow its quantification. Consequently, the major not being a logical proposition, the argument is not a logical argument I can only plead in extenuation that the limits of Traditional Logic are so extremely narrow, that a mere amateur cannot be expected to make a bull’s-eye at every shot. So few valid arguments are ‘logical,’ that one really ought to recognise that the more cogent an argument, the stronger the presumption that it is ‘not logical.’
This conclusion Logic forbids us to draw. Such an argument, such a conclusion, is illicit, illogical, illegitimate, and impossible. Why?
If logicians know of some particular quantities only, then they do not know all particular quantities; then there are some particular quantities of which they are ignorant; then their knowledge of particular quantities is imperfect, it is defective: and then there are quantities of which they do not know. These are all valid inferences from the premiss, but logicians are happily ignorant of them, and doubly ignorant; for, in the first place, the premise is inadmissible into Logic, and, in the second, the conclusions are arrived at by a process unknown to Traditional Logic. They are Immediate Inferences, but none of them is a converse, an obverse, a contrapositive, or an inverse.
If one particular quantity only is to be chosen to represent them all, the indefinite Some is the best that could be chosen, for it may be made, with little difficulty, to cover nearly all the rest. Many, a few, a very few, one, two, most, few of, three more, twice as many, the rest, the others, the first, the last, certain, this, those, enough, too many, and so forth, may all be included under Some; but apart altogether from these more definite interpretations, bare indefinite ‘Some’ is susceptible of twenty or thirty different meanings, as against the two that are recognised and the one that is used by Logic.
‘Some’ may mean an indefinite and unselected number taken distributively; or it may mean an indefinite but selected number taken distributively; or it may mean an indefinite unselected number taken collectively; or an indefinite selected number taken collectively; or it may mean an indefinite proportion, either selected or unselected, and in either case taken distributively or collectively; or it may mean an indefinite ordinal number, selected or unselected, and taken distributively or undistributively; and in any case it may mean the particular Some minimally, maximally, exactly, or vaguely, that is, without specifying whether it is minimal, maximal, or exact.
The exclusion from Traditional Logic of all these meanings, except one, of the indefinite Some, and of all the semi-definite and definite varieties of the particular quantity, in both the distributive and the collective quantities, and in all the three or four forms of this quantity, is not, however, as important as it seems at first sight; for already, by excluding Modals, and all Ratios except the copula, Traditional Logic had so narrowed the field of its operations as to deprive itself of almost all practical usefulness; and the addition of one more unnecessary restriction makes no important difference. Logicians declare that they can make no reference to any part of a class without assuming that that part may be the whole. Well, if their minds are so constituted that this is true of them, I have no more to say, except that they must not seek to impose the same restrictions upon others, whose minds are more capacious. The restrictions of Traditional Logic are so surprising, and are so discrepant from common experience, that if it should declare that it can entertain no argument, and accept no conclusion, unless the premisses are written in red ink, or in black letter, I should accept the statement without a murmur, as true of logicians themselves, as long as they are engaged in reasonings that they are pleased to call logical; but I should not accept it, any more than I accept the exclusion of Modals, of Ratios other than ‘is’ or of Particular quantities other than ‘Some at least,’ as true of myself, or of reasoners in general, or of logicians themselves, apart from their books.
The exclusion from Logic of all the semi-definite and definite Particular quantities, of all exact and maximal forms of quantity, of the Residual and Purposive quantities, not only deprives such reasonings as Logic can effect, of all precision, and therefore, in most cases, of all practical usefulness, but it also shuts out from Traditional Logic an immense range of arguments that are in constant use, but that this Logic cannot compass, for want of terms in which to express them. When these subordinate varieties of the Particular quantity are admitted into argument, it is found that the methods and principles of reasoning, as formulated by Traditional Logic, are not only defective, but also very erroneous. The rules of the Syllogism are applicable to deductive reasoning so long only as that reasoning is confined to the minimal indefinite distributive quantity. As soon as other quantities are admitted into statement and argument, every one of the Canons of the Syllogism is found to be false, and the Square of Opposition is blown to pieces. It will be necessary, therefore, even at the cost of some tedium, to examine in detail each of the Particular quantities that has been enumerated.
CHAPTER VII
PARTICULAR DISTRIBUTIVE QUANTITIES
THE ENUMERATIVE QUANTITY.
WHEN a term refers discriminately to all the individuals in a class, regarded as part of a larger class, and contemplates them simultaneously with respect to their number, the quantity is Enumerative, and answers the question, How many?
The Enumerative quantity may be Indefinite, Semi-definite, or Definite.
The Indefinite Enumerative.—The sign of this quantity is Some, which includes, not only the ‘Some at least it may be all’ of Traditional Logic, but also ‘Some only, it may be none,’ and ‘Some certainly, but not all.’ Of these, the minimal form, ‘Some at least,’ is sufficiently explained in the text books, and the exact form, ‘Some only, but some certainly,’ has been examined on a previous page. It remains to consider the maximal form, ‘Some only, and perhaps none.’ This maximal indefinite is not very often used, though the maximal semi-definite and definite, ‘Few if any,’ and ‘Not more than so many,’ frequently occur in reasoning. The maximal quantity is an uncertain quantity. It may or may not exist; and as its existence is uncertain, it may lead to an uncertain conclusion. It is excluded from Traditional Logic, therefore, on the double ground, that it is not ‘Some at least’ and that the proposition into which it enters may be Modal. Its exclusion from Traditional Logic need not, however, hinder us from employing it in statement and in reasoning. If Some only and perhaps none were drowned, then it is certain that some, and possible that all, may have been saved. If some only and perhaps none of the messengers arrived, some at least, and perhaps all, stopped or were diverted on the way, and all the messages were not delivered. If some only and perhaps none of the eggs hatch out, some at least and perhaps all will be addled; certainly some will not hatch out; there will be fewer chickens than eggs; the whole clutch may be a failure; and the money paid for it will be partly or wholly wasted. What is the matter with these inferences?
The Indefinite Enumerative quantity may qualify not the Subject-term only, but the Object-term, either instead of or together with the Subject-term. Vegetarianism is adopted by some men. Some rats were caught in some of the traps. This form of proposition is unknown to Traditional Logic, but the statements thus made are valid; and valid arguments may be deduced from them. If vegetarianism is adopted by some men only, it is not adopted by all men. If some of the rats only were caught in some of the traps, all the rats were not caught in those traps; and if some rats were caught in some only of the traps, there were other traps in which no rats were caught.
THE SEMI-DEFINITE QUANTITY.— Before entering on the examination of the Semi-definite Enumerative, it is necessary to consider certain features that are common to all semi-definite quantities. The first of these is the Emphatic or Unexpected form, the sign of which is ‘Such’ or ‘So.’ ‘Such a great many,’ ‘So many,’ ‘Such a few,’ ‘So few.’ The meaning conveyed by the addition of this qualification of the semi-definite is that the quantity is unexpected. It surprises us that there should be so many or so few as we find.
The next modification is more important. Every semi-definite, like every other quantity, exists in three forms, the minimal, maximal, and exact. These have already been described; but in addition to differences of form, and differences of emphasis, there is a third way in which semi-definite quantities may be classified. They may be referred to a certain standard or medium, and may be arranged in sets, according as they reach, exceed, or fall short of this standard. Moreover, the excess above the standard may be moderate or great, and the defect below it may be moderate or great, and this gives us a set of five degrees of semi-definite quantity, to one of which every such quantity may be referred. The degrees may be made more numerous than this, but five are enough for most purposes. This set of degrees applies to every semi-definite quantity, whether Enumerative, Proportional, Comparative, Ordinal or Selective. Omitting intermediate and excessive degrees, the degrees of semi-definite quantity include:—
Combining these degrees with the forms already discovered, and omitting the Indesignate form, we arrive at the following table:—
SEMI-DEFINITE ENUMERATIVE QUANTITIES.
‘A great many’ is a minimal form, though, like other terms, it is often misused. It means certainly a large number, and possibly all. ‘A great many men are honest’ states that certainly a large number of men are honest, and for aught we know, all may be. ‘There are a great many people on the cricket ground’ is an incorrect expression, for we know that the large number there are not all the people there are. The quantity referred to is an exact, though not a completely definite quantity. It means a large number certainly, but not all. It means a large number, no less and no more; and the proper sign of the exact semi-definite is not ‘a great many’ but ‘very many.’ ‘A great many specimens were damaged’ leaves in doubt whether all the specimens were or were not damaged. ‘Very many specimens were damaged’ clears up this doubt, and assures us that there were a very few that were not damaged.
The proper magnative minimal of this quantity is ‘A many,’ which meant ‘certainly many, and perhaps all’; but ‘A many’ is unfortunately become a vulgarism, and its ostracism from polite speech deprives us of the means of expressing a useful distinction.
No maximative or magnative quantity can be maximal. When we speak of a great many or of very many, or of many, we never make the reservation ‘Perhaps none.’ This reservation is confined to quantities less than the medium.
The sign of the medium semi-definite is ‘A good many.’ It may be minimal or exact, but can scarcely be maximal. We can scarcely have occasion to speak of ‘A good many only, and perhaps none.’
The parvative is the only degree that runs through all the forms, and the minimal form is a little incongruous, and is seldom employed. ‘A few at least, and it may be all’ is not often used or implied. In practice, the two forms are the maximal, ‘Few,’ meaning ‘ Few only, and perhaps none,’ and the exact ‘A Few’ meaning ‘A few, neither more nor less.’ If we do desire to express the minimal form, the addition of the article, which invests the quantity with positive character, and fixes it as ‘certainly a few,’ is not enough. We must add ‘at least,’ which effectually adds the meaning ‘perhaps more or even all,’ ‘A few men at least are honest’ leaves it possible that all men may be honest. ‘Few men are honest’ denies that all are, and leaves it possible that there may be no honest men. ‘A few men are honest’ declares that certainly a few, but not more than a few men are honest.
The sign of the minimative Enumerative is ‘Very few’; and it is evident that this quantity can have no minimal form. ‘A few at least, it may be all’ is a possible, though not a very frequently used quantity: but ‘A very few at least, it may be all’ is not a possible quantity. The addition of the ampliative, ‘Very,’ at once fixes a maximal limit to the quantity, and prevents its encroachment upwards, just as, at the other end of the scale, the addition of the ampliative to Many fixes a minimal limit. ‘Very few’ shouts ‘not many,’ says ‘few only,’ and whispers ‘perhaps none.’ In this, as in the other cases, the addition of the article at once invests the quantity with positive character, minimises it, and is equivalent to the addition of ‘certainly.’ ‘Very few’ may mean ‘perhaps none,’ but ‘A very few’ means ‘Certainly some, but a very small number,’ and is therefore exact in form.
The degrees that have been enumerated are not all the degrees there are. Beyond ‘A great many,’ there are ‘A very great many,’ ‘An enormous number,’ and so on; between ‘A good many’ and ‘A few’ there are ‘A good few’; and below ‘A very few’ are ‘A very few indeed,’ ‘Extremely few,’ and so on.
It is unnecessary to insist upon the frequency with which the semi-definite Enumerative quantities enter into statement and argument. That is at once apparent to everyone; but it is desirable to point out and insist that there are many arguments for the construction of which the semi-definite enumeratives are necessary—arguments that cannot be effected except by means of those quantities, and arguments that lead to conclusions utterly inaccessible to Traditional Logic. The following illustrate a few types of these arguments:—
If there are a great many books in the British Museum, then they must occupy a great deal of room, and it would take a very long time to read them all. If very few modes of argument are known to Traditional Logic, it does not know of many. If it is profitable to keep a hen when it lays a few eggs, it will certainly be profitable to keep it when it lays many. If a good many people go out for wool, and come home shorn, this unpleasant experience happens to more than a few people, and more than a few leave their wool behind them, and come home without it.
In the last example, it is the object-term that is semi-definitely quantified; yet the argument is valid.
The Definite Enumerative quantity is a cardinal number. Traditional Logic always assumes, though the text books do not definitely state, that Logic has nothing to do with the cardinal numbers, which are supposed to be susceptible of mathematical reasoning only, and to be outside the province of Logic. This doctrine, if it be a doctrine, is doubly erroneous. It is erroneous in supposing that numerical quantities are excluded from Logic, and it is erroneous in supposing that mathematical reasoning is distinct from Logical reasoning, and is not included in it. Still, although Logic is mistaken in supposing that its powers do not enable it to reason of numerical quantities, the numerical quantities of which Logic reasons are not the same as those employed in the reasonings of mathematics. Arithmetical numbers are doubly limited. They are exact in form. Logical numbers, unless formally stated to be so, are not limited, either minimally or maximally. In Arithmetic, three, or thirty, or three hundred, means that number exactly, neither more nor loss. No excess or deficiency is permitted, for if any were allowed, the operations of Arithmetic would be impossible. We could not satisfactorily add two and three together, or subtract two from three, if the two might be more or less than two, and the three less or more than three; but in Logic, the statement that two men entered a house does not preclude the possibility that three men entered it, and necessarily implies that one man entered it In Logic, a number means that number at least, or that number only; and if that number only, it may be that number exactly, or not more than that number. Since Mathematics cannot, except under very limited and rigidly defined conditions, reason of inexact numbers, such as more than seven, or fewer than five, it follows, if numerical reasoning. are excluded from Logic, that there is a field of reasoning that is neither Mathematical nor Logical. Where do such arguments belong? They belong to the New Logic here propounded.
If two men left the house, and three men entered it, Arithmetic assures us that the number of men in the house was augmented by one; and in so concluding, Arithmetic is reasoning according to its lights, and within its limitations. Granting the assumptions of Arithmetic, that two and three mean, respectively, two and three exactly, neither less nor more, Arithmetic is justified in coming to this conclusion. But such a conclusion would be wholly unwarranted in Logic. Before Logic would be justified in concluding that the number of men in the house was augmented by one, or was augmented at all, Logic must know whether the two men who left the house were two at least or two only, and whether the three who entered it were three only or three at least.
If ten men entered the house and two men left it, Arithmetic assures us that eight remained indoors. But if ten men only entered and two at least departed, Logic is not justified in concluding that any at all remained. The two at least may be the whole ten.
If ten men at least entered the house, and eight men only left it, then Logic can tell us that two men at least remained in the house, a conclusion that is within the scope of Arithmetic also; but Logic can conclude further, that more men entered the house than left it, and this conclusion is beyond the capacity of Arithmetic.
If ten men at least entered the house, and not more than three left it, Logic can derive the conclusion that seven at least remained; but Arithmetic knows no such number as ‘not more than three,’ and can derive no conclusion from these premisses. If not more than ten men entered, and seven at least departed, Arithmetic can arrive at no result, but Logic can tell us that there remained in the house not more than three, and perhaps none at all.
THE PROPORTIONAL QUANTITY.
When the individuals in a class, regarded as part of a larger class, are contemplated discriminately with respect to their proportion to the whole of the larger class, the quantity thus contemplated is the Proportional quantity, which may be an indefinite, a semi-definite, or a definite proportion of the whole.
The Indefinite Proportional.—This quantity, as well as the Indefinite Numerical, is signified by Some, and it is another disadvantage of this multiguous adjective that its enumerative meaning is not clearly distinguished from its meaning as a proportion of a class. It is true that it does not greatly matter, when ‘some’ is used, whether the ‘some’ is understood as enumerative or proportional, but it does matter that the constant use of ‘some’ in both senses tends to confuse the two quantities, to obscure the differences between them, and to lead to the employment of more definite signs ambiguously. Generally, ‘some’ simpliciter is enumerative. The proportional indefinite is signified by ‘Some of the.’ ‘Some men are honest’ means that an uncertain number of the whole class of men are honest. ‘Some of the men’ are honest means that an uncertain proportion of the men in a certain class are honest. The distinction is not made in Traditional Logic.
The Semi-definite Proportional.—This is subject to the same forms and the same set of degrees as other semi-definite quantities.
SEMI-DEFINITE PROPORTIONAL QUANTITIES.
It is scarcely necessary to go through these quantities and show that they answer generally to the description implied by their places in the table. The reader who has followed me thus far can do that for himself. The Proportional Parvative is, in its minimal form, ‘A few of,’ in its maximal form, ‘Few of.’ ‘A few of’ means ‘certainly a small, and perhaps a large proportion, and it may be the whole.’ ‘.Few of’ means ‘A small proportion only, and perhaps none.’ Commonly, the possibility of ‘Few of’ vanishing into none is emphasised by adding ‘if any,’ but this is not necessary. ‘Few of’ is already maximal.
The use of the proportional quantity enables us to conduct many arguments and reach many conclusions that are unattainable by Traditional Logic. If most men are honest, not only are all men not dishonest, but honest men are in the majority, and outnumber dishonest men. If nearly all were drowned, it follows that a few of them only, in fact scarcely any, were saved; and if nearly, but not quite all, were drowned, then a very few of them were certainly saved.
The Definite Proportion is, of course, a fraction. It is a half, two thirds, three fourths, or some other numerical proportion. Logic, of course, excludes such quantities, on the ground that, lying within the domain of Mathematics, they are necessarily outside the realm of Logic; but in examining Enumerative quantities, we found that the boundaries of Logic and Mathematics are not rigidly marked by the exclusion and inclusion of definite numerical quantities. An argument concerned with exact definite numerical quantities, that is with figures, whether integers or fractions, exactly limited in both directions, and with nothing else, is an arithmetical argument; but many arguments contain indefinite proportional as well as definite proportional quantities, and many contain inexact as well as exact proportions; and such mixed arguments belong to Logic, just as those which contain a mixture of definite and indefinite integers belong to Logic. I aver that if two thirds of the company at least were drunk, then, according to the New Logic here propounded, not half of them were sober. Traditional Logic, in order to reach this conclusion from this premiss, must call in the aid of a Mathematician; and the Mathematician is entitled to say ‘Mind your own business. “Not half” is not a mathematical quantity. The argument is not arithmetical, but logical. See ye to it, for I will be no judge of such matters.’
In Arithmetic, two thirds, three fourths, nineteen twentieths, and so forth, are exact fractions of an integer; in Logic, they are classes, each containing some definite proportion, that may be exact or inexact, of a larger class. In Arithmetic, each fraction is an exact form, and no maximal or minimal form is known: but in Logic, two thirds may be two thirds exactly; or it may be two thirds at least, and perhaps all; or it may be two thirds only, and perhaps less or none. If at least two thirds of the company were drunk, Arithmetic cannot tell us what proportion were sober; but Logic can conclude without difficulty that not more than one third were sober, and that not half of them were sober. If not more than one third were sober, Arithmetic cannot tell us what proportion were drunk, but Logic concludes without difficulty, that not only more than half, but at least two-thirds of them were drunk. The Definite Proportional quantity is, therefore, in strict propriety, a logical quantity; and Logic is very incomplete without it.
These are not arithmetical arguments. ‘Not half,’ ‘at least a third,’ ‘nearly a half,’ and ‘plenty,’ are not arithmetical quantities, and Arithmetic would rightly exclude the arguments from her domain. Traditional Logic is incapable of effecting them, and would shirk them on to Arithmetic if it could. Failing this effort, the only consistent course open to Traditional Logic is to deny that they are arguments at all. It would probably content itself, however, by saying that they are ‘not logical,’ and would then feel that its duty was done.
THE ORDINAL QUANTITY.
When we contemplate parts of classes in succession, with respect to the order in which they present themselves, or in which we choose to take them, the quantity is Ordinal, and, like the other quantities that we have examined, may be definite, semi-definite, or indefinite. It will be noted that the order is always definite; it is the quantity that varies in definition.
The Indefinite Ordinal quantity is signified by ‘the Former,’ ‘the Latter.’ It indicates the position in order of the individuals, but does not indicate their number; nor does it indicate with precision the ordinal position. It indicates the position with respect to one other part only of the class, not to every other part.
The Semi-definite Ordinal term is characterised by an ordinal number followed by the sign of a semi-definite quantity, which is almost always few—the first few, the last few. We do not speak of the first many, though there is no reason why we should not.
The Definite Ordinal is an ordinal number—the first, the second, etc. It does not specify an individual except by his position in the class, for the same individual may successively occupy more than one position, and the same position may be occupied successively by different individuals; but it indicates with precision the individual that for the time being occupies that place.
Predications and arguments with respect to Ordinal quantities are not infrequent; they cannot be expressed without the use of this quantity, and therefore Traditional Logic, which has not discovered the quantity, is incompetent to deal with them. If the first comer is first served, he is not put off until others are served, he is not served second, or last; he has the largest choice; he has the opportunity of getting first away; he does not have to wait so long as others; the second comer is not first served. If it is the last straw that breaks the camel’s back, it is not the first straw, nor the second, that produces this disaster; a single straw will not break a camel’s back; if the last straw is not added, the camel’s back will not be broken.
THE RESIDUAL QUANTITY.
When any part of a class is removed, set aside, selected, or distinguished in any way, the rest of the class outside of the distinguished part, regarded alternatively, forms a quantity of a special kind, which may be called the Residual quantity. As has already been pointed out, the Residual quantity is not divisible, as the quantities are that we have examined hitherto, into Indefinite, Semi-definite, and Definite quantities. When a portion of a class is removed, the residue is itself a class, and is susceptible of all the quantities, including even the residual, that could be attached to the whole class; and thus it is primarily divisible, not into Indefinite, Semi-definite, and Definite Residual, but into Universal Residual and Particular Residual.
The general sign of the Universal is All, which includes the three Universal quantities already examined; and correspondingly, we may predicate and reason of All the rest; of every one, each one, and any of the rest.
The general sign of the Particular quantity of the whole class is Some, but the general sign of the Residual Particular is Others; and there are as many classes, sub-classes and varieties of the Particular Residual as of the common Particular quantity.
Every Particular quantity has its complement in a Residuum, which remains undistinguished when the Particular is selected out of the class; but in the majority of cases, we do not take the residuum into account. It has already been shown that we possess the power of concentrating our contemplation on any aspect of a subject before us, and neglecting the rest; and when we select or distinguish a part of a class, we commonly ignore the remainder. But there are occasions on which the recognition and use of the Residual quantity are of the utmost service, and enable us to reach conclusions that would be utterly unattainable without it.
Every Particular quantity may of course leave its residuum; but it would seem that no Universal quantity can leave a residuum. If all are taken, there can be none left. This is true if all are taken simultaneously; but if all are taken in succession or alternately, there is at any rate a temporary residuum. If each man in turn takes a step forward, the rest remain for the time stationary; and though any stick may be taken out of a bundle, the rest may be left in it.
The Minimal Particular may or may not leave a residue, according as it does or does not extend to all; and one of the virtues of the residual quantity is that it settles decisively the form of the otherwise formless Some. If some men desire money, the some may, for aught that appears in the statement, be all there are: but if some men desire money and others do not, the some is fixed at its maximal form, and cannot be some, perhaps all.
Many arguments require the residual quantity for their statement, and cannot be effected without it. It differs from the quantities hitherto examined in that it must be preceded by a proposition containing some other quantity, to which it is residual, and therefore every argument containing the residual must be stated in the form of a compound proposition, and must be of the nature of what is called, in Traditional Logic, Mediate Inference. If some men desire money and others do not, then the desire for money is not universal among men; and with respect to the desire for money, men are divided. If two or three men were drowned, and the rest were saved, then neither were all drowned nor all saved; then all were saved but two or three; then very few were drowned; and then some at least were saved, and some only were drowned; and then, moreover, the fate of all was not the same.
The Residual may be preceded by more than one other quantity, and argument may be founded on a compound proposition of many members, containing many quantities. If some of the balls were red, others were blue, four were green, twice as many were black, certain of them were yellow, more were purple, and the rest crimson, then it is an irrefragable conclusion that none of them was white.
None of these reasonings can be conducted by any method known to Traditional Logic.
Traditional Logic, which knows not the Residual quantity, is incapable of examining this statement, and cannot tell us whether it is valid or not.
THE COMPARATIVE QUANTITY.
When the individuals in a class are contemplated discriminately, as part of a larger class, and alternatively with respect to other parts of the larger class, the comparison with the other part or parts results in the formation of the Comparative quantity, and the comparison may be indefinite, semi-definite, or definite, and the resulting quantity shares the degree of definition. The result of the comparison must be that the respective numbers are adjudged to be alike or different, and if different, the number in the class that is the chief object of attention, is adjudged to be more or less than that in the other. Thus the comparative quantities are primarily three, and, when the comparison is indefinite, are More, Fewer, and About as Many.
Without being completely definite, the comparison may be more definite than this.
On comparing these two lines of dots, it is at once apparent that the lower line contains two more dots than the upper. It is quite unnecessary to count each series, or to know how many dots in all each series contains. The excess of one, and the defect of the other, can be stated definitely, without the totals being definitely stated; and thus the signs of the Semi-definite Comparative are So many more, So many fewer, and As many.
The Definite Comparative is a multiple or sub-multiple. It is Twice as many, Two thirds as many, Half as many again, and so forth.
It seems incredible that Traditional Logic should have ignored the Comparative quantity, so frequently is it employed, so valuable are its uses, and so impossible is it to reach, without its aid, conclusions of the utmost practical importance.
If more money is put into the bank than is taken out of it, the balance will increase. If more money is withdrawn than is paid in, the balance will be diminished. If more geese than swans now live, more fools than wise, then the geese outnumber the swans, and the wise are rarer than fools. If more people were thrust into the Black Hole at Calcutta than the air would support, it was inevitable that some of them should die. If there are fewer teats than there are little pigs, one little pig must go without. If there are three more people than there are seats, then three people cannot sit down unless they share a seat with some one else. If there are twice as many carts as horses, at least half the carts must go unhorsed.
THE SELECTED QUANTITY.
As its name implies, the Selective term selects a certain individual or certain individuals out of a class, and the individuals selected may be definite or less than definite in two senses. They may be indefinitely selected, and indefinite, semi-definite, or definite in number; or they may be definitely selected, and indefinite, semi-definite, or definite in number.
The Indefinitely Selected Quantity is characterised by the sign ‘Certain,’ and Certain, standing alone, is a completely indefinite enumerative, and means some individuals selected out of a class, but neither indicates the individuals nor mentions their number. We may, however, select a proportion of a class, and indicate our selection by the sign ‘A certain proportion,’ and we may select a number out of the residue or a proportion of the residue, and signify our selection by ‘Certain others,’ or ‘A certain proportion of the others.’ These are all indefinitely selected indefinite quantities.
But the Indefinitely Selected quantity, whether Enumerative, Proportional, or Residual, may be semi-definite with respect to the number of individuals it indicates; and then is characterised by the signs, ‘Certain few,’ ‘A certain few,’ of the whole, or the rest, as the case may be.
Or this quantity may be completely definite as to the number or proportion selected, though this selection remains indefinite. The sign is then, ‘A certain’ as in ‘A certain one,’ ‘A certain three,’ ‘A certain half’ or ‘quarter.’
The individuals may be definitely selected and may or may not be definite in number or proportion. The signs of definite selection are This, These, That, Those. This, These, apply to a selected number or proportion near at hand or appropriated; That and Those apply to a selected number or proportion at a distance or repudiated.
‘A certain man drew a bow at a venture,’ selects one man out of the army, and places the bow in his hand. It was not any man in the army who drew the bow, it was a selected man; but beyond the fact that it was not any man taken at random, he is not identified. He may have been this man or that; he may have been the first in the army or the last, or in any intermediate position. He is not identified, but he is identifiable.
This selects, but does not identify, the people in question. They may be these people or those; they may be the first people you meet, or the last that you wish to meet; they may, for aught we are told, be you and me, reader; they may be but two in number, or they may be all the people in the world except Madam Albani; but they are selected as possessing some quality, and, though not identified, are identifiable, provided we know their distinctive quality.
This and These, That and Those, not only select, but Identify the individuals they characterise; and not only do they identify the individuals, but they indicate also the proximity or otherwise of those individuals to ourselves, or our attitude, as appropriative or repudiative, towards them.The Selective quantity is not free from the complication of ‘forms.’ Any Selective quantity may be Minimal, Maximal, or Exact. ‘Certain,’ ‘A certain proportion’ and ‘Certain others’ may be at least, only, or exactly; and similarly, the definitely selected or Demonstrative quantity may be This or These only, That or Those at least, or exactly.
Many predications, and therefore many arguments, can be conducted by means of the Selective quantity only; and many others by the Demonstrative quantity only. If certain people only should die before they sing, then other people should not die before they sing, and it is not true that any one taken at random ought to die before he sings. Moreover, it follows that certain other people should sing before they die.
If this little pig went to market, then there was a little pig who went to market; it was not that little pig who went to market; and this little pig did not stay at home. If this is the house that Jack built, it is not the barn that Jack built; it is not the house that Tom built; nor is that the house that Jack built. If there is no way but this, there is no other way. If that is the way the money goes, then we know in what way the money goes, and in what ways it does not go.
All of these arguments are beyond the range of Traditional Logic, and ultra vires of it.
THE PURPOSIVE QUANTITY.
The last quantity on our long list is the Purposive. In using this quantity, we regard the individuals with respect to the suitability or unsuitability of their number to the purpose in hand; and if it is unsuitable, is unsuitable by excess or by defect. The Suitable quantity has its minimal and exact forms, but the Unsuitable are known in the exact form only. We cannot say too many only, or too many at least, or too few only, or too few at least, or if we do, such expressions are meaningless; but we can say of a number that it is enough at least, or enough only, if by only we mean just enough, for though a certain quantity may be suitable to the purpose, the purpose may perhaps not be defeated if we have more than enough. There can be no maximal suitable quantity, however, for if there are less than enough there are not enough. We see, therefore, that there is a clear difference between more than enough and too many. More than enough is excessive, but does not defeat the purpose in hand. If there are more than enough stamps for all the letters, the letters can still all be stamped; but if there are too many plants for the pots, all the plants cannot be potted, and if there are too many letters for the stamps, all the letters cannot be stamped.
If the number is unsuitable, it may be deficient or excessive by an enumerative, a proportional, or a selective quantity, and the excessive or defective quantity may be definite, semi-definite, or indefinite. It may be one too many, or many too few, or more than enough; it may be too many by half, or deficient by a third; it may be this one too many, or that, or those, or may be defective by a certain few.As with other quantities, the Purposive has its own field of reasoning, and enters into predications and arguments that cannot be conducted without its aid. If there are enough glasses to go round, no one need go without a glass, and every one can have a glass to himself. If there are not enough glasses to go round, some one must go without, or share with some one else. If he is one too many, he is not wanted; he would be better away; there are enough without him. If too many cooks spoil the broth, it is a disadvantage to have too many cooks. If the Universal and Particular quantities of Traditional Logic are too few for the expression of all our thoughts, we must employ additional quantities or leave some of our thoughts unexpressed. If the number of quantities enumerated here is too many to be easily remembered, it will need an effort to commit them to memory, but it does not follow that they are more than enough to express our thoughts.
