a narrowing and cramping effect.

CAN ALL PROPOSITIONS BE REDUCED TO THE SYLLOGISTIC FORM? Probably: but
this is a purely scientific inquiry, collateral to Practical Logic.
The concern of Practical Logic is chiefly with forms of proposition
that favour inaccuracy or inexactness of thought. When there is no
room for ambiguity or other error, there is no virtue in artificial
syllogistic form. The attempt so to reduce any and every proposition
may lead, however, to the study of what Mr. Bosanquet happily calls
the "Morphology" of Judgment, Judgment being the technical name for
the mental act that accompanies the utterance of a proposition. Even
in such sentences as "How hot it is!" or "It rains," the rudiment of
subject and predicate may be detected. When a man says "How hot it
is," he conveys the meaning, though there is no definitely formed
subject in his mind, that the outer world at the moment of his
speaking has a certain quality or attribute. So with "It rains". The
study of such examples in their context, however, reveals the fact
that the same form of Common speech may cover different subjects and
predicates in different connexions. Thus in the argument:--

  "Whatever is, is best.
    It rains!"--

the Subject is _Rain_ and the Predicate _is now_, "is at the present
time," "is in the class of present events".

    [Footnote 1: Remember that when we speak of a general name,
    we do not necessarily mean a single word. A general name,
    logically viewed, is simply the name of a _genus_, kind, or
    class: and whether this is single-worded or many-worded
    is, strictly speaking, a grammatical question. "Man,"
    "man-of-ability," "man-of-ability-and-courage,"
    "man-of-ability-and-courage-and-gigantic-stature,"
    "man-who-fought-at-Marathon"--these are all general names
    in their logical function. No matter how the constitutive
    properties of the class are indicated, by one word or in
    combination, that word or combination is a general name. In
    actual speech we can seldom indicate by a single word the
    meaning predicated.]

    [Footnote 2: The objection taken to the word "indefinite,"
    that the quantity of particular propositions is indefinite,
    _some_ meaning any quantity less than all, is an example of
    the misplaced and frivolous subtlety that has done so much
    to disorder the tradition of Logic. By "indefinite" is
    simply meant not definitely expressed as either Universal
    or Particular, Total or Partial. The same objection might be
    taken to any word used to express the distinction: the degree
    of quantity in Some S is not "designate" any more than it is
    "definite" or "dioristic".]

    [Footnote 3: _Generally._ In this word we have an instance of
    the frequent conflict between the words of common speech and
    logical terminology. How it arises shall be explained in next
    chapter. A General proposition is a synonym for a Universal
    proposition (if the forms A and E are so termed): but
    "generally" in common speech means "for the most part," and is
    represented by the symbol of particular quantity, _Some_.]

    [Footnote 4: With some logicians it is a mechanical rule in
    reducing to syllogistic form to treat as I or O all sentences
    in which there is no formal expression of quantity. This is to
    err on the safe side, but common speakers are not so guarded,
    and it is to be presumed rather that they have a universal
    application in their minds when they do not expressly
    qualify.]

III.--SOME TECHNICAL DIFFICULTIES.

_The formula for_ EXCLUSIVE PROPOSITIONS. "None but the brave deserve
the fair": "No admittance except on business": "Only Protestants can
sit on the throne of England".

These propositions exemplify different ways in common speech of naming
a subject _exclusively_, the predication being made of all outside a
certain term. "None that are not brave, etc.;" "none that are not on
business, etc.;" "none that are not Protestants, etc.". No not-S is
P. It is only about all outside the given term that the universal
assertion is made: we say nothing universally about the individuals
within the term: we do not say that all Protestants are eligible, nor
that all persons on business are admitted, nor that every one of the
brave deserves the fair. All that we say is that the possession of the
attribute named is an indispensable condition: a person may possess
the attribute, and yet on other grounds may not be entitled to the
predicate.

The justification for taking special note of this form in Logic is
that we are apt by inadvertence to make an inclusive inference from
it. Let it be said that None but those who work hard can reasonably
expect to pass, and we are apt to take this as meaning that all who
work hard may reasonably expect to pass. But what is denied of every
Not-S is not necessarily affirmed of every S.

_The expression of_ TENSE or TIME _in the Syllogistic Forms_. Seeing
that the Copula in S is P or S is in P does not express time, but only
a certain relation between S and P, the question arises Where are we
to put time in the analytic formula? "Wheat is dear;" "All had fled;"
time is expressed in these propositions, and our formula should
render the whole content of what is given. Are we to include it in
the Predicate term or in the Subject term? If it must not be left out
altogether, and we cannot put it with the copula, we have a choice
between the two terms.

It is a purely scholastic question. The common technical treatment is
to view the tense as part of the predicate. "All had fled," All S is
P, _i.e._, the whole subject is included in a class constituted on the
attributes of flight at a given time. It may be that the Predicate is
solely a predicate of time. "The Board met yesterday at noon." S is P,
_i.e._, the meeting of the Board is one of the events characterised by
having happened at a certain time, agreeing with other events in that
respect.

But in some cases the time is more properly regarded as part of the
subject. _E.g._, "Wheat is dear". S does not here stand for wheat
collectively, but for the wheat now in the market, the wheat of the
present time: it is concerning this that the attribute of dearness is
predicated; it is this that is in the class of dear things.

_The expression of_ MODALITY _in the Syllogistic Forms_. Propositions
in which the predicate is qualified by an expression of necessity,
contingency, possibility or impossibility [_i.e._, in English by
_must_, _may_, _can_, or _cannot_], were called in Mediæval Logic
_Modal_ Propositions. "Two and two _must_ make four." "Grubs _may_
become butterflies." "Z _can_ paint." "Y _cannot_ fly."

There are two recognised ways of reducing such propositions to the
form S is P. One is to distinguish between the _Dictum_ and the
_Mode_, the proposition and the qualification of its certainty, and
to treat the _Dictum_ as the Subject and the _Mode_ as the Predicate.
Thus: "That two and two make four is necessary"; "That Y can fly is
impossible".

The other way is to treat the Mode as part of the predicate.
The propriety of this is not obvious in the case of Necessary
propositions, but it is unobjectionable in the case of the other three
modes. Thus: "Grubs are things that have the potentiality of becoming
butterflies"; "Z has the faculty of painting"; "Y has not the faculty
of flying".

The chief risk of error is in determining the quantity of the subject
about which the Contingent or Possible predicate is made. When it is
said that "Victories may be gained by accident," is the predicate made
concerning All victories or Some only? Here we are apt to confuse the
meaning of the contingent assertion with the matter of fact on which
in common belief it rests. It is true only that some victories have
been gained by accident, and it is on this ground that we assert in
the absence of certain knowledge concerning any victory that it
may have been so gained. The latter is the effect of the contingent
assertion: it is made about any victory in the absence of certain
knowledge, that is to say, formally about all.

The history of Modals in Logic is a good illustration of intricate
confusion arising from disregard of a clear traditional definition.
The treatment of them by Aristotle was simple, and had direct
reference to tricks of disputation practised in his time. He specified
four "modes," the four that descended to mediæval logic, and he
concerned himself chiefly with the import of contradicting these
modals. What is the true contradictory of such propositions as, "It
is possible to be" ([Greek: dynaton einai]), "It admits of being"
([Greek: endechetai einai]), "It must be" ([Greek: anankaion einai]),
"It is impossible to be" ([Greek: adynaton einai])? What is implied in
saying "No" to such propositions put interrogatively? "Is it possible
for Socrates to fly?" "No." Does this mean that it is not possible for
Socrates to fly, or that it is possible for Socrates not to fly?

A disputant who had trapped a respondent into admitting that it is
possible for Socrates not to fly, might have pushed the concession
farther in some such way as this: "Is it possible for Socrates not to
walk?" "Certainly." "Is it possible for him to walk?" "Yes." "When you
say that it is possible for a man to do anything do you not believe
that it is possible for him to do it?" "Yes." "But you have admitted
that it is possible for Socrates not to fly?"

It was in view of such perplexities as these that Aristotle set forth
the true contradictories of his four Modals. We may laugh at such
quibbles now and wonder that a grave logician should have thought them
worth guarding against. But historically this is the origin of the
Modals of Formal Logic, and to divert the names of them to signify
other distinctions than those between modes of qualifying the
certainty of a statement is to introduce confusion.

Thus we find "Alexander was a great general," given as an example of
a Contingent Modal, on the ground that though as a matter of fact
Alexander was so he might have been otherwise. It was not _necessary_
that Alexander should be a great general: therefore the proposition
is _contingent_. Now the distinction between Necessary truth and
Contingent truth may be important philosophically: but it is merely
confusing to call the character of propositions as one or the other by
the name of Modality. The original Modality is a mode of expression:
to apply the name to this character is to shift its meaning.

A more simple and obviously unwarrantable departure from tradition
is to extend the name Modality to any grammatical qualification of
a single verb in common speech. On this understanding "Alexander
conquered Darius" is given by Hamilton as a _Pure_ proposition, and
"Alexander conquered Darius honourably" as a _Modal_. This is a merely
grammatical distinction, a distinction in the mode of composing the
predicate term in common speech. In logical tradition Modality is a
mode of qualifying the certainty of an affirmation. "The conquest
of Darius by Alexander was honourable," or "Alexander in conquering
Darius was an honourable conqueror," is the syllogistic form of the
proposition: it is simply assertory, not qualified in any "mode".

There is a similar misunderstanding in Mr. Shedden's treatment of
"generally" as constituting a Modal in such sentences, as "Rivers
_generally_ flow into the sea". He argues that as _generally_ is not
part either of the Subject term or of the Predicate term, it must
belong to the Copula, and is therefore a _modal_ qualification of the
whole assertion. He overlooked the fact that the word "generally" is
an expression of Quantity: it determines the quantity of the Subject
term.

Finally it is sometimes held (_e.g._, by Mr. Venn) that the question
of Modality belongs properly to Scientific or Inductive Logic, and is
out of place in Formal Logic. This is so far accurate that it is
for Inductive Logic to expound the conditions of various degrees of
certainty. The consideration of Modality is pertinent to Formal Logic
only in so far as concerns special perplexities in the expression of
it. The treatment of it by Logicians has been rendered intricate by
torturing the old tradition to suit different conceptions of the end
and aim of Logic.

PART II.

DEFINITION.

CHAPTER I.

IMPERFECT UNDERSTANDING OF WORDS AND THE REMEDIES
THEREFOR.--DIALECTIC.--DEFINITION.

We cannot inquire far into the meaning of proverbs or traditional
sayings without discovering that the common understanding of general
and abstract names is loose and uncertain. Common speech is a
quicksand.

Consider how we acquire our vocabulary, how we pick up the words that
we use from our neighbours and from books, and why this is so soon
becomes apparent. Theoretically we know the full meaning of a name
when we know all the attributes that it connotes, and we are not
justified in extending it except to objects that possess all the
attributes. This is the logical ideal, but between the _ought to be_
of Logic and the _is_ of practical life, there is a vast difference.
How seldom do we conceive words in their full meaning! And who is to
instruct us in the full meaning? It is not as in the exact sciences,
where we start with a knowledge of the full meaning. In Geometry, for
example, we learn the definitions of the words used, _point_, _line_,
_parallel_, etc., before we proceed to use them. But in common speech,
we learn words first in their application to individual cases. Nobody
ever defined _good_ to us, or _fair_, or _kind_, or _highly educated_.
We hear the words applied to individual objects: we utter them in the
same connexion: we extend them to other objects that strike us as like
without knowing the precise points of likeness that the convention
of common speech includes. The more exact meaning we learn by
gradual induction from individual cases. _Ugly_, _beautiful_, _good_,
_bad_--we learn the words first as applicable to things and persons:
gradually there arises a more or less definite sense of what the
objects so designated have in common. The individual's extension of
the name proceeds upon what in the objects has most impressed him
when he caught the word: this may differ in different individuals; the
usage of neighbours corrects individual eccentricities. The child
in arms shouts _Da_ at the passing stranger who reminds him of his
father: for him at first it is a general name applicable to every man:
by degrees he learns that for him it is a singular name.

The mode in which words are learnt and extended may be studied most
simply in the nursery. A child, say, has learnt to say _mambro_ when
it sees its nurse. The nurse works a hand-turned sewing machine,
and sings to it as she works. In the street the child sees an
organ-grinder singing as he turns his handle: it calls _mambro_:
the nurse catches the meaning and the child is overjoyed. The
organ-grinder has a monkey: the child has an india-rubber monkey toy:
it calls this also _mambro_. The name is extended to a monkey in
a picture-book. It has a toy musical box with a handle: this also
becomes _mambro_, the word being extended along another line of
resemblance. A stroller with a French fiddle comes within the
denotation of the word: a towel-rail is also called _mambro_ from some
fancied resemblance to the fiddle. A very swarthy hunch-back _mambro_
frightens the child: this leads to the transference of the word to a
terrific coalman with a bag of coals on his back. In a short time
the word has become a name for a great variety of objects that have
nothing whatever common to all of them, though each is strikingly like
in some point to a predecessor in the series. When the application
becomes too heterogeneous, the word ceases to be of use as a sign and
is gradually abandoned, the most impressive meaning being the last
to go. In a child's vocabulary where the word _mambro_ had a run of
nearly two years, its last use was as an adjective signifying ugly or
horrible.

The history of such a word in a child's language is a type of what
goes on in the language of men. In the larger history we see similar
extensions under similar motives, checked and controlled in the same
way by surrounding usage.

It is obvious that to avoid error and confusion, the meaning or
connotation of names, the concepts, should somehow be fixed: names
cannot otherwise have an identical reference in human intercourse. We
may call this ideal fixed concept the LOGICAL CONCEPT: or we may call
it the SCIENTIFIC CONCEPT, inasmuch as one of the main objects of the
sciences is to attain such ideals in different departments of study.
But in actual speech we have also the PERSONAL CONCEPT, which varies
more or less with the individual user, and the POPULAR or VERNACULAR
CONCEPT, which, though roughly fixed, varies from social sect to
social sect and from generation to generation.

The variations in Popular Concepts may be traced in linguistic
history. Words change with things and with the aspects of things,
as these change in public interest and importance. As long as the
attributes that govern the application of words are simple, sensible
attributes, little confusion need arise: the variations are matters of
curious research for the philologist, but are logically insignificant.
Murray's Dictionary, or such books as Trench's _English Past and
Present_, supply endless examples, as many, indeed as there are
words in the language. _Clerk_ has almost as many connotations as
our typical _mambro_: clerk in holy orders, church clerk, town clerk,
clerk of assize, grocer's clerk. In Early English, the word meant "man
in a religious order, cleric, clergyman"; ability to read, write, and
keep accounts being a prominent attribute of the class, the word was
extended on this simple ground till it has ceased altogether to cover
its original field except as a formal designation. But no confusion is
caused by the variation, because the property connoted is simple.[1]
So with any common noun: street, carriage, ship, house, merchant,
lawyer, professor. We might be puzzled to give an exact definition of
such words, to say precisely what they connote in common usage; but
the risk of error in the use of them is small.

When we come to words of which the logical concept is a complex
relation, an obscure or intangible attribute, the defects of the
popular conception and its tendencies to change and confusion, are
of the greatest practical importance. Take such words as _Monarchy_,
_tyranny_, _civil freedom_, _freedom of contract_, _landlord_,
_gentleman_, _prig_, _culture_, _education_, _temperance_,
_generosity_. Not merely should we find it difficult to give an
analytic definition of such words: we might be unable to do so, and
yet flatter ourselves that we had a clear understanding of their
meaning. But let two men begin to discuss any proposition in which any
such word is involved, and it will often be found that they take the
word in different senses. If the relation expressed is complex, they
have different sides or lines of it in their minds; if the meaning
is an obscure quality, they are guided in their application of it by
different outward signs.

Monarchy, in its original meaning, is applied to a form of government
in which the will of one man is supreme, to make laws or break them,
to appoint or dismiss officers of state and justice, to determine
peace or war, without control of statute or custom. But supreme power
is never thus uncontrolled in reality; and the word has been extended
to cover governments in which the power of the titular head is
controlled in many different modes and degrees. The existence of a
head, with the title of King or Emperor, is the simplest and most
salient fact: and wherever this exists, the popular concept of a
monarchy is realised. The President of the United States has more real
power than the Sovereign of Great Britain; but the one government
is called a Republic and the other a Monarchy. People discuss the
advantages and disadvantages of monarchy without first deciding
whether they take the word in its etymological sense of unlimited
power, or its popular sense of titular kingship, or its logical sense
of power definitely limited in certain ways. And often in debate,
monarchy is really a singular term for the government of Great
Britain.

_Culture_, _religious_, _generous_, are names for inward states or
qualities: with most individuals some simple outward sign directs
the application of the word--it may be manner, or bearing, or routine
observances, or even nothing more significant than the cut of the
clothes or of the hair. Small things undoubtedly are significant, and
we must judge by small things when we have nothing else to go by: but
instead of trying to get definite conceptions for our moral epithets,
and suspending judgment till we know that the use of the epithet is
justified, the trifling superficial sign becomes for us practically
the whole meaning of the word. We feel that we must have a judgment of
some sort at once: only simple signs are suited to our impatience.

It was with reference to this state of things that Hegel formulated
his paradox that the true abstract thinker is the plain man who laughs
at philosophy as what he calls abstract and unpractical. He holds
decided opinions for or against this or the other abstraction,
_freedom_, _tyranny_, _revolution_, _reform_, _socialism_, but what
these words mean and within what limits the things signified are
desirable or undesirable, he is in too great a hurry to pause and
consider.

The disadvantages of this kind of "abstract" thinking are obvious.
The accumulated wisdom of mankind is stored in language. Until we have
cleared our conceptions, and penetrated to the full meaning of words,
that wisdom is a sealed book to us. Wise maxims are interpreted by
us hastily in accordance with our own narrow conceptions. All the
vocables of a language may be more or less familiar to us, and yet we
may not have learnt it as an instrument of thought. Outside the very
limited range of names for what we see and use in the daily routine of
life, food and clothes and the common occupations of men, words
have little meaning for us, and are the vehicles merely of thin
preconceptions and raw prejudices.

The remedy for "abstract" thinking is more thinking, and in pursuing
this two aims may be specified for the sake of clearness, though they
are closely allied, and progress towards both may often be made by
one and the same operation. (1) We want to reach a clear and full
conception of the meaning of names as used now or at a given time.
Let us call this the _Verification of the Meaning_. (2) We want to fix
such conceptions, and if necessary readjust their boundaries. This is
the province of _Definition_, which cannot be effectually performed
without _Scientific Classification_ or _Division_.

I.--VERIFICATION OF THE MEANING--DIALECTIC.

This can only be done by assembling the objects to which the words are
applied, and considering what they have in common. To ascertain the
actual connotation we must run over the actual denotation. And since
in such an operation two or more minds are better than one, discussion
or dialectic is both more fruitful and more stimulating than solitary
reflection or reading.

The first to practise this process on a memorable scale, and with
a distinct method and purpose, was Socrates. To insist upon the
necessity of clear conceptions, and to assist by his dialectic
procedure in forming them, was his contribution to philosophy.

His plan was to take a common name, profess ignorance of its meaning,
and ask his interlocutor whether he would apply it in such and such
an instance, producing one after another. According to Xenophon's
_Memorabilia_ he habitually chose the commonest names, _good_,
_unjust_, _fitting_, and so forth, and tried to set men thinking
about them, and helped them by his questions to form an intelligent
conception of the meaning.

For example, what is the meaning of injustice? Would you say that
the man who cheats or deceives is unjust? Suppose a man deceives his
enemies, is there any injustice in that? Can the definition be that
a man who deceives his friends is unjust? But there are cases where
friends are deceived for their own good: are these cases of injustice?
A general may inspirit his soldiers by a falsehood. A man may cajole
a weapon out of his friend's hand when he sees him about to commit
suicide. A father may deceive his son into taking medicine. Would you
call these men unjust? By some such process of interrogation we
are brought to the definition that a man is unjust who deceives his
friends to their hurt.

Observe that in much of his dialectic the aim of Socrates was merely
to bring out the meaning lying vague and latent, as it were, in
the common mind. His object was simply what we have called the
verification of the meaning. And a dialectic that confines itself to
the consideration of what is ordinarily meant as distinct from what
ought to be meant may often serve a useful purpose. Disputes about
words are not always as idle as is sometimes supposed. Mr. H. Sidgwick
truly remarks (_à propos_ of the terms of Political Economy) that
there is often more profit in seeking a definition than in finding it.
Conceptions are not merely cleared but deepened by the process. Mr.
Sidgwick's remarks are so happy that I must take leave to quote them:
they apply not merely to the verification of ordinary meaning but also
to the study of special uses by authorities, and the reasons for those
special uses.

    "The truth is--as most readers of Plato know, only it is a
    truth difficult to retain and apply--that what we gain by
    discussing a definition is often but slightly represented in
    the superior fitness of the formula that we ultimately adopt;
    it consists chiefly in the greater clearness and fulness in
    which the characteristics of the matter to which the formula
    refers have been brought before the mind in the process of
    seeking for it. While we are apparently aiming at definitions
    of terms, our attention should be really fixed on distinctions
    and relations of fact. These latter are what we are concerned
    to know, contemplate, and as far as possible arrange and
    systematise; and in subjects where we cannot present them to
    the mind in ordinary fulness by the exercise of the organs of
    sense, there is no way of surveying them so convenient as
    that of reflecting on our use of common terms.... In comparing
    different definitions our aim should be far less to decide
    which we ought to adopt, than to apprehend and duly consider
    the grounds on which each has commended itself to reflective
    minds. We shall generally find that each writer has noted some
    relation, some resemblance or difference, which others have
    overlooked; and we shall gain in completeness, and often
    in precision, of view by following him in his observations,
    whether or not we follow him in his conclusions."[2]

Mr. Sidgwick's own discussions of _Wealth_, _Value_, and _Money_ are
models. A clue is often found to the meaning in examining startlingly
discrepant statements connected with the same leading word. Thus
we find some authorities declaring that "style" cannot be taught or
learnt, while others declare that it can. But on trying to ascertain
what they mean by "style," we find that those who say it cannot be
taught mean either a certain marked individual character or manner of
writing--as in Buffon's saying, _Le style c'est l'homme même_--or a
certain felicity and dignity of expression, while those who say style
can be taught mean lucid method in the structure of sentences or in
the arrangement of a discourse. Again in discussions on the rank of
poets, we find different conceptions of what constitutes greatness in
poetry lying at the root of the inclusion of this or the other poet
among great poets. We find one poet excluded from the first rank of
greatness because his poetry was not serious; another because his
poetry was not widely popular; another because he wrote comparatively
little; another because he wrote only songs or odes and never
attempted drama or epic. These various opinions point to different
conceptions of what constitutes greatness in poets, different
connotations of "great poet". Comparing different opinions concerning
"education" we may be led to ask whether it means more than
instruction in the details of certain subjects, whether it does not
also import the formation of a disposition to learn or an interest in
learning or instruction in a certain method of learning.

Historically, dialectic turning on the use of words preceded the
attempt to formulate principles of Definition, and attempts at precise
definition led to Division and Classification, that is to systematic
arrangement of the objects to be defined. Attempt to define any such
word as "education," and you gradually become sensible of the needs in
respect of method that forced themselves upon mankind in the history
of thought. You soon become aware that you cannot define it by itself
alone; that you are beset by a swarm of more or less synonymous words,
_instruction_, _discipline_, _culture_, _training_, and so on; that
these various words represent distinctions and relations among things
more or less allied; and that, if each must be fixed to a definite
meaning, this must be done with reference to one another and to the
whole department of things that they cover.

The first memorable attempts at scientific arrangement were
Aristotle's treatises on Ethics and Politics, which had been the
subjects of active dialectic for at least a century before. That
these the most difficult of all departments to subject to scientific
treatment should have been the first chosen was due simply to their
preponderating interest: "The proper study of mankind is man". The
systems of what are known as the Natural Sciences are of modern
origin: the first, that of Botany, dates from Cesalpinus in the
sixteenth century. But the principles on which Aristotle proceeded in
dividing and defining, principles which have gradually themselves been
more precisely formulated, are principles applicable to all systematic
arrangements for purposes of orderly study. I give them in the precise
formulæ which they have gradually assumed in the tradition of Logic.
The principles of Division are often given in Formal Logic, and the
principles of Classification in Inductive Logic, but there is no
valid reason for the separation. The classification of objects in the
Natural Sciences, of animals, plants, and stones, with a view to
the thorough study of them in form, structure, and function, is
more complex than classifications for more limited purposes, and the
tendency is to restrict the word classification to these elaborate
systems. But really they are only a series of divisions and
subdivisions, and the same principles apply to each of the subordinate
divisions as well as to the division of the whole department of study.

II.--PRINCIPLES OF DIVISION OR CLASSIFICATION AND DEFINITION.

Confusion in the boundaries of names arises from confused ideas
regarding the resemblances and differences of things. As a protective
against this confusion, things must be clearly distinguished in their
points of likeness and difference, and this leads to their arrangement
in systems, that is, to division and classification. A name is not
secure against variation until it has a distinct place in such a
system as a symbol for clearly distinguished attributes. Nor must we
forget, further, that systems have their day, that the best system
attainable is only temporary, and may have to be recast to correspond
with changes of things and of man's way of looking at them.

The leading principles of DIVISION may be stated as follows:--

    I. Every division is made on the ground of differences in
    some attribute common to all the members of the whole to be
    divided.

This is merely a way of stating what a logical division is. It is
a division of a generic whole or _genus_, an indefinite number of
objects thought of together as possessing some common character or
attribute. All have this attribute, which is technically called the
_fundamentum divisionis_, or generic attribute. But the whole is
divisible into smaller groups (_species_), each of which possesses the
common character with a difference (_differentia_). Thus, mankind may
be divided into White men, Black men, Yellow men, on ground of the
differences in the colour of their skins: all have skins of some
colour: this is the _fundamentum divisionis_: but each subdivision or
species has a different colour: this is the _differentia_. Rectilineal
figures are divided into triangles, quadrangles, pentagons, etc., on
the ground of differences in the number of angles.

Unless there is a _fund. div._, _i.e._, unless the differences are
differences in a common character, the division is not a logical
division. To divide men into Europeans, opticians, tailors, blondes,
brunettes, and dyspeptics is not to make a logical division. This is
seen more clearly in connexion with the second condition of a perfect
division.

    II. In a perfect division, the subdivisions or species are
    mutually exclusive.

Every object possessing the common character should be in one or other
of the groups, and none should be in more than one.

Confusion between classes, or overlapping, may arise from two
causes. It may be due (1) to faulty division, to want of unity in
the _fundamentum divisionis_; (2) to the indistinct character of the
objects to be defined.

(1) Unless the division is based upon a single ground, unless each
species is based upon some mode of the generic character, confusion is
almost certain to arise. Suppose the field to be divided, the objects
to be classified, are three-sided rectilineal plane figures, each
group must be based upon some modification of the three sides. Divide
them into equilateral, isosceles, and scalene according as the three
sides are all of equal length, or two of equal length, or each of
different length, and you have a perfect division. Similarly you can
divide them perfectly according to the character of the angles into
acute-angled, right-angled and obtuse-angled. But if you do not keep
to a single basis, as in dividing them into equilateral, isosceles,
scalene, and right-angled, you have a cross-division. The same
triangle might be both right-angled and isosceles.

(2) Overlapping, however, may be unavoidable in practice owing to
the nature of the objects. There may be objects in which the dividing
characters are not distinctly marked, objects that possess the
differentiæ of more than one group in a greater or less degree. Things
are not always marked off from one another by hard and fast lines.
They shade into one another by imperceptible gradations. A clear
separation of them may be impossible. In that case you must allow a
certain indeterminate margin between your classes, and sometimes it
may be necessary to put an object into more than one class.

To insist that there is no essential difference unless a clear
demarcation can be made is a fallacy. A sophistical trick called the
_Sorites_ or Heap from the classical example of it was based upon this
difficulty of drawing sharp lines of definition. Assuming that it is
possible to say how many stones constitute a heap, you begin by asking
whether three stones form a heap. If your respondent says No, you
ask whether four stones form a heap, then five, and so on and he is
puzzled to say when the addition of a single stone makes that a heap
which was not a heap before. Or you may begin by asking whether twenty
stones form a heap, then nineteen, then eighteen, and so on, the
difficulty being to say when what was a heap ceases to be so.

Where the objects classified are mixed states or affections, the
products of interacting factors, or differently interlaced or
interfused growths from common roots, as in the case of virtues, or
emotions, or literary qualities, sharp demarcations are impossible.
To distinguish between wit and humour, or humour and pathos, or pathos
and sublimity is difficult because the same composition may partake
of more than one character. The specific characters cannot be made
rigidly exclusive one of another.

Even in the natural sciences, where the individuals are concrete
objects of perception, it may be difficult to decide in which of
two opposed groups an object should be included. Sydney Smith has
commemorated the perplexities of Naturalists over the newly
discovered animals and plants of Botany Bay, in especial with the
_Ornithorynchus_,--"a quadruped as big as a large cat, with the
eyes, colour, and skin of a mole, and the bill and web-feet of a
duck--puzzling Dr. Shaw, and rendering the latter half of his life
miserable, from his utter inability to determine whether it was a bird
or a beast".

    III. The classes in any scheme of division should be of
    co-ordinate rank.

The classes may be mutually exclusive, and yet the division imperfect,
owing to their not being of equal rank. Thus in the ordinary division
of the Parts of Speech, parts, that is, of a sentence, Prepositions
and Conjunctions are not co-ordinate in respect of function, which is
the basis of the division, with Nouns, Adjectives, Verbs, and Adverbs.
The preposition is a part of a phrase which serves the same function
as an adjective, _e.g._, _royal_ army, army _of the king_; it is thus
functionally part of a part, or a particle. So with the conjunction:
it also is a part of a part, _i.e._, part of a clause serving the
function of adjective or adverb.

    IV. The basis of division (_fundamentum divisionis_) should be
    an attribute admitting of important differences.

The importance of the attribute chosen as basis may vary with the
purpose of the division. An attribute that is of no importance in one
division, may be important enough to be the basis of another division.
Thus in a division of houses according to their architectural
attributes, the number of windows or the rent is of little importance;
but if houses are taxed or rated according to the number of windows
or the rent, these attributes become important enough to be a basis
of division for purposes of taxation or rating. They then admit of
important differences.

That the importance is relative to the purpose of the division should
be borne in mind because there is a tendency to regard attributes that
are of importance in any familiar or pre-eminent division as if they
had an absolute importance. In short, disregard of this relativity is
a fallacy to be guarded against.

In the sciences, the purpose being the attainment and preservation
of knowledge, the objects of study are divided so as to serve that
purpose. Groups must be formed so as to bring together the objects
that have most in common. The question, Who are to be placed together?
in any arrangement for purposes of study, receives the same answer
for individuals and for classes that have to be grouped into higher
classes, namely, Those that have most in common. This is what Dr. Bain
happily calls "the golden rule" of scientific classification: "Of the
various groupings of resembling things, preference is given to such as
have the greatest number of attributes in common". I slightly
modify Dr. Bain's statement: he says "the most numerous and the most
important attributes in common". But for scientific purposes number of
attributes constitutes importance, as is well recognised by Dr. Fowler
when he says that the test of importance in an attribute proposed as a
basis of classification is the number of other attributes of which it
is an index or invariable accompaniment. Thus in Zoology the
squirrel, the rat, and the beaver are classed together as Rodents, the
difference between their teeth and the teeth of other Mammalia being
the basis of division, because the difference in teeth is accompanied
by differences in many other properties. So the hedge-hog, the
shrew-mouse, and the mole, though very unlike in outward appearance
and habits, are classed together as Insectivora, the difference in
what they feed on being accompanied by a number of other differences.

_The Principles of_ DEFINITION. The word "definition" as used in Logic
shows the usual tendency of words to wander from a strict meaning and
become ambiguous. Throughout most of its uses it retains this much of
a common signification, the fixing or determining of the boundaries
of a class[3] by making clear its constituent attributes. Now in this
making clear two processes may be distinguished, a material process
and a verbal process. We have (1) the clearing up of the common
attributes by a careful examination of the objects included in the
class: and we have (2) the statement of these common attributes in
language. The rules of definition given by Dr. Bain, who devotes a
separate Book in his Logic to the subject of Definition, concern the
first of these processes: the rules more commonly given concern mainly
the second.

One eminent merit in Dr. Bain's treatment is that it recognises the
close connexion between Definition and Classification. His cardinal
rules are reduced to two.

    I. _Assemble for comparison representative individuals of the
    class._

    II. _Assemble for comparison representative individuals of the
    contrasted class or classes._

Seeing that the contrasted classes are contrasted on some basis of
division, this is in effect to recognise that you cannot clearly
define any class except in a scheme of classification. You must have
a wide _genus_ with its _fundamentum divisionis_, and, within this,
_species_ distinguished by their several _differentiæ_.

Next, as to the verbal process, rules are commonly laid down mostly
of a trifling and obvious character. That "a definition should state
neither more nor less than the common attributes of the class," or
than the attributes signified by the class-name, is sometimes given
as a rule of definition. This is really an explanation of what a
definition is, a definition of a definition. And as far as mere
statement goes it is not strictly accurate, for when the attributes
of a genus are known it is not necessary to give all the attributes of
the species, which include the generic attributes as well, but it is
sufficient to give the generic name and the differentia. Thus
Poetry may be defined as "a Fine Art having metrical language as its
instrument". This is technically known as definition _per genus
et differentiam_. This mode of statement is a recognition of the
connexion between Definition and Division.

The rule that "a definition should not be a synonymous repetition of
the name of the class to be defined," is too obvious to require formal
statement. To describe a Viceroy as a man who exercises viceregal
functions, may have point as an epigram in the case of a _faineant_
viceroy, but it is not a definition.

So with the rule that "a definition should not be couched in ambiguous
unfamiliar, or figurative language". To call the camel "the ship of
the desert" is a suggestive and luminous description of a property,
but it is not a definition. So with the noble description of Faith as
"the substance of things hoped for, the evidence of things not seen".
But if one wonders why so obvious a "rule" should be laid down, the
answer is that it has its historical origin in the caprices of
two classes of offenders, mystical philosophers and pompous
lexicographers.[4]

That "the definition should be simply convertible with the term for
the class defined," so that we may say, for example, either: "Wine is
the juice of the grape," or, "The juice of the grape is wine," is an
obvious corollary from the nature of definition, but should hardly be
dignified with the name of a "rule".

_The Principles of_ NAMING. Rules have been formulated for the choice
of names in scientific definition and classification, but it may be
doubted whether such choice can be reduced to precise rule. It is
enough to draw attention to certain considerations obvious enough on
reflection.

We may take for granted that there should be distinct names for every
defining attribute (a _Terminology_) and for every group or class (a
_Nomenclature_). What about the selection of the names? Suppose an
investigator is struck with likenesses and differences that seem to
him important enough to be the basis of a new division, how should he
be guided in his choice of names for the new groups that he proposes?
Should he coin new names, or should he take old names and try to fit
them with new definitions?

The balance of advantages is probably in favour of Dr. Whewell's
dictum that "in framing scientific terms, the appropriation of old
words is preferable to the invention of new ones". Only care must be
taken to keep as close as possible to the current meaning of the
old word, and not to run counter to strong associations. This is
an obvious precept with a view to avoiding confusion. Suppose, for
example, that in dividing Governments you take the distribution of
political power as your basis of division and come to the conclusion
that the most important differences are whether this power is vested
in a few or in the majority of the community. You want names to
fix this broad division. You decide instead of coining the new word
_Pollarchy_ to express the opposite of _Oligarchy_ to use the old
words _Republic_ and _Oligarchy_. You would find, as Sir George
Cornewall Lewis found, that however carefully you defined the word
Republic, a division under which the British Government had to be
ranked among Republics would not be generally understood and accepted.
Using the word in the sense explained above, Mr. Bagehot maintained
that the constitution of Great Britain was more Republican than that
of the United States, but his meaning was not taken except by a few.

The difficulty in choosing between new words and old words to express
new meanings is hardly felt in the exact sciences. It is at least at
a minimum. The innovator may encounter violent prejudice, but, arguing
with experts, he can at least make sure of being understood, if his
new division is based upon real and important differences. But in
other subjects the difficulty of transmitting truth or of expressing
it in language suited for precise transmission, is almost greater than
the difficulty of arriving at truth. Between new names and old
names redefined, the possessor of fresh knowledge, assuming it to be
perfectly verified, is in a quandary. The objects with which he deals
are already named in accordance with loose divisions resting on strong
prejudices. The names in current use are absolutely incapable of
conveying his meaning. He must redefine them if he is to use them. But
in that case he runs the risk of being misunderstood from people being
too impatient to master his redefinition. His right to redefine may
even be challenged without any reference to the facts to be expressed:
he may simply be accused of circulating false linguistic coin, of
debasing the verbal currency. The other alternative open to him is
to coin new words. In that case he runs the risk of not being read at
all. His contribution to verified knowledge is passed by as pedantic
and unintelligible. There is no simple rule of safety: between Scylla
and Charybdis the mariner must steer as best he may. Practically the
advantage lies with old words redefined, because thereby discussion is
provoked and discussion clears the air.

Whether it is best to attempt a formal definition or to use words in a
private, peculiar, or esoteric sense, and leave this to be gathered by
the reader from the general tenor of your utterances, is a question of
policy outside the limits of Logic. It is for the logician to expound
the method of Definition and the conditions of its application: how
far there are subjects that do not admit of its application profitably
must be decided on other grounds. But it is probably true that no
man who declines to be bound by a formal definition of his terms is
capable of carrying them in a clear unambiguous sense through a heated
controversy.

    [Footnote 1: Except, perhaps, in new offices to which the
    name is extended, such as _Clerk_ of School Board. The name,
    bearing its most simple and common meaning, may cause popular
    misapprehension of the nature of the duties. Any uncertainty
    in meaning may be dangerous in practice: elections have been
    affected by the ambiguity of this word.]

    [Footnote 2: Sidgwick's _Political Economy_, pp. 52-3. Ed.
    1883.]

    [Footnote 3: Some logicians, however, speak of defining a
    thing, and illustrate this as if by a thing they meant a
    concrete individual, the realistic treatment of Universals
    lending itself to such expressions. But though the authority
    of Aristotle might be claimed for this, it is better to
    confine the name in strictness to the main process of defining
    a class. Since, however, the method is the same whether it is
    an individual or a class that we want to make distinct, there
    is no harm in the extension of the word definition to both
    varieties. See Davidson's _Logic of Definition_, chap. ii.]

    [Footnote 4: See Davidson's _Logic of Definition_, chap. iii.]

CHAPTER II.

THE FIVE PREDICABLES.--VERBAL AND REAL PREDICATION.

We give a separate chapter to this topic out of respect for the space
that it occupies in the history of Logic. But except as an exercise in
subtle distinction for its own sake, all that falls to be said about
the Predicables might be given as a simple appendix to the chapter on
Definition.

Primarily, the Five Predicables or Heads of Predicables--Genus,
Species, Differentia, Proprium, and Accidens--are not predicables at
all, but merely a list or enumeration of terms used in dividing and
defining on the basis of attributes. They have no meaning except in
connexion with a fixed scheme of division. Given such a scheme, and
we can distinguish in it the whole to be divided (the _genus_), the
subordinate divisions (the _species_), the attribute or combination of
attributes on which each species is constituted (the _differentia_),
and other attributes, which belong to some or all of the individuals
but are not reckoned for purposes of division and definition
(_Propria_ and _Accidentia_). The list is not itself a logical
division: it is heterogeneous, not homogeneous; the two first are
names of classes, the three last of attributes. But corresponding to
it we might make a homogeneous division of attributes, as follows:--

                   Attributes
            ___________|____________
            |                      |
         Defining             Non-defining
       _____|______           ____|__________
       |          |           |             |
    Generic    Specific    Proprium      Accidens
             (Differentia)

The origin of the title Predicables as applied to these five terms is
curious, and may be worth noting as an instance of the difficulty of
keeping names precise, and of the confusion arising from forgetting
the purpose of a name. Porphyry in his [Greek: eisagôgê] or
Introduction explains the five words ([Greek: phônai]) simply as terms
that it is useful for various purposes to know, expressly mentioning
definition and division. But he casually remarks that Singular names,
"This man," "Socrates," can be predicated only of one individual,
whereas _Genera_, _Species_, _Differentiæ_, etc., are predicables
of many. That is to say he describes them as Predicables simply by
contradistinction from Singular names. A name, however, was wanted for
the five terms taken all together, and since they are not a logical
division, but merely a list of terms used in dividing and defining,
there was no apt general designation for them such as would occur
spontaneously. Thus it became the custom to refer to them as
the Predicables, a means of reference to them collectively being
desiderated, while the meaning of this descriptive title was
forgotten.

To call the five divisional elements or _Divisoria_ Predicables is to
present them as a division of Predicate Terms on the basis of their
relation to the Subject Term: to suggest that every Predicate Term
must be either a Genus or a Species, or a Differentia, or a Proprium,
or an Accidens of the Subject Term. They are sometimes criticised
as such, and it is rightly pointed out that the Predicate is never a
species of or with reference to the Subject. But, in truth, the five
so-called Predicables were never meant as a division of predicates
in relation to the subject: it is only the title that makes this
misleading suggestion.

To complete the confusion it so happens that Aristotle used three of
the Five terms in what was virtually a division of Predicates inasmuch
as it was a division of Problems or Questions. In expounding the
methods of Dialectic in the Topica he divided Problems into four
classes according to the relation of the Predicate to the Subject. The
Predicate must either be simply convertible with the subject or not.
If simply convertible, the two must be coextensive, and the
Predicate must be either a Proprium or the Definition. If not simply
convertible, the Predicate must either be part of the Definition or
not. If part of the Definition it must be either a Generic Property
or a Differentia (both of which in this connexion Aristotle includes
under Genus): if not part of the Definition, it is an Accident.
Aristotle thus arrives at a fourfold division of Problems or
Predicates:--[Greek: genos] (_Genus_, including _Differentia_,
[Greek: diaphora]); [Greek: horos] (Definition); [Greek: to idion]
(_Proprium_); and [Greek: to symbebêkos] (_Accidens_). The object of
it was to provide a basis for his systematic exposition; each of the
four kinds admitted of differences in dialectic method. For us it is
a matter of simple curiosity and ingenuity. It serves as a monument
of how much Greek dialectic turned on Definition, and it corresponds
exactly to the division of attributes into Defining and Non-defining
given above. It is sometimes said that Aristotle showed a more
scientific mind than Porphyry in making the Predicables four instead
of five. This is true if Porphyry's list had been meant as a division
of attributes: but it was not so meant.

The distinction between VERBAL or ANALYTIC and REAL or SYNTHETIC
Predication corresponds to the distinction between Defining and
Non-defining attributes, and also has no significance except with
reference to some scheme of Division, scientific and precise or loose
and popular.

When a proposition predicates of a subject something contained in the
full notion, concept, or definition of the subject term, it is called
Verbal, Analytic, or Explicative: _verbal_, inasmuch as it merely
explains the meaning of a name; _explicative_ for the same reason;
_analytic_, inasmuch as it unties the bundle of attributes held
together in the concept and pays out one, or all one by one.

When the attributes of the Predicate are not contained in the concept
of the Subject, the proposition is called _Real_, _Synthetic_, or
_Ampliative_, for parallel reasons.

Thus: "A triangle is a three-sided rectilinear figure" is Verbal or
Analytic; "Triangles have three angles together equal to two right
angles," or "Triangles are studied in schools," is Real or Synthetic.

According to this distinction, predications of the whole Definition
or of a Generic attribute or of a Specific attribute are Verbal;
predications of Accident are Real. A nice point is whether Propria are
Verbal or Real. They can hardly be classed with Verbal, inasmuch as
one may know the full meaning of the name without knowing them: but
it might be argued that they are Analytic, inasmuch as they are
implicitly contained in the defining attributes as being deducible
from them.

Observe, however, that the whole distinction is really valid only
in relation to some fixed or accepted scheme of classification or
division. Otherwise, what is Verbal or Analytic to one man may be
Real or Synthetic to another. It might even be argued that every
proposition is Analytic to the man who utters it and Synthetic to the
man who receives it. We must make some analysis of a whole of thought
before paying it out in words: and in the process of apprehending the
meaning of what we hear or read we must add the other members of the
sentence on to the subject. Whether or not this is super-subtle, it
clearly holds good that what is Verbal (in the sense defined) to the
learned man of science may be Real to the learner. That the horse has
six incisors in each jaw or that the domestic dog has a curly tail,
is a Verbal Proposition to the Natural Historian, a mere exposition
of defining marks; but the plain man has a notion of horse or dog into
which this defining attribute does not enter, and to him accordingly
the proposition is Real.

But what of propositions that the plain man would at once recognise as
Verbal? Charles Lamb, for example, remarks that the statement that "a
good name shows the estimation in which a man is held in the world" is
a verbal proposition. Where is the fixed scheme of division there?
The answer is that by a fixed scheme of division we do not necessarily
mean a scheme that is rigidly, definitely and precisely fixed. To make
such schemes is the business of Science. But the ordinary vocabulary
of common intercourse as a matter of fact proceeds upon schemes
of division, though the names used in common speech are not always
scientifically accurate, not always the best that could be devised for
the easy acquisition and sure transmission of thorough knowledge. The
plain man's vocabulary, though often twisted aside by such causes
as we have specified, is roughly moulded on the most marked
distinguishing attributes of things. This was practically recognised
by Aristotle when he made one of his modes of definition consist in
something like what we have called verifying the meaning of a name,
ascertaining the attributes that it signifies in common speech or in
the speech of sensible men. This is to ascertain the essence, [Greek:
ousia], or _Substantia_, of things, the most salient attributes that
strike the common eye either at once or after the closer inspection
that comes of long companionship, and form the basis of the ordinary
vocabulary. "Properly speaking," Mansel says,[1] "All Definition is an
inquiry into _Attributes_. Our complex notions of Substances can only
be resolved into various Attributes, with the addition of an unknown
_substratum_: a something to which we are compelled to regard
these attributes as belonging. _Man_, for example, is analysed
into Animality, Rationality, and the something which exhibits these
phenomena. Pursue the analysis and the result is the same. We have a
something corporeal, animated, sensible, rational. An unknown constant
must always be added to complete the integration." This "unknown
constant" was what Locke called the _Real_ Essence, as distinguished
from the _Nominal_ Essence, or complex of attributes. It is upon this
nominal essence, upon divisions of things according to attributes,
that common speech rests, and if it involves many cross-divisions,
this is because the divisions have been made for limited and
conflicting purposes.

    [Footnote 1: Aldrich's Compendium, Appendix, Note C. The
    reader may be referred to Mansel's Notes A and C for valuable
    historical notices of the Predicables and Definition.]

CHAPTER III.

ARISTOTLE'S CATEGORIES.

In deference to tradition a place must be found in every logical
treatise for Aristotle's Categories. No writing of the same length
has exercised a tithe of its influence on human thought. It governed
scholastic thought and expression for many centuries, being from its
shortness and consequent easiness of transcription one of the
few books in every educated man's library. It still regulates the
subdivisions of Parts of Speech in our grammars. Its universality of
acceptance is shown in the fact that the words _category_ ([Greek:
katêgoria]) and _predicament_, its Latin translation, have passed into
common speech.

The Categories have been much criticised and often condemned as a
division, but, strange to say, few have inquired what they originally
professed to be a division of, or what was the original author's
basis of division. Whether the basis is itself important, is another
question: but to call the division imperfect, without reference to the
author's intention, is merely confusing, and serves only to illustrate
the fact that the same objects may be differently divided on different
principles of division. Ramus was right in saying that the Categories
had no logical significance, inasmuch as they could not be made a
basis for departments of logical method; and Kant and Mill in saying
that they had no philosophical significance, inasmuch as they are
founded upon no theory of Knowing and Being: but this is to condemn
them for not being what they were never intended to be.

The sentence in which Aristotle states the objects to be divided, and
his division of them is so brief and bold that bearing in mind the
subsequent history of the Categories, one first comes upon it with a
certain surprise. He says simply:--

"Of things expressed without syntax (_i.e._, single words), each
signifies either substance, or quantity, or quality, or relation,
or place, or time, or disposition (_i.e._, attitude or internal
arrangement), or appurtenance, or action (doing), or suffering (being
done to)."[1]

The objects, then, that Aristotle proposed to classify were single
words (the _themata simplicia_ of the Schoolmen). He explains that by
"out of syntax" ([Greek: aneu symplokês]) he means without reference
to truth or falsehood: there can be no declaration of truth or
falsehood without a sentence, a combination, or syntax: "man runs" is
either true or false, "man" by itself, "runs" by itself, is neither.
His division, therefore, was a division of single words according to
their differences of signification, and without reference to the truth
or falsehood of their predication.[2]

Signification was thus the basis of division. But according to what
differences? The Categories themselves are so abstract that this
question might be discussed on their bare titles interminably. But
often when abstract terms are doubtful, an author's intention may be
gathered from his examples. And when Aristotle's examples are ranged
in a table, certain principles of subdivision leap to the eyes.
Thus:--

  Substance           Man                   } COMMON    { Substance
  ([Greek: ousia])    ([Greek: anthrôpos])  }  NOUN     {
  (_Substantia_)
  --------------------------------------------------------------
  Quantity            Five-feet-five        }           {
  ([Greek: poson])    ([Greek: tripêchu])   }           {
  (_Quantitas_)                             }           {
  Quality             Scholarly             }           { Permanent
  ([Greek: poion])    ([Greek: grammatikon])} ADJECTIVE  { Attribute
  (_Qualitas_)                              }           {
  Relation            Bigger                }           {
  ([Greek: pros ti])  ([Greek: meizon])     }           {
  (_Relatio_)                               }           {
  --------------------------------------------------------------
  Place               In-the-Lyceum         }           {
  ([Greek: pou])      ([Greek: en Lykeiô])  }           {
  (_Ubi_)                                   } ADVERB    { Temporary
  Time                Yesterday             }           { Attribute
  ([Greek: pote])     ([Greek: chthes])     }           {
  (_Quando_)                                }           {
  --------------------------------------------------------------
  Disposition         Reclines              }           {
  ([Greek: keisthai]) ([Greek: anakeitai])  }           {
  (_Positio_)                               }           {
  Appurtenance        Has-shoes-on          }           {
  ([Greek: echein])   ([Greek: hypodedetai])}           {
  (_Habitus_)                               } VERB      {
  Action              Cuts                  }           { Temporary
  ([Greek: poiein])   ([Greek: temnei])     }           { Attribute
  (_Actio_)                                 }           {
  Passion             Is cut                }           {
  ([Greek: paschein]) ([Greek: temnetai])   }           {
  (_Passio_)                                }           {

In looking at the examples, our first impression is that Aristotle has
fallen into a confusion. He professes to classify words out of syntax,
yet he gives words with the marks of syntax on them. Thus his division
is accidentally grammatical, a division of parts of speech, parts of
a sentence, into Nouns, Adjectives, Adverbs, and Verbs. And his
subdivisions of these parts are still followed in our grammars. But
really it is not the grammatical function that he attends to, but
the signification: and looking further at the examples, we see what
differences of signification he had in his mind. It is differences
relative to a concrete individual, differences in the words applied to
him according as they signify the substance of him or his attributes,
permanent or temporary.

Take any concrete thing, Socrates, this book, this table. It must
be some kind of a thing, a man, a book. It must have some size
or quantity, six feet high, three inches broad. It must have some
quality, white, learned, hard. It must have relations with other
things, half this, double that, the son of a father. It must be
somewhere, at some time, in some attitude, with some "havings,"
appendages, appurtenances, or belongings, doing something, or having
something done to it. Can you conceive any name (simple or composite)
applicable to any object of perception, whose signification does not
fall into one or other of these classes? If you cannot, the categories
are justified as an exhaustive division of significations. They are
a complete list of the most general resemblances among individual
things, in other words, of the _summa genera_, the _genera
generalissima_ of predicates concerning this, that or the other
concrete individual. No individual thing is _sui generis_: everything
is like other things: the categories are the most general likenesses.

The categories are exhaustive, but do they fulfil another requisite of
a good division--are they mutually exclusive? Aristotle himself
raised this question, and some of his answers to difficulties are
instructive. Particularly his discussion of the distinction between
Second Substances or Essences and Qualities. Here he approximates to
the modern doctrine of the distinction between Substance and Attribute
as set forth in our quotation from Mansel at p. 110. Aristotle's
Second Essences ([Greek: deuterai ousiai]) are common nouns or general
names, Species and Genera, _man_, _horse_, _animal_, as distinguished
from Singular names, _this man_, _this horse_, which he calls First
Substances ([Greek: prôtai ousiai]), essences _par excellence_, to
which real existence in the highest sense is attributed. Common nouns
are put in the First Category because they are predicated in answer to
the question, What is this? But he raises the difficulty whether they
may not rather be regarded as being in the Third Category, that of
Quality ([Greek: to poion]). When we say, "This is a man," do we not
declare what sort of a thing he is? do we not declare his Quality? If
Aristotle had gone farther along this line, he would have arrived
at the modern point of view that a man is a man in virtue of his
possessing certain attributes, that general names are applied in
virtue of their connotation. This would have been to make the line
of distinction between the First Category and the Third pass between
First Essence and Second, ranking the Second Essences with Qualities.
But Aristotle did not get out of the difficulty in this way. He solved
it by falling back on the differences in common speech. "Man" does not
signify the quality simply, as "whiteness" does. "Whiteness" signifies
nothing but the quality. That is to say, there is no separate name in
common speech for the common attributes of man. His further obscure
remark that general names "define quality round essence" [Greek: peri
ousian], inasmuch as they signify what sort a certain essence is, and
that genera make this definition more widely than species, bore fruit
in the mediæval discussions between Realists and Nominalists by which
the signification of general names was cleared up.

Another difficulty about the mutual exclusiveness of the Categories
was started by Aristotle in connexion with the Fourth Category,
Relation ([Greek: pros ti] _Ad aliquid_, _To something_). Mill remarks
that "that could not be a very comprehensive view of the nature of
Relation which would exclude action, passivity, and local situation
from that Category," and many commentators, from Simplicius down to
Hamilton, have remarked that all the last six Categories might be
included under Relation. This is so far correct that the word Relation
is one of the vaguest and most extensive of words; but the criticism
ignores the strictness with which Aristotle confined himself in
his Categories to the forms of common speech. It is clear from
his examples that in his Fourth Category he was thinking only of
"relation" as definitely expressed in common speech. In his meaning,
any word is a relative which is joined with another in a sentence by
means of a preposition or a case-inflection. Thus "disposition" is a
relative: it is the disposition _of_ something. This kind of relation
is perfect when the related terms reciprocate grammatically; thus
"master," "servant," since we can say either "the master of the
servant," or "the servant of the master". In mediæval logic the term
_Relata_ was confined to these perfect cases, but the Category had
a wider scope with Aristotle. And he expressly raised the question
whether a word might not have as much right to be put in another
Category as in this. Indeed, he went further than his critics in his
suggestions of what Relation might be made to include. Thus: "big"
signifies Quality; yet a thing is big with reference to something
else, and is so far a Relative. Knowledge must be knowledge of
something, and is a relative: why then should we put "knowing"
(_i.e._, learned) in the Category of Quality. "Hope" is a relative,
as being the hope _of_ a man and the hope of something. Yet we say,
"I have hope," and there hope would be in the category of Having,
Appurtenance. For the solution of all such difficulties, Aristotle
falls back upon the forms of common speech, and decides the place of
words in his categories according to them. This was hardly consistent
with his proposal to deal with separate words out of syntax, if by
this was meant anything more than dealing with them without reference
to truth or falsehood. He did not and could not succeed in dealing
with separate words otherwise than as parts of sentences, owing their
signification to their position as parts of a transient plexus of
thought. In so far as words have their being in common speech, and
it is their being in this sense that Aristotle considers in the
Categories, it is a transient being. What being they represent besides
is, in the words of Porphyry, a very deep affair, and one that needs
other and greater investigation.

    [Footnote 1: [Greek: tôn kata mêdemian symplokên legomenôn
    hekaston êtoi ousian sêmainei, ê poson, ê poion, ê pros ti,
    ê pou, ê pote, ê keisthai, ê echein, ê poiein, ê paschein.]
    (Categ. ii. 5.)]

    [Footnote 2: To describe the Categories as a grammatical
    division, as Mansel does in his instructive Appendix C to
    Aldrich, is a little misleading without a qualification.
    They are non-logical inasmuch as they have no bearing on any
    logical purpose. But they are grammatical only in so far as
    they are concerned with words. They are not grammatical in
    the sense of being concerned with the function of words
    in predication. The unit of grammar in this sense is the
    sentence, a combination of words in syntax; and it is
    expressly with words out of syntax that Aristotle deals, with
    single words not in relation to the other parts of a sentence,
    but in relation to the things signified. In any strict
    definition of the provinces of Grammar and Logic, the
    Categories are neither grammatical nor logical: the
    grammarians have appropriated them for the subdivision of
    certain parts of the sentence, but with no more right than the
    logicians. They really form a treatise by themselves, which
    is in the main ontological, a discussion of substances and
    attributes as underlying the forms of common speech. In saying
    this I use the word substance in the modern sense: but it
    must be remembered that Aristotle's [Greek: ousia], translated
    substantia, covered the word as well as the thing signified,
    and that his Categories are primarily classes of words. The
    union between names and things would seem to have been closer
    in the Greek mind than we can now realise. To get at it we
    must note that every separate word [Greek: to legomenon]
    is conceived as having a being or thing [Greek: to on]
    corresponding to it, so that beings or things [Greek: ta
    onta] are coextensive with single words: a being or thing is
    whatever receives a separate name. This is clear and simple
    enough, but perplexity begins when we try to distinguish
    between this nameable being and concrete being, which last is
    Aristotle's category of [Greek: ousia], the being signified by
    a Proper or a Common as distinguished from an Abstract Noun.
    As we shall see, it is relatively to the highest sense of this
    last kind of being, namely, the being signified by a Proper
    name, that he considers the other kinds of being.]

CHAPTER IV.

THE CONTROVERSY ABOUT UNIVERSALS.--DIFFICULTIES CONCERNING THE
RELATION OF GENERAL NAMES TO THOUGHT AND TO REALITY.

In the opening sentences of his Isagoge, before giving his simple
explanation of the Five Predicables, Porphyry mentions certain
questions concerning Genera and Species, which he passes over as being
too difficult for the beginner. "Concerning genera and species,"
he says, "the question whether they subsist (_i.e._, have real
substance), or whether they lie in the mere thoughts only, or whether,
granting them to subsist, they are corporeal or incorporeal, or
whether they subsist apart, or in sensible things and cohering round
them--this I shall pass over, such a question being a very deep affair
and one that needs other and greater investigation."

This passage, written about the end of the third century, A.D., is a
kind of isthmus between Greek Philosophy and Mediæval: it summarises
questions which had been turned over on every side and most
intricately discussed by Plato and Aristotle and their successors,
and the bald summary became a starting-point for equally intricate
discussions among the Schoolmen, among whom every conceivable variety
of doctrine found champions. The dispute became known as the dispute
about Universals, and three ultra-typical forms of doctrine
were developed, known respectively as Realism, Nominalism, and
Conceptualism. Undoubtedly the dispute, with all its waste of
ingenuity, had a clearing effect, and we may fairly try now what
Porphyry shrank from, to gather some simple results for the better
understanding of general names and their relations to thoughts and to
things. The rival schools had each some aspect of the general name in
view, which their exaggeration served to render more distinct.

What does a general name signify? For logical purposes it is
sufficient to answer--the points of resemblance as grasped in
the mind, fixed by a name applicable to each of the resembling
individuals. This is the signification of the general name
_logically_, its connotation or concept, the identical element of
objective reference in all uses of a general name.

But other questions may be asked that cannot be so simply answered.
What is this concept in thought? What is there in our minds
corresponding to the general name when we utter it? How is its
signification conceived? What is the signification _psychologically_?

We may ask, further, What is there in nature that the general name
signifies? What is its relation to reality? What corresponds to it in
the real world? Has the unity that it represents among individuals
no existence except in the mind? Calling this unity, this one in
the many, the Universal (_Universale_, [Greek: to pan]), what is the
Universal _ontologically_?

It was this ontological question that was so hotly and bewilderingly
debated among the Schoolmen. Before giving the ultra-typical answers
to it, it may be well to note how this question was mixed up with
still other questions of Theology and Cosmogony. Recognising that
there is a unity signified by the general name, we may go on to
inquire into the ground of the unity. Why are things essentially like
one another? How is the unity maintained? How is it continued? Where
does the common pattern come from? The question of the nature of
the Universal thus links itself with metaphysical theories of the
construction of the world, or even with the Darwinian theory of the
origin of species.

Passing by these remoter questions, we may give the answers of
the three extreme schools to the ontological question, What is a
Universal?

The answer of the Ultra-Realists, broadly put, was that a Universal is
a substance having an independent existence in nature.

Of the Ultra-Nominalists, that the Universal is a name and nothing
else, _vox et præterea nihil_; that this name is the only unity among
the individuals of a species, all that they have in common.

Of the Ultra-Conceptualists, that the individuals have more in common
than the name, that they have the name plus the meaning,
_vox_ + _significatio_, but that the Universals, the genera and
species, exist only in the mind.

Now these extreme doctrines, as literally interpreted by opponents,
are so easily refuted and so manifestly untenable, that it may be
doubted whether they were ever held by any thinker, and therefore I
call them Ultra-Realism, Ultra-Nominalism, and Ultra-Conceptualism.
They are mere exaggerations or caricatures, set up by opponents
because they can be easily knocked down.

To the Ultra-Realists, it is sufficient to say that if there existed
anywhere a substance having all the common attributes of a species
and only these, having none of the attributes peculiar to any of the
individuals of that species, corresponding to the general name as an
individual corresponds to a Proper or Singular name, it would not be
the Universal, the unity pervading the individuals, but only another
individual.

To the Ultra-Nominalists, it is sufficient to say that the individuals
must have more in common than the name, because the name is not
applied arbitrarily, but on some ground. The individuals must have in
common that on account of which they receive the common name: to call
them by the same name is not to make them of the same species.

To the Ultra-Conceptualists, it is sufficient to say that when we
employ a general name, as when we say "Socrates is a man," we do
not refer to any passing thought or state of mind, but to certain
attributes independent of what is passing in our minds. We cannot make
a thing of this or that species by merely thinking of it as such.

The ultra-forms of these doctrines are thus easily shown to
be inadequate, yet each of the three, Realism, Nominalism, and
Conceptualism, represents a phase of the whole truth.

Thus, take Realism. Although it is not true that there is anything
in reality corresponding to the general name such as there is
corresponding to the singular name, the general name merely signifying
attributes of what the singular name signifies, it does not follow, as
the opponents of Ultra-Realism hastily assume, that there is nothing
in the real world corresponding to the general name. Three senses may
be particularised in which Realism is justified.

(1) The points of resemblance from which the concept is formed are as
real as the individuals themselves. It is true in a sense that it
is our thought that gives unity to the individuals of a class, that
gathers the many into one, and so far the Conceptualists are right.
Still we should not gather them into one if they did not resemble one
another: that is the reason why we think of them together: and the
respects in which they resemble one another are as much independent of
us and our thinking as the individuals themselves, as much beyond the
power of our thought to change. We must go behind the activity of the
mind in unifying to the reason for the unification: and the ground of
unity is found in what really exists. We do not confer the unity:
we do not make all men or all dogs alike: we find them so. The curly
tails in a thousand domestic dogs, which serve to distinguish them
from wolves and foxes, are as real as the thousand individual domestic
dogs. In this sense the Aristotelian doctrine, _Universalia in re_,
expresses a plain truth.

(2) The Platonic doctrine, formulated by the Schoolmen as _Universalia
ante rem_, has also a plain validity. Individuals come and go, but
the type, the Universal, is more abiding. Men are born and die: man
remains throughout. The snows of last year have vanished, but snow is
still a reality to be faced. Wisdom does not perish with the wise
men of any generation. In this plain sense, at least, it is true that
Universals exist before Individuals, have a greater permanence, or, if
we like to say so, a higher, as it is a more enduring, reality.

(3) Further, the "idea," concept, or universal, though it cannot be
separated from the individual, and whether or not we ascribe to it
the separate suprasensual existence of the archetypal forms of Plato's
poetical fancy, is a very potent factor in the real world. Ideals of
conduct, of manners, of art, of policy, have a traditional life: they
do not pass away with the individuals in whom they have existed,
in whom they are temporarily materialised: they survive as potent
influences from age to age. The "idea" of Chaucer's Man of Law,
who always "seemed busier than he was," is still with us. Mediæval
conceptions of chivalry still govern conduct. The Universal enters
into the Individual, takes possession of him, makes of him its
temporary manifestation.

Nevertheless, the Nominalists are right in insisting on the importance
of names. What we call the real world is a common object of perception
and knowledge to you and me: we cannot arrive at a knowledge of it
without some means of communication with one another: our means of
communication is language. It may be doubted whether even thinking
could go far without symbols with the help of which conceptions may
be made definite. A concept cannot be explained without reference to
a symbol. There is even a sense in which the Ultra-Nominalist doctrine
that the individuals in a class have nothing in common but the name
is tenable. Denotability by the same name is the only respect in which
those individuals are absolutely identical: in this sense the name
alone is common to them, though it is applied in virtue of their
resemblance to one another.

Finally, the Conceptualists are right in insisting on the mind's
activity in connexion with general names. Genera and species are not
mere arbitrary subjective collections: the union is determined by the
characters of the things collected. Still it is with the concept in
each man's mind that the name is connected: it is by the activity of
thought in recognising likenesses and forming concepts that we are
able to master the diversity of our impressions, to introduce unity
into the manifold of sense, to reduce our various recollections to
order and coherence.

So much for the Ontological question. Now for the PSYCHOLOGICAL. What
is in the mind when we employ a general name? What is the Universal
psychologically? How is it conceived?

What breeds confusion in these subtle inquiries is the want of fixed
unambiguous names for the things to be distinguished. It is only by
means of such names that we can hold on to the distinctions, and
keep from puzzling ourselves. Now there are three things to be
distinguished in this inquiry, which we may call the Concept, the
Conception, and the Conceptual or Generic Image. Let us call them by
these names, and proceed to explain them.

By the Concept, I understand the meaning of the general name, what the
general name signifies: by the Conception, the mental act or state
of him who conceives this meaning. The concept of "triangle," _i.e._,
what you and I mean by the word, is not my act of mind or your act of
mind when we think or speak of a triangle. The Conception, which is
this act, is an event or incident in our mental history, a psychical
act or state, a distinct occurrence, a particular fact in time as much
as the battle of Waterloo. The concept is the objective reference of
the name, which is the same, or at least is understood to be the
same, every time we use it. I make a figure on paper with ink or on a
blackboard with chalk, and recognise or conceive it as a triangle: you
also conceive it as such: we do the same to-morrow: we did the same
yesterday: each act of conception is a different event, but the
concept is the same throughout.

Now the psychological question about the Universal is, What is this
conception? We cannot define it positively further than by saying that
it consists in realising the meaning of a general name: the act being
unique, we can only make it intelligible by producing an example of
it. But we may define it negatively by distinguishing it from the
conceptual image. Whenever we conceive anything, "man," "horse," there
is generally present to our minds an image of a man or horse, with
accidents of size, colour, position or other categories. But this
conceptual image is not the concept, and the mental act of forming it
is not conception.

This distinction between mental picturing or imaging and the
conception of common attributes is variously expressed. The
correlative terms _Intuitive_ and _Symbolical_ Thinking,
_Presentative_ and _Representative_ Knowledge have been employed.[1]
But whatever terms we use, the distinction itself is vital, and the
want of it leads to confusion.

Thus the fact that we cannot form a conceptual image composed
solely of common attributes has been used to support the argument of
Ultra-Nominalism, that the individuals classed under a common name
have nothing in common but the name. What the word "dog" signifies,
_i.e._, the "concept" of dog, is neither big nor little, neither black
nor tan, neither here nor there, neither Newfoundland, nor Retriever,
nor Terrier, nor Greyhound, nor Pug, nor Bulldog. The concept consists
only of the attributes common to all dogs apart from any that are
peculiar to any variety or any individual. Now we cannot form any such
conceptual image. Our conceptual image is always of some definite
size and shape. Therefore, it is argued, we cannot conceive what a dog
means, and dogs have nothing in common but the name. This, however,
does not follow. The concept is not the conceptual image, and
forming the image is not conception. We may even, as in the case of
a chiliagon, or thousand-sided figure, conceive the meaning without
being able to form any definite image.

How then, do we ordinarily proceed in conceiving, if we cannot picture
the common attributes alone and apart from particulars? We attend, or
strive to attend, only to those aspects of an image which it has in
common with the individual things denoted. And if we want to make our
conception definite, we pass in review an indefinite number of the
individuals, case after case.

A minor psychological question concerns the nature of the conceptual
image. Is it a copy of some particular impression, or a confused blur
or blend of many? Possibly neither: possibly it is something like
one of Mr. Galton's composite photographs, photographs produced by
exposing the same surface to the impressions of a number of different
photographs in succession. If the individuals are nearly alike,
the result is an image that is not an exact copy of any one of the
components and yet is perfectly distinct. Possibly the image that
comes into our mind's eye when we hear such a word as "horse" or "man"
is of this character, the result of the impressions of a number of
similar things, but not identical with any one. As, however, different
persons have different conceptual images of the same concept, so we
may have different conceptual images at different times. It is only
the concept that remains the same.

But how, it may be asked, can the concept remain the same? If the
universal or concept psychologically is an intellectual act, repeated
every time we conceive, what guarantee have we for the permanence
of the concept? Does this theory not do away with all possibility of
defining and fixing concepts?

This brings us back to the doctrine already laid down about the truth
of Realism. The theory of the concept is not exhausted when it is
viewed only psychologically, as a psychic act. If we would understand
it fully, we must consider the act in its relations to the real
experience of ourselves and others. To fix this act, we give it a
separate name, calling it the conception: and then we must go behind
the activity of the mind to the objects on which it is exercised.
The element of fixity is found in them. And here also the truth of
Nominalism comes in. By means of words we enter into communication
with other minds. It is thus that we discover what is real, and what
is merely personal to ourselves.

    [Footnote 1: The only objection to these terms is that they
    have slipped from their moorings in philosophical usage. Thus
    instead of Leibnitz's use of Intuitive and Symbolical, which
    corresponds to the above distinction between Imaging and
    Conception, Mr. Jevons employs the terms to express a
    distinction among conceptions proper. We can understand what
    a chiliagon means, but we cannot form an image of it in our
    minds, except in a very confused and imperfect way; whereas we
    can form a distinct image of a triangle. Mr. Jevons would call
    the conception of the triangle _Intuitive_, of the chiliagon
    _Symbolical_.

    Again, while Mansel uses the words Presentative and
    Representative to express our distinction, a more common
    usage is to call actual Perception Presentative Knowledge, and
    ideation or recollection in idea Representative.]

PART III.

THE INTERPRETATION OF PROPOSITIONS.--OPPOSITION AND IMMEDIATE
INFERENCE.

CHAPTER I.

THEORIES OF PREDICATION.--THEORIES OF JUDGMENT.

We may now return to the Syllogistic Forms, and the consideration of
the compatibility or incompatibility, implication, and interdependence
of propositions.

It was to make this consideration clear and simple that what we have
called the Syllogistic Form of propositions was devised. When are
propositions incompatible? When do they imply one another? When do
two imply a third? We have seen in the Introduction how such questions
were forced upon Aristotle by the disputative habits of his time.
It was to facilitate the answer that he analysed propositions into
Subject and Predicate, and viewed the Predicate as a reference to a
class: in other words, analysed the Predicate further into a Copula
and a Class Term.

But before showing how he exhibited the interconnexion of propositions
on this plan, we may turn aside to consider various so-called Theories
of Predication or of Judgment. Strictly speaking, they are not
altogether relevant to Logic, that is to say, as a practical science:
they are partly logical, partly psychological theories: some of
them have no bearing whatever on practice, but are matters of pure
scientific curiosity: but historically they are connected with the
logical treatment of propositions as having been developed out of
this.

The least confusing way of presenting these theories is to state
them and examine them both logically and psychologically. The logical
question is, Has the view any advantage for logical purposes? Does it
help to prevent error, to clear up confusion? Does it lead to firmer
conceptions of the truth? The psychological question is, Is this a
correct theory of how men actually think when they make propositions?
It is a question of _what is_ in the one case, and of _what ought to
be for a certain purpose_ in the other.

Whether we speak of Proposition or of Judgment does not materially
affect our answer. A Judgment is the mental act accompanying a
Proposition, or that may be expressed in a proposition and cannot be
expressed otherwise: we can give no other intelligible definition or
description of a judgment. So a proposition can only be defined as the
expression of a judgment: unless there is a judgment underneath them,
a form of words is not a proposition.

Let us take, then, the different theories in turn. We shall find that
they are not really antagonistic, but only different: that each is
substantially right from its own point of view: and that they seem to
contradict one another only when the point of view is misunderstood.

I. _That the Predicate term may be regarded as a class in or
from which the Subject is included or excluded._ Known as the
Class-Inclusion, Class-Reference, or Denotative view.

This way of analysing propositions is possible, as we have seen,
because every statement implies a general name, and the extension
or denotation of a general name is a class defined by the common
attribute or attributes. It is useful for syllogistic purposes:
certain relations among propositions can be most simply exhibited in
this way.

But if this is called a Theory of Predication or Judgment, and taken
psychologically as a theory of what is in men's minds whenever they
utter a significant Sentence, it is manifestly wrong. When discussed
as such, it is very properly rejected. When a man says "P struck Q,"
he has not necessarily a class of "strikers of Q" definitely in his
mind. What he has in his mind is the logical equivalent of this,
but it is not this directly. Similarly, Mr. Bradley would be quite
justified in speaking of Two Terms and a Copula as a superstition, if
it were meant that these analytic elements are present to the mind of
an ordinary speaker.

II. _That every Proposition may be regarded as affirming or denying
an attribute of a subject._ Known sometimes as the Connotative or
the Denotative-Connotative view. This also follows from the implicit
presence of a general name in every sentence. But it should not be
taken as meaning that the man who says: "Tom came here yesterday,"
or "James generally sits there," has a clearly analysed Subject and
Attribute in his mind. Otherwise it is as far wrong as the other view.

III. _That every proposition may be regarded as an equation between
two terms._ Known as the Equational View.

This is obviously not true for common speech or ordinary thought.
But it is a possible way of regarding the analytic components of
a proposition, legitimate enough if it serves any purpose. It is a
modification of the Class-Reference analysis, obtained by what is
known as Quantification of the Predicate. In "All S is in P," P
is undistributed, and has no symbol of Quantity. But since the
proposition imports that All S is a part of P, _i.e._, Some P, we may,
if we choose, prefix the symbol of Quantity, and then the proposition
may be read "All S = Some P". And so with the other forms.

Is there any advantage in this? Yes: it enables us to subject the
formulæ to algebraic manipulation. But any logical advantage--any
help to thinking? None whatever. The elaborate syllogistic systems of
Boole, De Morgan, and Jevons are not of the slightest use in helping
men to reason correctly. The value ascribed to them is merely an
illustration of the Bias of Happy Exercise. They are beautifully
ingenious, but they leave every recorded instance of learned
Scholastic trifling miles behind.

IV. _That every proposition is the expression of a comparison between
concepts._ Sometimes called the Conceptualist View.

"To judge," Hamilton says, "is to recognise the relation of congruence
or confliction in which two concepts, two individual things, or a
concept and an individual compared together stand to each other."

This way of regarding propositions is permissible or not according to
our interpretation of the words "congruence" and "confliction," and
the word "concept". If by concept we mean a conceived attribute of
a thing, and if by saying that two concepts are congruent or
conflicting, we mean that they may or may not cohere in the same
thing, and by saying that a concept is congruent or conflicting with
an individual that it may or may not belong to that individual, then
the theory is a corollary from Aristotle's analysis. Seeing that we
must pass through that analysis to reach it, it is obviously not
a theory of ordinary thought, but of the thought of a logician
performing that analysis.

The precise point of Hamilton's theory was that the logician does not
concern himself with the question whether two concepts are or are
not as a matter of fact found in the same subject, but only with the
question whether they are of such a character that they may be found,
or cannot be found, in the same subject. In so far as his theory
is sound, it is an abstruse and technical way of saying that we may
consider the consistency of propositions without considering whether
or not they are true, and that consistency is the peculiar business of
syllogistic logic.

V. _That the ultimate subject of every judgment is reality._

This is the form in which Mr. Bradley and Mr. Bosanquet deny the
Ultra-Conceptualist position. The same view is expressed by Mill when
he says that "propositions are concerned with things and not with our
ideas of them".

The least consideration shows that there is justice in the view thus
enounced. Take a number of propositions:--

          The streets are wet.
          George has blue eyes.
          The Earth goes round the Sun.
          Two and two make four.

Obviously, in any of these propositions, there is a reference beyond
the conceptions in the speaker's mind, viewed merely as incidents
in his mental history. They express beliefs about things and the
relations among things _in rerum natura_: when any one understands
them and gives his assent to them, he never stops to think of the
speaker's state of mind, but of what the words represent. When states
of mind are spoken of, as when we say that our ideas are confused, or
that a man's conception of duty influences his conduct, those states
of mind are viewed as objective facts in the world of realities. Even
when we speak of things that have in a sense no reality, as when we
say that a centaur is a combination of man and horse, or that centaurs
were fabled to live in the vales of Thessaly, it is not the passing
state of mind expressed by the speaker as such that we attend to or
think of; we pass at once to the objective reference of the words.

Psychologically, then, the theory is sound: what is its logical
value? It is sometimes put forward as if it were inconsistent with
the Class-reference theory or the theory that judgment consists in
a comparison of concepts. Historically the origin of its formal
statement is its supposed opposition to those theories. But really
it is only a misconception of them that it contradicts. It is
inconsistent with the Class-reference view only if by a class we
understand an arbitrary subjective collection, not a collection of
things on the ground of common attributes. And it is inconsistent with
the Conceptualist theory only if by a concept we understand not the
objective reference of a general name, but what we have distinguished
as a conception or a conceptual image. The theory that the ultimate
subject is reality is assumed in both the other theories, rightly
understood. If every proposition is the utterance of a judgment, and
every proposition implies a general name, and every general name has
a meaning or connotation, and every such meaning is an attribute of
things and not a mental state, it is implied that the ultimate subject
of every proposition is reality. But we may consider whether or not
propositions are consistent without considering whether or not they
are true, and it is only their mutual consistency that is considered
in the syllogistic formulæ. Thus, while it is perfectly correct to say
that every proposition expresses either truth or falsehood, or that
the characteristic quality of a judgment is to be true or false, it
is none the less correct to say that we may temporarily suspend
consideration of truth or falsehood, and that this is done in what is
commonly known as Formal Logic.

VI. _That every proposition may be regarded as expressing relations
between phenomena._

Bain follows Mill in treating this as the final import of Predication.
But he indicates more accurately the logical value of this view in
speaking of it as important for laying out the divisions of Inductive
Logic. They differ slightly in their lists of Universal Predicates
based upon Import in this sense--Mill's being Resemblance,
Coexistence, Simple Sequence, and Causal Sequence, and Bain's being
Coexistence, Succession, and Equality or Inequality. But both lay
stress upon Coexistence and Succession, and we shall find that the
distinctions between Simple Sequence and Causal Sequence, and between
Repeated and Occasional Coexistence, are all-important in the Logic of
Investigation. But for syllogistic purposes the distinctions have no
relevance.

CHAPTER II.

THE "OPPOSITION" OF PROPOSITIONS.--THE INTERPRETATION OF "NO".

Propositions are technically said to be "opposed" when, having the
same terms in Subject and Predicate, they differ in Quantity, or in
Quality, or in both.[1]

The practical question from which the technical doctrine has been
developed was how to determine the significance of contradiction.
What is meant by giving the answer "No" to a proposition put
interrogatively? What is the interpretation of "No"? What is the
respondent committed to thereby?

"Have all ratepayers a vote?" If you answer "No," you are bound to
admit that some ratepayers have not. O is the CONTRADICTORY of A. If A
is false, O must be true. So if you deny O, you are bound to admit A:
one or other must be true: either Some ratepayers have not a vote or
All have.

Is it the case that no man can live without sleep? Deny this, and you
commit yourself to maintaining that Some man, one at least, can live
without sleep. I is the Contradictory of E; and _vice versâ_.

Contradictory opposition is distinguished from CONTRARY, the
opposition of one Universal to another, of A to E and E to A. There is
a natural tendency to meet a strong assertion with the very reverse.
Let it be maintained that women are essentially faithless or that "the
poor in a lump is bad," and disputants are apt to meet this extreme
with another, that constancy is to be found only in women or true
virtue only among the poor. Both extremes, both A and E, may be false:
the truth may lie between: Some are, Some not.

Logically, the denial of A or E implies only the admission of O or I.
You are not committed to the full contrary. But the implication of the
Contradictory is absolute; there is no half-way house where the
truth may reside. Hence the name of EXCLUDED MIDDLE is applied to the
principle that "Of two Contradictories one or other must be true: they
cannot both be false".

While both CONTRARIES may be false, they cannot both be true.

It is sometimes said that in the case of Singular propositions, the
Contradictory and the Contrary coincide. A more correct doctrine is
that in the case of Singular propositions, the distinction is not
needed and does not apply. Put the question "Is Socrates wise?" or
"Is this paper white?" and the answer "No" admits of only one
interpretation, provided the terms remain the same. Socrates may
become foolish, or this paper may hereafter be coloured differently,
but in either case the subject term is not the same about which the
question was asked. Contrary opposition belongs only to general terms
taken universally as subjects. Concerning individual subjects an
attribute must be either affirmed or denied simply: there is no middle
course. Such a proposition as "Socrates is sometimes not wise," is
not a true Singular proposition, though it has a Singular term as
grammatical subject. Logically, it is a Particular proposition, of
which the subject-term is the actions or judgments of Socrates.[2]

Opposition, in the ordinary sense, is the opposition of incompatible
propositions, and it was with this only that Aristotle concerned
himself. But from an early period in the history of Logic, the word
was extended to cover mere differences in Quantity and Quality
among the four forms A E I O, which differences have been named
and exhibited symmetrically in a diagram known as: The Square of
Opposition.

  A______Contraries______E
  |\                    /|
  | \                  s |
  |  C                e  |
  |   o              i   |
  |    n            r    |
  |     t          o     |
  S      r        t      S
  u       a      c       u
  b        \    i        b
  a         \  d         a
  l          \/          l
  t          /\          t
  e         /  d         e
  r        /    i        r
  n       a      c       n
  s      r        t      s
  |     t          o     |
  |    n            r    |
  |   o              i   |
  |  C                e  |
  | /                  s |
  |/                    \|
  I____Sub-contraries____O

The four forms being placed at the four corners of the Square, and
the sides and diagonals representing relations between them thus
separated, a very pretty and symmetrical doctrine is the result.

_Contradictories_, A and O, E and I, differ both in Quantity and in
Quality.

_Contraries_, A and E, differ in Quality but not in Quantity, and are
both Universal.

_Sub-contraries_, I and O, differ in Quality but not in Quantity, and
are both Particular.

_Subalterns_, A and I, E and O, differ in Quantity but not in Quality.

Again, in respect of concurrent truth and falsehood there is a certain
symmetry.

Contradictories cannot both be true, nor can they both be false.

Contraries may both be false, but cannot both be true.

Sub-contraries may both be true, but cannot both be false.

Subalterns may both be false and both true. If the Universal is true,
its subalternate Particular is true: but the truth of the Particular
does not similarly imply the truth of its Subalternating Universal.

This last is another way of saying that the truth of the Contrary
involves the truth of the Contradictory, but the truth of the
Contradictory does not imply the truth of the Contrary.

There, however, the symmetry ends. The sides and the diagonals of the
Square do not symmetrically represent degrees of incompatibility, or
opposition in the ordinary sense.

There is no incompatibility between two Sub-contraries or a Subaltern
and its Subalternant. Both may be true at the same time. Indeed, as
Aristotle remarked of I and O, the truth of the one commonly implies
the truth of the other: to say that some of the crew were drowned,
implies that some were not, and _vice versâ_. Subaltern and
Subalternant also are compatible, and something more. If a man has
admitted A or E, he cannot refuse to admit I or O, the Particular of
the same Quality. If All poets are irritable, it cannot be denied
that some are so; if None is, that Some are not. The admission of the
Contrary includes the admission of the Contradictory.

Consideration of Subalterns, however, brings to light a nice ambiguity
in Some. It is only when I is regarded as the Contradictory of E,
that it can properly be said to be Subalternate to A. In that case the
meaning of Some is "not none," _i.e._, "Some at least". But when Some
is taken as the sign of Particular quantity simply, _i.e._, as meaning
"not all," or "some at most," I is not Subalternate to A, but opposed
to it in the sense that the truth of the one is incompatible with the
truth of the other.

Again, in the diagram Contrary opposition is represented by a side and
Contradictory by the diagonal; that is to say, the stronger form of
opposition by the shorter line. The Contrary is more than a denial: it
is a counter-assertion of the very reverse, [Greek: to enantion].
"Are good administrators always good speakers?" "On the contrary, they
never are." This is a much stronger opposition, in the ordinary sense,
than a modest contradictory, which is warranted by the existence of
a single exception. If the diagram were to represent incompatibility
accurately, the Contrary ought to have a longer line than the
Contradictory, and this it seems to have had in the diagram that
Aristotle had in mind (_De Interpret._, c. 10).

It is only when Opposition is taken to mean merely difference in
Quantity and Quality that there can be said to be greater opposition
between Contradictories than between Contraries. Contradictories
differ both in Quantity and in Quality: Contraries, in Quality only.

There is another sense in which the Particular Contradictory may be
said to be a stronger opposite than the Contrary. It is a stronger
position to take up argumentatively. It is easier to defend than a
Contrary. But this is because it offers a narrower and more limited
opposition.

We deal with what is called Immediate Inference in the next chapter.
Pending an exact definition of the process, it is obvious that two
immediate inferences are open under the above doctrines, (1) Granted
the truth of any proposition, you may immediately infer the falsehood
of its Contradictory. (2) Granted the truth of any Contrary, you may
immediately infer the truth of its Subaltern.[3]

    [Footnote 1: This is the traditional definition of Opposition
    from an early period, though the tradition does not start from
    Aristotle. With him opposition ([Greek: antikeisthai]) meant,
    as it still means in ordinary speech, incompatibility. The
    technical meaning of Opposition is based on the diagram (given
    afterwards in the text) known as the Square of Opposition, and
    probably originated in a confused apprehension of the reason
    why it received that name. It was called the Square of
    Opposition, because it was intended to illustrate the doctrine
    of Opposition in Aristotle's sense and the ordinary sense of
    repugnance or incompatibility. What the Square brings out is
    this. If the four forms A E I O are arranged symmetrically
    according as they differ in quantity, or quality, or both, it
    is seen that these differences do not correspond symmetrically
    to compatibility and incompatibility: that propositions may
    differ in quantity or in quality without being incompatible,
    and that they may differ in both (as Contradictories) and be
    less violently incompatible than when they differ in one only
    (as Contraries). The original purpose of the diagram was to
    bring this out, as is done in every exposition of it. Hence
    it was called the Square of Opposition. But as a descriptive
    title this is a misnomer: it should have been the Square of
    Differences in Quantity or Quality. This misnomer has been
    perpetuated by appropriating Opposition as a common name for
    difference in Quantity or Quality when the terms are the same
    and in the same order, and distinguishing it in this sense
    from Repugnance or Incompatibility (Tataretus in Summulas,
    _De Oppositionibus_ [1501], Keynes, _The Opposition of
    Propositions_ [1887]). Seeing that there never is occasion to
    speak of Opposition in the limited sense except in connexion
    with the Square, there is no real risk of confusion. A common
    name is certainly wanted in that connexion, if only to say
    that Opposition (in the limited or diagrammatic sense) does
    not mean incompatibility.]

    [Footnote 2: Cp. Keynes, pt. ii. ch. ii. s. 57. Aristotle laid
    down the distinction between Contrary and Contradictory to
    meet another quibble in contradiction, based on taking
    the Universal as a whole and indivisible subject like an
    Individual, of which a given predicate must be either affirmed
    or denied.]

    [Footnote 3: I have said that there is little risk of
    confusion in using the word Opposition in its technical or
    limited sense. There is, however, a little. When it is
    said that these Inferences are based on Opposition, or
    that Opposition is a mode of Immediate Inference, there is
    confusion of ideas unless it is pointed out that when this is
    said, it is Opposition in the ordinary sense that is meant.
    The inferences are really based on the rules of Contrary and
    Contradictory Opposition; Contraries cannot both be true, and
    of Contradictories one or other must be.]

CHAPTER III.

THE IMPLICATION OF PROPOSITIONS.--IMMEDIATE FORMAL INFERENCE.
--EDUCATION.

The meaning of Inference generally is a subject of dispute, and
to avoid entering upon debatable ground at this stage, instead of
attempting to define Inference generally, I will confine myself
to defining what is called Formal Inference, about which there is
comparatively little difference of opinion.

FORMAL INFERENCE then is the apprehension of what is implied in a
certain datum or admission: the derivation of one proposition,
called the CONCLUSION, from one or more given, admitted, or assumed
propositions, called the PREMISS or PREMISSES.

When the conclusion is drawn from one proposition, the inference is
said to be IMMEDIATE; when more than one proposition is necessary to
the conclusion, the inference is said to be MEDIATE.

Given the proposition, "All poets are irritable," we can immediately
infer that "Nobody that is not irritable is a poet"; and the one
admission implies the other. But we cannot infer immediately that "all
poets make bad husbands". Before we can do this we must have a second
proposition conceded, that "All irritable persons make bad husbands".
The inference in the second case is called Mediate.[1]

The modes and conditions of valid Mediate Inference constitute
Syllogism, which is in effect the reasoning together of separate
admissions. With this we shall deal presently. Meantime of Immediate
Inference.

To state all the implications of a certain form of proposition, to
make explicit all that it implies, is the same thing with showing
what immediate inferences from it are legitimate. Formal inference, in
short, is the eduction of all that a proposition implies.

Most of the modes of Immediate Inference formulated by logicians are
preliminary to the Syllogistic process, and have no other practical
application. The most important of them technically is the process
known as Conversion, but others have been judged worthy of attention.

ÆQUIPOLLENT OR EQUIVALENT FORMS--OBVERSION.

Æquipollence or Equivalence ([Greek: Isodynamia]) is defined as the
perfect agreement in sense of two propositions that differ somehow in
expression.[2]

The history of Æquipollence in logical treatises illustrates two
tendencies. There is a tendency on the one hand to narrow a theme
down to definite and manageable forms. But when a useful exercise is
discarded from one place it has a tendency to break out in another
under another name. A third tendency may also be said to be specially
well illustrated--the tendency to change the traditional application
of logical terms.

In accordance with the above definition of Æquipollence or
Equivalence, which corresponds with ordinary acceptation, the term
would apply to all cases of "identical meaning under difference of
expression". Most examples of the reduction of ordinary speech into
syllogistic form would be examples of æquipollence; all, in fact,
would be so were it not that ordinary speech loses somewhat in the
process, owing to the indefiniteness of the syllogistic symbol for
particular quality, Some. And in truth all such transmutations
of expression are as much entitled to the dignity of being called
Immediate Inferences as most of the processes so entitled.

Dr. Bain uses the word with an approach to this width of application
in discussing all that is now most commonly called Immediate Inference
under the title of Equivalent Forms. The chief objection to this usage
is that the Converse _per accidens_ is not strictly equivalent. A
debater may want for his argument less than the strict equivalent, and
content himself with educing this much from his opponent's admission.
(Whether Dr. Bain is right in treating the Minor and Conclusion of
a Hypothetical Syllogism as being equivalent to the Major, is not so
much a question of naming.)

But in the history of the subject, the traditional usage has been
to confine Æquipollence to cases of equivalence between positive and
negative forms of expression. "Not all are," is equivalent to "Some
are not": "Not none is," to "Some are". In Pre-Aldrichian text-books,
Æquipollence corresponds mainly to what it is now customary to call
(_e.g._, Fowler, pt. iii. c. ii., Keynes, pt. ii. c. vii.) Immediate
Inference based on Opposition. The denial of any proposition involves
the admission of its contradictory. Thus, if the negative particle
"Not" is placed before the sign of Quantity, All or Some, in
a proposition, the resulting proposition is equivalent to the
Contradictory of the original. Not all S is P = Some S is not P.
Not any S is P = No S is P. The mediæval logicians tabulated these
equivalents, and also the forms resulting from placing the negative
particle after, or both before and after, the sign of Quantity. Under
the title of Æquipollence, in fact, they considered the interpretation
of the negative particle generally. If the negative is placed after
the universal sign, it results in the Contrary: if both before and
after, in the Subaltern. The statement of these equivalents is a
puzzling exercise which no doubt accounts for the prominence given it
by Aristotle and the Schoolmen. The latter helped the student with the
following Mnemonic line: _Præ Contradic., post Contrar., præ postque
Subaltern._[3]

To Æquipollence belonged also the manipulation of the forms known
after the _Summulæ_ as _Exponibiles_, notably _Exclusive_ and
_Exceptive propositions_, such as None but barristers are eligible,
The virtuous alone are happy. The introduction of a negative particle
into these already negative forms makes a very trying problem in
interpretation. The æquipollence of the Exponibiles was dropped from
text-books long before Aldrich, and it is the custom to laugh at them
as extreme examples of frivolous scholastic subtlety: but most modern
text-books deal with part of the doctrine of the _Exponibiles_ in
casual exercises.

Curiously enough, a form left unnamed by the scholastic logicians
because too simple and useless, has the name Æquipollent appropriated
to it, and to it alone, by Ueberweg, and has been adopted under
various names into all recent treatises.

Bain calls it the FORMAL OBVERSE,[4] and the title of OBVERSION (which
has the advantage of rhyming with CONVERSION) has been adopted by
Keynes, Miss Johnson, and others.

Fowler (following Karslake) calls it PERMUTATION. The title is not a
happy one, having neither rhyme nor reason in its favour, but it is
also extensively used.

This immediate inference is a very simple affair to have been honoured
with such a choice of terminology. "This road is long: therefore, it
is not short," is an easy inference: the second proposition is the
Obverse, or Permutation, or Æquipollent, or (in Jevons's title) the
Immediate Inference by Privative Conception, of the first.

The inference, such as it is, depends on the Law of Excluded Middle.
Either a term P, or its contradictory, not-P, must be true of any
given subject, S: hence to affirm P of all or some S, is equivalent
to denying not-P of the same: and, similarly, to deny P, is to affirm
not-P. Hence the rule of Obversion;--Substitute for the predicate term
its Contrapositive,[5] and change the Quality of the proposition.

       All S is P = No S is not-P.
        No S is P = All S is not-P.
      Some S is P = Some S is not not-P.
  Some S is not P = Some S is not-P.

CONVERSION.

The process takes its name from the interchange of the terms. The
Predicate-term becomes the Subject-term, and the Subject-term the
Predicate-term.

When propositions are analysed into relations of inclusion or
exclusion between terms, the assertion of any such relation between
one term and another, implies a Converse relation between the second
term and the first. The statement of this implied assertion is
technically known as the CONVERSE of the original proposition, which
may be called the _Convertend_.

Three modes of Conversion are commonly recognised:--(_a_) SIMPLE
CONVERSION; (_b_) CONVERSION _per accidens_ or by limitation; (_c_)
CONVERSION BY CONTRAPOSITION.

(_a_) E and I can be simply converted, only the terms being
interchanged, and Quantity and Quality remaining the same.

If S is wholly excluded from P, P must be wholly excluded from S. If
Some S is contained in P, then Some P must be contained in S.

(_b_) A cannot be simply converted. To know that All S is contained in
P, gives you no information about that portion of P which is outside
S. It only enables you to assert that Some P is S; that portion of P,
namely, which coincides with S.

O cannot be converted either simply or _per accidens_. Some S is not P
does not enable you to make any converse assertion about P. All P
may be S, or No P may be S, or Some P may be not S. All the three
following diagrams are compatible with Some S being excluded from P.

  Concentric circles of S and P - P in centre

  S in one circle and P in another circle.

  S and P each in a circle, overlapping circle.
]

(_c_) Another mode of Conversion, known by mediæval logicians
following Boethius as _Conversio per contra positionem terminorum_, is
useful in some syllogistic manipulations. This Converse is obtained
by substituting for the predicate term its Contrapositive or
Contradictory, not-P, making the consequent change of Quality, and
simply converting. Thus All S is P is converted into the equivalent No
not-P is S.[6]

Some have called it "Conversion by Negation," but "negation"
is manifestly too wide and common a word to be thus arbitrarily
restricted to the process of substituting for one term its opposite.

Others (and this has some mediæval usage in its favour, though not the
most intelligent) would call the form All not-P is not-S (the Obverse
or Permutation of No not-P is S), the Converse by Contraposition. This
is to conform to an imaginary rule that in Conversion the Converse
must be of the same Quality with the Convertend. But the essence of
Conversion is the interchange of Subject and Predicate: the Quality is
not in the definition except by a bungle: it is an accident. No
not-P is S, and Some not-P is S are the forms used in Syllogism,
and therefore specially named. Unless a form had a use, it was left
unnamed, like the Subalternate forms of Syllogism: Nomen habent
nullum: nec, si bene colligis, usum.

TABLE OF CONTRAPOSITIVE CONVERSES.

                             Con. Con.
  All S is P                 No not-P is S
  No S is P                  Some not-P is S
  Some S is not P            Some not-P is S
  Some S is P                None.

When not-P is substituted for P, Some S is P becomes Some S is not
not-P, and this form is inconvertible.

OTHER FORMS OF IMMEDIATE INFERENCE.

I have already spoken of the Immediate Inferences based on the rules
of Contradictory and Contrary Opposition (see p. 145 - Part III, Ch.
II).

Another process was observed by Thomson, and named _Immediate
Inference by Added Determinants_. If it is granted that "A negro is
a fellow-creature," it follows that "A negro in suffering is a
fellow-creature in suffering". But that this does not follow for every
attribute[7] is manifest if you take another case:--"A tortoise is
an animal: therefore, a fast tortoise is a fast animal". The form,
indeed, holds in cases not worth specifying: and is a mere handle for
quibbling. It could not be erected into a general rule unless it
were true that whatever distinguishes a species within a class, will
equally distinguish it in every class in which the first is included.

MODAL CONSEQUENCE has also been named among the forms of Immediate
Inference. By this is meant the inference of the lower degrees of
certainty from the higher. Thus _must be_ is said to imply _may be_;
and _None can be_ to imply _None is_.

Dr. Bain includes also _Material Obversion_, the analogue of _Formal
Obversion_ applied to a Subject. Thus Peace is beneficial to commerce,
implies that War is injurious to commerce. Dr. Bain calls this
Material Obversion because it cannot be practised safely without
reference to the matter of the proposition. We shall recur to the
subject in another chapter.

    [Footnote 1: I purposely chose disputable propositions to
    emphasise the fact that Formal Logic has no concern with the
    truth, but only with the interdependence of its propositions.]

    [Footnote 2: Mark Duncan, _Inst. Log._, ii. 5, 1612.]

    [Footnote 3: There can be no doubt that in their doctrine of
    Æquipollents, the Schoolmen were trying to make plain a real
    difficulty in interpretation, the interpretation of the force
    of negatives. Their results would have been more obviously
    useful if they had seen their way to generalising them.
    Perhaps too they wasted their strength in applying it to the
    artificial syllogistic forms, which men do not ordinarily
    encounter except in the manipulation of syllogisms. Their
    results might have been generalised as follows:--

    (1) A "not" placed before the sign of Quantity contradicts
    the whole proposition. Not "All S is P," not "No S is P,"
    not "Some S is P," not "Some S is not P," are equivalent
    respectively to contradictories of the propositions thus
    negatived.

    (2) A "not" placed after the sign of Quantity affects the
    copula, and amounts to inverting its Quality, thus denying
    the predicate term of the same quantity of the subject term of
    which it was originally affirmed, and _vice versâ_.

           All S is "not" P = No S is P.
            No S is "not" P = All S is P.
          Some S is "not" P = Some S is not P.
      Some S is "not" not P = Some S is P.

    (3) If a "not" is placed before as well as after, the
    resulting forms are obviously equivalent (under Rule 1) to the
    assertion of the contradictories of the forms on the right (in
    the illustration of Rule 2).

    Not | All S is "not" P      = No S is P       | = Some S is P.
    Not | No S is "not" P       = All S is P      | = Some S is not P.
    Not | Some S is "not" P     = Some S is not P | = All S is P.
    Not | Some S is "not" not P = Some S is P     | = No S is P.
    ]

    [Footnote 4: _Formal_ to distinguish it from what he called
    the _Material Obverse_, about which more presently.]

    [Footnote 5: The mediæval word for the opposite of a term, the
    word Contradictory being confined to the propositional form.]