Deductive Logic by St. George Stock

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§ 115. The number of things, it is clear, is infinite. For, granting
that the physical universe consists of a definite number of
atoms--neither one more nor one less--still we are far from having
exhausted the possible number of things. All the manifold material
objects, which are made up by the various combinations of these atoms,
constitute separate objects of thought, or things, and the mind has
further an indefinite power of conjoining and dividing these objects,
so as to furnish itself with materials of thought, and also of fixing
its attention by abstraction upon attributes, so as to regard them as
things, apart from the substances to which they belong.

§ 116. This being so, it is only a very small number of things, which
are constantly obtruding themselves upon the mind, that have singular
terms permanently set apart to denote them. Human beings, some
domestic animals, and divisions of time and place, have proper names
assigned to them in most languages, e.g. 'John,' 'Mary,' 'Grip,'
'January,' 'Easter,' 'Belgium,' 'Brussels,' 'the Thames,' 'Ben-Nevis.'
Besides these, all abstract terms, when used without reference to
lower notions, are of the nature of proper names, being permanently
set apart to denote certain special attributes, e.g. 'benevolence,'
'veracity,' 'imagination,' 'indigestibility, 'retrenchment.'

§ 117. But the needs of language often require a singular term to
denote some thing which has not had a proper name assigned to it. This
is effected by taking a common term, and so limiting it as to make it
applicable, under the given circumstances, to one thing only. Such a
limitation may be effected in English by prefixing a demonstrative or
the definite article, or by appending a description, e.g. 'this pen,'
'the sofa,' 'the last rose of summer.' When a proper name is unknown,
or for some reason, unavailable, recourse may be had to a designation,
e.g. 'the honourable member who spoke last but one.'

_Collective Terms_.

§ 118. The division of terms into singular and common being, like
those which have preceded it, fundamental and exhaustive, there is
evidently no room in it for a third class of Collective Terms. Nor is
there any distinct class of terms to which that name can be given. The
same term may be used collectively or distributively in different
relations. Thus the term 'library,' when used of the books which
compose a library, is collective; when used of various collections of
books, as the Bodleian, Queen's library, and so on, it is
distributive, which, in this case, is the same thing as being a common
term.

§ 119, The distinction between the collective and distributive use of
a term is of importance, because the confusion of the two is a
favourite source of fallacy. When it is said 'The plays of Shakspeare
cannot be read in a day,' the proposition meets with a very different
measure of acceptance according as its subject is understood
collectively or distributively. The word 'all' is perfectly ambiguous
in this respect. It may mean all together or each separately--two
senses which are distinguished in Latin by 'totus' or 'cunctus,' for
the collective, and 'omnis' for the distributive use.

§ 120. What is usually meant however when people speak of a collective
term is a particular kind of singular term.

§ 121. From this point of view singular terms may be subdivided into
Individual and Collective, by an Individual Term being meant the name
of one object, by a Collective Term the name of several considered as
one. 'This key' is an individual term; 'my bunch of keys' is a
collective term.

§ 122. A collective term is quite as much the name of one thing as an
individual term is, though the thing in question happens to be a
group. A group is one thing, if we choose to think of it as one. For
the mind, as we have already seen, has an unlimited power of forming
its own things, or objects of thought. Thus a particular peak in a
mountain chain is as much one thing as the chain itself, though,
physically speaking, it is inseparable from it, just as the chain
itself is inseparable from the earth's surface. In the same way a
necklace is as much one thing as the individual beads which compose
it.

§ 123. We have just seen that a collective term is the name of a group
regarded as one thing: but every term which is the name of such a
group is not necessarily a collective term. 'London,' for instance, is
the name of a group of objects considered as one thing. But 'London'
is not a collective term, whereas 'flock,' 'regiment,' and 'senate'
are. Wherein then lies the difference? It lies in this--that flock,
regiment and senate are groups composed of objects which are, to a
certain extent, similar, whereas London is a group made up of the most
dissimilar objects--streets and squares and squalid slums, fine
carriages and dirty faces, and so on. In the case of a true collective
term all the members of the group will come under some one common
name. Thus all the members of the group, flock of sheep, come under
the common name 'sheep,' all the members of the group 'regiment' under
the common name, 'soldier,' and so on.

§ 124. The subdivision of singular terms into individual and
collective need not be confined to the names of concrete things. An
abstract term like 'scarlet,' which is the name of one definite
attribute, may be reckoned 'individual,' while a term like 'human
nature,' which is the name of a whole group of attributes, would more
fitly be regarded as collective.

§ 126. The main division of terms, which we have been discussing, into
singular and collective, is based upon their Quantity in
Extension. This phrase will be explained presently.

§ 126. We come now to a threefold division of terms into Positive,
Privative and Negative. It is based upon an implied two-fold division
into positive and non-positive, the latter member being subdivided
into Privative and Negative.

Term
_______________|_______________
| |
Positive Non-Positive
_____________|____________
| |
Privative Negative

If this division be extended, as it sometimes is, to terms in general,
a positive term must be taken to mean only the definite, or
comparatively definite, member of an exhaustive division in accordance
with the law of excluded middle (§ 25). Thus 'Socrates' and 'man' are
positive, as opposed to 'not-Socrates' and 'not-man.'

§ 127. The chief value of the division, however, and especially of the
distinction drawn between privative and negative terms, is in relation
to attributives.

From this point of view we may define the three classes of terms as
follows:

A Positive Term signifies the presence of an attribute, e.g.: 'wise,'
'full.'

A Negative Term signifies merely the absence of an attribute,
e.g. 'not-wise,' 'not-full.'

A Privative Term signifies the absence of an attribute in a subject
capable of possessing it, e.g. 'unwise,' 'empty'. [Footnote: A
privative term is usually defined to mean one which signifies the
absence of an attribute where it was once possessed, or might have
been expected to be present, e.g. 'blind.' The utility of the slight
extension of meaning here assigned to the expression will, it is
hoped, prove its justification.]

§ 128. Thus a privative term stands midway in meaning between the
other two, being partly positive and partly negative--negative in so
far as it indicates the absence of a certain attribute, positive in so
far as it implies that the thing which is declared to lack that
attribute is of such a nature as to be capable of possessing it. A
purely negative term conveys to the mind no positive information at
all about the nature of the thing of which it is predicated, but
leaves us to seek for it among the universe of things which fail to
exhibit a given attribute.

A privative term, on the other hand, restricts us within a definite
sphere. The term 'empty' restricts us within the sphere of things
which are capable of fulness, that is, if the term be taken in its
literal sense, things which possess extension in three dimensions.

§ 129. A positive and a negative term, which have the same matter,
must exhaust the universe between them, e.g. 'white' and 'not-white,'
since, according to the law of excluded middle, everything must be
either one or the other. To say, however, that a thing is 'not-white'
is merely to say that the term 'white' is inapplicable to it.
'Not-white' may be predicated of things which do not possess extension
as well as of those which do. Such a pair of terms as 'white' and
'not-white,' in their relation to one another, are called
Contradictories.

§ 130. Contrary terms must be distinguished from
contradictory. Contrary terms are those which are most opposed under
the same head. Thus 'white' and 'black' are contrary terms, being the
most opposed under the same head of colour. 'Virtuous' and 'vicious'
again are contraries, being the most opposed under the same head of
moral quality.

§ 131. A positive and a privative term in the same matter will always
be contraries, e.g. 'wise' and 'unwise,' 'safe' and 'unsafe': but
contraries do not always assume the shape of positive and privative
terms, but may both be positive in form, e.g. 'wise' and 'foolish,'
'safe' and 'dangerous.'

§ 132. Words which are positive in form are often privative in
meaning, and vice versâ. This is the case, for instance, with the word
'safe,' which connotes nothing more than the absence of danger. We
talk of a thing involving 'positive danger' and of its being
'positively unsafe' to do so and so. 'Unhappy,' on the other hand,
signifies the presence of actual misery. Similarly in Latin 'inutilis'
signifies not merely that there is no benefit to be derived from a
thing, but that it is _positively injurious_. All such questions,
however, are for the grammarian or lexicographer, and not for the
logician. For the latter it is sufficient to know that corresponding
to every term which signifies the presence of some attribute there may
be imagined another which indicates the absence of the same attribute,
where it might be possessed, and a third which indicates its absence,
whether it might be possessed or not.

§ 133. Negative terms proper are formed by the prefix 'not-' or
'non-,' and are mere figments of logic. We do not in practice require
to speak of the whole universe of objects minus those which possess a
given attribute or collection of attributes. We have often occasion to
speak of things which might be wise and are not, but seldom, if ever,
of all things other than wise.

§ 134. Every privative attributive has, or may have, a corresponding
abstract term, and the same is the case with negatives: for the
absence of an attribute, is itself an attribute. Corresponding to
'empty,' there is 'emptiness'; corresponding to 'not-full' there may
be imagined the term 'not-fulness.'

§ 135. The contrary of a given term always involves the contradictory,
but it involves positive elements as well. Thus 'black' is
'not-white,' but it is something more besides. Terms which, without
being directly contrary, involve a latent contradiction, are called
Repugnant, e.g. 'red' and 'blue.' All terms whatever which signify
attributes that exclude one another may be called Incompatible.

§ 136. The preceding division is based on what is known as the Quality
of terms, a positive term being said to differ in quality from a
non-positive one.

_Univocal and Equivocal Terms_.

§ 137. A term is said to be Univocal, when it has one and the same
meaning wherever it occurs. A term which has more than one meaning is
called Equivocal. 'Jam-pot,' 'hydrogen' are examples of univocal
terms; 'pipe' and 'suit' of equivocal.

§ 138. This division does not properly come within the scope of logic,
since it is a question of language, not of thought. From the
logician's point of view an equivocal term is two or more different
terms, for the definition in each sense would be different.

§ 139. Sometimes a third member is added to the same division under
the head of Analogous Terms. The word 'sweet,' for instance, is
applied by analogy to things so different in their own nature as a
lump of sugar, a young lady, a tune, a poem, and so on. Again, because
the head is the highest part of man, the highest part of a stream is
called by analogy 'the head.' It is plainly inappropriate to make a
separate class of analogous terms. Rather, terms become equivocal by
being extended by analogy from one thing to another.

_Absolute and Relative Terms_.

§ 140. An Absolute term is a name given to a thing without reference
to anything else.

§ 141. A Relative term is a name given to a thing with direct
reference to some other thing.

§ 142. 'Hodge' and 'man' are absolute terms. 'Husband' 'father,'
'shepherd' are relative terms. 'Husband' conveys a direct reference to
'wife,' 'father' to 'Child,' 'shepherd' to 'sheep.' Given one term of
a relation, the other is called the correlative, e.g. 'subject' is
the correlative of 'ruler,' and conversely 'ruler' of 'subject.' The
two terms are also spoken of as a pair of correlatives.

§ 143. The distinction between relative and absolute applies to
attributives as well as subject-terms. 'Greater,' 'near, 'like,' are
instances of attributives which everyone would recognise as relative.

§ 144. A relation, it will be remembered, is a kind of attribute,
differing from a quality in that it necessarily involves more
substances than one. Every relation is at bottom a fact, or series of
facts, in which two or more substances play a part. A relative term
connotes this fact or facts from the point of view of one of the
substances, its correlative from that of the other. Thus 'ruler' and
'subject' imply the same set of facts, looked at from opposite points
of view. The series of facts itself, regarded from either side, is
denoted by the corresponding abstract terms, 'rule 'and 'subjection.'

§ 145. It is a nice question whether the abstract names of relations
should themselves be considered relative terms. Difficulties will
perhaps be avoided by confining the expression 'relative _term_'
to names of concrete things. 'Absolute,' it must be remembered, is a
mere negative of 'relative,' and covers everything to which the
definition of the latter does not strictly apply. Now it can hardly be
said that 'rule' is a name given to a certain abstract thing with
direct reference to some other thing, namely, subjection. Rather
'rule' and 'subjection' are two names for identically the same series
of facts, according to the side from which we look at them. 'Ruler'
and 'subject,' on the other hand, are names of two distinct
substances, but each involving a reference to the other.

§ 146. This division then may be said to be based on the number of
things involved in the name.

_Connotative and Non-Connotative Terms._

§ 147. Before explaining this division, it is necessary to treat of
what is called the Quantity of Terms.

_Quantity of Terms._

§ 148. A term is possessed of quantity in two ways--

(1) In Extension;

(2) In Intension.

§ 149. The Extension of a term is the number of things to which it
applies.

§ 150. The Intension of a term is the number of attributes which it
implies.

§ 151. It will simplify matters to bear in mind that the intension of
a term is the same thing as its meaning. To take an example, the term
'man' applies to certain things, namely, all the members of the human
race that have been, are, or ever will be: this is its quantity in
extension. But the term 'man' has also a certain meaning, and implies
certain attributes--rationality, animality, and a definite bodily
shape: the sum of these attributes constitutes its quantity in
intension.

§ 152. The distinction between the two kinds of quantity possessed by
a term is also conveyed by a variety of expressions which are here
appended.

Extension = breadth = compass = application = denotation.

Intension = depth = comprehension = implication = connotation.

Of these various expressions, 'application' and 'implication' have the
advantage of most clearly conveying their own meaning. 'Extension' and
'intension,' however, are more usual; and neither 'implication' nor
'connotation' is quite exact as a synonym for 'intension.' (§ 164.)

§ 153. We now return to the division of terms into connotative and
non-connotative.

§ 154. A term is said to connote attributes, when it implies certain
attributes at the same time that it applies to certain things distinct
therefrom. [Footnote: Originally 'connotative' was used in the same
sense in which we have used 'attributive,' for a word which directly
signifies the presence of an attribute and indirectly applies to a
subject. In this, its original sense, it was the subject which was
said to be connoted, and not the attribute.]

§ 155. A term which possesses both extension and intension, distinct
from one another, is connotative.

§ 156. A term which possesses no intension (if that be possible) or in
which extension and intension coincide is non-connotative.

§ 157. The subject-term, 'man,' and its corresponding attributive,
'human,' have both extension and intension, distinct from one
another. They are therefore connotative. But the abstract term,
'humanity,' denotes the very collection of attributes, which was
before connoted by the concrete terms, 'man' and 'human.' In this
case, therefore, extension and intension coincide, and the term is
non-connotative.

§ 158. The above remark must be understood to be limited to abstract
terms in their singular sense. When employed as common terms, abstract
terms possess both extension and intension distinct from one
another. Thus the term 'colour' applies to red, blue, and yellow, and
at the same time implies (i.e. connotes), the power of affecting the
eye.

§ 159. Since all terms are names of things, whether substances or
attributes, it is clear that all terms must possess extension, though
the extension of singular terms is the narrowest possible, as being
confined to one thing.

§ 160. Are there then any terms which possess no intension? To ask
this, is to ask--Are there any terms which have absolutely no meaning?
It is often said that proper names are devoid of meaning, and the
remark is, in a certain sense, true. When we call a being by the name
'man,' we do so because that being possesses human attributes, but
when we call the same being by the name, 'John,' we do not mean to
indicate the presence of any Johannine attributes. We simply wish to
distinguish that being, in thought and language, from other beings of
the same kind. Roughly speaking, therefore, proper names are devoid of
meaning or intension. But no name can be entirely devoid of
meaning. For, even setting aside the fact, which is not universally
true, that proper names indicate the sex of the owner, the mere act of
giving a name to a thing implies at least that the thing exists,
whether in fact or thought; it implies what we may call 'thinghood':
so that every term must carry with it some small amount of intension.

§ 161. From another point of view, however, proper names possess more
intension than any other terms. For when we know a person, his name
calls up to our minds all the individual attributes with which we are
familiar, and these must be far more numerous than the attributes
which are conveyed by any common term which can be applied to
him. Thus the name 'John' means more to a person who knows him than
'attorney,' 'conservative,' 'scamp,' of 'vestry-man,' or any other
term which may happen to apply to him. This, however, is the acquired
intension of a term, and must be distinguished from the original
intension. The name 'John' was never meant to indicate the attributes
which its owner has, as a matter of fact, developed. He would be John
all the same, if he were none of these.

§ 162. Hitherto we have been speaking only of christening-names, but
it is evident that family names have a certain amount of connotation
from the first. For when we dub John with the additional appellation
of Smith, we do not give this second name as a mere individual mark,
but intend thereby to indicate a relationship to other persons. The
amount of connotation that can be conveyed by proper names is very
noticeable in the Latin language. Let us take for an example the full
name of a distinguished Roman--Publius Cornelius Scipio Æmilianus
Africanus minor. Here it is only the prænomen, Publius, that can be
said to be a mere individual mark, and even this distinctly indicates
the sex of the owner. The nomen proper, Cornelius, declares the wearer
of it to belong to the illustrious gens Cornelia. The cognomen,
Scipio, further specifies him as a member of a distinguished family in
that gens. The agnomen adoptivum indicates his transference by
adoption from one gens to another. The second agnomen recalls the
fact of his victory over the Carthaginians, while the addition of the
word 'minor' distinguishes him from the former wearer of the same
title. The name, instead of being devoid of meaning, is a chapter of
history in itself. Homeric epithets, such as 'The Cloud-compeller,'
'The Earth-shaker' are instances of intensive proper names. Many of
our own family names are obviously connotative in their origin,
implying either some personal peculiarity, e.g. Armstrong, Cruikshank,
Courteney; or the employment, trade or calling of the original bearer
of the name, Smith, Carpenter, Baker, Clark, Leach, Archer, and so on;
or else his abode, domain or nationality, as De Caen, De Montmorency,
French, Langley; or simply the fact of descent from some presumably
more noteworthy parent, as Jackson, Thomson, Fitzgerald, O'Connor,
Macdonald, Apjohn, Price, Davids, etc. The question, however, whether
a term is connotative or not, has to be decided, not by its origin,
but by its use. We have seen that there are some proper names which,
in a rough sense, may be said to possess no intension.

§ 163. The other kind of singular terms, namely, designations (§ 113)
are obviously connotative. We cannot employ even the simplest of them
without conveying more or less information about the qualities of the
thing which they are used to denote. When, for instance, we say 'this
table,' 'this book,' we indicate the proximity to the speaker of the
object in question. Other designations have a higher degree of
intension, as when we say 'the present prime minister of England,'
'the honourable member who brought forward this motion to-night.'
Such terms have a good deal of significance in themselves, apart from
any knowledge we may happen to possess of the individuals they denote.

§ 164. We have seen that, speaking quite strictly, there are no terms
which are non-connotative: but, for practical purposes, we may apply
the expression to proper names, on the ground that they possess no
intension, and to singular abstract terms on the ground that their
extension and intension coincide. In the latter case it is indifferent
whether we call the quantity extension or intension. Only we cannot
call it 'connotation,' because that implies two quantities distinct
from one another. A term must already denote a subject before it can
be said to connote its attributes.

§ 165. The division of terms into connotative and non-connotative is
based on their possession of one quantity or two.

CHAPTER IV.

_Of the Law of Inverse Variation of Extension and Intension._

§ 166. In a series of terms which fall under one another, as the
extension decreases, the intension increases, and vice versâ. Take for
instance the following series--

Thing
|
Substance
|
Matter
|
Organism
|
Animal
|
Vertebrate
|
Mammal
|
Ruminant
|
Sheep
|
This sheep.

Here the term at the top possesses the widest possible extension,
since it applies to everything. But at the same time it possesses the
least possible amount of intension, implying nothing more than mere
existence, whether in fact or thought. On the other hand, the term at
the bottom possesses the greatest amount of intension, since it
implies all the attributes of, an individual superadded to those of
the class to which it belongs: but its extension is the narrowest
possible, being limited to one thing.

§ 167. At each step in the descent from the term at the top, which is
called the 'Summum genus,' to the individual, we decrease the
extension by increasing the intension. Thus by adding on to the bare
notion of a thing the idea of independent existence, we descend to the
term 'substance,' This process is known as Determination, or
Specialisation.

§ 168. Again, by withdrawing our attention from the individual
characteristics of a particular sheep, and fixing it upon those which
are common to it with other animals of the same kind, we arrive at the
common term, 'sheep.' Here we have increased the extension by
decreasing the intension. This process is known as Generalisation.

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