Logic: Deductive and Inductive Part 31

We saw also that cla.s.sification is not merely the determination of isolated groups of things, but a systematic arrangement of such groups in relation to one another. Hence, again, Definitions are not independent, but relative to one another; and, of course, in the same way as cla.s.ses are relative. That is to say, as a cla.s.s is placed in subordination to higher or more comprehensive groups, so the definition of its name is subordinate to that of their names; and as a cla.s.s stands in contrast with co-ordinate cla.s.ses (those that are in the same degree of subordination to the same higher groups), so the definition of its name is in contrast or co-ordination with the definitions of their names. Lion is subordinate to _Felis_, to Digitigrade, to Carnivore and so on up to Animal; and, beyond the Animal Kingdom, to Phenomenon; it is co-ordinate with tiger, puma, etc.; and more remotely it is co-ordinate with dog, jackal, wolf, which come under _Canis_--a genus co-ordinate with _Felis_. The definition of lion, therefore, is subordinate to that of _Felis_, and to all above it up to Phenomenon; and is co-ordinate with that of tiger, and with all species in the same grade. This is the ground of the old method of definition _per genus et differentiam_.

The genus being the next cla.s.s above any species, the _differentia_ or Difference consists of the qualities which mark that species in addition to those that mark the genus, and which therefore distinguish it from all other species of the same genus. In the above definition of lion, for example, all the properties down to "light and muscular in build"

are generic, that is, are possessed by the whole genus, _Felis_; and the remaining four (size, colour, tufted tail, and mane in the male) are the Difference or specific properties, because in those points the lion contrasts with the other species of that genus. Differences may be exhibited thus:

_Lion._ _Tiger._ SIZE: about 9-1/2 feet from nose | About 10 feet.

to tip of tail. | COLOUR: tawny. | Warm tawny, striped with black.

TAIL: tufted. | Tapering.

MANE: present in the male. | Both s.e.xes maneless.

There are other differences in the shape of the skull. In defining lion, then, it would have been enough to mention the genus and the properties making up the Difference; because the properties of the genus may be found by turning to the definition of the genus; and, on the principle of economy, whatever it is enough to do it is right to do. To define 'by genus and difference' is a point of elegance, when the genus is known; but the only way of knowing it is to compare the individuals comprised in it and in co-ordinate genera, according to the methods of scientific cla.s.sification. It may be added that, as the genus represents ancestral derivation, the predication of genus in a definition indicates the remote causes of the phenomena denoted by the name defined. And this way of defining corresponds with the method of double naming by genus and species: _Felis leo_, _Felis tigris_, etc.; _Vanessa Atalanta_, _Vanessa Io_, etc.

The so-called Genetic Definition, chiefly used in Mathematics, is a rule for constructing that which a name denotes, in such a way as to ensure its possessing the tributes connoted by the name. Thus, for a circle: Take any point and, at any constant distance from it, trace a line returning into itself. In Chemistry a genetic definition of any compound might be given in the form of directions for the requisite synthesis of elements.

-- 6. The chief difficulty in the definition of scientific names consists in determining exactly the nature of the things denoted by them, as in cla.s.sifying plants and animals. If organic species are free growths, continually changing, however gradually, according as circ.u.mstances give some advantage to one form over others, we may expect to find such species branching into varieties, which differ considerably from one another in some respects, though not enough to const.i.tute distinct species. This is the case; and, consequently, there arises some uncertainty in collecting from all the varieties those attributes which are common to the species as a whole; and, therefore, of course, uncertainty in defining the species. The same difficulty may occur in defining a genus, on account of the extent to which some of its species differ from others, whilst having enough of the common character to deter the cla.s.sifier from forming a distinct genus on their account. On the other hand the occurrence of numerous intermediate varieties may make it difficult to distinguish genera or species at all. Even the Kingdoms of plants and animals are hard to discriminate at the lowest levels of organisation. Now, where there is a difficulty of cla.s.sification there must be a corresponding difficulty of definition.

It has been proposed in such cases to subst.i.tute a Type for a Definition; to select some variety of a species, or species of a genus, as exhibiting its character in an eminent degree, and to regard other groups as belonging to the same species or genus, according as they agree more with this type than with other types representing other species or genera. But the selection of one group as typical implies a recognition of its attributes as prevailing generally (though not universally) throughout the species or genus; and to recognise these attributes and yet refuse to enumerate them in a definition, seems to be no great gain. To enumerate the attributes of the type as an Approximate Definition of the species or genus, true of _most_ of the groups const.i.tuting the species or genus, answers the same purpose, is more explicit, and can mislead no one who really attends to the exposition.

An approximate definition is, indeed, less misleading than the indication of a type; for the latter method seems to imply that the group which is now typical has a greater permanence or reality than its co-ordinate groups; whereas, for aught we know, one of the outside varieties or species may even now be superseding and extinguishing it.

But the statement of a definition as approximate, is an honest confession that both the definition and the cla.s.sification are (like a provisional hypothesis) merely the best account we can give of the matter according to our present knowledge.

-- 7. The limits of Definition are twofold: (a) A name whose meaning cannot be a.n.a.lysed cannot be defined. This limitation meets us only in dealing with the names of the metaphysical parts or simple qualities of objects under the second requisite of a Terminology. Resistance and weight, colour and its modes, many names of sounds, tastes, smells, heat and cold--in fact, whatever stands for an una.n.a.lysable perception, cannot be made intelligible to any one who has not had experience of the facts denoted; they cannot be defined, but only exemplified. A sort of genetic definition may perhaps be attempted, as if we say that colour is the special sensation of the cones of the retina, or that blue is the sensation produced by a ray of light vibrating about 650,000,000,000,000 times a second; but such expressions can give no notion of our meaning to a blind man, or to any one who has never seen a blue object. Nor can we explain what heat is like, or the smell of tobacco, to those who have never experienced them; nor the sound of C 128 to one who knows nothing of the musical scale.

If we distinguish the property of an object from the sensation it excites in us, we may define any simple property as 'the power of producing the sensation'; the colour of a flower as the power of exciting the sensation of colour in us. Still, this gives no information to the blind nor to the colour-blind. Abstract names may be defined by defining the corresponding concrete: the definition of 'human nature' is the same as of 'man.' But if the corresponding concrete be a simple sensation (as blue), this being indefinable, the abstract (blueness) is also indefinable.

(b) The second limit of Definition is the impossibility of exhausting infinity, which would be necessary in order to convey the meaning of the name of any individual thing or person. For, as we saw in chap. iv., if in attempting to define a proper name we stop short of infinity, our list of qualities or properties may possibly be found in two individuals, and then it becomes the definition of a cla.s.s-name or general name, however small the actual cla.s.s. Hence we can only give a Description of that which a proper name denotes, enumerating enough of its properties to distinguish it from everything else as far as our knowledge goes.

-- 8. The five Predicables (Species, Genus, Difference, Proprium, Accident) may best be discussed in connection with Cla.s.sification and Definition; and in giving an account of Cla.s.sification, most of what has to be said about them has been antic.i.p.ated. Their name, indeed, connects them with the doctrine of Propositions; for Predicables are terms that may be predicated, cla.s.sified according to their connotative relation to the subject of a proposition (that is, according to the relation in which their connotation stands to the connotation of the subject): nevertheless, the significance of the relations of such predicates to a subject is derivative from the general doctrine of cla.s.sification.

For example, in the proposition 'X is Y,' Y must be one of the five sorts of predicables in relation to X; but of what sort, depends upon what X (the subject) is, or means. The subject of the proposition must be either a definition, or a general connotative name, or a singular name.

If X be a definition, Y must be a species; for nothing but a general name can be predicated of a definition: and, strictly speaking, it is only in relation to a definition (as subject) that species can be a predicable; when it is called _Species predicabilis_ (1).

If X be a connotative name, it is itself a species (_Species subjicibilis_); and the place of the subject of a proposition is the usual one for species. The predicate, Y, may then be related to the species in three different ways. First, it may be a definition, exactly equivalent to the species;--in fact, nothing else than the species in an explicit form, the a.n.a.lysis of its connotation. Secondly, the predicate may be, or connote, some _part only_ of the definition or connotation of the species; and then it is either genus (2), or difference (3).

Thirdly, the predicate may connote _no part_ of the definition, and then it is either derivable from it, being a proprium (4), or not derivable from it, being an accident (5). These points of doctrine will be expanded and ill.u.s.trated in subsequent pages.

If X be a singular name, deriving connotation from its const.i.tuent terms (chap. iv. -- 2), as 'The present Emperor of China,' it may be treated as a _Species subjicibilis_. Then that he is 'an absolute monarch,'

predicates a genus; because that is a genus of 'Emperor,' a part of the singular name that gives it connotation. That he wears a yellow robe is a proprium, derivable from the ceremonial of his court. That he is thirty years of age is an accident.

But if X be a proper name, having no connotation, Y must always be an accident; since there can then be no definition of X, and therefore neither species, genus, difference, nor proprium. Hence, that 'John Doe is a man' is an accidental proposition: 'man' is not here a _Species predicabilis_; for the name might have been given to a dog or a mountain. That is what enables the proposition to convey information: it would be useless if the proper name implied 'humanity.'

'Species' is most frequently used (as in Zoology) for the _cla.s.s denoted_ by a general name; but in Logic it is better to treat it as a general name used connotatively for the attributes possessed in common by the things denoted, and on account of which they are regarded as a cla.s.s: it is sometimes called the Essence (-- 9). In this connotative sense, a species is implicitly what the definition is explicitly; and therefore the two are always simply convertible. Thus, 'A plane triangle' (species) is 'a figure enclosed by three straight lines'

(definition): clearly we may equally say, 'A figure enclosed by three straight lines is a plane triangle.' It is a simple ident.i.ty.

A genus is also commonly viewed denotatively, as a cla.s.s containing smaller cla.s.ses, its species; but in Logic it is, again, better to treat it connotatively, as a name whose definition is part of the definition of a given species.

A difference is the remainder of the definition of any species after subtracting a given genus. Hence, the genus and difference together make up the species; whence the method of definition _per genus et differentiam_ (_ante_, -- 5).

Whilst in Botany and Zoology the species is fixed at the lowest step of the cla.s.sification (varieties not being reckoned as cla.s.ses), and the genus is also fixed on the step next above it, in Logic these predicables are treated as movable up and down the ladder: any lower cla.s.s being species in relation to any higher; which higher cla.s.s, wherever taken, thus becomes a genus. Lion may logically be regarded as a species of digitigrade, or mammal, or animal; and then each of these is a genus as to lion: or, again, digitigrade may be regarded as a species of mammal, or mammal as a species of animal. The highest cla.s.s, however, is never a species; wherefore it is called a _Summum Genus_: and the lowest cla.s.s is never a genus; wherefore it is called an _Infima Species_. Between these two any step may be either species or genus, according to the relation in which it is viewed to other cla.s.ses, and is then called Subaltern. The _summum genus_, again, may be viewed in relation to a _given_ universe or _suppositio_ (that is, any limited area of existence now the object of attention), or to the _whole_ universe. If we take the animal kingdom as our _suppositio_, Animal is the _summum genus_; but if we take the whole universe, 'All things' is the _summum genus_.

"Porphyry's tree" is used to ill.u.s.trate this doctrine. It begins with a _summum genus_, 'Substance,' and descends by adding differences, step by step, to the _infima species_, 'Man.' It also ill.u.s.trates Division by Dichotomy.

SUBSTANCE / CORPOREAL || INCORPOREAL || BODY / ANIMATE || INANIMATE || LIVING BODY / SENSIBLE || INSENSIBLE || ANIMAL / RATIONAL || IRRATIONAL || MAN //|| // || // || // || // || // || // || _Socrates_ _Plato_ _Aristotle_

Beginning with 'Substance,' as _summum genus_, and adding the difference 'Corporeal,' we frame the species 'Body.' Taking 'Body' as the genus and adding the difference 'Animate,' we frame the species 'Living Body;' and so on till 'Man' is reached; which, being _infima species_, is only subdivisible into individuals. But the division of Man into individuals involves a change of principle; it is a division of the denotation, not an increase of the connotation as in the earlier steps.

Only one side of each dichotomy is followed out in the 'tree': if the other side had been taken, Incorporeal Substance would be 'Spirit'; which might be similarly subdivided.

Genus and species, then, have a double relation. In denotation the genus includes the species; in connotation the species includes the genus.

Hence the doctrine that by increasing the connotation of a name we decrease its denotation: if, for example, to the definition of 'lion' we add 'inhabiting Africa,' Asiatic lions are no longer denoted by it. On the other hand, if we use a name to denote objects that it did not formerly apply to, some of the connotation must be dropped: if, for example, the name 'lion' be used to include 'pumas,' the tufted tail and mane can no longer be part of the meaning of the word; since pumas have not these properties.

This doctrine is logically or formally true, but it may not always be true in fact. It is logically true; because wherever we add to the connotation of a name, it is possible that some things to which it formerly applied are now excluded from its denotation, though we may not know of any such things. Still, as a matter of fact, an object may be discovered to have a property previously unknown, and this property may be fundamental and co-extensive with the denotation of its name, or even more widely prevalent. The discovery that the whale is a mammal did not limit the cla.s.s 'whale'; nor did the discovery that lions, dogs, wolves, etc., walk upon their toes, affect the application of any of these names.

Similarly, the extension of a name to things not previously denoted by it, may not in fact alter its definition; for the extension may be made on the very ground that the things now first denoted by it have been found to have the properties enumerated in its definition, as when the name 'mammal' was applied to whales, dolphins, etc. If, however, 'mammal' had formerly been understood to apply only to land animals, so that its definition included (at least, popularly) the quality of 'living on the land,' this part of the connotation was of course lost when the denotation came to include certain aquatic animals.

A proprium is an attribute derived from the definition: being either (a) implied in it, or deducible from it, as 'having its three angles equal to two right angles' may be proved from the definition of a triangle; or (b) causally dependent on it, as being 'dangerous to flocks' results from the nature of a wolf, and as 'moving in an ellipse' results from the nature of a planet in its relation to the sun.

An accident is a property accompanying the defining attributes without being deducible from them. The word suggests that such a property is merely 'accidental,' or there 'by chance'; but it only means that we do not understand the connection.

Proprium and Accident bear the same relation to one another as Derivative and Empirical Laws: the predication of a proprium is a derivative law, and the predication of an accident is an empirical law.

Both accidents and empirical laws present problems, the solution of which consists in reducing them, respectively, to propria and derivative laws. Thus the colour of animals was once regarded as an accident for which no reason could be given; but now the colour of animals is regarded as an effect of their nature and habits, the chief determinants of it being the advantage of concealment; whilst in other cases, as among brightly coloured insects and snakes, the determinant may be the advantage of advertising their own noxiousness. If such reasoning is sound, colour is a proprium (and if so, it cannot _logically_ be included in a definition; but it is better to be judicious than formal).

If the colour of animals is a proprium, we must recognise a distinction between Inseparable and Separable Propria, according as they do, or do not, always accompany the essence: for mankind is regarded as one species; but each colour, white, black or yellow, is separable from it under different climatic conditions; whilst tigers are everywhere coloured and striped in much the same way; so that we may consider their colouring as inseparable, in spite of exceptional specimens black or white or clouded.

The same distinction may be drawn between accidents. 'Inhabiting Asia'

is an Inseparable Accident of tiger, but a Separable Accident of lion.

Even the occasional characteristics and occupations of individuals are sometimes called separable accidents of the species; as, of man, being colour-blind, carpentering, or running.

A proprium in the original signification of the term ?d??? was peculiar to a species, never found with any other, and was therefore convertible with the subject; but this restriction is no longer insisted on.

-- 9. Any predication of a genus, difference or definition, is a verbal, a.n.a.lytic, or essential proposition: and any predication of a proprium or accident, is a real, synthetic, or accidental proposition (chap. v. -- 6). A proposition is called verbal or a.n.a.lytic when the predicate is a part, or the whole, of the meaning of the subject; and the subject being species, a genus or difference is part, and a definition is the whole, of its meaning or connotation. Hence such a proposition has also been called explicative. Again, a proposition is called real or synthetic when the predicate is no part of the meaning of the subject; and, the subject being species, a proprium or accident is no part of its meaning or connotation. Hence such a proposition has been called ampliative.

As to Essential and Accidental, these terms are derived from the doctrine of Realism. Realists maintain that the essence of a thing, or that which makes a thing to be what (or of what kind) it is, also makes everything else of the same kind to be what it is. The essence, they say, is not proper to each thing or separately inherent in it, but is an 'Universal' common to all things of that kind. Some hold that the universal nature of things of any kind is an Idea existing (apart from the things) in the intelligible world, invisible to mortal eye and only accessible to thought; whence the Idea is called a noumenon: that only the Idea is truly real, and that the things (say, trees, bedsteads and cities) which appear to us in sense-perception, and which therefore are called phenomena, only exist by partic.i.p.ating in, or imitating, the Idea of each kind of them. The standard of this school bears the legend _Universalia ante rem_.

But others think that the Universal does not exist apart from particular things, but is their present essence; gives them actuality as individual substances; "informs" them, or is their formal cause, and thus makes them to be what they are of their kind according to the definition: the universal lion is in all lions, and is not merely similar, but identical in all; for thus the Universal Reason thinks and energises in Nature.

This school inscribes upon its banners, _Universalia in re_.

To define anything, then, is to discover its essence, whether transcendent or immanent; and to predicate the definition, or any part of it (genus or difference), is to enounce an essential proposition. But a proprium, being no part of a definition, though it always goes along with it, does not show what a thing is; nor of course does an accident; so that to predicate either of these is to enounce an accidental proposition.

Another school of Metaphysicians denies the existence of Universal Ideas or Forms; the real things, according to them, are individuals; which, so far as any of them resemble one another, are regarded as forming cla.s.ses; and the only Universal is the cla.s.s-name, which is applied universally in the same sense. Hence, they are called Nominalists. The sense in which any name is applied, they say, is derived from a comparison of the individuals, and by abstraction of the properties they have in common; and thus the definition is formed. _Universalia post rem_ is their motto. Some Nominalists, however, hold that, though Universals do not exist in nature, they do in our minds, as Abstract Ideas or Concepts; and that to define a term is to a.n.a.lyse the concept it stands for; whence, these philosophers are called Conceptualists.

Such questions belong to Metaphysics rather than to Logic; and the foregoing is a commonplace account of a subject upon every point of which there is much difference of opinion.

-- 10. The doctrine of the Predicaments, or Categories, is so interwoven with the history of speculation and especially of Logic that, though its vitality is exhausted, it can hardly be pa.s.sed over unmentioned. The predicaments of Aristotle are the heads of a cla.s.sification of terms as possible predicates of a particular thing or individual. Hamilton (_Logic_: Lect. xi.) has given a cla.s.sification of them; which, if it cannot be found in Aristotle, is an aid to the memory, and may be thrown into a table thus:

Substance ??s?a (1) {Quant.i.ty p?s?? (2) Attribute {Quality p???? (3) {Relation p??s t? (4)

{ Where p?? (5) { When p?te (6)

{ Action p??e?? (7) Modes of Relation { Pa.s.sion p?s?e?? (8)

{ Posture ?e?s?a? (9) { Habit ??e?? (10)