very first concern to explain fully the nature of those primordial data
that are relations. Indeed he seems himself to be fully aware of this
natural expectation for he says: “What then is a relation; what is a
concept or notion? To this double question an answer is necessary and a
precise answer not consisting in the substitution of one form of words
for another form of words bearing the same meaning or no meaning at all.”
We cannot however find that he has done this. Instead of it and almost
while saying that every concept and notion ought to resolve in a group
of relations he announces, “What I have to examine is the constitution,
the structure of these _relation-generating groups_.” Thus he starts with
a synthesis when what is needed is analysis. He starts with supposing
a group of relational elements indeterminate in number and proceeds to
inquire as to the conditions that must subsist with regard to them,
respectively and in combination, in order that a _definite_ relation may
subsist as to a pair of the relational elements. These conditions he
finds to be four in number. First, he finds that—
“It is necessary that every one of the terms of the group should be
connected one to the other by _definite relations_; that between any two
terms there must always be intermediate terms that connect them in a
continuous way.” This he calls the condition of “Solidarity.” Secondly,
he finds that there must obtain the condition of _Co-Existence_. By
co-existence he intends—
“Not a definite co-existence in time, that is to say, a relation of
simultaneity or concomitance, nor yet that established co-existence
which constitutes the causal relation, but an indefinite co-existence
independent of the order of its terms and of all consideration of time or
duration.”
Having thus supposed his group all well stocked with relations, he
proceeds to relegate most of them to the limbo of inconsequence by
invoking a _principle_ which he calls the principle of _indetermination_.
By virtue of this principle in every particular case the _particular
determinations_ of all the terms become indeterminate and those of the
intermediate terms doubly so. Thus the supposed facts of the case become
fit for the existence of the _Third_ condition, that of _Abstraction_,
and for the arising of a general concept or notion.
But corresponding to every concept or notion is its negative or opposite
concept or notion. As this negative depends necessarily upon certain
_particular determinations_ of the same terms that bear the _particular
determinations_ and which being singly indeterminate admit of the
positive concept or notion, there necessarily must obtain two systems
of singly indeterminate _particular determinations_ relative to but
incompatible with one another, and so relative that the negation of one
set entails the obtaining of the other set, or in other words either
set being negated entails the obtaining of the other set. These facts
constitute what M. Mouret calls the _principle_ of _incompatibility_ and
involve his _Fourth_ condition of _Relativity_ stated by him as follows:
“All the particular determinations of the extreme terms must
not be compatible with the system and the negation of certain
relations of the system must entail the negation of the
relation which they make between the extreme terms.”
V. REFLECTIONS ON M. MOURET’S THEORY.
Now we cannot regard this as a successful attempt to explain the nature
and characteristics of relations, or to unfold the involutions of
relation-lore.
We fully realise that if every concept or relation resolves into a group
of relations we must in some form or other take what are relations in
reality as data to begin with, but this does not prevent us from taking
our datum terms for our turn of explanation as not requiring at present
any recognition other than as relational elements. What is needful as
a prime requisite on the very start of any research in relation-lore
is to obtain a clear idea of what is meant by a relation. Meanings are
primarily matters of mental status. We have to determine the relation
that subsists between the mind and the object through the mediating
interpretation of a word, and the mental affection lies nearest and
logically comes first. It may very well be that the mental affection
requires correction, but this cannot take place until its faults are
observed, and these cannot become evident until the mental affection is
itself duly understood.
The disciples of that school of logic in which DeMorgan and Boole,
both eminent mathematicians, hold so exalted a rank as discoverers,
regard cognition as arising in consequence of brain functioning or
_mental operation_ and study the results of this operation as yielding
their import in dependence upon and only in dependence upon the proper
operation in virtue of which they arise.
Now no cognition whatever, even of the most elementary sort, arises
except in connection with and in consequence of that operation of the
sensibility which is _distinction_. Distinction is of multitudinous and
manifold aspects. In all its phases whether it be passive or active
it is naught else than the arising or the assigning of relations. The
attempt to posit an unqualifiedly absolute—that is, an unqualifiedly
unrelated—universe of discourse must be futile and blank, necessarily
and insuperably. Any form of notation that pretends to express such a
universe of discourse, is only saved, if at all, from being unqualifiedly
nonsensical, by standing as antithetical—that is, by being _related_
to forms of notation that express relation and nothing else than
relation. This rigorously prime operation of distinction is not only pure
relation-ing but it is of that sort of relation-ing that is at once a
distinguishing and a conjoining. The “One and the Many” are insuperably
implicit therein. Distinction having operated to various extents, and
thereby various relations having come into view, we become aware of those
items of experience that are objects or facts. Each and every one of
these objects or facts are in truth distinguished and are therefore in
no strict sense _indiscernible_ from each other, but since no science
can possibly obtain in relation to mere particulars we find it useful to
disregard various points of distinction that obtain in respect to various
objects and facts and to converge our regard upon the points wherein
distinction, not absolutely vanishes, but _tends to vanish_.
By this operation, which is _abstraction_, various objects and facts
become in mental regard fit and useful to be taken as copies of one
another and as indifferent for use in most of the turns of mental life.
There are indeed various relations, objects, and facts, with respect to
which no further operation or operations of distinction than the mere
distinctions of the time, the place, or the occasion of their various
manifestations have been applied nor can without great difficulty be
applied. But we are therefore by no means entitled to say that such
are in truth irresolvable. Contrarily, and reasoning inductively we are
justified in concluding that every relation, object, and fact will under
analysis of adequate power resolve without limit into other relations,
objects, and facts.
We are not yet prepared to see that the ultimate components of relations,
objects, and facts resolve into relations, and nothing but relations,
because we are not yet prepared with an explicit idea of the nature and
characteristics of these elementary objects.
The study of M. Mouret since it starts with relations combining them
under the conditions of Solidarity Co-Existence, Abstraction and
Relativity, (which are nothing else than other relations or compounds
of relations,) does not seem to us to advance us at all in the most
fundamental requisite. He says no more than to say that in order for the
groups of relations to generate further definite relations the relations
thus grouped together must be related to one another and then that most
of these relations must be disregarded.
M. Mouret distinguishes, with respect to a relation three factors, the
_Matter_, or the relational elements grouped together, the _Form_, or the
order in which the relational elements are arrayed, and the _Foundation_.
As his study of the topic of relations is professedly for the purpose
of enabling him the better to solve the nature of the relation of
mathematical equality, his success may be estimated by reference to
his conclusions in regard to that relation. These are as follows: “The
relation of equality is formed of undetermined matter, it possesses a
binary form, and has for a condition a relation of indiscernibility
between the two elements.”
Such conclusions appear to us to be impotent not to say erroneous. If
two things obtain at all, they obtain as two and not one, in very virtue
of being distinguished the one from the other. Except with regard to
some more or less arbitrary distinctions, like the distinction between
coincident points, all distinctions obtain only in virtue of some
relation that can be nothing else than a point of _discernibility_.
Numbers and other mathematical things are taken as not-different not
because they are in truth indiscernible but because for the turn in hand
their points of difference are irrelevant.
Concerning the much mooted question of the proper field of logic as a
science M. Mouret holds it to be the “science of relations and general
concepts.” Although we hold that logic is particularly concerned with the
lore that is more directly related to the phenomena of erroneous thinking
and its correction, the view of M. Mouret is not unacceptable. “Reasoning
consists in the observation that where certain relations subsist certain
others are found,” as Mr. Peirce has remarked.
VI. CONCERNING MEANINGS AND EXPLANATIONS.
As preliminary to our account of relations we will make an observation
which seems to us of considerable use in connection therewith. It is
not without its bearing on the theory of definition or rather upon the
broader theory of explanation. With M. Mouret we hold that every concept
and relation resolves into a compound of relations. Since relations are
data that are absolutely elementary at least so far as we are at present
instructed they are of course not subsumable under any other sort of
data that are better known. Moreover, whatever explanation we here make
must needs be made by means of written words. Thus an important question
arises as to what method is to be pursued in this special exigency. The
theory of definition leads us to the same difficulty, for although the
meanings of many words can be defined in terms that are more proximate
to the elementary relations, we will always come at last to terms that
admit of no improved explanation by such a method. There is no device
of words that can evade or supersede the ultimate recourse to things.
Now the significations of words are learned in most cases not so much by
definitions and verbal descriptions as by _the observation of the various
applications of the words_. Indeed this is the primitive way in which the
meanings of words are found out. The child knows nothing of what, say,
the word _horse_ means until some one shows it an actual horse and may
be pointing to it says repeatedly, _horse_, in such a way as to excite
the observation of the child to the intended application of the word to
the thing. This is because the relation of every general sign to its
object subsists only in consequence of a mental association, and until
this mental association is created the sign has no meaning. The methods
of evoking these mental associations are at present quite unmethodical
and do not receive the attention which their importance merits. One
feasible method is to present or to state a number of scenes that shall
present the object in various ways in connection with the sign thereof,
and thus to excite attention to the proper application of the sign. The
geometer does this by means of his diagrams without which or their mental
counterparts all his mere words would be in vain.
Mr. Edward T. Dixon has lately published a work on the “Foundations of
Geometry” in which he would introduce as a fundamental datum what is
really an altogether new and exceedingly abstract conception which he
calls by an old name, that of _direction_. The old term has never been
as yet taken in any abstract universal sense because apart from definite
right lines showing it as an attribute any abstract universal meaning
is wholly unassignable. But the conception that Mr. Dixon would instal
is removed in abstractness from such a universal yet one more step in
universality. A three-fold infinity of right lines differing in direction
can be drawn in ordinary space to each of which pertain two corresponding
universals of direction, one converse to the other. Now the conception
of direction that Mr. Dixon proposes for service as an elementary
geometrical datum is the universal that subsumes all these lower ranking
universals as particulars. Of course he has difficulty in even trying to
explain what he means. Realising the impossibility of subsuming it in
any way he takes a method which if it were more thoroughly applied, and
wholly emancipated from the lingering notion of definition, might have
been more successful. As he actually left the matter his real meaning
can only be drawn from close study of the way in which he applies the
term in his discourse in general.
Owing to its excessive abstractness his conception is wholly unfit for
service in elementary geometry. One has to become a good geometer before
the conception can even be approached.
VII. ANOTHER THEORY OF RELATIONS.
We shall proceed to explain what we regard as a true and adequate notion
of a relation by stating some scenes that display the same. We do not
regard it as needful to state many of them and we take for our first one,
the common transaction of making a donation. We have here for relational
elements or terms as they are usually called a set of three. Separately,
or as not yet brought into relation in virtue of the giving, there may
be, say _G_ an owner of _W_ a watch and _R_ the intended beneficiary. The
plural fact of the giving is the _relationship_ or the _foundation_ of
the _relations_ that arise in virtue of said giving. This _foundation_
becomes to be in virtue of the creation of such relations by the giving.
Either one of the set of three may be taken as the datum of reference and
according to the election in this respect, the relations may differ and
the technical names we are about to give will vary in their application.
Since simplicity will be gained thereby and also our present turn fully
subserved we will take _G_ as the datum term of reference. So taking
it _G_ is called the _relate_ and both _W_ and _R_ are called the
_correlates_. For this present turn and in very virtue of the giving and
only in virtue thereof _G_ becomes related to _W_ and _R_ in a certain
relation one of the names of which is giver. When _W_ or _R_ are taken
as relates certain other relations appear, some of the names of which
are respectively _present_ and _recipient_. In relation to the relation
of giver the relations of present and recipient are named _converse_
relations, as are likewise the relations of present and giver to the
relation of recipient and the relations of giver and recipient to the
relation of present. Here are three distinct relations growing out of
the same relationship or foundation. As each relational element has its
corresponding negation, the true logical system of a set of three terms
involves not less than eight relations.
We take for our second scene the case of a boundary. This might be a
surface or a point but we will take the special case of a line on a
surface. Here we have again a set of three, the spread on one side, _A_,
the opposite spread, _B_, and the line _L_. _A_ has a certain relation,
say above, to _L_ and _B_, _B_ has the certain relation, below, to _A_
and _L_, which relation is converse to the relation above: and _L_ has
the certain relation, boundary, to _A_ and _B_ which relation is converse
to the relations above and below.
The two examples now given are cases of the _conjugative_ kind. The
relationship is a conjugative one and the relations are conjugative
relations. The distinguishing characteristic of a conjugative case is the
fact essentially involved of the mediation between relational elements
by another certain element, or in other words the bringing of diverse
relational elements into relation by the function of another relational
element. Without the mediation or function of this conjugating element
neither the relationship nor the conjugative relations can exist. There
is reason to believe that all conjugative cases can be certified as cases
of three relational elements or as compounds of a number of such sets of
three. To ordinary uncritical thought which is largely constrained by
the trammels of ordinary language the most abundant sort of relations
appears to be of that sort that are taken to involve only two relational
elements. These are cases of what are called _dual_ relationships and the
relations that arise out of them are called _dual_ relations. Such are
those like father, son, husband, wife, etc. Strictly viewed they ought to
be regarded as _degenerate_ relationships and relations just as a pair of
lines is regarded as a degenerate conic.
VIII. CAUTIONS AND APPLICATIONS.
Now besides the error of confounding relations with relationship, it is
a very common fault to think and speak of a relation as being _between_
two or more terms. This imports into thought the thoroughly misleading
idea of an intervening independent existence for relations. Relations are
attributive predicates of terms and each one of them pertains strictly to
its proper term or combination of terms, in the same sense for this turn
(_pro hac vice_) that qualities are held to pertain to their so-called
substances. And yet relations so pertain to their proper terms not in
virtue of such terms separately but in virtue of their membership in
the plural fact which obtains as the _relationship_ or _foundation_.
The notion of a relation as a “betweenness” has perhaps been fostered
by the exact coincidence of relations pertaining to the several members
of the same relationship. When on contemplating the connection, say
of two points, we observe that the distance of one from the other is
apparently indistinguishable from the distance of the other from the one,
we naturally overlook the fact that we are truly to regard the connection
as the coincidence of two really distinct relations, and regarding the
pair of relations as one thing and finding it not attributable to one
point more than to the other we dissociate it from both. But when we
consider a pair of relations that are converse to one another and that
arise out of a dual relationship like that of husband and wife we may see
that there is _no_ betweenness, no single relation that interlies, but
two relations, one the relation of husband and the other the converse
relation of wife.
An interaction, say like that of approach under the influence of
gravitation, is a relationship. Each body stands in the relation of a
_puller_ of the other and the mediating term which we find impossible to
argue out of the account we call the attraction of gravitation. In this
case the relation of action of the one body is not usually distinguished
from the relation of action of the other one. Indeed this is the case in
all cases of mechanical action and we lay it down as a maxim that action
and reaction are equal but they are not alike since their directions are
opposite. Sensation is a relationship, since it is our interaction.
The object interacts with the brain. As to the conjugating term we are
as yet in the dark and so we are in the habit of regarding this case
as a dual relation. The relation of the brain to the object is that of
a _knower_ and that of the object to the brain that of a _stimulater_.
Each character or mark of the object that becomes apparent gives rise to
relations and their respective converses each correlative pair of which
are respectively so many distinct interactions of detail in the entire
interaction. Whatever an object as known to us is, it is in virtue of
those relations of brain action and detailed object stimulation, which
are relations and always relations. Since consciousness exists only
by the arising of relations of distinction, supposably in consequence
of internal brain interaction, is it presumptuous to allege that
consciousness consists of relations or a complex of relations?
IX. NATURE OF OBJECTS.
With regard to the object no one can prevent whoever may be so disposed,
from imputing to it various points of possession that do not and cannot
interact with the brain. So far as such imputed points are regarded
as merely not yet interacting but possible to interact with knowing
substance such points in no wise differ in essential nature from the
known attributes. They are potential relations and nothing else. But in
so far as they are regarded as essentially impossible of ever interacting
with knowing substance in any possible stage of its development such
regard is pure nonsense and utterly without any assignable meaning. There
is no occasion whatever for such an imputation, for the existence of
interaction actual or potential is fully adequate to explain all that
will ever present itself to be explained.
At this point let us instruct ourselves with an example of the reasonings
of a much and deservedly honored philosopher. He says:
“In the most general predicate which is determined Being
or existence—for all things in the universe are determined
beings—we have an evident two-foldness (a composite nature)
which allows of a further analysis into pure Being and
determination.”
We will parallel this analysis. For the sake of simplicity we will take
a limited right line. It has the determinations straight and long, not
length in the sense of measure, for length is ambiguous in its intent,
but length in its qualitative sense—its linearity so to speak. Now
separate from it first its straightness without however giving to it
any other determination, and then its quality of longness. We have then
a _pure_ line, that is neither straight, nor long, nor anything else.
Such is an example of “Pure Being.” _We_ say however that its very being
as a line is absolutely dependent on its determination as a long line;
that such a determination alone constitutes it a line, is at once its
determination and its being, that there is no two-foldness at all but
only two names, and that as one-fold its determination as long and its
being vanish together. What is true of a line is true of all relations
and compounds of relations whatever.
Thus not only all knowledge but all existence so far as that term can
ever have any meaning is relative; relative to all intents and for every
possible turn.
To those who accept the essentials of this account of relations it will
be easy to see what is the nature of an object and that of a concept
or notion. An object is a relation or some congeries of relations that
usually present themselves as a coherent whole to our sensibility or to
consciousness. This is primarily effected in virtue of some efficacy
which we cannot appropriate to ourselves, and so we distinguish our own
personality from that manifold that we call the objective world. It is
pure self-stultification after having made this most useful distinction
to try and abolish it. Nothing but an utter abolition of all useful
thought can result from so taking the data of experience.
X. NATURE OF A CONCEPT.