(4) If an event may coincide with two or more other independent events, the probability that they will together be a sign of it, is found by multiplying together the fractions representing the improbability that each is a sign of it, and subtracting the product from unity.
This is the rule for estimating the cogency of circ.u.mstantial evidence and a.n.a.logical evidence; or, generally, for combining approximate generalisations "self-corroboratively." If, for example, each of two independent circ.u.mstances, A and B, indicates a probability of 6 to 1 in favour of a certain event; taking 1 to represent certainty, 1-6/7 is the improbability of the event, notwithstanding each circ.u.mstance. Then 1/7 1/7 = 1/49, the improbability of both proving it. Therefore the probability of the event is 48 to 1. The matter may be plainer if put thus: A"s indication is right 6 times in 7, or 42 in 49; in the remaining 7 times in 49, B"s indication will be right 6 times.
Therefore, together they will be right 48 times in 49. If each of two witnesses is truthful 6 times in 7, one or the other will be truthful 48 times in 49. But they will not be believed unless they agree; and in the 42 cases of A being right, B will contradict him 6 times; so that they only concur in being right 36 times. In the remaining 7 times in which A is wrong, B will contradict him 6 times, and once they will both be wrong. It does not follow that when both are wrong they will concur; for they may tell very different stories and still contradict one another.
If in an a.n.a.logical argument there were 8 points of comparison, 5 for and 3 against a certain inference, and the probability raised by each point could be quantified, the total value of the evidence might be estimated by doing similar sums for and against, and subtracting the unfavourable from the favourable total.
When approximate generalisations that have not been precisely quantified combine their evidence, the cogency of the argument increases in the same way, though it cannot be made so definite. If it be true that most poets are irritable, and also that most invalids are irritable, a still greater proportion will be irritable of those who are both invalids and poets.
On the whole, from the discussion of probabilities there emerge four princ.i.p.al cautions as to their use: Not to make a pedantic parade of numerical probability, where the numbers have not been ascertained; Not to trust to our feeling of what is likely, if statistics can be obtained; Not to apply an average probability to special cla.s.ses or individuals without inquiring whether they correspond to the average type; and Not to trust to the empirical probability of events, if their causes can be discovered and made the basis of reasoning which the empirical probability may be used to verify.
The reader who wishes to pursue this subject further should read a work to which the foregoing chapter is greatly indebted, Dr. Venn"s _Logic of Chance_.
CHAPTER XXI
DIVISION AND CLa.s.sIFICATION
-- 1. Cla.s.sification, in its widest sense, is a mental grouping of facts or phenomena according to their resemblances and differences, so as best to serve some purpose. A "mental grouping": for although in museums we often see the things themselves arranged in cla.s.ses, yet such an arrangement only contains specimens representing a cla.s.sification. The cla.s.sification itself may extend to innumerable objects most of which have never been seen at all. Extinct animals, for example, are cla.s.sified from what we know of their fossils; and some of the fossils may be seen arranged in a museum; but the animals themselves have disappeared for many ages.
Again, things are cla.s.sed according to their resemblances and differences: that is to say, those that most closely resemble one another are cla.s.sed together on that ground; and those that differ from one another in important ways, are distributed into other cla.s.ses. The more the things differ, the wider apart are their cla.s.ses both in thought and in the arrangements of a museum. If their differences are very great, as with animals, vegetables and minerals, the cla.s.sing of them falls to different departments of thought or science, and is often represented in different museums, zoological, botanical, mineralogical.
We must not, however, suppose that there is only one way of cla.s.sifying things. The same objects may be cla.s.sed in various ways according to the purpose in view. For gardening, we are usually content to cla.s.sify plants into trees, shrubs, flowers, gra.s.ses and weeds; the ordinary crops of English agriculture are distinguished, in settling their rotation, into white and green; the botanist divides the higher plants into gymnosperms and angiosperms, and the latter into monocotyledons and dicotyledons. The principle of resemblance and difference is recognised in all these cases; but what resemblances or differences are important depends upon the purpose to be served.
Purposes are either (a) special or practical, as in gardening or hunting, or () general or scientific, as in Botany or Zoology. The scientific purpose is merely knowledge; it may indeed subserve all particular or practical ends, but has no other end than knowledge directly in view. And whilst, even for knowledge, different cla.s.sifications may be suitable for different lines of inquiry, in Botany and Zoology the Morphological Cla.s.sification is that which gives the most general and comprehensive knowledge (see Huxley, _On the Cla.s.sification of Animals_, ch. 1). Most of what a logician says about cla.s.sification is applicable to the practical kind; but the scientific (often called "Natural Cla.s.sification"), as the most thorough and comprehensive, is what he keeps most constantly before him.
Scientific cla.s.sification comes late in human history, and at first works over earlier cla.s.sifications which have been made by the growth of intelligence, of language, and of the practical arts. Even in the distinctions recognised by animals, may be traced the grounds of cla.s.sification: a cat does not confound a dog with one of its own species, nor water with milk, nor cabbage with fish. But it is in the development of language that the progress of instinctive cla.s.sification may best be seen. The use of general names implies the recognition of cla.s.ses of things corresponding to them, which form their denotation, and whose resembling qualities, so far as recognised, form their connotation; and such names are of many degrees of generality. The use of abstract names shows that the objects cla.s.sed have also been a.n.a.lysed, and that their resembling qualities have been recognised amidst diverse groups of qualities.
Of the cla.s.ses marked by popular language it is worth while to distinguish two sorts (_cf._ chap. xix. -- 4): Kinds, and those having but few points of agreement.
But the popular cla.s.sifications, made by language and the primitive arts, are very imperfect. They omit innumerable things which have not been found useful or noxious, or have been inconspicuous, or have not happened to occur in the region inhabited by those who speak a particular language; and even things recognised and named may have been very superficially examined, and therefore wrongly cla.s.sed, as when a whale or porpoise is called a fish, or a slowworm is confounded with snakes. A scientific cla.s.sification, on the other hand, aims at the utmost comprehensiveness, ransacking the whole world from the depths of the earth to the remotest star for new objects, and scrutinising everything with the aid of crucible and dissecting knife, microscope and spectroscope, to find the qualities and const.i.tution of everything, in order that it may be cla.s.sed among those things with which it has most in common and distinguished from those other things from which it differs. A scientific cla.s.sification continually grows more comprehensive, more discriminative, more definitely and systematically coherent. Hence the uses of cla.s.sification may be easily perceived.
-- 2. The first use of cla.s.sification is the better understanding of the facts of Nature (or of any sphere of practice); for understanding consists in perceiving and comprehending the likeness and difference of things, in a.s.similating and distinguishing them; and, in carrying out this process systematically, new correlations of properties are continually disclosed. Thus cla.s.sification is closely a.n.a.logous to explanation. Explanation has been shown (chap. xix. -- 5) to consist in the discovery of the laws or causes of changes in Nature; and laws and causes imply similarity, or like changes under like conditions: in the same way cla.s.sification consists in the discovery of resemblances in the things that undergo change. We may say (subject to subsequent qualifications) that Explanation deals with Nature in its dynamic, Cla.s.sification in its static aspect. In both cases we have a feeling of relief. When the cause of any event is pointed out, or an object is a.s.signed its place in a system of cla.s.ses, the gaping wonder, or confusion, or perplexity, occasioned by an unintelligible thing, or by a mult.i.tude of such things, is dissipated. Some people are more than others susceptible of this pleasure and fastidious about its purity.
A second use of cla.s.sification is to aid the memory. It strengthens memory, because one of the conditions of our recollecting things is, that they resemble what we last thought of; so that to be accustomed to study and think of things in cla.s.ses must greatly facilitate recollection. But, besides this, a cla.s.sification enables us easily to run over all the contrasted and related things that we want to think of.
Explanation and cla.s.sification both tend to rationalise the memory, and to organise the mind in correspondence with Nature.
Every one knows how a poor mind is always repeating itself, going by rote through the same train of words, ideas, actions; and that such a mind is neither interesting nor practical. It is not practical, because the circ.u.mstances of life are rarely exactly repeated, so that for a present purpose it is rarely enough to remember only one former case; we need several, that by comparing (perhaps automatically) their resemblances and differences with the one before us, we may select a course of action, or a principle, or a parallel, suited to our immediate needs. Greater fertility and flexibility of thought seem naturally to result from the practice of explanation and cla.s.sification.
But it must be honestly added, that the result depends upon the spirit in which such study is carried on; for if we are too fond of finality, too eager to believe that we have already attained a greater precision and comprehension than are in fact attainable, nothing can be more petrific than "science," and our last state may be worse than the first.
Of this, students of Logic have often furnished examples.
-- 3. Cla.s.sification may be either Deductive or Inductive; that is to say, in the formation of cla.s.ses, as in the proof of propositions, we may, on the whole, proceed from the more to the less, or from the less to the more general; not that these two processes are entirely independent.
If we begin with some large cla.s.s, such as "Animal," and subdivide it deductively into Vertebrate and Invertebrate, yet the principle of division (namely, central structure) has first been reached by a comparison of examples and by generalisation; if, on the other hand, beginning with individuals, we group them inductively into cla.s.ses, and these again into wider ones (as dogs, rats, horses, whales and monkeys into mammalia) we are guided both in special cases by hypotheses as to the best grounds of resemblance, and throughout by the general principle of cla.s.sification--to a.s.sociate things that are alike and to separate things that are unlike. This principle holds implicitly a place in cla.s.sification similar to that of causation in explanation; both are principles of intelligence. Here, then, as in proof, induction is implied in deduction, and deduction in induction. Still, the two modes of procedure may be usefully distinguished: in deduction, we proceed from the idea of a whole to its parts, from general to special; in induction, from special (or particular) to general, from parts to the idea of a whole.
-- 4. The process of Deductive Cla.s.sification, or Formal Division, may be represented thus:
A-------------------A B A b---------- ----------A B C A B c A b C A b c
Given any cla.s.s (A) to be divided:
1. Select one important character, attribute, or quality (B), not common to all the individuals comprehended in the cla.s.s, as the basis of division (_fundamentum divisionis_).
2. Proceed by Dichotomy; that is, cut the given cla.s.s into two, one having the selected attribute (say, B), the other not having it (b).
This, like all formal processes, a.s.sumes the principles of Contradiction and Excluded Middle, that "No A is both B and not-B," and that "Every A is either B or not-B" (chap. vi. -- 3); and if these principles are not true, or not applicable, the method fails.
When a cla.s.s is thus subdivided, it may be called, in relation to its subcla.s.ses, a Genus; and in relation to it, the subcla.s.ses may be called Species: thus--genus A, species AB and Ab, etc.
3. Proceed gradually in the order of the importance of characters; that is, having divided the given cla.s.s, subdivide on the same principle the two cla.s.ses thence arising; and so again and again, step by step, until all the characters are exhausted: _Divisio ne fiat per saltum_.
Suppose we were to attempt an exhaustive cla.s.sification of things by this method, we must begin with "All Things," and divide them (say) into phenomenal and not-phenomenal, and then subdivide phenomena, and so on, thus:
All Things------------------Phenomenal Not-phenomenal--------------------Extended Unextended(e.g., Pleasure and Pain)----------------Resistant Not-resistant (Matter) (s.p.a.ce)-------------------Gravitating Not-gravitating-----------Simple Compound
Having subdivided "Simple" by all possible characters, we must then go back and similarly subdivide Not-phenomenal, Unextended, Not-resistant, Not-gravitating, and Compound. Now, if we knew all possible characters, and the order of their importance, we might prepare _a priori_ a cla.s.sification of all possible things; at least, of all things that come under the principles of Contradiction and Excluded Middle. Many of our compartments might contain nothing actual; there may, for example, be nothing that is not phenomenal to some mind, or nothing that is extended and not-resistant (no vacuum), and so forth. This would imply a breach of the rule, that the dividing quality be not common to the whole cla.s.s; but, in fact, doubts have been, and are, seriously entertained whether these compartments are filled or not. If they are not, we have concepts representing nothing, which have been generated by the mere force of grammatical negation, or by the habit of thinking according to the principle of Excluded Middle; and, on the strength of these empty concepts, we have been misled into dividing by an attribute, which (being universal) cannot be a _fundamentum divisionis_. But though in such a cla.s.sification places might be empty, there would be a place for everything; for whatever did not come into some positive cla.s.s (such as Gravitating) must fall under one of the negative cla.s.ses (the "Nots") that run down the right-hand side of the Table and of its subdivisions.
This is the ideal of cla.s.sification. Unfortunately we have to learn what characters or attributes are possible, by experience and comparison; we are far from knowing them all: and we do not know the order of their importance; nor are we even clear what "important" means in this context, whether "widely prevalent," or "ancient," or "causally influential," or "indicative of others." Hence, in cla.s.sifying actual things, we must follow the inductive method of beginning with particulars, and sorting them according to their likeness and difference as discovered by investigation. The exceptional cases, in which deduction is really useful, occur where certain limits to the number and combination of qualities happen to be known, as they may be in human inst.i.tutions, or where there are mathematical conditions. Thus, we might be able to cla.s.sify orders of Architecture, or the cla.s.sical metres and stanzas of English poetry; though, in fact, these things are too free, subtle and complex for deductive treatment: for do not the Arts grow like trees? The only sure cases are mathematical; as we may show that there are possible only three kinds of plane triangles, four conic sections, five regular solids.
-- 5. The rules for _testing_ a Division are as follows:
1. Each Sub-cla.s.s, or Species, should comprise less than the Cla.s.s, or Genus, to be divided. This provides that the division shall be a real one, and not based upon an attribute common to the whole cla.s.s; that, therefore, the first rule for making a division shall have been adhered to. But, as in -- 4, we are here met by a logical difficulty. Suppose that the cla.s.s to be divided is A, and that we attempt to divide upon the attribute B, into AB and Ab; is this a true division, if we do not know any A that is not B? As far as our knowledge extends, we have not divided A at all. On the other hand, our knowledge of concrete things is never exhaustive; so that, although we know of no A that is not B, it may yet exist, and we have seen that it is a logical caution not to a.s.sume what we do not know. In a deductive cla.s.sification, at least, it seems better to regard every attribute as a possible ground of division.
Hence, in the above division of "All Things,"--"Not-phenomenal,"
"Extended-Not-resistant," "Resistant-Not-gravitating," appear as negative cla.s.ses (that is, cla.s.ses based on the negation of an attribute), although their real existence may be doubtful. But, if this be justifiable, we must either rewrite the first test of a division thus: "Each sub-cla.s.s should _possibly_ comprise less than the cla.s.s to be divided"; or else we must confine the test to (a) thoroughly empirical divisions, as in dividing Colour into Red and Not-red, where we know that both sub-cla.s.ses are real; and (b) divisions under demonstrable conditions--as in dividing the three kinds of triangles by the quality equilateral, we know that it is only applicable to acute-angled triangles, and do not attempt to divide the right-angled or obtuse-angled by it.
2. The Sub-cla.s.ses taken together should be equal to the Cla.s.s to be divided: the sum of the Species const.i.tutes the Genus. This provides that the division shall be exhaustive; which dichotomy always secures, according to the principle of Excluded Middle; because whatever is not in the positive cla.s.s, must be in the negative: Red and Not-red include all colours.
3. The Sub-cla.s.ses must be opposed or mutually exclusive: Species must not overlap. This again is secured by dichotomy, according to the principle of Contradiction, provided the division be made upon one attribute at a time. But, if we attempt to divide simultaneously upon two attributes, as "Musicians" upon "nationality" and "method," we get what is called a Cross-division, thus "German Musicians." "Not-German,"
"Cla.s.sical," "Not-Cla.s.sical;" for these cla.s.ses may overlap, the same men sometimes appearing in two groups--Bach in "German" and "Cla.s.sical,"
Pergolesi in "Not-German" and "Cla.s.sical." If, however, we divide Musicians upon these attributes successively, cross division will be avoided, thus:
Musicians------------------------Cla.s.sical Not-cla.s.sical------------ ------------German Not-German German Not-German
Here no Musician will be found in two cla.s.ses, _unless_ he has written works in two styles, _or unless_ there are works whose style is undecided. This "unless--or unless" may suggest caution in using dichotomy as a short cut to the cla.s.sification of realities.
4. No Sub-cla.s.s must include anything that is not comprised in the cla.s.s to be divided: the Genus comprises all the Species. We must not divide Dogs into fox-terriers and dog-fish.
-- 6. The process of Inductive Cla.s.sification may be represented thus:
Given any mult.i.tude of individuals to be cla.s.sified:
(1) Place together in groups (or in thought) those things that have in common the most, the most widely diffused and the most important qualities.
(2) Connect those groups which have, as groups, the greater resemblance, and separate those that have the greater difference.
(3) Demarcate, as forming higher or more general cla.s.ses, those groups of groups that have important characters in common; and, if possible, on the same principle, form these higher cla.s.ses into cla.s.ses higher still: that is to say, graduate the cla.s.sification upwards.