Part 29
We opine that when M. Mouret shall have consulted the great memoir of DeMorgan in the tenth volume of the “Cambridge Philosophical Transactions” or better, when he shall have become acquainted with the more developed work of Mr. C. S. Peirce, to whom beyond question relation-lore is most indebted for its present state of progress, he will have a better esteem for the value of the “tentatives” of DeMorgan and Boole and their disciples. Mr. Peirce has published three principal papers on the subject in question. The first of these was published in 1870 in the ninth volume of the “Memoirs of the American Academy of Arts and Sciences.” Then in 1880 and in 1884, while Mr. Peirce was lecturer on logic at Johns Hopkins University, he published in the _American Journal of Mathematics_ two papers dealing more or less extensively with relation-lore. One of these his “_Hauptwerk_” as it is called by Professor Schroeder of Carlsruhe, appears in the third volume of the journal mentioned and the other in the seventh volume of the same. Mr. Peirce was one of the contributors to our new American “Century Dictionary,” and in that work under the definitions of Relation and Relativity there appears a summary treatment of the subject which is as we take it the work of Mr. Peirce and which might have given to M. Mouret hints which he would have appreciated. Also the editor of this magazine in his article “Are there Things in Themselves?” in the January 1892 number thereof incidentally touches upon the topic under consideration in such a way as to correct some of the more inveterate misconceptions.
IV. M. MOURET’S THEORY OF RELATIONS.
The article of M. Mouret is in so many points so excellent a discourse that the chief reflection one is inclined to make is that upon its own principles it ought to have been better. He seems to have been widely awake to the primordial nature of relations as philosophical data. He says: “Every notion or relation is a _function_ of relations more directly known and enters as a relational element into other relations less proximate to the common sources,” and also that “every concept or notion ought to resolve into a group of relations.”
By such tokens as these we naturally look to see M. Mouret making it his 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: