Deductive Logic
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St. George Stock >> Deductive Logic
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328. It would be less irrelevant to point out how the classification
breaks down in relation to the singular term as subject. When, for
instance, we say 'Socrates is an animal,' 'Socrates is a man,' there
is nothing in the proposition to show us whether the predicate is a
genus or a species: for we have not here the relation of class to
class, which gives us genus or species according to their relative
extension, but the relation of a class to an individual.
329. Again, when we say
(1) Some animals are men,
(2) Some men are black,
what is there to tell us that the predicate is to be regarded in the
one case as a species and in the other as an accident of the subject?
Nothing plainly but the assumption of a definition already known.
330. But if this assumption be granted, the classification seems to
admit of a more or less complete defense by logic.
For, given any subject, we can predicate of it either a class or an
attribute.
When the predicate is a class, the term predicated is called a Genus,
if the subject itself be a class, or a Species, if it be an
individual.
When, on the other hand, the predicate is an attribute, the attribute
predicated may be either the very attribute which distinguishes the
subject from other members of the same class, in which case it is
called the Difference, or it may be some attribute connected with the
definition, i.e. Property, or not connected with it, i.e. Accident.
331. These results may be exhibited in the following scheme--
Predicate
________________|_________________
| |
Class Attribute
______|_______ __________|________
| | | |
(Subject a (Subject a (The (Not the
common singular distinguishing distinguishing
term) term) Attribute) attribute)
Genus Species Difference
|___________________
| |
(Connected (Not connected
with the with the
definition) definition)
Property Accident
332. The distinction which underlies this division between
predicating a class and predicating an attribute (in quid or in quale)
is a perfectly intelligible one, corresponding as it does to the
grammatical distinction between the predicate being a noun substantive
or a noun adjective. Nevertheless it is a somewhat arbitrary one,
since, even when the predicate is a class-name, what we are concerned
to convey to the mind, is the fact that the subject possesses the
attributes which are connoted by that class-name. We have not here the
difference between extensive and intensive predication, since, as we
have already seen ( 264), that is not a difference between one
proposition and another, but a distinction in our mode of interpreting
any and every proposition. Whatever proposition we like to take may be
read either in extension or in intension, according as we fix our
minds on the fact of inclusion in a class or the fact of the
possession of attributes.
333. It will be seen that the term 'species,' as it appears in the
scheme, has a wholly different meaning from the current acceptation in
which it was defined above. Species, in its now accepted meaning,
signifies the relation of a smaller class to a larger one: as it was
originally intended in the heads of predicables it signifies a class
in reference to individuals.
334. Another point which requires to be noticed with regard to this
five-fold list of heads of predicables, if its object be to classify
the relations of a predicate to a subject, is that it takes no account
of those forms of predication in which class and attribute are
combined. Under which of the five heads would the predicates in the
following propositions fall?
(1) Man is a rational animal.
(2) Man is a featherless biped.
In the one case we have a combination of genus and difference; in the
other we have a genus combined with an accident.
335. The list of heads of predicables which we have been discussing
is not derived from Aristotle, but from the 'Introduction' of
Porphyry, a Greek commentator who lived more than six centuries later.
_Aristotle's Heads of Predicables_.
336. Aristotle himself, by adopting a different basis of division,
has allowed room in his classification for the mixed forms of
predication above alluded to. His list contains only four heads,
namely,
Genus ([Greek: genos])
Definition ([Greek: orismos])
Proprium ([Greek: idion])
Accident ([Greek: sumbebekos])
337. Genus here is not distinguished from difference. Whether we
say 'Man is an animal' or 'Man is rational,' we are equally understood
to be predicating a genus.
338. There is no account taken of species, which, when predicated,
resolves itself either into genus or accident. When predicated of an
individual, it is regarded as a genus, e.g. 'Socrates is a man'; when
predicated of a class, it is regarded as an accident, e.g. 'Some
animals are men.'
339. Aristotle's classification may easily be seen to be
exhaustive. For every predicate must either be coextensive with its
subject or not, i.e. predicable of the same things. And if the two
terms coincide in extension, the predicate must either coincide also
in intension with the subject or not.
A predicate which coincides both in extension and intension with its
subject is exactly what is meant by a definition. One which coincides
in extension without coinciding in intension, that is, which applies
to the same things without expressing the whole meaning, of the
subject, is what is known as a Proprium or Peculiar Property.
If, on the other hand, the two terms are not co-extensive, the
predicate must either partially coincide in intension with the subject
or not. [Footnote: The case could not arise of a predicate which was
entirely coincided in intension with a subject with which it was not
co-extensive. For, if the extension of the predicate were greater than
that of the subject, its intension would be less, and if less,
greater, in accordance with the law of inverse variation of the two
quantities ( 166).] This is equivalent to saying that it must either
state part of the definition of the subject or not. Now the definition
is made up of genus and difference, either of which may form the
predicate: but as the two are indistinguishable in relation to a
single subject, they are lumped together for the present purpose under
the one head, genus. When the predicate, not being co-extensive, is
not even partially co-intensive with its subject, it is called an
Accident.
340. Proprium, it will be seen, differs from property. A proprium
is an attribute which is possessed by all the members of a class, and
by them alone, e.g. 'Men are the only religious animals.'
341. Under the head of definition must be included all propositions
in which the predicate is a mere synonym of the subject, e.g. 'Naso is
Ovid,' 'A Hebrew is a Jew,' 'The skipper is the captain.' In such
propositions the predicate coincides in extension with the subject,
and may be considered to coincide in intension where the intension of
both subject and predicate is at zero, as in the case of two proper
names.
342. Designations and descriptions will fall under the head of
'proprium' or peculiar property, e.g. 'Lord Salisbury is the present
prime minister of England,' 'Man is a mammal with hands and without
hair.' For here, while the terms are coincident in extension, they are
far from being so in intension.
343. The term 'genus' must be understood to include not only genus
in the accepted sense, but difference and generic property as well.
344. These results may be exhibited in the following
scheme--
Predicate
___________________|______________
| |
Coextensive with not
the subject coextensive
________|_________ _____|________
| | | |
Co-intensive not partially not at all
with the subject cointensive cointensive [Greek: sumbubekos]
[Greek: orismos] [Greek: idion] [Greek: genos] Accident
______|_____ ______|_____________ |________________
| | | | | | | |
Defini- Synonym Designa- Descrip- Peculiar Genus Differ- Generic
tion tion tion Property ence Property
345. Thus Aristotle's four heads of predicables may be split up, if
we please, into nine--
1. Definition \
> [Greek: orismos].
2. Synonym /
3. Designation \
|
4. Description > [Greek: idion].
|
5. Peculiar Property/
6. Genus \
|
7. Difference > [Greek: genos].
|
8. Generic Property/
9. Accident--[Greek: sumbebekos].
346. We now pass on to the two subjects of Definition and Division,
the discussion of which will complete our treatment of the second part
of logic. Definition and division correspond respectively to the two
kinds of quantity possessed by terms.
Definition is unfolding the quantity of a term in intension.
Division is unfolding the quantity of a term in extension.
CHAPTER VII.
_Of Definition._
347. To define a term is to unfold its intension, i.e. to explain
its meaning.
348. From this it follows that any term which possesses no intension
cannot be defined.
349. Hence proper names do not admit of definition, except just in
so far as they do possess some slight degree of intension: Thus we can
define the term 'John' only so far as to say that 'John' is the name
of a male person. This is said with regard to the original intension
of proper names; their acquired intension will be considered later.
350. Again, since definition is unfolding the intension of a term,
it follows that those terms will not admit of being defined whose
intension is already so simple that it cannot be unfolded further. Of
this nature are names of simple attributes, such as greenness,
sweetness, pleasure, existence. We know what these things are, but we
cannot define them. To a man who has never enjoyed sight, no language
can convey an idea of the greenness of the grass or the blueness of
the sky; and if a person were unaware of the meaning of the term
'sweetness,' no form of words could convey to him an idea of it. We
might put a lump of sugar into his mouth, but that would not be a
logical definition.
351. Thus we see that, for a thing to admit of definition, the idea
of it must be complex. Simple ideas baffle definition, but at the same
time do not require it. In defining we lay out the simpler ideas
which are combined in our notion of something, and so explain that
complex notion. We have defined 'triangle,' when we analyse it into
'figure' and 'contained by three lines.' Similarly we have defined
'substance' when we analyse it into 'thing' and 'which can be
conceived to exist by itself.'
352. But when we get to 'thing' we have reached a limit. The Summum
Genus, or highest class under which all things fall, cannot be defined
any more than a simple attribute; and for the very good reason that it
connotes nothing but pure being, which is the simplest of all
attributes. To say that a thing is an 'object of thought' is not
really to define it, but to explain its etymology, and to reclaim a
philosophical term from its abuse by popular language, in which it is
limited to the concrete and the lifeless. Again, to define it
negatively and to say that a thing is 'that which is not nothing' does
not carry us any further than we were before. The law of contradiction
warrants us in saying as much as that.
353. Definition is confined to subject-terms, and does not properly
extend to attributives. For definition is of things through names, and
an attributive out of predication is not the name of anything. The
attributive is defined, so far as it can be, through the corresponding
abstract term.
354. Common terms, other than attributives, ought always to admit of
definition. For things are distributed by the mind into classes owing
to their possessing certain attributes in common, and the definition
of the class-name can be effected by detailing these attributes, or at
least a sufficient number of them.
355. It is different with singular terms. Singular terms, when
abstract, admit of definition, in so far as they are not names of
attributes so simple as to evade analysis. When singular terms are
concrete, we have to distinguish between the two cases of proper names
and designations. Designations are connotative singular terms. They
are formed by limiting a common term to the 'case in hand.' Whatever
definition therefore fits the common term will fit also the
designation which is formed from it, if we add the attributes implied
by the limitations. Thus whatever definition fits the common term
'prime minister' will fit also the singular term 'the present prime
minister of England' by the addition to it of the attributes of place
and time which are indicated by the expression. Such terms as this
have a definite amount of intension, which can therefore be seized
upon and expounded by a definition.
356. But proper names, having no original intension of their own,
cannot be defined at all; whereas, if we look upon them from the point
of view of their acquired intension, they defy definition by reason of
the very complexity of their meaning. We cannot say exactly what
'John' and 'Mary' mean, because those names, to us who know the
particular persons denoted by them, suggest all the most trifling
accidents of the individual as well as the essential attributes of the
genus.
357. Definition serves the practical purpose of enabling us mentally
to distinguish, or, as the name implies, 'mark off' the thing defined
from all other things whatsoever. This may seem at first an endless
task, but there is a short cut by which the goal may be reached. For,
if we distinguish the thing in hand from the things which it is most
like, we shall, 'a fortiori,' have distinguished it from things to
which it bears a less resemblance.
358. Hence the first thing to do in seeking for a definition is to
fix upon the class into which the thing to be defined most naturally
falls, and then to distinguish the thing in question from the other
members of that class. If we were asked to define a triangle, we
would not begin by distinguishing it from a hawser, but from a square
and other figures with which it is more possible to confound it. The
class into which a thing falls is called its Genus, and the attribute
or attributes which distinguish it from other members of that class
are called its Difference.
359. If definition thus consists in referring a thing to a class, we
see a further reason why the summum genus of all things cannot be
defined.
360. We have said that definition is useful in enabling us to
distinguish things from one another in our minds: but this must not be
regarded as the direct object of the process. For this object may be
accomplished without giving a definition at all, by means of what is
called a Description. By a description is meant an enumeration of
accidents with or without the mention of some class-name. It is as
applicable to proper names as to common terms. When we say 'John Smith
lives next door on the right-hand side and passes by to his office
every morning at nine o'clock,' we have, in all probability,
effectually distinguished John Smith from other people: but living
next, &c., cannot be part of the intension of John Smith, since John
Smith may change his residence or abandon his occupation without
ceasing to be called by his name. Indirectly then definition serves
the purpose of distinguishing things in the mind, but its direct
object is to unfold the intension of terms, and so impart precision to
our thoughts by setting plainly before us the meaning of the words we
are using.
361. But when we say that definition is unfolding the intension of
terms, it must not be imagined that we are bound in defining to unfold
completely the intension of terms. This would be a tedious, and often
an endless, task. A term may mean, or convey to the mind, a good many
more attributes than those which are stated in its definition. There
is no limit indeed to the meaning which a term may legitimately
convey, except the common attributes of the things denoted by it. Who
shall say, for instance, that a triangle means a figure with three
sides, and does not mean a figure with three angles, or the surface of
the perpendicular bisection of a cone? Or again, that man means a
rational, and does not mean a speaking, a religious, or an aesthetic
animal, or a biped with two eyes, a nose, and a mouth? The only
attributes of which it can safely be asserted that they can form no
part of the intension of a term are those which are not common to all
the things to which the name applies. Thus a particular complexion,
colour, height, creed, nationality cannot form any part of the
intension of the term 'man.' But among the attributes common to a
class we cannot distinguish between essential and unessential, except
by the aid of definition itself. Formal logic cannot recognise any
order of priority between the attributes common to all the members of
a class, such as to necessitate our recognising some as genera and
differentiae and relegating others to the place of properties or
inseparable accidents.
362. The art of giving a good definition is to seize upon the
salient characteristics of the thing defined and those wherefrom the
largest number of other attributes can be deduced as consequences. To
do this well requires a special knowledge of the thing in question,
and is not the province of the formal logician.
363. We have seen already, in treating of the Heads of Predicables
( 325), that the difference between genus and difference on the one
hand and property on the other is wholly relative to some assumed
definition. Now definitions are always to a certain extent arbitrary,
and will vary with the point of view from which we consider the thing
required to be defined. Thus 'man' is usually contrasted with 'brute,'
and from this point of view it is held a sufficient definition of him
to say that he is 'a rational animal,' But a theologian might be more
anxious to contrast man with supposed incorporeal intelligences, and
from this point of view man would be defined as an 'embodied spirit.'
364. In the two definitions just given it will be noticed that we
have really employed exactly the same attributes, only their place as
genus and difference has been reversed. It is man's rational, or
spiritual, nature which distinguishes him from the brutes: but this is
just what he is supposed to have in common with incorporeal
intelligences, from whom he is differentiated by his animal nature.
[Illustration]
This illustration is sufficient to show us that, while there is no
absolute definition of anything, in the sense of a fixed genus and
difference, there may at the same time be certain attributes which
permanently distinguish the members of a given class from those of all
other classes.
365. The above remarks will have made it clear that the intension of
a term is often much too wide to be conveyed by any definition; and
that what a definition generally does is to select certain attributes
from the whole intension, which are regarded as being more typical of
the thing than the remainder. No definition can be expected to exhaust
the whole intension of a term, and there will always be room for
varying definitions of the same thing, according to the different
points of view from which it is approached.
366. Names of attributes lend themselves to definition far more
easily than names of substances. The reason of this is that names of
attributes are primarily intensive in force, whereas substances are
known to us in extension before they become known to us in
intension. There is no difficulty in defining a term like 'triangle'
or 'monarchy,' because these terms were expressly invented to cover
certain attributes; but the case is different with such terms as
'dog,' 'tree,' 'plant,' 'metal,' and other names of concrete
things. We none of us have any difficulty in recognising a dog or
tree, when we see them, or in distinguishing them from other animals
or plants respectively. We are therefore led to imagine that we know
the meaning of these terms. It is not until we are called upon for a
definition that we discover how superficial our knowledge really is of
the common attributes possessed by the things which these names
denote.
367. It might be imagined that a common name would never be given to
things except in virtue of our knowledge of their common
attributes. But as a matter of fact, the common name was first given
from a confused notion of resemblance, and we had afterwards to detect
the common attributes, when sometimes the name had been so extended
from one thing to another like it, that there were hardly any definite
attributes possessed in common by the earlier and later members of the
class.
368. This is especially the case where the meaning of terms has been
extended by analogy, e.g. head, foot, arm, post, pole, pipe, &c.
369. But in the progress of thought we come to form terms in which
the intensive capacity is everything. Of this kind notably are
mathematical conceptions. Terms of this kind, as we said before, lend
themselves readily to definition.
370. We may lay down then roughly that words are easy or difficult
of definition according as their intensive or extensive capacity
predominates.
371. There is a marked distinction to be observed between the
classes made by the mind of man and the classes made by nature, which
are known as 'real kinds.' In the former there is generally little or
nothing in common except the particular attribute which is selected as
the ground of classification, as in the case of red and white things,
which are alike only in their redness or whiteness; or else their
attributes are all necessarily connected, as in the case of circle,
square and triangle. But the members of nature's classes agree in
innumerable attributes which have no discoverable connection with one
another, and which must therefore, provisionally at least, be regarded
as standing in the relation of inseparable accidents to any particular
attributes which we may select for the purposes of definition. There
is no assignable reason why a rational animal should have hair on its
head or a nose on its face, and yet man, as a matter of fact, has
both; and generally the particular bodily configuration of man can
only be regarded as an inseparable accident of his nature as a
rational animal.
372. 'Real kinds' belong to the class of words mentioned above in
which the extension predominates over the intension. We know well
enough the things denoted by them, while most of us have only a dim
idea of the points of resemblance between these things. Nature's
classes moreover shade off into one another by such imperceptible
degrees that it is often impossible to fix the boundary line between
one class and another. A still greater source of perplexity in dealing
with real kinds is that it is sometimes almost impossible to fix upon
any attribute which is common to every individual member of the class
without exception. All that we can do in such cases is to lay down a
type of the class in its perfect form, and judge of individual
instances by the degree of their approximation to it. Again, real
kinds being known to us primarily in extension, the intension which we
attach to the names is hable to be affected by the advance of
knowledge. In dealing therefore with such terms we must be content
with provisional definitions, which adequately express our knowledge
of the things denoted by them, at the time, though a further study of
their attributes may induce us subsequently to alter the
definition. Thus the old definition of animal as a sentient organism
has been rendered inadequate by the discovery that so many of the
phenomena of sensation can be exhibited by plants,
373. But terms in which intension is the predominant idea are more
capable of being defined once for all. Aristotle's definitions of
'wealth' and 'monarchy' are as applicable now as in his own day, and
no subsequent discoveries of the properties of figures will render
Euclid's definitions unavailable.
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