GENERAL ANALYSIS OF PROPOSITIONS
Sec. 1. Since Logic discusses the proof or disproof, or (briefly) the
testing of propositions, we must begin by explaining their nature. A
proposition, then, may first be described in the language of grammar as
_a sentence indicative_; and it is usually expressed in the present
tense.
It is true that other kinds of sentences, optative, imperative,
interrogative, exclamatory, if they express or imply an assertion, are
not beyond the view of Logic; but before treating such sentences, Logic,
for greater precision, reduces them to their equivalent sentences
indicative. Thus, _I wish it were summer_ may be understood to mean,
_The coming of summer is an object of my desire_. _Thou shalt not kill_
may be interpreted as _Murderers are in danger of the judgment_.
Interrogatories, when used in argument, if their form is affirmative,
have negative force, and affirmative force if their form is negative.
Thus, _Do hypocrites love virtue?_ anticipates the answer, _No_. _Are
not traitors the vilest of mankind?_ anticipates the answer, _Yes_. So
that the logical form of these sentences is, _Hypocrites are not lovers
of virtue_; _Traitors are the vilest of mankind_. Impersonal
propositions, such as _It rains_, are easily rendered into logical forms
of equivalent meaning, thus: _Rain is falling_; or (if that be
tautology), _The clouds are raining_. Exclamations may seem capricious,
but are often part of the argument. _Shade of Chatham!_ usually means
_Chatham, being aware of our present foreign policy, is much
disgusted_. It is in fact, an appeal to authority, without the
inconvenience of stating what exactly it is that the authority declares.
Sec. 2. But even sentences indicative may not be expressed in the way most
convenient to logicians. _Salt dissolves in water_ is a plain enough
statement; but the logician prefers to have it thus: _Salt is soluble in
water_. For he says that a proposition is analysable into three
elements: (1) a Subject (as _Salt_) about which something is asserted or
denied; (2) a Predicate (as _soluble in water_) which is asserted or
denied of the Subject, and (3) the Copula (_is_ or _are_, or _is not_ or
_are not_), the sign of relation between the Subject and Predicate. The
Subject and Predicate are called the Terms of the proposition: and the
Copula may be called the sign of predication, using the verb 'to
predicate' indefinitely for either 'to affirm' or 'to deny.' Thus _S is
P_ means that the term _P_ is given as related in some way to the term
_S_. We may, therefore, further define a Proposition as 'a sentence in
which one term is predicated of another.'
In such a proposition as _Salt dissolves_, the copula (_is_) is
contained in the predicate, and, besides the subject, only one element
is exhibited: it is therefore said to be _secundi adjacentis_. When all
three parts are exhibited, as in _Salt is soluble_, the proposition is
said to be _tertii adjacentis_.
For the ordinary purposes of Logic, in predicating attributes of a thing
or class of things, the copula _is_, or _is not_, sufficiently
represents the relation of subject and predicate; but when it is
desirable to realise fully the nature of the relation involved, it may
be better to use a more explicit form. Instead of saying
_Salt--is--soluble_, we may say _Solubility--coinheres with--the nature
of salt_, or _The putting of salt in water--is a cause of--its
dissolving_: thus expanding the copula into a full expression of the
relation we have in view, whether coinherence or causation.
Sec. 3. The sentences of ordinary discourse are, indeed, for the most
part, longer and more complicated than the logical form of propositions;
it is in order to prove them, or to use them in the proof of other
propositions, that they are in Logic reduced as nearly as possible to
such simple but explicit expressions as the above (_tertii adjacentis_).
A Compound Proposition, reducible to two or more simple ones, is said to
be exponible.
The modes of compounding sentences are explained in every grammar-book.
One of the commonest forms is the copulative, such as _Salt is both
savoury and wholesome_, equivalent to two simple propositions: _Salt is
savoury; Salt is wholesome. Pure water is neither sapid nor odorous_,
equivalent to _Water is not sapid; Water is not odorous_. Or, again,
_Tobacco is injurious, but not when used in moderation_, equivalent to
_Much tobacco is injurious; a little is not_.
Another form of Exponible is the Exceptive, as _Kladderadatsch is
published daily, except on week-days_, equivalent to _Kladderadatsch is
published on Sunday; it is not published any other day_. Still another
Exponible is the Exclusive, as _Only men use fire_, equivalent to _Men
are users of fire; No other animals are_. Exceptive and exclusive
sentences are, however, equivalent forms; for we may say,
_Kladderadatsch is published only on Sunday_; and _No animals use fire,
except men_.
There are other compound sentences that are not exponible, since, though
they contain two or more verbal clauses, the construction shows that
these are inseparable. Thus, _If cats are scarce, mice are plentiful_,
contains two verbal clauses; but _if cats are scarce_ is conditional,
not indicative; and _mice are plentiful_ is subject to the condition
that _cats are scarce_. Hence the whole sentence is called a Conditional
Proposition. For the various forms of Conditional Propositions see chap.
v. Sec. 4.
But, in fact, to find the logical force of recognised grammatical forms
is the least of a logician's difficulties in bringing the discourses of
men to a plain issue. Metaphors, epigrams, innuendoes and other figures
of speech present far greater obstacles to a lucid reduction whether for
approval or refutation. No rules can be given for finding everybody's
meaning. The poets have their own way of expressing themselves;
sophists, too, have their own way. And the point often lies in what is
unexpressed. Thus, "barbarous nations make, the civilised write
history," means that civilised nations do not make history, which none
is so brazen as openly to assert. Or, again, "Alcibiades is dead, but X
is still with us"; the whole meaning of this 'exponible' is that X would
be the lesser loss to society. Even an epithet or a suffix may imply a
proposition: _This personage_ may mean _X is a pretentious nobody_.
How shall we interpret such illusive predications except by cultivating
our literary perceptions, by reading the most significant authors until
we are at home with them? But, no doubt, to disentangle the compound
propositions, and to expand the abbreviations of literature and
conversation, is a useful logical exercise. And if it seem a laborious
task thus to reduce to its logical elements a long argument in a speech
or treatise, it should be observed that, as a rule, in a long discourse
only a few sentences are of principal importance to the reasoning, the
rest being explanatory or illustrative digression, and that a close
scrutiny of these cardinal sentences will frequently dispense us from
giving much attention to the rest.
Sec. 4. But now, returning to the definition of a Proposition given in Sec. 2,
that it is 'a sentence in which one term is predicated of another,' we
must consider what is the import of such predication. For the
definition, as it stands, seems to be purely Nominalist. Is a
proposition nothing more than a certain synthesis of words; or, is it
meant to correspond with something further, a synthesis of ideas, or a
relation of facts?
Conceptualist logicians, who speak of judgments instead of
propositions, of course define the judgment in their own language.
According to Hamilton, it is "a recognition of the relation of
congruence or confliction in which two concepts stand to each other." To
lighten the sentence, I have omitted one or two qualifications
(Hamilton's _Lectures on Logic_, xiii.). "Thus," he goes on "if we
compare the thoughts _water_, _iron_, and _rusting_, we find them
congruent, and connect them into a single thought, thus: _water rusts
iron_--in that case we form a judgment." When a judgment is expressed in
words, he says, it is called a proposition.
But has a proposition no meaning beyond the judgment it expresses? Mill,
who defines it as "a portion of discourse in which a predicate is
affirmed or denied of a subject" (_Logic_, Book 1., chap. iv. Sec. 1.),
proceeds to inquire into the import of propositions (Book 1., chap. v.),
and finds three classes of them: (a) those in which one proper name is
predicated of another; and of these Hobbes's Nominalist definition is
adequate, namely, that a proposition asserts or denies that the
predicate is a name for the same thing as the subject, as _Tully is
Cicero_.
(b) Propositions in which the predicate means a part (or the whole) of
what the subject means, as _Horses are animals_, _Man is a rational
animal_. These are Verbal Propositions (see below: chap. v. Sec. 6), and
their import consists in affirming or denying a coincidence between the
meanings of names, as _The meaning of 'animal' is part of the meaning of
'horse.'_ They are partial or complete definitions.
But (c) there are also Real Propositions, whose predicates do not mean
the same as their subjects, and whose import consists in affirming or
denying one of five different kinds of matter of fact: (1) That the
subject exists, or does not; as if we say _The bison exists_, _The great
auk is extinct_. (2) Co-existence, as _Man is mortal_; that is, _the
being subject to death coinheres with the qualities on account of which
we call certain objects men_. (3) Succession, as _Night follows day_.
(4) Causation (a particular kind of Succession), as _Water rusts iron_.
(5) Resemblance, as _The colour of this geranium is like that of a
soldier's coat_, or _A = B_.
On comparing this list of real predications with the list of logical
relations given above (chap. i. Sec. 5 (a)), it will be seen that the two
differ only in this, that I have there omitted simple Existence. Nothing
simply exists, unrelated either in Nature or in knowledge. Such a
proposition as _The bison exists_ may, no doubt, be used in Logic
(subject to interpretation) for the sake of custom or for the sake of
brevity; but it means that some specimens are still to be found in N.
America, or in Zoological gardens.
Controversy as to the Import of Propositions really turns upon a
difference of opinion as to the scope of Logic and the foundations of
knowledge. Mill was dissatisfied with the "congruity" of concepts as the
basis of a judgment. Clearly, mere congruity does not justify belief. In
the proposition _Water rusts iron_, the concepts _water_, _rust_ and
_iron_ may be congruous, but does any one assert their connection on
that ground? In the proposition _Murderers are haunted by the ghosts of
their victims_, the concepts _victim_, _murderer_, _ghost_ have a high
degree of congruity; yet, unfortunately, I cannot believe it: there
seems to be no such cheap defence of innocence. Now, Mill held that
Logic is concerned with the grounds of belief, and that the scope of
Logic includes Induction as well as Deduction; whereas, according to
Hamilton, Induction is only Modified Logic, a mere appendix to the
theory of the "forms of thought as thought." Indeed, Mill endeavoured in
his _Logic_ to probe the grounds of belief deeper than usual, and
introduced a good deal of Metaphysics--either too much or not
enough--concerning the ground of axioms. But, at any rate, his great
point was that belief, and therefore (for the most part) the Real
Proposition, is concerned not merely with the relations of words, or
even of ideas, but with matters of fact; that is, both propositions and
judgments point to something further, to the relations of things which
we can examine, not merely by thinking about them (comparing them in
thought), but by observing them with the united powers of thought and
perception. This is what convinces us that _water rusts iron_: and the
difficulty of doing this is what prevents our feeling sure that
_murderers are haunted by the ghosts of their victims_. Hence, although
Mill's definition of a proposition, given above, is adequate for
propositions in general; yet that kind of proposition (the Real) with
regard to which Logic (in Mill's view) investigates the conditions of
proof, may be more explicitly and pertinently defined as 'a predication
concerning the relation of matters of fact.'
Sec. 5. This leads to a very important distinction to which we shall often
have to refer in subsequent pages--namely, the distinction between the
Form and the Matter of a proposition or of an argument. The distinction
between Form and Matter, as it is ordinarily employed, is easily
understood. An apple growing in the orchard and a waxen apple on the
table may have the same shape or form, but they consist of different
materials; two real apples may have the same shape, but contain distinct
ounces of apple-stuff, so that after one is eaten the other remains to
be eaten. Similarly, tables may have the same shape, though one be made
of marble, another of oak, another of iron. The form is common to
several things, the matter is peculiar to each. Metaphysicians have
carried the distinction further: apples, they say, may have not only the
same outward shape, but the same inward constitution, which, therefore,
may be called the Form of apple-stuff itself--namely, a certain
pulpiness, juiciness, sweetness, etc.; qualities common to all dessert
apples: yet their Matter is different, one being here, another
there--differing in place or time, if in nothing else. The definition of
a species is the form of every specimen of it.
To apply this distinction to the things of Logic: it is easy to see how
two propositions may have the same Form but different Matter: not using
'Form' in the sense of 'shape,' but for that which is common to many
things, in contrast with that which is peculiar to each. Thus, _All male
lions are tawny_ and _All water is liquid at 50 deg. Fahrenheit_, are two
propositions that have the same form, though their matter is entirely
different. They both predicate something of the whole of their subjects,
though their subjects are different, and so are the things predicated of
them. Again, _All male lions have tufted tails_ and _All male lions have
manes_, are two propositions having the same form and, in their
subjects, the same matter, but different matter in their predicates. If,
however, we take two such propositions as these: _All male lions have
manes_ and _Some male lions have manes_, here the matter is the same in
both, but the form is different--in the first, predication is made
concerning _every_ male lion; in the second of only _some_ male lions;
the first is _universal_, the second is _particular_. Or, again, if we
take _Some tigers are man-eaters_ and _Some tigers are not man-eaters_,
here too the matter is the same, but the form is different; for the
first proposition is _affirmative_, whilst the second is _negative_.
Sec. 6. Now, according to Hamilton and Whately, pure Logic has to do only
with the Form of propositions and arguments. As to their Matter, whether
they are really true in fact, that is a question, they said, not for
Logic, but for experience, or for the special sciences. But Mill desired
so to extend logical method as to test the material truth of
propositions: he thought that he could expound a method by which
experience itself and the conclusions of the special sciences may be
examined.
To this method it may be objected, that the claim to determine Material
Truth takes for granted that the order of Nature will remain unchanged,
that (for example) water not only at present is a liquid at 50 deg.
Fahrenheit, but will always be so; whereas (although we have no reason
to expect such a thing) the order of Nature may alter--it is at least
supposable--and in that event water may freeze at such a temperature.
Any matter of fact, again, must depend on observation, either directly,
or by inference--as when something is asserted about atoms or ether. But
observation and material inference are subject to the limitations of our
faculties; and however we may aid observation by microscopes and
micrometers, it is still observation; and however we may correct our
observations by repetition, comparison and refined mathematical methods
of making allowances, the correction of error is only an approximation
to accuracy. Outside of Formal Reasoning, suspense of judgment is your
only attitude.
But such objections imply that nothing short of absolute truth has any
value; that all our discussions and investigations in science or social
affairs are without logical criteria; that Logic must be confined to
symbols, and considered entirely as mental gymnastics. In this book
prominence will be given to the character of Logic as a formal science,
and it will also be shown that Induction itself may be treated formally;
but it will be assumed that logical forms are valuable as representing
the actual relations of natural and social phenomena.
Sec. 7. Symbols are often used in Logic instead of concrete terms, not only
in Symbolic Logic where the science is treated algebraically (as by Dr.
Venn in his _Symbolic Logic_), but in ordinary manuals; so that it may
be well to explain the use of them before going further.
It is a common and convenient practice to illustrate logical doctrines
by examples: to show what is meant by a Proposition we may give _salt is
soluble_, or _water rusts iron:_ the copulative exponible is exemplified
by _salt is savoury and wholesome_; and so on. But this procedure has
some disadvantages: it is often cumbrous; and it may distract the
reader's attention from the point to be explained by exciting his
interest in the special fact of the illustration. Clearly, too, so far
as Logic is formal, no particular matter of fact can adequately
illustrate any of its doctrines. Accordingly, writers on Logic employ
letters of the alphabet instead of concrete terms, (say) _X_ instead of
_salt_ or instead of _iron_, and (say) _Y_ instead of _soluble_ or
instead of _rusted by water_; and then a proposition may be represented
by _X is Y_. It is still more usual to represent a proposition by _S is
(or is not) P, S_ being the initial of Subject and _P_ of Predicate;
though this has the drawback that if we argue--_S is P_, therefore _P is
S_, the symbols in the latter proposition no longer have the same
significance, since the former subject is now the predicate.
Again, negative terms frequently occur in Logic, such as _not-water_, or
_not-iron_, and then if _water_ or _iron_ be expressed by _X_, the
corresponding negative may be expressed by _x_; or, generally, if a
capital letter stand for a positive term, the corresponding small letter
represents the negative. The same device may be adopted to express
contradictory terms: either of them being _X_, the other is _x_ (see
chap. iv., Sec.Sec. 7-8); or the contradictory terms may be expressed by _x_
and _[x]_, _y_ and _[y]_.
And as terms are often compounded, it may be convenient to express them
by a combination of letters: instead of illustrating such a case by
_boiling water_ or _water that is boiling_, we may write _XY_; or since
positive and negative terms may be compounded, instead of illustrating
this by _water that is not boiling_, we may write _Xy_.
The convenience of this is obvious; but it is more than convenient; for,
if one of the uses of Logic be to discipline the power of abstract
thought, this can be done far more effectually by symbolic than by
concrete examples; and if such discipline were the only use of Logic it
might be best to discard concrete illustrations altogether, at least in
advanced text-books, though no doubt the practice would be too severe
for elementary manuals. On the other hand, to show the practical
applicability of Logic to the arguments and proofs of actual life, or
even of the concrete sciences, merely symbolic illustration may be not
only useless but even misleading. When we speak of politics, or poetry,
or species, or the weather, the terms that must be used can rarely have
the distinctness and isolation of X and Y; so that the perfunctory use
of symbolic illustration makes argument and proof appear to be much
simpler and easier matters than they really are. Our belief in any
proposition never rests on the proposition itself, nor merely upon one
or two others, but upon the immense background of our general knowledge
and beliefs, full of circumstances and analogies, in relation to which
alone any given proposition is intelligible. Indeed, for this reason, it
is impossible to illustrate Logic sufficiently: the reader who is in
earnest about the cogency of arguments and the limitation of proofs, and
is scrupulous as to the degrees of assent that they require, must
constantly look for illustrations in his own knowledge and experience
and rely at last upon his own sagacity.