Deductive Logic
by
St. George Stock
Part 2 out of 6
Affirmative
Negative
_Simple and Complex Propositions_.
§ 207. A Simple Proposition is one in which a predicate is directly
affirmed or denied of a subject, e.g. 'Rain is falling.'
§ 208. A simple proposition is otherwise known as Categorical.
§ 209. A Complex Proposition is one in which a statement is made
subject to some condition, e.g. 'If the wind drops, rain will fall.'
§ 210. Hence the complex proposition is also known as Conditional.
§ 211. Every complex proposition consists of two parts--
(1) Antecedent;
(2) Consequent.
§ 212. The Antecedent is the condition on which another statement is
made to depend. It precedes the other in the order of thought, but may
either precede or follow it in the order of language. Thus we may say
indifferently--'If the wind drops, we shall have rain' or 'We shall
have rain, if the wind drops.'
§ 213. The Consequent is the statement which is made subject to some
condition.
§ 214. The complex proposition assumes two forms,
(1) If A is B, C is D.
This is known as the Conjunctive or Hypothetical proposition.
(2) Either A is B or C is D.
This is known as the Disjunctive proposition.
§ 215. The disjunctive proposition may also appear in
the form
A is either B or C,
which is equivalent to saying
Either A is B or A is C;
or again in the form
Either A or B is C,
which is equivalent to saying
Either A is C or B is C.
§ 216. As the double nomenclature may cause some confusion, a scheme
is appended.
Proposition
____________|_____________
| |
Simple Complex
(Categorical) (Conditional)
___________|__________
| |
Conjunctive Disjunctive.
(Hypothetical)
§ 217. The first set of names is preferable. 'Categorical' properly
means 'predicable' and 'hypothetical' is a mere synonym for
'conditional.'
§ 218. Let us examine now what is the real nature of the statement
which is made in the complex form of proposition. When, for instance,
we say 'If the sky falls, we shall catch larks,' what is it that we
really mean to assert? Not that the sky will fall, and not that we
shall catch larks, but a certain connection between the two, namely,
that the truth of the antecedent involves the truth of the
consequent. This is why this form of proposition is called
'conjunctive,' because in it the truth of the consequent is conjoined
to the truth of the antecedent.
§ 219. Again, when we say 'Jones is either a knave or a fool,' what is
really meant to be asserted is--'If you do not find Jones to be a
knave, you may be sure that he is a fool.' Here it is the falsity of
the antecedent which involves the truth of the consequent; and the
proposition is known as 'disjunctive,' because the truth of the
consequent is disjoined from the truth of the antecedent.
§ 220. Complex propositions then turn out to be propositions about
propositions, that is, of which the subject and predicate are
themselves propositions. But the nature of a proposition never varies
in thought. Ultimately every proposition must assume the form 'A is,
or is not, B.' 'If the sky falls, we shall catch larks' may be
compressed into 'Sky-falling is lark-catching.'
§ 221. Hence this division turns upon the form of expression, and may
be said to be founded on the simplicity or complexity of the terms
employed in a proposition.
§ 222. In the complex proposition there appears to be more than one
subject or predicate or both, but in reality there is only a single
statement; and this statement refers, as we have Seen, to a certain
connection between two propositions.
§ 223. If there were logically, and not merely grammatically, more
than one subject or predicate, there would be more than one
proposition. Thus when we say 'The Jews and Carthaginians were Semitic
peoples and spoke a Semitic language,' we have four propositions
compressed into a single sentence for the sake of brevity.
§ 224. On the other hand when we say 'Either the Carthaginians were of
Semitic origin or argument from language is of no value in ethnology,'
we have two propositions only in appearance.
§ 225. The complex proposition then must be distinguished from those
contrivances of language for abbreviating expression in which several
distinct statements are combined into a single sentence.
_Verbal and Real Propositions_.
§ 226. A Verbal Proposition is one which states nothing more about the
subject than is contained in its definition, e.g. 'Man is an animal';
'Men are rational beings.'
§ 227. A Real Proposition states some fact not contained in the
definition of the subject, e.g. 'Some animals have four feet.'
§ 228. It will be seen that the distinction between verbal and real
propositions assumes a knowledge of the precise meaning of terms, that
is to say, a knowledge of definitions.
§ 229. To a person who does not know the meaning of terms a verbal
proposition will convey as much information as a real one. To say 'The
sun is in mid-heaven at noon,' though a merely verbal proposition,
will convey information to a person who is being taught to attach a
meaning to the word 'noon.' We use so many terms without knowing their
meaning, that a merely verbal proposition appears a revelation to many
minds. Thus there are people who are surprised to hear that the lion
is a cat, though in its definition 'lion' is referred to the class
'cat.' The reason of this is that we know material objects far better
in their extension than in their intension, that is to say, we know
what things a name applies to without knowing the attributes which
those things possess in common.
§ 230. There is nothing in the mere look of a proposition to inform us
whether it is verbal or real; the difference is wholly relative to,
and constituted by, the definition of the subject. When we have
accepted as the definition of a triangle that it is 'a figure
contained by three sides,' the statement of the further fact that it
has three angles becomes a real proposition. Again the proposition
'Man is progressive' is a real proposition. For though his
progressiveness is a consequence of his rationality, still there is no
actual reference to progressiveness contained in the usually accepted
definition, 'Man is a rational animal.'
§ 231. If we were to admit, under the term 'verbal proposition,' all
statements which, though not actually contained in the definition of
the subject, are implied by it, the whole body of necessary truth
would have to be pronounced merely verbal, and the most penetrating
conclusions of mathematicians set down as only another way of stating
the simplest axioms from which they started. For the propositions of
which necessary truth is composed are so linked together that, given
one, the rest can always follow. But necessary truth, which is arrived
at 'a priori,' that is, by the mind's own working, is quite as real as
contingent truth, which is arrived at 'a posteriori,' or by the
teachings of experience, in other words, through our own senses or
those of others.
§ 232. The process by which real truth, which is other than deductive,
is arrived at 'a priori' is known as Intuition. E.g. The mind sees
that what has three sides cannot but have three angles.
§ 233. Only such propositions then must be considered verbal as state
facts expressly mentioned in the definition.
§ 234. Strictly speaking, the division of propositions into verbal and
real is extraneous to our subject: since it is not the province of
logic to acquaint us with the content of definitions.
§ 235, The same distinction as between verbal and real proposition, is
conveyed by the expressions 'Analytical' and 'Synthetical,' or
'Explicative' and 'Ampliative' judgements.
§ 236. A verbal proposition is called analytical, as breaking up the
subject into its component notions.
§ 237. A real proposition is called synthetical, as attaching some new
notion to the subject.
§ 238. Among the scholastic logicians verbal propositions were known
as 'Essential,' because what was stated in the definition was
considered to be of the essence of the subject, while real
propositions were known as 'Accidental.'
_Universal AND PARTICULAR Propositions_.
§ 239. A Universal proposition is one in which it is evident from the
form that the predicate applies to the subject in its whole extent.
§ 240. When the predicate does not apply to the subject in its whole
extent, or when it is not clear that it does so, the proposition is
called Particular.
§ 241. To say that a predicate applies to a subject in its whole
extent, is to say that it is asserted or denied of all the things of
which the subject is a name.
§ 242. 'All men are mortal' is a universal proposition.
§ 243. 'Some men are black' is a particular proposition. So also is
'Men are fallible;' for here it is not clear from the form whether
'all' or only 'some' is meant.
§ 244. The latter kind of proposition is known as Indefinite, and must
be distinguished from the particular proposition strictly so called,
in which the predicate applies to part only of the subject.
§ 245. The division into universal and particular is founded on the
Quantity of propositions.
§ 246. The quantity of a proposition is determined by the quantity in
extension of its subject.
§ 247. Very often the matter of an indefinite proposition is such as
clearly to indicate to us its quantity. When, for instance, we say
'Metals are elements,' we are understood to be referring to all
metals; and the same thing holds true of scientific statements in
general. Formal logic, however, cannot take account of the matter of
propositions; and is therefore obliged to set down all indefinite
propositions as particular, since it is not evident from the form that
they are universal.
§ 248. Particular propositions, therefore, are sub-divided into such
as are Indefinite and such as are Particular, in the strict sense of
the term.
§ 249. We must now examine the sub-division of universal propositions
into Singular and General.
§ 250. A Singular proposition is one which has a singular term for its
subject, e.g. 'Virtue is beautiful.'
§ 251. A General proposition is one which has for its subject a common
term taken in its whole extent.
§ 252. Now when we say 'John is a man' or 'This table is oblong,' the
proposition is quite as universal, in the sense of the predicate
applying to the whole of the subject, as when we say 'All men are
mortal.' For since a singular term applies only to one thing, we
cannot avoid using it in its whole extent, if we use it at all.
§ 253. The most usual signs of generality in a proposition are the
words 'all,' 'every,' 'each,' in affirmative, and the words 'no,'
'none,' 'not one,' &c. in negative propositions.
§ 254. The terminology of the division of propositions according to
quantity is unsatisfactory. Not only has the indefinite proposition to
be set down as particular, even when the sense manifestly declares it
to be universal; but the proposition which is expressed in a
particular form has also to be construed as indefinite, _so_ that
an unnatural meaning is imparted to the word 'some,' as used in
logic. If in common conversation we were to say 'Some cows chew the
cud,' the person whom we were addressing would doubtless imagine us to
suppose that there were some cows which did not possess this
attribute. But in logic the word 'some' is not held to express more
than 'some at least, if not all.' Hence we find not only that an
indefinite proposition may, as a matter of fact, be strictly
particular, but that a proposition which appears to be strictly
particular may be indefinite. So a proposition expressed in precisely
the same form 'Some A is B' may be either strictly particular, if some
be taken to exclude all, or indefinite, if the word 'some' does not
exclude the possibility of the statement being true of all. It is
evident that the term 'particular' has become distorted from its
original meaning. It would naturally lead us to infer that a statement
is limited to part of the subject, whereas, by its being opposed to
universal, in the sense in which that term has been defined, it can
only mean that we have nothing to show us whether part or the whole is
spoken of.
§ 255. This awkwardness of expression is due to the indefinite
proposition having been displaced from its proper position. Formerly
propositions were divided under three heads--
(1) Universal,
(2) Particular,
(3) Indefinite.
But logicians anxious for simplification asked, whether a predicate in
any given case must not either apply to the whole of the subject or
not? And whether, therefore, the third head of indefinite propositions
were not as superfluous as the so-called 'common gender' of nouns in
grammar?
§ 256. It is quite true that, as a matter of fact, any given predicate
must either apply to the whole of the subject or not, so that in the
nature of things there is no middle course between universal and
particular. But the important point is that we may not know whether
the predicate applies to the whole of the subject or not. The primary
division then should be into propositions whose quantity is known and
propositions whose quantity is unknown. Those propositions whose
quantity is known may be sub-divided into 'definitely universal' and
'definitely particular,' while all those whose quantity is unknown are
classed together under the term 'indefinite.' Hence the proper
division is as follows--
Proposition
__________|____________
| |
Definite Indefinite
_____|_______
| |
Universal Particular.
§ 257. Another very obvious defeat of terminology is that the word
'universal' is naturally opposed to 'singular,' whereas it is here so
used as to include it; while, on the other hand, there is no obvious
difference between universal and general, though in the division the
latter is distinguished from the former as species from genus.
_Affirmative and Negative Propositions._
§ 258. This division rests upon the Quality of propositions.
§ 259. It is the quality of the form to be affirmative or negative:
the quality of the matter, as we saw before (§ 204), is to be true or
false. But since formal logic takes no account of the matter of
thought, when we speak of 'quality' we are understood to mean the
quality of the form.
§ 260. By combining the division of propositions
according to quantity with the division according to quality,
we obtain four kinds of proposition, namely--
(1) Universal Affirmative (A).
(2) Universal Negative (E).
(3) Particular Affirmative (I).
(4) Particular Negative (O).
§ 261. This is an exhaustive classification of propositions, and any
proposition, no matter what its form may be, must fall under one or
other of these four heads. For every proposition must be either
universal or particular, in the sense that the subject must either be
known to be used in its whole extent or not; and any proposition,
whether universal or particular, must be either affirmative or
negative, for by denying modality to the copula we have excluded
everything intermediate between downright assertion and denial. This
classification therefore may be regarded as a Procrustes' bed, into
which every proposition is bound to fit at its proper peril.
§ 262. These four kinds of propositions are represented respectively
by the symbols A, E, I, O.
§ 263. The vowels A and I, which denote the two affirmatives, occur in
the Latin words 'affirmo' and 'aio;' E and O, which denote the two
negatives, occur in the Latin word 'nego.'
_Extensive and Intensive Propositions._
§ 264. It is important to notice the difference between Extensive and
Intensive propositions; but this is not a division of propositions,
but a distinction as to our way of regarding them. Propositions may be
read either in extension or intension. Thus when we say 'All cows are
ruminants,' we may mean that the class, cow, is contained in the
larger class, ruminant. This is reading the proposition in
extension. Or we may mean that the attribute of chewing the cud is
contained in, or accompanies, the attributes which make up our idea of
'cow.' This is reading the proposition in intension. What, as a matter
of fact, we do mean, is a mixture of the two, namely, that the class,
cow, has the attribute of chewing the cud. For in the ordinary and
natural form of proposition the subject is used in extension, and the
predicate in intension, that is to say, when we use a subject, we are
thinking of certain objects, whereas when we use a predicate, we
indicate the possession of certain attributes. The predicate, however,
need not always be used in intension, e.g. in the proposition 'His
name is John' the predicate is not intended to convey the idea of any
attributes at all. What is meant to be asserted is that the name of
the person in question is that particular name, John, and not
Zacharias or Abinadab or any other name that might be given him.
§ 265. Let it be noticed that when a proposition is read in extension,
the predicate contains the subject, whereas, when it is read in
intension, the subject contains the predicate.
_Exclusive Propositions._
§ 266. An Exclusive Proposition is so called because in it all but a
given subject is excluded from participation in a given predicate,
e.g. 'The good alone are happy,' 'None but the brave deserve the
fair,' 'No one except yourself would have done this.'
§ 267. By the above forms of expression the predicate is declared to
apply to a given subject and to that subject only. Hence an exclusive
proposition is really equivalent to two propositions, one affirmative
and one negative. The first of the above propositions, for instance,
means that some of the good are happy, and that no one else is so. It
does not necessarily mean that all the good are happy, but asserts
that among the good will be found all the happy. It is therefore
equivalent to saying that all the happy are good, only that it puts
prominently forward in addition what is otherwise a latent consequence
of that assertion, namely, that some at least of the good are happy.
§ 268. Logically expressed the exclusive proposition when universal
assumes the form of an E proposition, with a negative term for its
subject
No not-A is B.
§ 269. Under the head of exclusive comes the strictly particular
proposition, 'Some A is B,' which implies at the same time that 'Some
A is not B.' Here 'some' is understood to mean 'some only,' which is
the meaning that it usually bears in common language. When, for
instance, we say 'Some of the gates into the park are closed at
nightfall,' we are understood to mean 'Some are left open.'
_Exceptive Propositions._
§ 270. An Exceptive Proposition is so called as affirming the
predicate of the whole of the subject, with the exception of a certain
part, e.g. 'All the jury, except two, condemned the prisoner.'
§ 271. This form of proposition again involves two distinct
statements, one negative and one affirmative, being equivalent to 'Two
of the jury did not condemn the prisoner; and all the rest did.'
§ 272. The exceptive proposition is merely an affirmative way of
stating the exclusive--
No not-A is B = All not-A is not-B.
No one but the sage is sane = All except the sage are mad.
_Tautologous or Identical Propositions_
§ 273. A Tautologous or Identical proposition affirms the subject of
itself, e.g. 'A man's a man,' 'What I have written, I have written,'
'Whatever is, is.' The second of these instances amounts formally to
saying 'The thing that I have written is the thing that I have
written,' though of course the implication is that the writing will
not be altered.
CHAPTER IV.
_Of the Distribution of Terms._
§ 274. The treatment of this subject falls under the second part of
logic, since distribution is not an attribute of terms in themselves,
but one which they acquire in predication.
§ 275. A term is said to be distributed when it is known to be used in
its whole extent, that is, with reference to all the things of which
it is a name. When it is not so used, or is not known to be so used,
it is called undistributed.
§ 276. When we say 'All men are mortal,' the subject is distributed,
since it is apparent from the form of the expression that it is used
in its whole extent. But when we say 'Men are miserable' or 'Some men
are black,' the subject is undistributed.
§ 277. There is the same ambiguity attaching to the term
'undistributed' which we found to underlie the use of the term
'particular.' 'Undistributed' is applied both to a term whose quantity
is undefined, and to one whose quantity is definitely limited to a
part of its possible extent.
§ 278. This awkwardness arises from not inquiring first whether the
quantity of a term is determined or undetermined, and afterwards
proceeding to inquire, whether it is determined as a whole or part of
its possible extent. As it is, to say that a term is distributed,
involves two distinct statements--
(1) That its quantity is known;
(2) That its quantity is the greatest possible.
The term 'undistributed' serves sometimes to contradict one of these
statements and sometimes to contradict the other.
§ 279. With regard to the quantity of the subject of a proposition no
difficulty can arise. The use of the words 'all' or 'some,' or of a
variety of equivalent expressions, mark the subject as being
distributed or undistributed respectively, while, if there be nothing
to mark the quantity, the subject is for that reason reckoned
undistributed.
§ 280. With regard to the predicate more difficulty may arise.
§ 281. It has been laid down already that, in the ordinary form of
proposition, the subject is used in extension and the predicate in
intension. Let us illustrate the meaning of this by an example. If
someone were to say 'Cows are ruminants,' you would have a right to
ask him whether he meant 'all cows' or only 'some.' You would not by
so doing be asking for fresh information, but merely for a more
distinct explanation of the statement already made. The subject being
used in extension naturally assumes the form of the whole or part of a
class. But, if you were to ask the same person 'Do you mean that cows
are all the ruminants that there are, or only some of them?' he would
have a right to complain of the question, and might fairly reply, 'I
did not mean either one or the other; I was not thinking of ruminants
as a class. I wished merely to assert an attribute of cows; in fact, I
meant no more than that cows chew the cud.'
§ 282. Since therefore a predicate is not used in extension at all, it
cannot possibly be known whether it is used in its whole extent or
not.
§ 283. It would appear then that every predicate is necessarily
undistributed; and this consequence does follow in the case of
affirmative propositions.
§ 284. In a negative proposition, however, the predicate, though still
used in intension, must be regarded as distributed. This arises from
the nature of a negative proposition. For we must remember that in any
proposition, although the predicate be not meant in extension, it
always admits of being so read. Now we cannot exclude one class from
another without at the same time wholly excluding that other from the
former. To take an example, when we say 'No horses are ruminants,' the
meaning we really wish to convey is that no member of the class,
horse, has a particular attribute, namely, that of chewing the
cud. But the proposition admits of being read in another form, namely,
'That no member of the class, horse, is a member of the class,
ruminant.' For by excluding a class from the possession of a given
attribute, we inevitably exclude at the same time any class of things
which possess that attribute from the former class.
§ 285. The difference between the use of a predicate in an affirmative
and in a negative proposition may be illustrated to the eye as
follows. To say 'All A is B' may mean either that A is included in B
or that A and B are exactly co-extensive.
[Illustration]
§ 286. As we cannot be sure which of these two relations of A to B is
meant, the predicate B has to be reckoned undistributed, since a term
is held to be distributed only when we know that it is used in its
whole extent.
§ 287. To say 'No A is B,' however, is to say that A falls wholly
outside of B, which involves the consequence that B falls wholly
outside of A.
[Illustration]
§ 288. Let us now apply the same mode of illustration to the
particular forms of proposition.
§ 289. If I be taken in the strictly particular sense, there are, from
the point of view of extension, two things which may be meant when we
say 'Some A is B'--
(1) That A and B are two classes which overlap one another, that is
to say, have some members in common, e.g. 'Some cats are black.'
[Illustration]
(2) That B is wholly contained in A, which is an inverted way of
saying that all B is A, e.g. 'Some animals are men.'
[Illustration]
§ 290. Since we cannot be sure which of these two is meant, the
predicate is again reckoned undistributed.
§ 291. If on the other hand 1 be taken in an indefinite sense, so as
to admit the possibility of the universal being true, then the two
diagrams which have already been used for A must be extended to 1, in
addition to its own, together with the remarks which we made in
connection with them (§§ 285-6).
§ 292. Again, when we say 'Some A is not B,' we mean that some, if not
the whole of A, is excluded from the possession of the attribute B. In
either case the things which possess the attribute B are wholly
excluded either from a particular part or from the whole of A. The
predicate therefore is distributed.
[Illustration]
From the above considerations we elicit the following--
§ 293. Four Rules for the Distribution of Terms.
(1) All universal propositions distribute their subject.
(2) No particular propositions distribute their subject,
(3) All negative propositions distribute their predicate.
(4) No affirmative propositions distribute their predicate.
§ 294. The question of the distribution or non-distribution of the
subject turns upon the quantity of the proposition, whether universal
or particular; the question of the distribution or non-distribution of
the predicate turns upon the quality of the proposition, whether
affirmative or negative.
CHAPTER V.
_Of the Quantification of the Predicate._
§ 295. The rules that have been given for the distribution of terms,
together with the fourfold division of propositions into A, E, 1, 0,
are based on the assumption that it is the distribution or
non-distribution of the subject only that needs to be taken into
account in estimating the quantity of a proposition.
§ 296. But some logicians have maintained that the predicate, though
seldom quantified in expression, must always be quantified in
thought--in other words, that when we say, for instance, 'All A is B,'
we must mean either that 'All A is all B' or only that 'All A is some
B.'
§ 297. If this were so, it is plain that the number of possible
propositions would be exactly doubled, and that, instead of four
forms, we should now have to recognise eight, which may be expressed
as follows--
1. All A is all B. ([upsilon]).
2. All A is some B. ([Lambda]).
3. No A is any B. ([Epsilon]).
4. No A is some B. ([eta]).
5. Some A is all B. ([Upsilon]).
6. Some A is some B. ([Iota]).
7. Some A is not any B. ([Omega]).
8. Some A is not some B. ([omega]).
§ 298. It is evident that it is the second of the above propositions
which represents the original A, in accordance with the rule that 'No
affirmative propositions distribute their predicate' (§ 293).
§ 299. The third represents the original E, in accordance with the
rule that 'All negative propositions distribute their predicate.'
§ 300. The sixth represents the original I, in accordance with the
rule that 'No affirmative propositions distribute their predicate.'
§ 301. The seventh represents the original O, in accordance with the
rule that 'All negative propositions distribute their predicate.'
§ 302. Four new symbols are required, if the quantity of the predicate
as well as that of the subject be taken into account in the
classification of propositions. These have been supplied, somewhat
fancifully, as follows--
§ 303. The first, 'All A is all B,' which distributes both subject and
predicate, has been called [upsilon], to mark its extreme
universality.
§ 304. The fourth, 'No A is some B,' is contained in E, and has
therefore been denoted by the symbol [eta], to show its connection
with E.
§ 305. The fifth, 'Some A is all B,' is the exact converse of the
second, 'All A is some B,' and has therefore been denoted by the
symbol [Upsilon], which resembles an inverted A.
§ 306. The eighth is contained in O, as part in whole, and has
therefore had assigned to it the symbol [omega],
§ 307. The attempt to take the predicate in extension, instead of, as
it should naturally be taken, in intension, leads to some curious
results. Let us take, for instance, the u proposition. Either the sign
of quantity 'all' must be understood as forming part of the predicate
or not. If it is not, then the u proposition 'All A is all B' seems
to contain within itself, not one proposition, but two, namely, 'All A
is B' and 'All B is A.' But if on the other hand 'all' is understood
to form part of the predicate, then u is not really a general but a
singular proposition. When we say, 'All men are rational animals,' we
have a true general proposition, because the predicate applies to the
subject distributively, and not collectively. What we mean is that
'rational animal' may be affirmed of every individual in the class,
man. But when we say 'All men are all rational animals,' the predicate
no longer applies to the subject distributively, but only
collectively. For it is obvious that 'all rational animals' cannot be
affirmed of every individual in the class, man. What the proposition
means is that the class, man, is co-extensive with the class, rational
animal. The same meaning may be expressed intensively by saying that
the one class has the attribute of co-extension with the other.
§ 308. Under the head o u come all propositions in which both subject
and predicate are singular terms, e.g. 'Homer was the author of the
Iliad,' 'Virtue is the way to happiness.'
§ 309. The proposition [eta] conveys very little information to the
mind. 'No A is some B' is compatible with the A proposition in the
same matter. 'No men are some animals' may be true, while at the same
time it is true that 'All men are animals.' No men, for instance, are
the particular animals known as kangaroos.
§ 310. The [omega] proposition conveys still less information than the
[eta]. For [omega] is compatible, not only with A, but with
[upsilon]. Even though 'All men are all rational animals,' it is still
true that 'Some men are not some rational animals': for no given human
being is the same rational animal as any other.
§ 311. Nay, even when the [upsilon] is an identical proposition,
[omega] will still hold in the same matter. 'All rational animals are
all rational animals': but, for all that, 'Some rational animals are
not some others.' This last form of proposition therefore is almost
wholly devoid of meaning.
§ 312. The chief advantage claimed for the quantification of the
predicate is that it reduces every affirmative proposition to an exact
equation between its subject and predicate. As a consequence every
proposition would admit of simple conversion, that is to say, of
having the subject and predicate transposed without any further change
in the proposition. The forms also of Reduction (a term which will be
explained later on) would be simplified; and generally the
introduction of the quantified predicate into logic might be attended
with certain mechanical advantages. The object of the logician,
however, is not to invent an ingenious system, but to arrive at a true
analysis of thought. Now, if it be admitted that in the ordinary form
of proposition the subject is used in extension and the predicate in
intension, the ground for the doctrine is at once cut away. For, if
the predicate be not used in its extensive capacity at all, we plainly
cannot be called upon to determine whether it is used in its whole
extent or not.
CHAPTER VI.
_Of the Heads of Predicables_.
§ 313. A predicate is something which is stated of a subject.
§ 314. A predicable is something which can be stated of a subject.
§ 315. The Heads of Predicables are a classification of the various
things which can be stated of a subject, viewed in their relation to
it.
§ 316. The treatment of this topic, therefore, as it involves the
relation of a predicate to a subject, manifestly falls under the
second part of logic, which deals with the proposition. It is
sometimes treated under the first part of logic, as though the heads
of predicables were a classification of universal notions, i.e. common
terms, in relation to one another, without reference to their place in
the proposition.
§ 317. The heads of predicables are commonly reckoned
as five, namely,
(1) Genus.
(2) Species.
(3) Difference.
(4) Property.
(5) Accident.
§ 318. We will first define these terms in the sense in which they are
now used, and afterwards examine the principle on which the
classification is founded and the sense in which they were originally
intended.
(1) A Genus is a larger class containing under it smaller
classes. Animal is a genus in relation to man and brute.
(2) A Species is a smaller class contained under a larger one. Man
is a species in relation to animal.
(3) Difference is the attribute, or attributes, which distinguish
one species from others contained under the same genus. Rationality
is the attribute which distinguishes the species, man, from the
species, brute.
N.B. The genus and the difference together make up the Definition of
a class-name, or common term.
(4) A Property is an attribute which is not contained in the
definition of a term, but which flows from it.
A Generic Property is one which flows from the genus.
A Specific Property is one which flows from the difference.
It is a generic property of man that he is mortal, which is a
consequence of his animality. It is a specific property of man that
he is progressive, which is a consequence of his rationality.
(5) An Accident is an attribute, which is neither contained in the
definition, nor flows from it.
§ 319. Accidents are either Separable or Inseparable.
A Separable Accident is one which belongs only to some members of a
class.
An Inseparable Accident is one which belongs to all the members of a
class.
Blackness is a separable accident of man, an inseparable accident of
coals.
§ 320. The attributes which belong to anything may be distinguished
broadly under the two heads of essential and non-essential, or
accidental. By the essential attributes of anything are meant those
which are contained in, or which flow from, the definition. Now it may
be questioned whether there can, in the nature of things, be such a
thing as an inseparable accident. For if an attribute were found to
belong invariably to all the members of a class, we should suspect
that there was some causal connection between it and the attributes
which constitute the definition, that is, we should suspect the
attribute in question to be essential and not accidental. Nevertheless
the term 'inseparable accident' may be retained as a cloak for our
ignorance, whenever it is found that an attribute does, as a matter of
fact, belong to all the members of a class, without there being any
apparent reason why it should do so. It has been observed that animals
which have horns chew the cud. As no one can adduce any reason why
animals that have horns should chew the cud any more than animals
which have not, we may call the fact of chewing the cud an inseparable
accident of horned animals.
§ 321. The distinction between separable and inseparable accidents is
sometimes extended from classes to individuals.
An inseparable accident of an individual is one which belongs to him
at all times. A separable accident of an individual is one which
belongs to him at one time and not at another.
§ 322. It is an inseparable accident of an individual that he was born
at a certain place and on a certain date. It is a separable accident
of an individual that he resides at a certain place and is of a
certain age.
§ 323. There are some remarks which it may be well to make about the
above five terms before we pass on to investigate the principle upon
which the division is based.
§ 324. In the first place, it must of course be borne in mind that
genus and species are relative terms. No class in itself can be either
a genus or a species; it only becomes so in reference to some other
class, as standing to it in the relation of containing or contained.
§ 325. Again, the distinction between genus and difference on the one
hand and property on the other is wholly relative to an assumed
definition. When we say 'Man is an animal,' 'Man is rational,' 'Man is
progressive,' there is nothing in the nature of these statements
themselves to tell us that the predicate is genus, difference, or
property respectively. It is only by a tacit reference to the accepted
definition of man that this becomes evident to us, Similarly, we
cannot know beforehand that the fact of a triangle having three sides
is its difference, and the fact of its having three angles a
property. It is only when we assume the definition of a triangle as a
three-sided figure that the fact of its having three angles sinks into
a property. Had we chosen to define it, in accordance with its
etymological meaning, as a figure with three angles, its
three-sidedness would then have been a mere property, instead of being
the difference; for these two attributes are so connected together
that, whichever is postulated, the other will necessarily follow.
§ 326. Lastly, it must be noticed that we have not really defined the
term 'accident,' not having stated what it is, but only what it is
not. It has in fact been reserved as a residual head to cover any
attribute which is neither a difference nor a property.
§ 327. If the five heads of predicables above given were offered to us
as an exhaustive classification of the possible relations in which the
predicate can stand to the subject in a proposition, the first thing
that would strike us is that they do not cover the case in which the
predicate is a singular term. In such a proposition as 'This man is
John,' we have neither a predication of genus or species nor of
attribute: but merely the identification of one term with another, as
applying to the same object. Such criticism as this, however, would be
entirely erroneous, since no singular term was regarded as a
predicate. A predicable was another name for a universal, the common
term being called a predicable in one relation and a universal in
another-a predicable, extensively, in so far as it was applicable to
several different things, a universal, intensively, in so far as the
attributes indicated were implied in several other notions, as the
attributes indicated by 'animal' are implied in 'horse,' 'sheep,'
'goat,' &c.
§ 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: génos])
Definition ([Greek: ňrismós])
Proprium ([Greek: îdion])
Accident ([Greek: sumbebekós])
§ 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: sumbubekós]
[Greek: ňrismós] [Greek: îdion] [Greek: génos] 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: ňrismós].
2. Synonym /
3. Designation \
|
4. Description > [Greek: îdion].
|
5. Peculiar Property/
6. Genus \
|
7. Difference > [Greek: génos].
|
8. Generic Property/
9. Accident--[Greek: sumbebekós].
§ 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.
§ 374. We may distinguish therefore between two kinds of definition,
namely,
(1) Final.
(2) Provisional.
§ 375. A distinction is also observed between Real and Nominal
Definitions. Both of these explain the meaning of a term: but a real
definition further assumes the actual existence of the thing
defined. Thus the explanation of the term 'Centaur' would be a
nominal, that of 'horse' a real definition.
It is useless to assert, as is often done, that a nominal definition
explains the meaning of a term and a real definition the nature of a
thing; for, as we have seen already, the meaning of a term is whatever
we know of the nature of a thing.
§ 376. It now remains to lay down certain rules for correct
definition.
§ 377. The first rule that is commonly given is that a definition
should state the essential attributes of the thing defined. But this
amounts merely to saying that a definition should be a definition;
since it is only by the aid of definition that we can distinguish
between essential and non-essential among the common attributes
exhibited by a class of things. The rule however may be retained as a
material test of the soundness of a definition, in the sense that he
who seeks to define anything should fix upon its most important
attributes. To define man as a mammiferous animal having two hands, or
as a featherless biped, we feel to be absurd and incongruous, since
there is no reference to the most salient characteristic of man,
namely, his rationality. Nevertheless we cannot quarrel with these
definitions on formal, but only on material grounds. Again, if anyone
chose to define logic as the art of thinking, all we could say is that
we differ from him in opinion, as we think logic is more properly to
be regarded as the science of the laws of thought. But here also it is
on material grounds that we dissent from the definition.
§ 378. Confining ourselves therefore to the sphere with which we are
properly concerned, we lay down the following
_Rules for Definition._
(1) A definition must be co-extensive with the term defined.
(2) A definition must not state attributes which imply one another.
(3) A definition must not contain the name defined, either directly
or by implication.
(4) A definition must be clearer than the term defined.
(5) A definition must not be negative, if it can be affirmative.
Briefly, a definition must be adequate (1), terse (2), clear (4); and
must not be tautologous (3), or, if it can be avoided, negative (5).
§ 379. It is worth while to notice a slight ambiguity in the term
'definition' itself. Sometimes it is applied to the whole proposition
which expounds the meaning of the term; at other times it is confined
to the predicate of this proposition. Thus in stating the first four
rules we have used the term in the latter sense, and in stating the
fifth in the former.
§ 380. We will now illustrate the force of the above rules by giving
examples of their violation.
Rule 1. Violations. A triangle is a figure with three equal sides.
A square is a four-sided figure having all its sides equal.
In the first instance the definition is less extensive than the term
defined, since it applies only to equilateral triangles. This fault
may be amended by decreasing the intension, which we do by eliminating
the reference to the equality of the sides.
In the second instance the definition is more extensive than the term
defined. We must accordingly increase the intension by adding a new
attribute 'and all its angles right angles.'
Rule 2. Violation. A triangle is a figure with three sides and three
angles.
One of the chief merits of a definition is to be terse, and this
definition is redundant, since what has three sides cannot but have
three angles.
Rule 3. Violations. A citizen is a person both of whose parents were
citizens.
Man is a human being.
Rule 4. Violations. A net is a reticulated fabric, decussated at
regular intervals.
Life is the definite combination of heterogeneous changes, both
simultaneous and successive, in correspondence with external
co-existences and sequences.
Rule 5. Violations. A mineral is that which is neither animal nor
vegetable.
Virtue is the absence of vice.
§ 381. The object of definition being to explain what a thing is, this
object is evidently defeated, if we confine ourselves to saying what
it is not. But sometimes this is impossible to be avoided. For there
are many terms which, though positive in form, are privative in force.
These terms serve as a kind of residual heads under which to throw
everything within a given sphere, which does not exhibit certain
positive attributes. Of this unavoidably negative nature was the
definition which we give of 'accident,' which amounted merely to
saying that it was any attribute which was neither a difference nor a
property.
§ 382. The violation of Rule 3, which guards against defining a thing
by itself, is technically known as 'circulus in definiendo,' or
defining in a circle. This rule is often apparently violated, without
being really so. Thus Euclid defines an acute-angled triangle as one
which has three acute angles. This seems a glaring violation of the
rule, but is perfectly correct in its context; for it has already been
explained what is meant by the terms 'triangle' and 'acute angle,' and
all that is now required is to distinguish the acute-angled triangle
from its cognate species. He might have said that an acute-angled
triangle is one which has neither a right angle nor an obtuse angle:
but rightly preferred to throw the same statement into a positive
form.
§ 383. The violation of Rule 4 is known as 'ignotum per ignotius' or
'per aeque ignotum.' This rule also may seemingly be violated when it
is not really so. For a definition may be correct enough from a
special point of view, which, apart from that particular context,
would appear ridiculous. From the point of view of conic sections, it
is correct enough to define a triangle as that section of a cone which
is formed by a plane passing through the vertex perpendicularly to the
base, but this could not be expected to make things clearer to a
person who was inquiring for the first time into the meaning of the
word triangle. But a real violation of the fourth rule may arise, not
only from obscurity, but from the employment of ambiguous language or
metaphor. To say that 'temperance is a harmony of the soul' or that
'bread is the staff of life,' throws no real light upon the nature of
the definiend.
§ 384. The material correctness of a definition is, as we have already
seen, a matter extraneous to formal logic. An acquaintance with the
attributes which terms imply involves material knowledge quite as much
as an acquaintance with the things they apply to; knowledge of the
intension and of the extension of terms is alike acquired by
experience. No names are such that their meaning is rendered evident
by the very constitution of our mental faculties; yet nothing short of
this would suffice to bring the material content of definition within
the province of formal logic.
CHAPTER VIII.
_Of Division._
§ 385. To divide a term is to unfold its extension, that is, to set
forth the things of which it is a name.
§ 386. But as in definition we need not completely unfold the
intension of a term, so in division we must not completely unfold its
extension.
§ 387. Completely to unfold the extension of a term would involve
stating all the individual objects to which the name applies, a thing
which would be impossible in the case of most common terms. When it is
done, it is called Enumeration. To reckon up all the months of the
year from January to December would be an enumeration, and not a
division, of the term 'month.'
§ 388. Logical division always stops short at classes. It may be
defined as the statement of the various classes of things that can be
called by a common name. Technically we may say that it consists in
breaking up a genus into its component species.
§ 389. Since division thus starts with a class and ends with classes,
it is clear that it is only common terms which admit of division, and
also that the members of the division must themselves be common terms.
§ 390. An 'individual' is so called as not admitting of logical
division. We may divide the term 'cow' into classes, as Jersey,
Devonshire, &c., to which the name 'cow' will still be applicable, but
the parts of an individual cow are no longer called by the name of the
whole, but are known as beefsteaks, briskets, &c.
§ 391. In dividing a term the first requisite is to fix upon some
point wherein certain members of the class differ from others. The
point thus selected is called the Fundamentum Divisionis or Basis of
the Division.
§ 392. The basis of the division will of course differ according to
the purpose in hand, and the same term will admit of being divided on
a number of different principles. Thus we may divide the term 'man,'
on the basis of colour, into white, black, brown, red, and yellow; or,
on the basis of locality, into Europeans, Asiatics, Africans,
Americans, Australians, New Zealanders, and Polynesians; or again, on
a very different principle, into men of nervous, sanguine, bilious,
lymphatic and mixed temperaments.
§ 393. The term required to be divided is known as the Totum Divisum
or Divided Whole. It might also be called the Dividend.
§ 394. The classes into which the dividend is split up are called the
Membra Dividentia, or Dividing Members.
§ 395. Only two rules need be given for division--
(1) The division must be conducted on a single basis.
(2) The dividing members must be coextensive with the divided whole.
§ 396. More briefly, we may put the same thing thus--There must be no
cross-division (1) and the division must be exhaustive (2).
§ 397. The rule, which is commonly given, that each dividing member
must be a common term, is already provided for under our definition of
the process.
§ 398. The rule that the dividend must be predicable of each of the
dividing members is contained in our second rule; since, if there were
any term of which the dividend were not predicable, it would be
impossible for the dividing members to be exactly coextensive with it.
It would not do, for instance, to introduce mules and donkeys into a
division of the term horse.
§ 399. Another rule, which is sometimes given, namely, that the
constituent species must exclude one another, is a consequence of our
first; for, if the division be conducted on a single principle, the
constituent species must exclude one another. The converse, however,
does not hold true. We may have a division consisting of mutually
exclusive members, which yet involves a mixture of different bases,
e.g. if we were to divide triangle into scalene, isosceles and
equiangular. This happens because two distinct attributes may be found
in invariable conjunction.
§ 400. There is no better test, however, of the soundness of a
division than to try whether the species overlap, that is to say,
whether there are any individuals that would fall into two or more of
the classes. When this is found to be the case, we may be sure that we
have mixed two or more different fundamenta divisionis. If man, for
instance, were to be divided into European, American, Aryan, and
Semitic, the species would overlap; for both Europe and America
contain inhabitants of Aryan and Semitic origin. We have here members
of a division based on locality mixed up with members of another
division, which is based on race as indicated by language.
§ 401. The classes which are arrived at by an act of division may
themselves be divided into smaller classes. This further process is
called Subdivision.
§ 402. Let it be noticed that Rule 1 applies only to a single act of
division. The moment that we begin to subdivide we not only may, but
must, adopt a new basis of division; since the old one has, 'ex
hypothesi,' been exhausted. Thus, having divided men according to the
colour of their skins, if we wish to subdivide any of the classes, we
must look out for some fresh attribute wherein some men of the same
complexion differ from others, e.g. we might divide black men into
woolly-haired blacks, such as the Negroes, and straight-haired blacks,
like the natives of Australia.
§ 403. We will now take an instance of division and
subdivision, with a view to illustrating some of the
technical terms which are used in connection with the
process. Keeping closely to our proper subject, we will
select as an instance a division of the products of thought,
which it is the province of logic to investigate.
Product of thought
_______________|____________________________
| | |
Term Proposition Inference
____|___ ______|_____ _____|______
| | | | | |
Singular Common Universal Particular Immediate Mediate
___|___ ___|___
| | | |
A E I O
Here we have first a threefold division of the products of thought
based on their comparative complexity. The first two of these, namely,
the term and the proposition, are then subdivided on the basis of
their respective quantities. In the case of inference the basis of the
division is again the degree of complexity. The subdivision of the
proposition is carried a step further than that of the others. Having
exhausted our old basis of quantity, we take a new attribute, namely,
quality, on which to found the next step of subdivision.
§ 404. Now in such a scheme of division and subdivision as the
foregoing, the highest class taken is known as the Summum Genus. Thus
the summum genus is the same thing as the divided whole, viewed in a
different relation. The term which is called the divided whole with
reference to a single act of division, is called the summum genus
whenever subdivision has taken place.
§ 405. The classes at which the division stops, that is, any which are
not subdivided, are known as the Infimae Species.
§ 406. All classes intermediate between the summum genus and the
infimae species are called Subaltern Genera or Subaltern Species,
according to the way they are looked at, being genera in relation to
the classes below them and species in relation to the classes above
them.
§ 407. Any classes which fall immediately under the same genus are
called Cognate Species, e.g. singular and common terms are cognate
species of term.
§ 408. The classes under which any lower class successively falls are
called Cognate Genera. The relation of cognate species to one another
is like that of children of the same parents, whereas cognate genera
resemble a line of ancestry.
§ 409. The Specific Difference of anything is the attribute or
attributes which distinguish it from its cognate species. Thus the
specific difference of a universal proposition is that the predicate
is known to apply to the whole of the subject. A specific difference
is said to constitute the species.
§ 410. The specific difference of a higher class becomes a Generic
Difference with respect to the class below it. A generic difference
then may be said to be the distinguishing attribute of the whole class
to which a given species belongs. The generic difference is common to
species that are cognate to one another, whereas the specific
difference is peculiar to each. It is the generic difference of an A
proposition that it is universal, the specific difference that it is
affirmative.
§ 411. The same distinction is observed between the specific and
generic properties of a thing. A Specific Property is an attribute
which flows from the difference of a thing itself; a Generic Property
is an attribute which flows from the difference of the genus to which
the thing belongs. It is a specific property of an E proposition that
its predicate is distributed, a generic property that its contrary
cannot be true along with it (§ 465); for this last characteristic
flows from the nature of the universal proposition generally.
§ 412. It now remains to say a few words as to the place in logic of
the process of division. Since the attributes in which members of the
same class differ from one another cannot possibly be indicated by
their common name, they must be sought for by the aid of experience;
or, to put the same thing in other words, since all the infimae
species are alike contained under the summum genus, their distinctive
attributes can be no more than separable accidents when viewed in
relation to the summum genus. Hence division, being always founded on
the possession or non-possession of accidental attributes, seems to
lie wholly outside the sphere of formal logic. This however is not
quite the case, for, in virtue of the Law of Excluded Middle, there is
always open to us, independently of experience, a hypothetical
division by dichotomy. By dichotomy is meant a division into two
classes by a pair of contradictory terms, e.g. a division of the
class, man, into white and not-white. Now we cannot know,
independently of experience, that any members of the class, man,
possess whiteness; but we may be quite sure, independently of all
experience, that men are either white or not. Hence division by
dichotomy comes strictly within the province of formal logic. Only it
must be noticed that both sides of the division must be hypothetical.
For experience alone can tell us, on the one hand, that there are any
men that are white, and on the other, that there are any but white
men.
§ 413. What we call a division on a single basis is in reality the
compressed result of a scheme of division and subdivision by
dichotomy, in which a fresh principle has been introduced at every
step. Thus when we divide men, on the basis of colour, into white,
black, brown, red and yellow, we may be held to have first divided men
into white and not-white, and then to have subdivided the men that are
not-white into black and not-black, and so on. From the strictly
formal point of view this division can only be represented as
follows--
Men
___________________|_____
| |
White (if any) Not-white (if any)
_________________|_____
| |
Black (if any) Not-black (if any)
__________________|____
| |
Brown (if any) Not-brown (if any)
____________________|____
| |
Red (if any) Not-red (if any).
§ 414. Formal correctness requires that the last term in such a series
should be negative. We have here to keep the term 'not-red' open, to
cover any blue or green men that might turn up. It is only experience
that enables us to substitute the positive term 'yellow' for
'not-red,' since we know as a matter of fact that there are no men but
those of the five colours given in the original division.
§ 415. Any correct logical division always admits of being arrived at
by the longer process of division and subdivision by dichotomy. For
instance, the term quadrilateral, or four-sided rectilinear figure, is
correctly divided into square, oblong, rhombus, rhomboid and
trapezium. The steps of which this division consists are as follows--
Quadrilateral
__________|_________
| |
Parallelogram Trapezium
_____|_____________________
| |
Rectangle Non-rectangle
___|___ _____|_____
| | | |
Square Oblong Rhombus Rhomboid.
§ 416. In reckoning up the infimae species in such a scheme, we must
of course be careful not to include any class which has been already
subdivided; but no harm is done by mixing an undivided class, like
trapezium, with the subdivisions of its cognate species.
§ 417. The two processes of definition and division are intimately
connected with one another. Every definition suggests a division by
dichotomy, and every division supplies us at once with a complete
definition of all its members.
§ 418. Definition itself, so far as concerns its content, is, as we
have already seen, extraneous to formal logic: but when once we have
elicited a genus and difference out of the material elements of
thought, we are enabled, without any further appeal to experience, to
base thereon a division by dichotomy. Thus when man has been defined
as a rational animal, we have at once suggested to us a division of
animal into rational and irrational.
§ 419. Again, the addition of the attributes, rational and irrational
respectively, to the common genus, animal, ipso facto supplies us with
definitions of the species, man and brute. Similarly, when we
subdivided rectangle into square and oblong on the basis of the
equality or inequality of the adjacent sides, we were by so doing
supplied with a definition both of square and oblong--'A square is a
rectangle having all its sides equal,' and 'An oblong is a rectangle
which has only its opposite sides equal.'
§ 420. The definition of a square just given amounts to the same thing
as Euclid's definition, but it complies with a rule which has value as
a matter of method, namely, that the definition should state the
Proximate Genus of the thing defined.
§ 421. Since definition and division are concerned with the intension
and extension of terms, they are commonly treated of under the first
part of logic: but as the treatment of the subject implies a
familiarity with the Heads of Predicables, which in their turn imply
the proposition, it seems more desirable to deal with them under the
second.
§ 422. We have already had occasion to distinguish division from
Enumeration. The latter is the statement of the individual things to
which a name applies. In enumeration, as in division, the wider term
is predicable of each of the narrower ones.
§ 423. Partition is the mapping out of a physical whole into its
component parts, as when we say that a tree consists of roots, stem,
and branches. In a partition the name of the whole is not predicable
of each of the parts.
§ 424. Distinction is the separation from one another of the various
meanings of an equivocal term. The term distinguished is predicable
indeed of each of the members, but of each in a different sense. An
equivocal term is in fact not one but several terms, as would quickly
appear, if we were to use definitions in place of names.
§ 425. We have seen that a logical whole is a genus viewed in relation
to its underlying species. From this must be distinguished a
metaphysical whole, which is a substance viewed in relation to its
attributes, or a class regarded in the same way. Logically, man is a
part of the class, animal; metaphysically, animal is contained in
man. Thus a logical whole is a whole in extension, while a
metaphysical whole is a whole in intension. From the former point of
view species is contained in genus; from the latter genus is contained
in species.
PART III.--OF INFERENCES.
CHAPTER I.
_Of Inferences in General_.
§ 426. To infer is to arrive at some truth, not by direct experience,
but as a consequence of some truth or truths already known. If we see
a charred circle on the grass, we infer that somebody has been
lighting a fire there, though we have not seen anyone do it. This
conclusion is arrived at in consequence of our previous experience of
the effects of fire.
§ 427. The term Inference is used both for a process and for a product
of thought.
As a process inference may be defined as the passage of the mind from
one or more propositions to another.
As a product of thought inference may be loosely declared to be the
result of comparing propositions.
§ 428. Every inference consists of two parts--
(1) the truth or truths already known;
(2) the truth which we arrive at therefrom.
The former is called the Antecedent, the latter the Consequent. But
this use of the terms 'antecedent' and 'consequent' must be carefully
distinguished from the use to which they were put previously, to
denote the two parts of a complex proposition.
§ 429. Strictly speaking, the term inference, as applied to a product
of thought, includes both the antecedent and consequent: but it is
often used for the consequent to the exclusion of the
antecedent. Thus, when we have stated our premisses, we say quite
naturally, 'And the inference I draw is so and so.'
§ 430. Inferences are either Inductive or Deductive. In induction we
proceed from the less to the more general; in deduction from the more
to the less general, or, at all events, to a truth of not greater
generality than the one from which we started. In the former we work
up to general principles; in the latter we work down from them, and
elicit the particulars which they contain.
§ 431. Hence induction is a real process from the known to the
unknown, whereas deduction is no more than the application of
previously existing knowledge; or, to put the same thing more
technically, in an inductive inference the consequent is not contained
in the antecedent, in a deductive inference it is.
§ 432. When, after observing that gold, silver, lead, and other
metals, are capable of being reduced to a liquid state by the
application of heat, the mind leaps to the conclusion that the same
will hold true of some other metal, as platinum, or of all metals, we
have then an inductive inference, in which the conclusion, or
consequent, is a new proposition, which was not contained in those
that went before. We are led to this conclusion, not by reason, but by
an instinct which teaches us to expect like results, under like
circumstances. Experience can tell us only of the past: but we allow
it to affect our notions of the future through a blind belief that
'the thing that hath been, it is that which shall be; and that which
is done is that which shall be done; and there is no new thing under
the sun.' Take away this conviction, and the bridge is cut which
connects the known with the unknown, the past with the future. The
commonest acts of daily life would fail to be performed, were it not
for this assumption, which is itself no product of the reason. Thus
man's intellect, like his faculties generally, rests upon a basis of
instinct. He walks by faith, not by sight.
§ 433. It is a mistake to talk of inductive reasoning, as though it
were a distinct species from deductive. The fact is that inductive
inferences are either wholly instinctive, and so unsusceptible of
logical vindication, or else they may be exhibited under the form of
deductive inferences. We cannot be justified in inferring that
platinum will be melted by heat, except where we have equal reason for
asserting the same thing of copper or any other metal. In fact we are
justified in drawing an individual inference only when we can lay down
the universal proposition, 'Every metal can be melted by heat.' But
the moment this universal proposition is stated, the truth of the
proposition in the individual instance flows from it by way of
deductive inference. Take away the universal, and we have no logical
warrant for arguing from one individual case to another. We do so, as
was said before, only in virtue of that vague instinct which leads us
to anticipate like results from like appearances.
§ 434. Inductive inferences are wholly extraneous to the science of
formal logic, which deals only with formal, or necessary, inferences,
that is to say with deductive inferences, whether immediate or
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