Chapter
III.
THE JUDGMENT AND THE PROPOSITION.
5. The Fourfold Scheme of Propositions
6.
Analytic and Synthetic Propositions
8. Compound Categorical Propositions
10.
Reduction of Propositions to Logical Form
§ 1. The
Proposition. As the Term is the external expression of the
Concept, so the Proposition is the expression of the Judgment. The Proposition may be defined as a verbal
expression in which we affirm or deny an attribute of a subject (de
Interp. c. 6, §1). It is also
sometimes defined as a verbal expression enunciating a
truth or falsity (de Interp. c. i): for it is characteristic of
every proposition that it must be either true or false. The form of the proposition is S is (or
is not) P, e.g. 'The lion is vertebrate', 'Caesar is not alive'.
A proposition
of the kind we have described, is commonly known as the Categorical
Proposition, to distinguish it from Conditional
Propositions. In these we do not assert the attribute of the subject
absolutely: we merely affirm that, given certain conditions, it belongs to
it. In regard to the Categorical
proposition, the following points are to be noted: -- (1) It is always in the
indicative mood. In the other grammatical moods, the mind does not judge that
the attribute belongs to the subject, but expresses a wish that it may
be so, or gives an injunction that it should be so. In the indicative
alone we affirm (or deny) the attribute of the subject. Thus we have, "The
messenger is speaking", "May the messenger speak?" "Speak,
messenger!" So too it is only when the attribute is affirmed of the
subject that the mind reaches truth or falsity. For truth is attained when the
mind assigns to the subject an attribute which belongs to it in the real
order. (2) The logical proposition is
always stated in the present tense. Our purpose in Logic, as we
have seen, is to study the mode in which the mind represents the real order.
As regards this the question of present, past or future is purely accidental.
The time-determination does not affect the mental representation as such. Hence
differences of tense so necessary in the use of language for practical ends
have no place in Logic. (3) The logical predicate is always separated
from the copula. In the language of common life, we frequently express them in
one word, as for instance, 'The bird flies'. In Logic we must say, 'The bird is
flying.'
This same
process must be performed whenever we get mutilated expressions, such as
'Wolf!' 'Fire!' 'Rain!' For Logic demands that every sentence, whatever its
grammatical form, shall be so analysed and expressed, as to represent as
closely as possible the intellectual act.
Sir W. Hamilton stated this in what is sometimes termed Hamilton's
Postulate, viz. 'Logic postulates to be allowed to state explicitly in
language, whatever is implicitly contained in thought.' Our three mutilated
expressions may be respectively resolved into, 'A wolf is near,' 'A fire is
burning,' 'Rain is falling.'
We must
carefully distinguish between the 'is' of the copula, and the same word when it
means 'to exist'. This point will be treated at length later. Here it will be
sufficient to note that the copula does not necessarily imply that the subject
exists [N1]. Its office in affirmative
sentences is to denote the objective identity of the subject and predicate: that
they are expressions representing one and the same object. In negative
sentences the copula 'is not' signifies the objective diversity of the terms:
the predicate is not applicable to the object denoted by the subject. I may
say, 'The lion is vertebrate,' because the term 'vertebrate' is rightly applied
to the same object as the term 'lion'. I cannot say 'The octopus is
vertebrate'. This relation between the subject and predicate of the
proposition, arises immediately from the nature of the mental act which the
proposition represents -- the judgment. In every affirmative judgment the two
terms are different mental expressions of the same object. The same object is expressed in one concept
as 'lion', in another as 'vertebrate'. But the object which I conceive as
'octopus', I cannot conceive also as 'vertebrate'. Hence the judgment, 'The
octopus is vertebrate', is impossible.
Here we see how totally Scholastic Logic differs from Formal
Logic. Strictly, Formal Logic should
take no account of the content of the subject and predicate. To it every judgment is simply S is
P. Scholastic Logic rejects any
judgments in which the concepts do not represent the same objective reality,
e.g. 'Men are circles', 'Cows are lions'. In these the two notions are repugnant the
one to the other. The judgment is impossible.
§
2. Analysis of the Judgment. Although as we have seen, the subject and
predicate of the judgment are different concepts of the same thing, it is
important to bear in mind that it is the subject which directly expresses the
thing, i.e. that to which attributes belong. The predicate expresses the
thing as qualified by a particular attribute or form [N2].
Whenever we fix our attention on a thing, our mind immediately commences
to abstract the attributes from the object of thought, and affirm them of it
one by one. It judges: 'The thing is hard -- is black -- is brittle, etc.'
[N3]. Here the predicate in each
case is the attribute or form —not
indeed the attribute considered in separation from the object in which it
inheres, i.e. hardness, blackness, etc., but considered as qualifying the
thing. It is easy to see how such
judgments develop into more complex ones. The hard, brittle thing will be
called 'coal': and the judgment will take predicates of a less primary
character [N4].
A judgment is
said to be true when the form expressed by the predicate is really found
in the object denoted by the subject. Thus, if I see some object, e.g.
Socrates, and I judge 'Socrates is walking,' my judgment is a true one if the
attribute 'walking', which I affirm of Socrates in thought, does in fact belong
to him in the real order. Hence truth
is defined as the conformity of the mind with its object. For in every
true affirmative judgment the mental concept expressed in the predicate is in
conformity with a real attribute belonging to the external object. As regards negative judgments the case is
somewhat different. In them we declare that the form expressed in the predicate
is not to be found in the object to which the judgment refers. Yet in a
somewhat wide sense we can say that in negative judgments also the mind is
conformed to its object. In judging a form not to belong to an object which in fact
does not possess it, my mind is in correspondence with reality. But negation is
a secondary and subsidiary form of truth. In affirmation there is perfect
correspondence between the mental form expressed in the predicate and the
objective reality. (
Grammatically
the subject does not always take the first place. It is the meaning of the proposition, not the
arrangement of the words, which tells us which is the subject and which the
predicate. The term which qualifies or
defines the other, whether it comes first or last, is the predicate. Thus in
the words, 'Blessed are the meek,' it is the meek who form the logical subject.
What has just
been said will throw light on the nature of the third element of the proposition
-- the copula. While the predicate expresses one of the
forms which determine the subject, thus telling us what the thing is,
the copula expresses the being which is thus determined. Unless
the subject were conceived as possessed of being, we could not attribute any
predicates to it: for its predicates are so many determinations of its being. A
thing which is not cannot be determined. It is non ens - a nonentity.
These relative functions
of copula and predicate are clearly seen, if we consider the judgment in its
primary form, viz, that in which the subject is a concrete singular substance,
and the predicate a form apprehended as belonging to it, e.g. 'Socrates is a
man',
'Bucephalus
is black'. In the
first of these propositions the being of Socrates is characterized as human; in
the second the being of Bucephalus is determined by the accidental form of a
particular colour. Again, if we wish to
affirm that the subject possesses real being - that it exists - though without
indicating how that being is determined, we use the same verb 'to be', which
is employed in the copula. We say, 'Socrates is'.
It will have been noticed
that we said it was requisite that the subject should be conceived as possessed
of being. We did not say that it must actually possess real being. In fact there are many true propositions in
which the subject has no real being: many in which the predicate is not a real
form. Our words, as we have often said, are the expression of our thoughts:
they signify things as mentally represented.
Hence the being of which the copula is the direct and immediate
expression is being in the conceptual order: and the forms signified by the
predicate are forms as they conceived.
In the propositions 'Socrates is a man', 'Bucephalus is black', there
is correspondence between the real order and the conceptual. But this is not
always the case. We can give conceptual existence to things which have never
actually existed, and many of our judgments relate, not to something which has
really existed, but to some creation of the mind. Thus, when speaking of Greek
mythology, I may say, 'A centaur is an animal half man and half horse'. The
judgment is true because it corresponds with the object of thought. But that object only existed in the mind: it
had no real existence. The same
principle accounts for those judgments in which the predicate is a second
intention, e.g.'Man is a universal nature.' The nature 'man' viewed as
universal, exists only conceptually.
It is thus manifest that
the copula does not necessarily affirm the existence of the subject in rerum
natura. Its essential function is to signify the objective identity of
subject and predicate. This it does in every proposition without exception. For
in every proposition its office is to declare that the term employed in the
predicate signifies the same object which is denoted by the subject.
Just as the copula
frequently expresses, not real being, but what is mentally expressed as being,
so the predicate is often not a real quality, but something the mind expresses
as a quality. We saw when discussing
Negative terms that we are able to conceive as real entities things which are
in fact simply the negations of entities.
Our predicates are often of this kind, as e.g. in the proposition, 'The
horse is riderless'. Sometimes both subject and predicate are of this
character: for instance, 'Blindness deprives men of much happiness.' Here not
merely is a privation conceived as if it were a real subject; but a purely
negative result is conceived as a positive action.
Mill
gravely informs us that his father was "the first among philosophers to
notice that 'to be' in the sense of 'to exist' has not the same signification
as when it means to be some specified thing, as 'to be a man'; and adds that
"Aristotle and all the ancients believed it to have a common meaning
wherever used." He added: "The
fog which rose from this narrow spot diffused itself at an early period over
the whole surface of Metaphysics" (Logic, Bk. I. c. 4, § 1). Mill
frequently falls into error when criticizing the philosophy of Aristotle and
his followers, with whose writings he was but imperfectly acquainted. Nowhere
perhaps is he more astray than here. Not merely was the distinction carefully
noted by Aristotle: but the various senses of 'Being' was one of the points
most canvassed in the writings of the Scholastics.
It remains to be noticed
that a judgment is a single act of the mind. No mistake could be greater than
to represent it as three separate acts, corresponding respectively to subject,
copula and predicate. Such a view might
seem to be implied, when it is said that in affirmation we have the
conjunction, in negation the separation of two concepts. But it is manifest
that a synthesis, in which we recognize a relation of identity, and a
separation in which we judge that such a relation is absent, are alike single
acts [N5].
Certain points which
remain to be considered in regard to the Scholastic theory of the judgment,
must be dealt with later [N6].
*
The
following two citations will be sufficient by way of illustration
Sciendum est quod Esse dicitur tripliciter … Tertio modo dicitur Esse
quod significat veritatem compositionis in propositionibus, secundum quod est
dicitur copula: et secundum hoc est in intellectu componente et dividente
quantum ad suum complementum; sed fundatur in esse rei, quod est actus
essentiae. I Sent. dist. 33, Q. 1, Art. 1, ad. 1 [Latin].
Secundum Avicennam (tract II Metaph. c. 1) de eo quod nullo modo
est non potest aliquid enuntiari: ad minus enim oportet quad illud de quo
aliquid enunciatur sit apprehensum: et ita habet aliquod esse saltem in
intellectu apprehendente: et ita constat quod semper veritati respondet aliquod
esse nec oportet quod semper respondeat sibi esse in re extra animam, cum
ratio veritatis compleatur in ratione animae. I. Sent. dist. 19, Q. 5, Art. 1, ad. 5
[Latin]
§
3. Quality of Propositions. In every proposition P must be either
affirmed or denied of S. This alternative determines the Quality of the
proposition, which must be either (I) affirmative, or (2) negative. This
division is ultimate. Some logicians
have, it is true, endeavoured to reduce all propositions to the affirmative
form by writing S is not-P. But the difference cannot be thus
bridged. S is not-P is, of
course, equivalent to S is not P. But they differ the one from the
other: since in S is not P we deny the positive concept P of S, and in S
is not-P we affirm the negative concept not-P of S. The negative
and affirmative forms remain radically distinct.
Kant admits three forms,
Affirmative, Negative, Infinite, S is P. S is not P, S is not-P.
His motive in assigning the Infinite judgment to a separate class, instead of
reckoning them with the Affirmatives to which they rightly belong, seems to
have been the desire that his scheme of Categories should present an harmonious
appearance. A triple division was required in its other portions, and a triple
division must perforce be found for the Quality of judgments.
§
4. Quantity of Propositions. In any affirmation or negation, P may be affirmed
or denied, (1) of all the objects denoted by the subject-term, e.g. 'All men
are mortal'; or (2) of only some of these objects, e.g. 'Some men are negroes':
or (3) there may be no sign to mark whether the predicate refers to some only
or to all, e.g. 'Pleasure is not a good' : or (4) the subject may be a singular
term, e.g. 'Socrates is wise,' 'The highest of the Alps has been scaled.' These
various alternatives lead to the division of propositions according to
quantity.
A Universal proposition is one, in which the predicate is affirmed (or
denied) of a subject, taken in ita whole extension and distributively.
We have already explained that when a subject
is employed distributively, the predicate is affirmed of every individual
denoted by the subject. When we say, 'All sparrows are winged,' we mean that
every individual sparrow is possessed of wings.
A proposition in which the subject is understood collectively is not
universal. Thus the proposition, 'All the slates covered the roof,' is not a
universal proposition. The predicate is not affirmable of each individual
denoted by the subject, but of the individuals as forming one group. Hence,
whenever the word All (and not Every) is employed to qualify the
subject, care must be taken to observe whether it be understood collectively or
distributively.
It is plain that though the Affirmative
Universal is of the form All S is P, the Negative Universal will not be All
S is not P, but No S is P. The form All S is not P does not
exclude P from each and every individual S, as at once appears in the
proposition 'All soldiers are not generals'. If, however, I say, 'No
Englishmen are negroes,' I deny the attribute of every Englishman.
The employment of the plural in a universal
proposition, e.g. 'All men are mortal,' may possibly mislead the student into
supposing that in the subject the intellect conceives a number of individuals.
This is, of course, impossible. Mentally the whole class is expressed by the
universal concept 'man.' But the grammatical form 'All men are mortal,' shows
that we have under our consideration, not the universal nature viewed in
abstraction from particulars, but the concrete individuals. Man is mortal' is
the purely logical form. 'All men are mortal' puts us in touch with concrete
reality.
A Particular proposition is one in which the predicate is affirmed (or
denied) of a part only of the extension of the subject.
The form of the Particular proposition is Some
S are (or are not) P; for instance, 'Some soldiers are
brave,' 'Some rich men are not generous'. The sense, in which the word 'some'
is here used, differs in certain respects from that in which it is ordinarily
employed. In ordinary use, when we speak, e.g. of 'some' men, we are under
stood to mean more than one, and also to exclude the supposition that what we
say may be true of all men. 'Some' means
'several but not all.' In Logic, the 'some' of a particular proposition, may be
used even where the predicate might be truly affirmed of all: and it may be
used also even if there be but one individual to whom it could be applied. Thus I may say, 'Some birds have wings,' even
though it be the case that all birds possess them: and 'Some men are eight feet
high,' though in fact there be but one such man. 'Some' leaves the extension to
which reference is made indeterminate [N7].
The essential distinction then between
Universal and Particular propositions lies in this, that Universals deal with
the whole class, Particulars with an indeterminate portion of the class.
And here it is well to call attention to the
fact, that universal propositions are of two sorts. The majority of them cannot
be attained by mere enumeration of instances. Some indeed can. I can arrive at
the universal truth, that 'All the apostles were Jews,' by a process of counting.
But propositions of this character are of minor moment. Enumeration will not
serve me in regard to such propositions as, 'All men are mortal,' 'All birds
are oviparous.' Here, I refer not merely to an incalculable number of past
instances, but also to the future. All laws of nature known to science are
propositions of this character. The aim
and object of scientific enquiry is to establish such universal truths.
How is it that we can affirm a predicate of
individuals, which have not come within our experience? The explanation lies in
the fact, that in these propositions we know the predicate to be invariably
connected with the universal class-notion employed in the subject. In a later
part of Logic, we shall consider how we reach this knowledge. It is sufficient
here to observe that to what ever individuals the notion 'man ' is applicable,
the predicate 'mortal' is applicable also. In virtue of their being men, they
possess the attribute of mortality. The universality of these propositions
rests not on enumeration, but on our knowledge of the constant connection
between the concepts of the subject and predicate.
It remains to consider Indesignate and
Singular pro positions.
Indesignate propositions are such as have no sign
of quantity. As far as form is concerned, they may be universal, or they may be
particular. If I say, 'Old men are melancholy,' it does not appear, whether I
am speaking of all old men, or of some only. Hence indesignate propositions
have no place in Logic, until a sign of quantity is affixed to them. In some
cases indeed the Indesignate is used to signify that the predicate is connected
necessarily with the subject, e.g. 'Man is mortal'. Here the proposition is of course equivalent
to a universal. For these judgments in which the Indesignate form stands not
for individuals, but for the class-nature, some authors employ the convenient
term Generic judgments. But it should be observed that we do not know
their universal character from the logical form, but from our previous
acquaintance with the matter under consideration. Very often the Indesignate is used for what
are termed moral universals, as in the example already given', Old men
are melancholy'. A moral universal
admits exceptions, and hence is logically a particular.
The Singular proposition is, as we have
said, one whose subject is either a significant Singular term or a proper name.
These propositions present some anomalies. On the one hand, the individual
object is a member of a class, and it appears incongruous to treat it as though
it were itself a class. On the other, the definition of a Universal proposition
is applicable to them, for the predicate is affirmed of the subject in its
whole extension, the extension in this case being restricted to a single
individual.
Modern logicians have resolved to treat this
proposition as a Universal, and it will be convenient to adhere to that
arrangement.
The older logicians classify the Singular
proposition separately, and assign it neither to the Universal nor to the
Particular [N8]. This was, it
would seem, the more scientific course. For the Universal and Particular are
distinguished by the manner in which the concept employed as subject is
understood in regard to extension. But as we have explained above (Ch. 2, §1) we have
no singular concepts. Hence there is a fundamental difference between such a
proposition as, 'All men are mortal,' and 'Socrates is a philosopher.'
Propositions whose subject is a Collective term
are Singular propositions. Thus if I say,' All the apples filled the bowl,' it
is clear that I refer to this group of apples considered as a single object.
§
5. The Fourfold Scheme of Propositions. The last paragraph has shown us that the two
fundamental forms of the proposition are the Universal and the Particular. In one of these two, every known truth can be
expressed. For the assertion made is either known to hold good of the subject
in its whole extension, or not. If it is
known to hold good, we use the Universal proposition. If it does not hold good
as regards the whole extension of the subject, or if, though it holds good, we
do not know this to be the case, we use the Particular. This distinction, combined with that based on
quality, gives us the fourfold scheme, viz. Universal Affirmative, Particular
Affirmative, Universal Negative, Particular Negative. These are respectively given by the letters,
A.I.E.O. These letters are the vowels of the two Latin words, Affirmo (I
affirm) and Nego (I deny). The first vowel in each stands for the
Universal, the second vowel for the Particular. Another notation, which is
found convenient, is SaP, SiP, SeP, SoP: this
notation has symbols for the subject and predicate, as well as for quantity and
quality. Hence, our four propositions may be thus expressed.
All
S are P. A. SaP.
Some
S are P. I. SiP.
No
S are P. E. SeP.
Some
S are not P. 0. SoP.
§
6. Analytic
and Synthetic Propositions. This distinction is
based on the fact that each of our Judgments is based on one or other of two
very different motives. The point will
best be elucidated by a few examples. If we consider the following
propositions, 'The angles of every triangle are equal to two right angles,'
'The whole is greater than its part,' 'Every square has four sides,' and
compare them with such propositions as 'Water freezes at 320o
Fahrenheit,' 'Some cows are black,' we shall at once recognize that there is a
difference between the two classes. We are,
indeed, certain of the truth of all these propositions. But our certainty has a
different motive in the first class, and in the second. In the case of the
first class of Judgments, as soon as we consider the concepts of the subject
and predicate, we see that they are necessarily bound together. A triangle must have its angles equal to two
right angles; otherwise it would not be what we mean by a triangle. Were we
told of any figure that its interior angles were greater or less than two right
angles, we should be justified in affirming that it was not, and could not
under any circumstances be a rectilinear triangle. In the same way the
intension of the concepts 'whole' and 'part' excludes the supposition of a
whole that is not greater than its part; for the meaning of the term 'whole,'
is 'that which consists of parts.' In regard to the second class, the motive of
our assent is very different. It is experience that has led to my conviction
that water freezes at 320 F., and that certain cows are black. There is nothing
in my notion of ' cows which prescribes 'blackness,' nor in my notion of'
water,' which compels me to think of it as possessing this particular
freezing-point at the sea-level. In neither of these propositions are the two
concepts linked together in virtue of their intension.
The former class of propositions is termed
Analytic, the latter Synthetic.
The definition of Analytic and Synthetic
propositions is differently given by Scholastic philosophers on the one hand,
and by the greater number of Logicians since the days of Kant, on the
other. The difference is of primary
importance in philosophy. We place the Scholastic definitions first.
An Analytic proposition is one, in which either the predicate is
contained in the intension of the subject, or the subject in the intension of
the predicate.
A Synthetic proposition is one in which the connection of subject and
predicate is not involved in the intension of the terms.
It will be seen that Analytic propositions are
of two kinds. The first kind consists of those in which the predicate is a term
signifying either the whole intension, or part of the intension of the subject.
Such is the proposition, 'Every square has four sides.' The second kind
consists of those in which the predicate is an attribute which results
necessarily from the nature of the subject [N9]. For where this is the case the subject is
found in the intension of the predicate. An example is furnished by the
proposition, 'A triangle is a figure having its interior angles equal to two
right angles.' The predicate here is not found in the intension or definition
of 'triangle'. But it is an attribute which necessarily results from and is
involved in the characteristics of a triangle. And if we desire to define the
attribute 'having its interior angles equal to two right angles,' we can only
do so by stating that it is a quantitative measure proper to the angles of a
triangle [N10].
It is not however necessary that the
connection of the attribute with the subject should be evident on the first
consideration. Many steps may be necessary.
Every geometrical theorem gives us an Analytical pro position as its
conclusion. The connection between
subject and predicate is involved in the intension of the terms : but we must
often take a long series of steps before the necessity of that connection
becomes manifest to us [N11].
The modern definitions are as follows
An Analytic Proposition is one, in which the predicate is contained in
the definition of the subject.
A Synthetic proposition is one, in which the predicate is not contained
in the notion of the subject.
So prevalent
have these definitions become, that in any public examination at the present
day, a question, involving these terms, would certainly employ them in this
latter sense.
* (a) We have
spoken of the differences between the two definitions as of vital moment. The
Kantian division of Analytic and Synthetic propositions relegates to one class
propositions, our knowledge of which depends wholly on experience of the
individual case, such as 'This book is bound in cloth,' and propositions such
as, 'The square on the hypotenuse of a right-angled triangle is equal to the
sum of the squares on the remaining sides'. Neither of these, Kant tells us,
can be discovered by analysis. For he considers only the case, in which the
predicate is found by the analysis of the subject, and entirely ignores the
case, in which the subject-term is revealed by an analysis of the predicate.
What account then is to be given of our conviction as to the truth of such
propositions as that relating to the square on the hypotenuse? If they are not
analytic, two hypotheses only are possible. Either (1) we accept them on a
dictate of our understanding, of which no account can be given. They are synthetic a priori. This is Kant's
solution. Or (2) their truth is a conclusion, at which we arrive from an
examination of individual instances, but we possess no ground for saying, that
they must be true, that e.g. every right-angled triangle has the property
described. This is Mill's solution.
(b) There have
been three theories as to the object of Analytic propositions. Mill (following Hobbes) holds that they are
concerned with the meaning of names only, and terms them Verbal propositions. Leibniz held that they are concerned with our
concepts. The Scholastics taught that
they are truths relating to things, though known through our concepts, and
expressed in words. Professor Case
wisely says, "The division of propositions into verbal and real is
defective. A verbal is not necessarily opposed to a real proposition, a
predicate does not cease to be characteristic of a thing by becoming the
meaning of its name, and there are some propositions which are verbal and real,
such as, all bodies are extended, the whole is greater than its part. . . .
Sometimes the same analytical judgment is at once 'real, notional and verbal,
e.g. the whole is, is conceived, and means that which is greater than its
part" (Physical Realism, p. 340).
(c) It is sometimes said,
that every synthetic judgment becomes analytic with the growth of our
knowledge, that e.g. 'George III. died in 1820,' is an analytic judgment to one
who knows the history of that period. The argument is quite fallacious. The
facts, which occur to an individual member of a class, are not necessary notes
of his nature, forming the connotation of the concept which expresses it. In
the first place, of individuals as such we have no concepts: all concepts are
universal. And secondly, even were it possible
to have a concept expressing the essential nature of the individual, the purely
contingent facts relating to him would not be part of it. Of course, if I form
a complex concept applicable to George III, such as e.g. 'The King of England
at the beginning of the nineteenth century,' and to this add the note 'who died
in 1820,' then I may form the analytic proposition, 'The King of England at the
beginning of the nineteenth century, who died in 1820, died in 1820.' But the
value of an analytic proposition of this kind is not great.
(d) Analytic propositions
are also termed Essential, Explicative, a priori, Verbal; and
correspondingly, synthetic propositions are known as Accidental, Ampliative, a
posteriori, Real. On this whole subject, see Professor Case's Physical
Realism, pp. 334 - 353.
§
7. Complex Propositions. Complex propositions are such as have a
complex term for their subject or their predicate. By a complex term is
understood a many-worded term, consisting of two or more distinct parts, so
that it expresses, not merely the nature of the thing denoted, but also one or
more qualifications belonging to it, e.g. 'the white knight,' 'the roller which
is in my garden,' and the like. These qualifications are often (as in the
second of the instances just given), expressed by subordinate clauses,
introduced by a relative. Yet it is manifest that, even if a complex term
involves two or three such clauses, the term is but one, and constitutes a
single subject or predicate, as the case may be.
Two forms of complex propositions are
ordinarily distinguished by logicians. The distinction is grammatical, not
logical, and is given in order to put us on our guard against ambiguity.
(1) Propositions with an explicative
qualification. In these the qualification belongs to every individual signified
by the general name, to which it belongs. Thus in the, proposition, 'Whales,
which are mammals, are aquatic animals,' the relative clause is applicable to
every individual, that is signified by the general name 'whales.'
(2) Propositions with a restrictive (or determinative)
qualification. In these, the qualification restricts the signification of the
general name to a certain part of its denotation. Thus in the sentence, 'All
nations, that have been civilized, have cultivated philosophy,' the
qualification does not belong to all the members of the class indicated. Not
all nations are civilized.
The time determination involved in the use of
the past and future tenses of the verb, is a special form of complexity in the
proposition. This however, as we have noted, Logic is enabled to disregard.
Another constantly recurring form is that produced by the employment of
transitive verbs, followed by an object, e.g. 'Brutus slew his benefactor,'
which gives as the logical predicate, the complex term 'a slayer of his benefactor.'
§ 8. Compound Categorical Propositions. It often happens that what grammatically is a
single assertion, is resolvable into two or more propositions, each with its
own subject and predicate. In such cases, we have the Compound Categorical
Proposition. Propositions of this kind are divided into two classes - those
whose character is apparent from their grammatical structure (aperté compositi), and those in which
the grammatical form does not manifest their composite nature (occulté compositi). These latter are
termed Exponibles.
(a) Propositions compound in form. Of these there are three classes: --
(i) Copulative
propositions. These are affirmative propositions, in which there are two or
more subjects or predicates or both. Hence they are resolvable into a number of
independent affirmative propositions: e.g. 'Peter and Paul ended their days at
(ii). Remotive
propositions. These are negations similarly united. The conjunctions
employed will be such as the negative form demands. For example, 'Neither
riches nor honours can banish anxiety'; this sentence may be resolved, 'Riches
cannot banish anxiety. Honours cannot banish anxiety.' 'No Mohammedan will eat
swine's flesh or drink wine.' This, in its logical expression, becomes, 'No
Mohammedan will eat swine's flesh. No Mohammedan will drink wine.'
(iii).
Discretive or Adversative propositions.
Here we have either two affirmative propositions, or an affirmative and a
negative proposition, connected by an adversative conjunction, such as but, although, yet. Thus: 'William I.
was brave but not magnanimous.' This
gives us the two propositions 'William I. was brave. William I. was not magnanimous.'
(b)
Exponible propositions. In these, as we have
said, there is nothing in the grammatical structure of the sentence to indicate
that it is equivalent to more than one logical proposition. Here also, three
classes are ordinarily enumerated.
(i) Exclusive
Propositions. These contain a word, such as 'alone,' attached to the
subject, and thus excluding the predicate from any other subject than this one.
Hence two propositions are necessary to declare the full meaning, one to affirm
the predicate of this subject, and another to deny it of all others. For
instance, 'God alone is omnipotent.' This is equivalent to 'God is omnipotent.
No other is omnipotent.'
(ii) Exceptive
Propositions. In these, the subject term is restricted in its application
by a word such as except - save, which excludes a portion of its
denotation -e.g. 'All the crew save one were drowned.' Here
again, two exponent propositions are needed, the one denying the predicate of
the excepted part, the other affirming it of the remainder. The example just
given will become, 'One of the crew was not drowned. The remaining members were
drowned.'
If
the order of the terms is altered, then both Exclusives and Exceptives may be
expressed by a single exponent. 'Only God is omnipotent,' will become 'All that
is omnipotent is God'; and 'All the crew save one were drowned,' will be 'The
portion of the crew that was not drowned was one man.' But if the original
order is to be preserved, two propositions are necessary. The reasons which
justify a change of order will be dealt with in Ch. 5.
(3) Inceptive and Desitive Propositions.
In these a statement is made as to the commencement or ending of something; e.g.
'Printing became customary after 'the fifteenth century,' 'Paganism ceased in
England 'about the year 700 A D.' These are resolved by two propositions, one
relating to the state of things before the time indicated, and one relating to
what occurred subsequently. Thus the first example will become 'Printing was
not customary before the close of the fifteenth century. Printing was customary
after that date.'
§ 9. Modal Propositions. The Modal proposition affords us another case
in which the traditional terminology differs from that in vogue since the days
of Kant. Here too we shall first explain modality as understood by the
Scholastic philosophers, and then deal with the Kantian account.
The characteristic of the Modal, is that the
copula undergoes modification, in order to express the manner in which the
predicate belongs to the subject. There are propositions, in which the
attribute affirmed belongs to the subject by strict necessity. Thus 'mortality'
is an attribute that is necessarily connected with the subject 'man.' In other
cases the element of necessity is absent. 'To be learned' is affirmable of some
men only. It is not an attribute belonging necessarily to the nature
'man.' The pure categorical draws no distinction
between these cases. We employ the same copula 'is,' whether the connection is
necessary or contingent. But in the Modal proposition, the nature of the
connection between attribute and subject receives expression.
It has been frequently objected, that this
whole question belongs not to Logic but to Metaphysics. Thus Sir W. Hamilton
says, "Necessity, Possibility, etc. do not relate to the connection of the
subject and predicate . . . as terms in thought, but as realities in existence:
they are metaphysical, not logical condi'tions." This objection rests on a misconception as to
the
The relation
of the attribute to the subject is, objectively, determined by one of three
modes. These are (i) the Necessary,
in which the attribute belongs necessarily to the subject. This is expressed by
a proposition of the form, 'Men are necessarily mortal,' 'Equilateral triangles
are necessarily equiangular.' (ii) The Impossible:
in this case the predicate is repugnant to the subject, e.g. 'It is impossible
for irrational creatures to exercise free will.' And (iii) the Possible
(or Contingent). In this case the predicate belongs to the subject in
some instances, while in other instances it is not found with it. Thus e.g. 'It
is possible for a man to be a grammarian.' The conjunction of the two
attributes involves no impossibility, but on the other hand is not necessary.
This relation may be asserted from two points of view. We may assert the
possibility of the connection between subject and predicate, and express the
proposition as it is expressed above. Or we may declare the possibility of
their separation: and in this case the proposition will take the form, 'It is
possible for a man not to be a grammarian.' Hence though there are but three
modes, there are four fundamental forms of Modal propositions, as there are
four fundamental forms of Categorical.
A difficulty is occasioned by the fact that
ambiguity attaches to the word 'possible.'
'Possible' may have the sense in which we have just explained it. It may
however, include in its signification the Necessary also; for if a predicate
belongs necessarily to a subject, we can say with truth that that subject is
capable of receiving it [N12] . If all triangles must have three angles, it is
true to say that it is possible for a triangle to have three angles. And similarly the assertion that it is
possible for a subject not to have such and such a predicate, may have a sense
in which it includes the Impossible.
The Modal may be expressed in two forms. In
the first of these, the mode itself constitutes the predicate, having for its
subject the proposition whose copula it affects, e.g. 'That man should be
mortal is necessary,' 'that a bird should have gay plumage is possible.' Modals of this form are all singular, since
the subject is not a term, but a proposition taken as a whole. Nevertheless the
modes of necessity and impossibility are a sure sign that the proposition in
question is universal; while on the other hand the mode of possibility, in the
sense of merely possible (as distinguished from the case in which possibility
is predicated of a necessary judgment) is indicative of a particular
proposition. In the second forms of the Modal the mode qualifies the copula
itself: e.g. All triangles are necessarily three-angled. Modals of this form
are not singular but take their quantity from their subject.
* Kant's division of
Modals is based, not on the objective relation of the predicate to the subject,
but on the subjective certainty of the thinker. He divides judgments into the Problematic,
i.e. 'S may be P,' the Assertoric, i.e. 'S is P.' and the Apodictic,
i.e. 'S must be P.' Of the problematic judgment he says that it expresses
"a free choice of admitting such a proposition, and a purely optional
admission of it into the under 'standing.' The assertoric judgment "
implies logical reality or 'truth." The apodictic gives us the same
judgment as the assertoric, when it is recognized as determined by the formal
laws of the understanding, and therefore as subjectively necessary (see Ch. i. note
(5)). In regard to this division it may
be said in the first place that such a proposition as 'S may be P' is of no
value to the logician. It is a mere declaration of ignorance, and not a
judgment at all. Secondly, since the
apodictic judgment enunciates the same truth as the assertoric, merely
involving that the speaker recognizes more clearly the subjective necessity
under which he lies of thus judging, there is no reason why he should not
express the assertoric in the same form as the apodictic 'S must be P.' The root error of this view is the failure to
see that the copula is not a mere mental act of union, but expresses the
objective connection between the subject and its attribute in the real order
(see Ch. 9, § 4).
The influence of the
Kantian system is to be seen in many recent logicians. We are not infrequently
told that when a truth is styled 'necessary,' nothing more is meant than a
'necessity of thought,' and that the term has no reference to the real order.
Mr. Bradley tells us, "a necessary truth is really an inference, and an
inference is a necessary truth " (Principles, p. 225). Similarly
Mr. Bosanquet writes, "Every necessary truth must, in so far as it is
necessary, present itself as the conclusion from an antecedent" (Logic,
II. 222). Such a view as this must needs
be fatal to any hope of attaining certitude in philosophy or science. The existence of any necessary first
principles is denied. But where there are no necessary principles, there can be
no necessity in the conclusion derived from them [N13].
§ 10. Reduction of Propositions to Logical Form. The sentences employed in literature and in
ordinary conversation exhibit considerable variety of form and complexity of
structure. It is possible however to analyse them and express them in the shape
of A E 1 0 propositions. This process is styled their reduction to logical
form. By submitting sentences to this analysis we reach the simple elements of
thought, which are contained in them. It is plain that this is very different
from grammatical analysis into parts of speech. That process is concerned not
with thoughts but with words. The preceding paragraphs should have rendered the
task of reduction comparatively easy. Its essential feature is to obtain propositions,
consisting of (1) a subject with the sign of quantity attached; (2) a copula,
which must be of the form is or are (or is not, are not), and (3)
a predicate.
·
We find the subject by
putting to ourselves the question, Of what or of whom is this statement made?
·
We find the quantity of
the subject by asking, Is the assertion made of the whole extension of the
subject, or of but part of it?
·
We find the predicate by
enquiring, What is it that is asserted of the subject?
These three points must
always be considered, whenever the analysis of a sentence is attempted. Two
other cautions may be added. First, that it is well, whenever it is possible,
to express the predicate as an attribute, i.e. adjectivally, in order to bring
out the true meaning of the proposition: e.g. the form 'All flattery is to be
avoided' is better than 'All flattery is a thing to be avoided.' Secondly, that
wherever it is necessary to introduce a time determination, this must be done
in the predicate as in No. (7) below. The copula must always be in the present
tense.
A few examples will
illustrate the process: --
(1)
'Fools despise wisdom.'
This will become, 'All
fools are despisers of wisdom' (A).
(2)
'All's well that ends well.'
This will be, 'All that
ends well is well' (A).
(3) 'Firm
at his dangerous post he stood.'
This in logical form is,
'He is standing firm at his dangerous post' (A).
(4)
'As a man sows, so shall he reap.'
Here we have a relative
sentence. The two clauses of these propositions give us the terms of a
relation. Where the words 'As . . . so' are employed to introduce the clauses,
the relation is one of likeness. The analysis gives us, 'In every instance, the
character of a man's harvest is like the character of his sowing' (A).
If the words 'Where . . .
there' are used, we have a relation of place: if 'When . . . then,' a relation
of time.
(5)
'Where thy treasure is, there will thy heart be also.'
Logically, this is, 'In
every instance, the place of your treasure is the place of your heart' (A).
(6)
'Love is akin to madness.'
Here the subject is used
without any sign of quantity, but clearly stands for the whole denotation of
the term. The pro position becomes, 'All cases of love are akin to
madness.'
Where we have compound or
exponible propositions, they need resolving into their component parts, e.g.
(7)
'Lions and tigers once lived wild in
This gives us four
propositions.
'Some
lions are animals, that once lived wild in
'Some
tigers are animals, that once lived wild in
'No
lions are living wild in
'No
tigers are living wild in
(8) 'Only the just enjoy
peace of mind.'
This is resolved into:
'Some
of the just are enjoying peace of mind' (I).
'None,
who are not just, are enjoying peace of mind' (E).
(9)
'All save he had fled.'
Here we have a case,
where the full force of the proposition cannot be brought out in the analysis,
since we have no universal term by which to designate all the remainder. The
reduction gives: --
'He is
not fleeing' (E).
'Some
(the rest) are fleeing' (I).
(10)
'The great is not good, but the good is great.'
Notice should be taken of
the 'reduplicative' use of the word 'great' in the first clause. It signifies 'the
great as such,' or 'the great, just in so far as it is great.' This must be
expressed in the analysis
'The
great, merely in virtue of its greatness, is not good' (E).
'The
good is great' (A).
Other Signs of Quantity. It will be
useful to mention a few other modes of expressing Quantity besides those we
have already noticed.
A.
The universal affirmative is occasionally denoted by the expressions, Any,
Whoever, He who, Always, In every case.
I
may be denoted by A few, Certain, Often. Generally, Most.
E
may be expressed by the word Never.
O
has equivalents in A few . . . not, Not all . . . are, All are not, Few,
Certain . . . not.
The word Most has
been placed as one of the equivalents of I. The proposition 'Most S's are P'
signifies that 'Some (more than half) S's are P,' but does not necessarily
imply in addition that 'Some S's are not P.' It merely signifies that the
majority of instances have been examined, and found to possess the attribute
P. Thus we might say, 'Most English
flowering-plants are dicotyledonous,' without desiring to commit ourselves to
any opinion as to the whole flora: or again, after looking at seven cards out
of a hand at whist, we might say, 'Most of the cards in this hand are
court-cards,' knowing that it was possible they might all prove to be so. Similarly we might say, 'Few English
flowering-p]ants are monocotyledonous,' even if we were ignorant whether there
were any of that character. Hence Few is commonly reckoned as merely a
sign of the proposition O [N14]. The words Hardly any, Scarcely any
are also regarded as equivalent to O. The use of All with a negative to
signify O should be carefully noticed. 'Not all the crew were lost,' will be
expressed 'Some of the crew were not lost.'
Special note should be
taken as to whether the terms, to which words such as All, A few,
etc. etc. are attached, are used distributively or collectively (Ch. 2, § 15). Wherever the use is collective the
proposition is singular. 'All the men built a raft' is a case in point. The
proposition may be expressed, 'The whole body of men is building a
raft.'
§
11. Hypothetical Propositions. Besides the Categorical propositions which we
have hitherto been considering, there is another class of judgments called Conditional.
These are distinguished from Categoricals by the fact that in them the
predicate is not asserted absolutely of the subject. They are divided into two
classes, termed Hypothetical and Disjunctive. In the present section we are
concerned with the Hypothetical.
A Hypothetical Proposition is one in which the predication made in one
proposition, is asserted as a consequence from that made in another. The proposition on which the truth of the
other depends, is called the Antecedent: that which follows on its
admission, is called the Consequent.
Thus in the proposition, 'If the shepherd be negligent, the sheep go
astray,' the antecedent is ' If the shepherd be negligent ' ; the consequent is
'the sheep go astray.' It will be seen
that neither part of the proposition is independently asserted as true. We do
not affirm that 'the shepherd is negligent,' nor yet that ' the sheep go
astray.' It is the nexus between the two, the dependence of consequent on
antecedent, which is affirmed.
There are two forms in which the hypothetical
sentence may be expressed. These are (1) If A is B, C is D, and (2) If S is M,
it is P. Judgments constructed according to the first formula, may usually by a
little manipulation be expressed in the second form also. But it is incorrect
to say that the latter is a more fundamental type than the former.
Hypotheticals of the second form, can be
expressed categorically, by substituting in the place of 'If S is M, it is P,'
the form 'All S that is M is P,' or 'All SM is P.' Similarly for the
categorical 'All S is P,' we may write, ' If anything is S. it is P.'
Some writers on Logic
have maintained that the categorical and hypothetical propositions are in fact
equivalent. There can be no doubt that this opinion is erroneous. In the categorical
we state unconditionally that S is P. In the hypothetical we state that S is P,
if certain conditions are fulfilled. The constituent parts of the
categorical are related as subject and attribute: the parts of a hypothetical
are related as reason and consequent.
Nor is it only the mental forms that are different. The fact to be
expressed positively demands one form to the exclusion of the other. Such
propositions as 'Gold is yellow,' and 'If the King comes, a salute will be
fired,' are distorted when they are expressed as 'If anything is gold, it is
yellow,' and 'The case of the King's arrival is a case of firing a
salute.' In regard to the employment of
the one form in place of the other, Professor Case has well said: "Taking
the carelessly expressed propositions of ordinary life logicians do not
perceive that similar propositions are often differently expressed, e.g. ''I
being a man am mortal,' and 'If I am a man I am mortal': and conversely that
different judgments are often similarly expressed. In ordinary life we may say
'All men are mortal, . 'All candidates arriving five minutes late are fined.'
But of these universal propositions, the first expresses a categorical belief .
. . the other is a slipshod expression of the 'hypothetical belief, 'If any
candidates arrive late, they are 'fined.'" Encycl. Brit. (10th
ed.), vol. 30, p. 333, Art. Logic.
Quantity and Quality of Hypotheticals. All
hypothetical propositions are affirmative. If we desire to meet a hypothetical with
its negation, we must deny what it affirms. That is to say we must deny the
nexus between the antecedent and consequent. This is done by the form 'Although
S is M, it need not be P.' The negative of 'If he is poor, he is uneducated,'
is 'Although he is poor, he may not be uneducated.' These negative forms,
however, are not themselves hypotheticals: for they do not assert the
dependence of consequent on antecedent.
There can be no differences of quantity in
hypotheticals, because there is no question of extension. The affirmation, as
we have seen, relates solely to the nexus between the two members of the
proposition. Hence every hypothetical is
singular.
§
12. Disjunctive Propositions.
A Disjunctive Proposition is one which makes an alternative predication.
Disjunctives like Hypotheticals are of two
forms: (1) Either A is B, or C is D; and (2) S is either P or Q, e.g. 'Either
the general was incompetent or his subordinates were disobedient,' 'Religions
are either false or true.'
It has been much disputed whether the
alternatives in a disjunctive are mutually exclusive or not in other words,
whether we not only know that one must be true, but also that if the one is
true, the other is certainly false. Thus supposing we are aware that 'S is
either P or Q,' and are then informed that 'S is P,' can we conclude that S is
not Q? We shall consider this point in a subsequent chapter (see below, Ch. 14,
§4).
The Disjunctive can be expressed by means of
Hypothetical propositions. If it be maintained that the disjunction is
exclusive, we need two hypothetical propositions to represent a disjunctive,
viz., (1) If S is P, it is not Q. (2) If S is not P, it is Q. If the mutual exclusiveness be denied, a
single hypothetical will suffice. viz., ' If S is not P, it is Q.'
Quantity and Quality of Disjunctives. By virtue of
their form all disjunctives are affirmative. The alternative is necessarily
asserted. However, a difference in quantity is possible. The proposition may be
of the form 'All S are P or Q'; or it may be particular, as, 'Some S are P or
Q'.
A form of proposition termed by the
Scholastics Conjunctive gives us what is practically the negative form
of the Disjunctive. Its formula is 'S is
not both P and Q,' 'The King is not both at
The whole terminology of Conditionals is in
confusion. We have followed that preferred by
Footnotes
[N1]
The question of the 'implication of existence' in the copula will be
more fully dealt with in Ch. 7. [Not
available yet – ed.]
[N2] Cf. St. Thomas, Summa Theol. I. Q. 53, Art. 12 [? English, Latin].
[N3] "C'est donc grace a.
1'abstraction intellectuelle que les choses sont affirmables les unes des
autres, et peuvent faire fonction de predicat dans les propositions."
Mercier, § 31. And Themistius says on Arist.: "The sense-faculty gives us the phantasm of Socrates
walking as a single whole: the intellect abstracts, and separates Socrates
on the one hand from is walking on
the other." Them. 202, 10 cited in Rodier, Traite de I'ame. II. 471.
[N4] When the
predicate is not something perceived by the senses, the process of judgment is
not by direct abstraction. Experience stores the mind with concepts. On one
ground or another we recognize one of these in the subject, and judge, e.g.,
'Caius is a coward,' 'Balbus is wise.'
[N5] On the unity of the
act of judgment Aristotle is explicit. He calls it "a synthesis of
concepts as though they were but one" and de Anima, III. c. 6, §
41 [English].
[N6]
See below, Ch. 7, §§ 1,4, Ch.9, §4, Ch.10, §5 [Not
available yet – ed.].
[N7] The reason for this is easy to see, when
the word has the significance "some only." it is really equivalent to
two propositions, one affirmative, one negative. When it is used in reference
to certain definite individuals, A. B. C, it is equivalent to so many singular
judgments. It is only in its indeterminate reference that it is an independent
and elementary thought. form.
[N8] Cf. St. Thomas, Opusc. 44 Summa
Totius Logicae, de Interp. C 6 [Latin]. It is of his
work but not in his hand.
[N9] An attribute which is thus connected with
the subject by necessary resultancy is termed a property of that subject. The
term will be fully discussed in Ch. 8, §1.
[N10]
Analytic Propositions were termed by the Scholastics 'Propositiones
per se notae'. cf. Arist. An. Post. I. C. 4, § 3 [English],
and
[N11] If without axioms it is impossible to
infer," says Mr. Bradley, "I wonder where all the axioms can have
come from" (Principles of Logic, p. 227). There is no mystery about axioms. They are
Analytic propositions in which the connection of subject and predicate is
immediately evident. Cf.
[N12] Summa Totius Logicae, Tract. 6,
c.13 [Latin].
"Notandum quod possibile dupliciter potest sumi: vel in toto suo
significato, et tunc comprehendit necessarium et contingens. . . . AIio modo,
sumitur solum pro contingentibus." Similarly Aristotle tells us that
'possible' when used in regard of what is necessary, is employed in a distinct
sense'
[N13]
Cf. Rickaby, General Metaphysics, p. x8o.
[N14]
The difference between A few and Few is to be observed. A
few is equivalent to some. "Few," says Mr. Keynes,
" has a negative force. And 'Few S's are P' may be regarded as equivalent
to 'Most S's are not P.'"
Copyright © E.D.Buckner 2005.