Section 9

Section 9 — Mathematical Formulas and Metaphysical Reflections of G. Reinin

a) Personal attitude

When I first heard about Reinin in 1987 — the man after whom Aushra Augustinavichiute named the differential traits (DTs) — I was filled with genuine respect for him. Rumors circulated that he had made an important discovery in socionics, mathematically showing that if there exists a set of four independent traits which generate a sixteen-element typology and thereby serve as its basis, then eleven further traits must exist. These “additional” traits, though dependent on the basis, make it possible to see non-obvious parameters of the types. And Aushra said that when Reinin, one of her first students, told her of the existence of these “hidden” traits, she carried out the necessary work and discovered the octaves of socionic types that possess these traits. Aushra gave the discovered traits a substantive interpretation and assigned them more or less provisional names — which subsequently became both the foundation of scientific research and discussion and the cause of numerous but superficial misunderstandings, including over whether these traits exist at all. Indeed, people appeared who set about defending the utterly incomprehensible and, in my view, simply wild position that no DTs exist besides the four Jungian ones. To this day there is no consensus on the matter. Getting ahead of myself, and seeking to restore authorial priority, I insist that these traits should at the very least bear a double name — the Augustinavichiute–Reinin traits, or ARTs for short. I discussed this with Grigory Romanovich, and he agreed that this is right. One can understand Aushra’s selfless position and her quixotic modesty, but let us nonetheless be scrupulous in these matters. Assessing Reinin’s contribution, one might say that his name could perhaps be given to the differentiating traits of abstract sixteen-element sets — but even then they ought to be named after Reinin–Ekman, in whose work the first sketch appears of a formula linking the number of types with the number of traits that can describe them. As for the socion, ignoring the name of Aushra Augustinavichiute here would be simply improper.

Later I read Reinin’s articles, saw the lists of his conference talks on typological themes, heard his presentations, and thus had some grounds for forming a direct opinion of his high scientific competence. But in 2007 I bought his book Socionics, published in 2005, and slowly began working through it. Here are my impressions. I speak of them frankly, because I do not want to hide anything, much less lie. After a first pass through the book, my opinion of it split exactly in half. As soon as I discovered that in the first, properly socionic part Reinin uses Model “K” (Kalinauskas’s), resting on the MQS (method of qualitative structures), his rating in my eyes immediately fell. By that time I had already worked out Kalinauskas’s paradigm — he was the first to impose the MQS on socionics and thereby obtained a theoretical construct he called constructive socionics. This constructive socionics differed so much from classical socionics that nothing remained in common between them except terms with reworked meanings and graphic symbols. My critique of Kalinauskas can be found in the corresponding chapter of this book, and anyone interested can not only read it but discuss the questions that arise with me. But I warn you at once that I am interested only in the truth. And if, on the path of seeking it, manipulative ethical insinuations are put before me, or unscrupulous lampoons are written about me, let them remain on the conscience of the manipulators. I violate no ethics, because telling the truth and conscientiously investigating nature is no such violation.

Assessing the first part of Reinin’s book, I also noticed that Reinin, while reproducing Kalinauskas’s methodology and terminology almost in full, introduced certain additions to Model “K,” so that his model should naturally be called Model “K-R,” “Kalinauskas–Reinin.” However, I believe the additions Reinin made only aggravate the distortion of classical socionics. The point is that Kalinauskas, unfolding “his helm,” supplemented the plus-functions with minus-functions in which the sign carries a fully mathematical semantics, since it changes the value of the plus-function to its opposite — “good” to “bad,” “must” to “must not,” problems to their solution, “want” to “don’t want.” Thus his model, though it looks eight-functional, is in fact only four-functional. It comes closest to Model “Yu” — or rather “A-Yu” — but compared with Model “A-Yu” the vertness of its PLR (Place of Least Resistance) is changed, which means that in interpreting Model “K” we will get a systematic error, as indeed turned out to be the case in his book.

Reinin designated the minus-functions of Model “K” by the aspects of the plus-functions but with the opposite vertness. This creates the illusion that Model “K-R” coincides with Model “A,” but of course it does not, because a change of vertness and a change of sign are not identical. The sign change in Kalinauskas does not contradict the meaning of his model, but the vertness change in Model “K-R” leads to a semantic error. Having then read several of Reinin’s type descriptions, I was deeply disappointed that formally “correct” interpretations were made from a distorted model, while interpretations more or less correct in content followed from nothing and looked like translations from other sources — first of all, of course, from Aushra. The result gives the impression of an eclectic mixture in which correct descriptions of types and relations are whimsically combined with interpretations that, from the standpoint of classical socionics, are distorted or meaningless.

Looking through the second part of the book, with its methodological and mathematical sections, I thought: well, this part, at least, is surely of high value. Skimming the formulas, somewhere on a subconscious level I noted that formulas are formulas, and since they are written down, everything there is automatically in order. And suddenly, glancing at page 188 [1], I noticed an unusual arrangement of the coordinate axes — the X axis pointed upward. On the one hand, being a mathematician and a liberal, I thought this really doesn’t matter much; but being an editor, I began scrutinizing the page and suddenly realized that in formula (18) [1] the notation does not correspond to the drawing. I began reading the text carefully, pencil in hand, and found gross errors in formulas (19) and (20), in the variable notation on page 189, and in formula (22) on p. 191 [1]. My trust in the text then fell sharply. I began reading the second part of the book from the very beginning, and did so meticulously and attentively — now as an editor. But what matters here is that all this is easily verifiable — one need only look at the corresponding page of the book.

I will add that the subsequent analysis often goes beyond the tasks Reinin set himself in his book, and in many cases my comments do not concern Reinin but only the general-scientific concepts encountered in his work. They too have not all been brought to perfection. These are my methodological improvisations on the themes of Reinin’s texts, and they in no way cast a shadow on the undoubtedly subtle and perceptive intellect of Grigory Romanovich, whom I regard with deep and genuine respect.

b) Methodology and metaphysics

Fragment of a letter from Churyumov to Reinin

“Hello, Grigory!

I write to you honestly and openly, as befits one scientist writing to another, and I would very much not want you to take offense. Still, ‘truth is dearer…’ And besides, what could be offensive in a discussion of scientific problems?

As long as I had not read closely into your text, it remained a sacred cow for me and, as such, was perceived as a full-fledged source of information. It was with that feeling that I read the first part of the book, but I very quickly sensed its superficiality — behind the correct words hides a highly unstable meaning: diffuse, elusive, rootless. Just one more subjective variation on the primary, but very solid and conscientious, material created by Aushra. By and large a writer, and all the more a scientist, should write when his material becomes a value that must be passed on to people and staying silent is impossible. Unfortunately, of the several dozen books written on socionics, the greater part falls short of this and tends toward graphomania.

Socionics needs an axiomatics — above all, a semantic one. There is none in your text. There is a rehashing of others’ leavings. I was especially disappointed by your turn to Model “K” (Kalinauskas’s), which in his rendition at least looks authentic — you can feel that he invented it himself. Nevertheless, it is a ‘tricked-out’ and distorted version of what AA called Model “Yu” — which of course is really Model “A-Yu” (Aushra–Jung), because Jung had not a model but a schema that only in first approximation pointed to the structure of the psyche. Igor distorted it by changing the vertness of the third function, as a result of which all his interpretations contain a systematic error, and his ‘helm,’ once the aesthetic and metaphysical embellishments are stripped away, reads — compared with Model “A” — as an altered and degenerate construct. I have described this in detail in the corresponding chapter of this book. In brief, the MQS is a metaphysical construct inadequate to the psyche, which clarifies nothing and only obscures the essence of the matter. And in general, why use an external, artificial contrivance when there exists Model “A,” which fully reflects the structure of the psyche? Yet Kalinauskas, while partially preserving the terminology, completely changed the conceptual structure of socionics, effectively substituting his own notions of it for Aushra’s socionics.

You introduce into Model “K” a certain improvement — something K did not have. Namely, instead of the minus-ends of the ‘helm’ (want – don’t want, must – must not…), you draw aspect symbols with changed vertness. And this distorts the meaning of Model “A” even more, since a change in an aspect’s vertness does not correspond to and is not equivalent to a change of sign. ‘Minus black intuition’ is not ‘white’ — [its variations] remain internal variations of ‘black.’ As a result, your attempts to ‘improve’ a heavily deformed construct make it even worse. All this creates a different apparatus of interpretation with substantially altered meaning, and the partial coincidences with Aushra’s type descriptions look suspicious — like fitting to a known answer. In short, the impression is that the author has the most superficial notions of scientific methodology, though he is undoubtedly intelligent and experienced. Igor, at least, spells out his foundations clearly: the I-concept instead of the structure of the psyche, Simonov’s physiological substrate instead of IM, and so on. Reinin, however, only listlessly reproduces others’ ideas, apparently neither fully understood nor accepted by him. As a result he writes something plausible-ish, with a certain quantity of small ornaments. As a result a great many people will be misled about socionics. Here it is easy to hide behind the relativity of subjective exposition, and behind relativity in general, but that would be no more than a positivist trick. Incidentally, that is exactly what K does when, at the very end of his book, he writes about the subjectivity of any description. Of course, the first part of your book deserves a more detailed examination — not only ‘for the edification of posterity’ but for working through methodological subtleties — but I will give only an analysis of the fragment on typology from the second part of your book. And, properly speaking, this may bear no relation to the author’s intent. But I am using the occasion to ‘work through’ the topic, so you need not take my discussion personally.

As I understood from various conversations, you at one time wrote a diploma thesis toward the qualification of psychologist (I would like to know the details from the primary source). It appears this work entered your book without changes. Unfortunately, the fragment under consideration turned out to contain many errors. At first I noticed only a few carelessnesses in the formulas, but when I looked closer… It would be fine if you would, as far as possible, respond to my remarks, which, in any case, have the character of editorial corrections and in no way claim absolute truth.”

As a student work, the fragment in question from Reinin’s book [1] is quite understandable — the more so since, when spoken through quickly, it gives the impression of a deep and many-sided investigation. It is also possible that the psychologist-instructors who examined it did not know mathematics very deeply — the theory of abelian groups in particular — while a cursory glance at mathematical formulas leaves an impression of their natural correctness. On closer examination, gross errors in the mathematical derivations become visible, along with the unjustified use of certain mathematical concepts, the hasty and simplified introduction of generalizing constructs, and subtle logical inaccuracies that make the text not merely dubious but misleading. A condescending attitude toward a student work is quite understandable, but it cannot be transferred to a serious theoretical claim — all the more so one placed in a published book. It is of course a pity that the edition’s editors noticed neither the theoretical defects of the fragment nor the gross errors in the mathematical derivations — and apparently the author, for whatever reasons, had no opportunity to proofread the galleys carefully. It should be added that, because of all this, plausible assertions in this text acquire the status of correct conclusions from incorrect premises. To this category belongs, for instance, the conclusion that “all whales are vertebrates” (which is true) on the grounds that “all whales are fish” (which is false) and “all fish are vertebrates” (which is true).

Reinin begins the corresponding section of the book with the task of clarifying the concepts “classification” and “typology.” He conscientiously surveys the literature, on which basis he concludes that “in the interpretation of these two concepts by psychologists, sociologists, and representatives of other fields of knowledge, there are differences.” Let us cite examples of such interpretations from his book [1], which is methodologically instructive in itself — though note that the author gave them no principled assessment, limiting himself to quotation, which is also not bad. Once more I stress that my analysis bears no relation to the author’s theoretical intentions but simply represents an opportunity to work with the material since it has come to hand. And, it goes without saying, we are dealing here with a specimen of high theoretical thinking, deserving of every respect and attentive treatment — and what the editor perceives as defects may not be defects at all.

Reinin (drawn from literary sources, see [1]):

“Classification is a special case of applying the logical operation of dividing the object of a concept. Classification by essential traits is called typology. Classification is a way of organized representation of a phenomenological field (motivational factors, personality properties, properties of the objects under study). Classification by types is a process of discovering combinations of traits. Typology is a partition of the studied population into groups of same-type objects. Objects are called same-type when the difference between them, in the direction of interest to the researcher, is explained by factors random with respect to the phenomenon under study and to the mechanisms determining its character. In the same work, classification is defined as ‘the partition of an initial population of objects into classes by means of some formal method.’ A partition whose classes are described through the traits used in constructing it is called a typology, or typological grouping. If a typological grouping is described also in terms of some other system of traits pertaining to the given phenomenon (i.e., establishes in explicit form theoretical connections between different sides of the studied system), then it is called a classification.”

Commentary. None of the cited interpretations of the concepts under study satisfies the rules of definition. Let us see why.

In the first case, the rule of definiteness of the genus concept is violated: it is designated as “a special case,” but it is not said in what the specialness consists. In the second part of the first example, typology is defined as a particular case of classification, but obviously such a definition is incorrect, since the first part is incorrect.

In the second case, the rule of defining the unknown through the known is not observed, since the predicate “a way of organized representation of a phenomenological field” can hardly be considered self-evident. All the more so since “a way of organized representation of a phenomenological field” is the proximate genus, while the concrete variety of this way is not indicated — that is, it is one such way, but what its concretizing peculiarity is, the definition does not say.

The third interpretation sounds still more enigmatic: when one says “classification by types,” the suspicion creeps in that there also exists a “classification by classes,” and the predicate “a process of discovering combinations of traits” is likewise not something self-evident.

The fourth case contains the error of a “circle in the definition,” since typology is defined through type and same-typeness. This error, however, is easily removed here if the term “same-typeness” is replaced by its explication. It could then read: “Typology is a partition of the studied population into groups of objects the differences between which, in the direction of interest to the researcher, are explained by factors random with respect to the phenomenon under study and to the mechanisms determining its character.” Of course, one must understand here that the talk is of a set of empirical objects, which limits the scope of the definition — likewise among the errors of definition. A “circle in the definition” is present in the fifth case too, but there it is no longer compensated: classification is defined through class.

The sixth definition is a special case. Here typology is defined through classes, which also leads to a disguised “circle in the definition,” since the concepts “class” and “type” have considerable semantic commonality and there exists a domain of their use in which they are synonyms. The presence of the “circle in the definition” is further strengthened when the typology so defined is, under certain conditions, renamed a classification.

Having made a fairly complete survey, Reinin arrives at a correct conclusion: “From the multitude of formulations available in different fields of knowledge it is evident that some authors consider the concepts T and C coordinate, others establish relations of inclusion between them, including now T in C, now C in T. Moreover, in many works T and C are not defined at all, or are used as synonyms.”

Commentary. As far as I understand, the concepts of type and class as taxonomic units are synonymous, but by convention they may be assigned to different levels of a system’s organization — as, for example, in biology. The subtle semantic difference is that “class” points directly at a set, whereas type points at an individual object entering the set and possessing definite properties. In addition, there are peculiarities of word usage in different subject areas — in psychology, for instance, as applied to describing personality, the concept “type” took root, and, correspondingly, typology. It should also be borne in mind that classification is the process of partitioning into classes, and hence is a methodological instrument. Typology, by contrast, is the finished theoretical product of partitioning a set into classes or types — which within certain limits is synonymous. Classification could have been paired with “typologization,” but in the language this did not happen. Another term, “typing” (tipirovanie), proposed by the Dnepropetrovsk socionists, acquired the more concrete meaning of socionic diagnostics, i.e., determining the TIM of a subject.

It is not difficult to give a tautological definition of typology as the process or result of partitioning an empirical set into types. This definition, however, becomes identically correct once a definition of type is given. For example, thus: a type is a homeomorphic model of a concrete subset of empirical objects. “Homeomorphic” here means that the model reproduces at least part of the parameters of the object under study. An isomorphic model would reproduce all the essential parameters. Analogously one can transform the tautological definition of classification as the process or result of partitioning an empirical set into classes. Here, for example, a class could be defined as a subset of empirical objects identical or indistinguishable by some trait or by an arbitrary set of traits. Despite the somewhat different verbal dress of these two definitions, they are, in essence, identical. A large number of definitions of type, class, typology, and classification is possible — identical in meaning but differing terminologically.

Closely adjoining the two epistemological categories — classification and typology — is the concept of “systematics.” Establishing this concept’s synonymic closeness to classification and typology, however, is somewhat harder. The general-scientific concept of systematics is the bringing of existing knowledge into a system; but in biology this concept acquired the more concrete meaning of bringing into a system the principal objects of study, namely plants and animals. In biology the concept of systematics is identical to the concepts of classification and typology, since it constitutes a hierarchical system of several taxa. A taxon in biology is a set of plants or animals possessing traits of a definite degree of commonality. As names of taxa such terms serve as species, genus, family, class, type/phylum, order, division, etc. — words containing the seme of “set” — and their hierarchy in biology took shape more or less spontaneously and bears no direct relation to their primary meaning.

Concluding the survey, Reinin writes: “Such discord in formulations leads to terminological confusion and methodological errors.”

Commentary. This is undoubtedly a serious statement, showing that the author is aware of the measure of responsibility connected with methodological errors.

After this the author sets himself the task of “analyzing the concept ‘type’ in more detail.” The analysis begins with the phrase: “Consider the semantic field of some quality (e.g., human, tree, snow), represented by the set of its essential traits:

A = {A₁, A₂, A₃, … Aₘ} (2)

where m is the number of existing essential traits characterizing the given quality.”

Commentary. Of course, calling a human or a tree a quality — the more so when the author says this “quality” possesses a set of essential traits — is, at minimum, carelessness, since these are objects possessing qualities. How would you like the expression “This quality possesses the following qualities…“? Clearly, in that case the conditions of semantic discrimination deteriorate. But in some very generalized sense, or as a working term, such word usage could be accepted temporarily — pending possible consequences in the form of contradictions or paradoxes. Moreover, since the author made no corresponding reservations and clarifications, anything at all could be assigned to the category of such “qualities,” including, for example, the points of a continuous set: of any curve or segment of it, of any plane figure and any geometric formation whatsoever — which renders many of the subsequent arguments dubious or incorrect.

Reinin: “Every trait from A distinguishes, in a certain subject sphere S which our thinking connects with this quality, a definite number of nonempty homogeneous subsets.”

Commentary. In this phrase, to my editorial eye, it would be better to write not “a certain” but a concrete subject sphere, or to drop “certain” altogether — but let us chalk that up as a minor stylistic flaw. And, of course, one should write “nonempty subsets homogeneous with respect to the given trait.” It is harder to agree that a single trait is responsible for the appearance of “a definite number of nonempty homogeneous subsets”: since “homogeneity in this case means the impossibility of distinguishing objects by means of this trait,” it ought to guarantee the identity of objects at each of the trait’s poles, and exactly two subsets should be distinguished. Objections are possible here pointing to multipolar traits, but that concept must be introduced separately. Yet the very next assertion speaks of bipolarity, not multipolarity.

Reinin: “Henceforth, without loss of generality, by a trait we will understand a certain bipolar construct (a, ā). Classification of the elements of S by any such trait from A generates a partition into two homogeneous subsets Sₐ and S_ā.”

Commentary. Thus the author admits that a trait — or rather one of its poles — generates one subset, which contradicts his preceding assertion. But already in the first phrase the author makes two errors. First, using the standard mathematical turn of phrase “without loss of generality” (another variant: “without restriction of generality”), he immediately restricts the concept of trait, proposing to understand it as a bipolar construct. Alas, logic has two concepts for designating opposition: contrariety and contradictoriness. The first relates to polar concepts between which discrete or continuous transitions are possible — for example, black–white; the second to opposite concepts between which there are no transitions. One of the latter concepts is something concrete, and the second is its negation and can be arbitrarily blurred — for example, black–non-black. In the first case a trait built on a contrary pair will be bipolar. But a trait built on a contradictory pair, though it can, if one wishes, be considered bipolar, is in fact monopolar, since selection by it is carried out not by distributing into two sets but by gathering suitable objects into their set — that is, in this case one pole is active and the other reduced.

However, traits can have any number of extrema. For example, the choice of language on a computer may be made from a considerable set of available options. Contrary-dichotomous traits constitute an important but not the only class of traits. Already in socionics we deal with at least three types of traits: dichotomous, which divide the socion into regular octaves (Augustinavichiute — there are 30 of them); tetratomous, which divide the socion into Reinin’s regular quadruples (there are 35, but since each quadruple contains four quadruples, this number must be multiplied by four, giving 140); and octotomous, which divide the socion into Churyumov’s regular dyads (this is a joke, but I really did count the number of regular dyads and have partially studied them — there are 120).

Reinin: “Such a partition can be written as an ordered pair:

X = (Sₐ, S_ā) (3)”

Commentary. Understandably, the partition — both as an operation and as its result, i.e., the set of the two subsets into which S is partitioned — can be denoted by the letter X; but obviously X as a set is exactly equal to S, i.e., X = S. And if we partition S by one of the traits from A, why not denote the resulting pair Aₘ = (Sₐ, S_ā)? In that case the trait Aₘ acts as the partition operator. Such an approach would reduce the number of variables, introducing no extra confusion, and would read as “the bipolar trait A partitions the set of types S into two subsets, Sₐ and S_ā.” Moreover, a direct correspondence would be established between the trait and the dichotomous pair of sets. But suppose the author here distinguishes the trait A by which the set is partitioned, the partition operation X itself, and the partitioned set S. This has its point, of course, but it masks the organic unity of the operational elements, which matters for the reader’s understanding of the text. Of course, the author may choose his system of notation as he sees fit. I only want to draw attention here to the author’s theoretical style, to the shades of his intellectual activity — and, naturally, my view of set partitions in no way compromises Reinin; it merely shows the difference in approach between two intuitives.

Reinin: “We will continue the partitioning procedure by traits as long as at least one of the subsets remains heterogeneous in at least one of the traits. As a result of this procedure we obtain a certain number of non-intersecting subsets homogeneous with respect to the trait set A, and a certain set of sections:

R = {X₁, X₂, X₃, … X_q} (4)

where q is the number of sections.”

Commentary. True, the author has only just called this procedure a partition, which can be assessed either as a distinction between two close concepts, or as carelessness, or as a free use of synonyms. Still, one may ask a perhaps naive question: are partition and section one and the same, or is there a definite difference? Moreover, since there are m traits, the number of sections will also be m, and it is entirely unclear why the quantitative index m should be changed to q. Had the quantitative index been kept, it would, in particular, have been clear that the partitioning by traits Aₘ continues “as long as…“.

Reinin: “Since several traits from A may correspond to one and the same section from R, …”

Commentary. This assertion is, at the least, not obvious. Let us try an example: the set (apples, pears, quinces…) — they can be partitioned by species, by variety, by color, by ripeness, by worminess… If we now take the section “all ripe fruit,” does that mean it corresponds to several traits? For this set to satisfy at least two traits, it is necessary that all the fruit either belong to one species, or be of one variety, or of one color. Such a coincidence can happen, but for a concrete empirical set it will be random in character. As for the socion, such a postulate will not hold for individually taken traits, though it may hold for pairs of traits, since all traits beyond the basis ones are linearly dependent. But the author is speaking of polar traits and of a partition in half, and in that case each of the 15 ARTs will generate only two sets, each corresponding to one pole. Still, the assertion under consideration may prove valid if inessential traits are present in the initial set of traits.

Reinin: “…the set R is, in the general case, a homomorphic image of the set A.”

Commentary. Two questions here: 1) what is “in the general case”? 2) Can one presuppose, or construct, a case in which these two sets are not homeomorphic? The point is that if by the general case we understand that all objects without exception possess the indicated property, then the assertion under consideration is trivial and contributes nothing to the understanding of the problem of typology.

Reinin: “The classes obtained in this way ((apples, ripe, not wormy, Calville, yellow — editor’s insertion)), in contrast to heterogeneous groupings, we will call types. Thus a type is a totality of the objects under study homogeneous with respect to a given set of traits.”

Commentary. Such is the author’s definition, but he fails to notice that here he uses the concept “type” as the concept “class” — a set of objects indistinguishable by some totality of traits. From the standpoint of the identity of the concepts “type” and “class” this can be accepted; but from the standpoint of their difference, a type is better defined as an object’s possession of a functional system characterized by a definite set of properties, parameters, or possibilities. This is equivalent to the preceding but preserves the seme of the singularity of the object possessing the type.

Reinin: “Clearly, type here is a relative concept, since the homogeneity of a set is relative.”

Commentary. In empirical sets both type and homogeneity should obviously be presumed relative until the contrary is theoretically proven — as, for example, turned out to be the case in crystallography.

Reinin: “Typology, accordingly, is the totality of all the types distinguished on the set of objects under study by means of a definite set of traits.”

Commentary. Such a definition of typology is, first, tautological, since it is defined through type; and if the author needs such a definition for solving some special tasks, that is his business — but in that case he loses sight of the qualitative difference between a typology and an extensive collection of types. A typology is an essential part of a theory of types, not simply a “heap” of types. In the concept of “typology” one can even discern a shade of ordering of the empirical set; this, though optional, leads to a homonymic shift in the understanding of the term.

Reinin: “From these definitions it is evident that the concept ‘typology’ is rigidly connected with a certain set of traits.”

Commentary. Here the qualifier “rigidly” is perceived as redundant and even misleading, since an empirical typology always deals with fuzzy traits and variables whose domain of existence can be specified only with a certain accuracy and, if need be, redefined. As for deductive typologies, an illusion of unambiguous definiteness may arise there, but only because from the very start the level of discrimination is accepted by default.

Reinin: “A type manifests one of the possible logical realizations of the set of traits corresponding to the quality under study, and a typology is the totality of all possible realizations.”

Commentary. In this assertion it is unclear what “logical realization” means. Why not mathematical, combinatorial, or set-theoretic? In a theoretical text there should be no accidental terms; the concept “logical realization” was not previously introduced by the author, nor is it self-evident. “Logical” can be understood as conforming to logical laws and syllogisms, but also as corresponding to certain rules. The first understanding is more concrete and correspondingly has a smaller scope. The second is more blurred and has a larger scope, since it includes the first case. Furthermore, as already noted, the term quality is used in this text in an altered sense, since the talk is of a typology of objects — a quality, too, can act as an object of typology, but that would be a particular case.

Reinin: “Differences within a type (analogous to intraspecific ones), having a random, quantitative character.”

Commentary. The author simplifies — these “random” differences may be quite essential, but belong to another sphere of existence or subject area — like the diversity within a TIM in socionics.

Reinin: “Henceforth we will call the set of all intertype differences the content of the typology, and the number of types composing it the volume of the typology.”

Commentary. Of course, the author may call anything whatever he likes. However, this proposal has an analogue in the definition of concepts in logic and distorts the concepts of the content and scope of a concept accepted in logic, by analogy with which Reinin introduces previously nonexistent concepts, assigning them to typology. The content of a concept is the trait on whose basis objects are united into a class. The scope is the totality — the class — of objects possessing the trait constituting the concept’s content. These concepts pertain to the concept, and their transfer to typology requires correction. Of course, the content of a typology comprises both the types and the traits, as well as the intertype relations (if any), while its volume will be the cardinalities of the sets of elements composing the content, and their combined number. I personally have managed to show that in socionics the sets of TIMs, IRs, and DTs are not merely equinumerous but isomorphic, and moreover strictly ordered, and that between each two of these sets there exists a relation of duality (UChK) — a direct semantic correspondence between the elements of these sets. Thereby dual statements make sense, such as “a DT is the result of the opposition of sets of TIMs, and a TIM is the result of the opposition of sets of DT poles,” or “an IR can be represented as the opposition of two TIMs, and a TIM can be represented as the opposition of two IRs,” which is not at all trivial. On this principle I built models of the relations, and I am now working on a model of the DT. The relation model allows one to interpret the peculiarities of relations directly, without turning to TIM models. The DT model allows one to build its semantics in terms of the IAs and the structure of Model “A.” Analogously for a relation: its semantics can be built from its model, without turning to TIM models. However, both the theory of the TIM and the theory of the IR proved somewhat more complex than seemed at the outset. After Reinin distinguished the semantically definite set of regular quadruples (140 in all), it became clear that they can be interpreted as types defined by three DTs. In this light the metaphysics of types and traits unfolded in the work under analysis, though it contains principled defects, can still be understood as a pointer to the existence, in special cases, of combined and derivative types — hence traits, and also relations, if present in the typology. I managed to show that in socionics, besides the tetratypes — types uniquely defined by four linearly independent traits and designated in socionics as TIMs — one can distinguish sets of tritypes, which are the regular quadruples of TIMs (Reinin’s tetrads); ditypes, defined by two DTs (Churyumov’s dyads — 120 in all); and monotypes, each defined by a single DT pole (30 of them). Apparently similar derivative sets may be found among the DTs and among the IRs. I have partially done this work, showing that what Aushra called intertype relations (IRs) are in essence interfunctional relations (IFRs), or types of IR, whereas an IR is the totality of relations between all TIMs — and there turn out to be not 16 of them: for different concepts of IR there will be 136, 256, and with gender variations 1520. Analogously, subsets of DTs will be distinguished, but their semantics may become so generalized that its verbal formulation will be even harder than formulating the semantics of the ARTs.

The author’s adoption of the concepts VT and CT [volume and content of the typology] looks, in light of the above, like the introduction of artificial constructs only formally answering the sense of typology, but constituting metaphysical categories that in the present case have merely decorative meaning. This is a very unpleasant metaphysical error, connected with the uncritical — and moreover mistaken — transfer of categories from one subject area to another. It appears to have been done hastily.

Reinin: “So, a typology is the set of concrete realizations of the quality under consideration, corresponding to that quality. Or, in other words, a typology is a description of a quality by means of a finite number of its realizations.”

Commentary. A little above the author already gave a definition of typology, so it is not very clear why it needs doing again. True, here the definition is preceded by the introductory word “so,” which looks like a summing-up — possibly the final version is meant. But the author gives two versions which, in his opinion, differ only in verbal dress. Nonetheless these are two different definitions, though the author considers them identical.

1. “A typology is the set of concrete realizations of some quality.” This is a bad definition, because a problem arises with defining a quality — is a human being a quality, for example? It is also unclear how this one quality — something whole, monistic, definitely expressed — can realize itself in different ways. After all, earlier the talk was of a set of bipolar constructs and of the combinatorics of poles, and there it was clear that in a typology all the admissible combinations of the set of poles are realized. Is not each of the poles a quality? And earlier the author identified the concept of quality with such classes of objects as human, tree, or snow. If one of these “qualities” — say, human — is substituted into the given definition, we get: the typology of people is the set of concrete realizations of the human — which is obviously wrong. Wrong, because the set of concrete realizations of the human is all concrete people with their concrete characters, whereas a typology is a theoretical construct relating to groups of people, not the set of people itself.

2. “A typology is a description of a quality by means of a finite number of its realizations.” This definition cannot be identified with the previous one, since here typology is called a description, not a set — that is, it is assigned to a different proximate genus than before. Here it turns out that the typology of the human is a description of the human by means of a finite number of his realizations. But now a problem arises with the concept of realization, and — chiefly — the notion of the criterion by which the typology is built disappears, and the definition itself becomes too broad: as with the previous definition, the entire set of people falls under it.

Reinin: “A typology arises at once, as soon as we use a set of its essential traits to describe some quality. If there are traits, the possibility of distinguishing objects appears, and hence a typology appears. A typology, by its very construction, contains in folded form all the available information about the given quality or concept, and is a description adequate to the factor description. These are two complementary ways of describing one and the same semantic field.”

Commentary. In this paragraph the author’s concrete knowledge of the question at hand is concentrated, yet a certain confusion is felt — given, of course, everything said earlier. How is one to understand that a typology arises at once? Does this mean the time needed to create a typology equals zero, or is at least extremely small? Or is Reinin using “at once” in some figurative sense? Further, the author says that if “there are traits, the possibility of distinguishing objects appears” — objects, then, after all, and not qualities. The assertion that “a typology, by its very construction, contains in folded form all the available information about the given quality or concept…” needs at least some explication, but there is none. One could probably accept by default that a typology contains some information about the set of objects under study — in virtue of that being a particular-affirmative judgment, whereas the original assertion is a universal-affirmative one, for whose verification one must investigate, or at least assess, all (without exception) elements of the set, without which a categorical assertion is incorrect. And though the assertion specifies that the information is present “in folded form,” this does not change its overall assessment, since the operator IFF (“in folded form”) is likewise used without explication — after all, for example, all the information IFF is contained already in the object itself, without any typology. I am simply drawing attention here to the author’s blurred use of concepts.

And finally, from the author’s words there follows “the adequacy of the typological and factor descriptions of personality.” However, in the context of the theoretical problem connected with the concept of typology, corresponding basic and clarifying definitions are of course required. The author has proposed his definition of typology, but has given no definition of a factor. In the psychological literature available to me I found no direct definition of this concept. It appears that in the American primary sources of the era of Cattell and Hathaway this term is used by default as a general-scientific one, i.e., in the sense of an acting cause. The concept of factor became sense-forming in factor analysis, originally developed at the beginning of the 19th century precisely for the needs of psychology, and only later used in sociology, medicine, economics, and other sciences dealing with large numbers of variables. The task of factor analysis is to discover the leading variables, establish the degree of connection between variables, and find the “principal factors” determining the phenomenology under study. From this last one may conclude that the principal factors are the principal variables — with which one may well agree. In fact, however, judging by the contextual use of the term, in factor analysis the concept of factor underwent a homonymic shift and, besides its primary meaning, came to be used in the sense of a generalized parameter, or a generalization of several parameters. The level of generalization was designated by ordinal words — a factor of the first, second, etc., order. With such an understanding of the term “factor” there are no visible grounds for identifying the factor and typological descriptions of personality with respect to their equifinality — i.e., that one and the same phenomenology can be reconstructed from either. Granting that a type is the integral effect of several parameters, one may say these are different ways of packaging one and the same information. One may also speak here of the adequacy of these descriptions, but for that a corresponding definition must be given. If, for example, adequacy is understood as a correspondence, or homeomorphism, of two descriptions, definitions are still needed. Are the physiological and the shamanic descriptions homeomorphic, for instance? There is no getting by here without deep comparativist research.

And one more remark, on the word combination “information about the given quality or concept.” Here the terms “quality” and “concept” appear as objects of study with equal standing. But the study of qualities and the study of concepts are two entirely different subject areas. And the failure to distinguish this is methodologically inadmissible.

Reinin: “Does a dependence exist between the number of types and the number of traits?”

Commentary. According to the author’s definitions this must mean that the talk is of the connection between volume and content, since for him the set of types constitutes the volume of the typology and the set of traits its content. But of course the question posed makes sense regardless of the author’s incorrect definitions of these concepts. However, when correct and incorrect assertions meet within a single text, a semantic leveling and semantic interference between them arises, as a result of which the incorrect may be perceived as having positive sense while the correct falls under suspicion — and this hampers the reader’s understanding of the text.

Reinin: “We were, unfortunately, unable to find answers to these questions in the literature. However, a large number of studies are devoted, to one degree or another, to the problem of the relation between typological descriptions and descriptions by means of properties.”

Commentary. This phrase is an argument in favor of a typology having two contents — they are complementary and therefore interchangeable, and, on the author’s definitions, each of them can be considered the volume for the other, which in that case turns out to be the content. Clearly the question of the volume and content of the concept of classification is resolved differently. But it is unclear why the author says that “answers to these questions could not be found in the literature,” and “unfortunately” at that — for in the subsequent references to J. Ekman he directly cites a formula linking the number of types and the number of traits, and all this may perplex the attentive reader.

Reinin: “The question of the relation between the number of types and the number of properties was posed in [57, 74]. J. Ekman, in the work ‘A comparison of the applicability of a system of types and a system of properties for describing personality’ [74], defines a type as a certain pole of a measuring axis — a ‘pure’ type — and describes concrete objects by means of a certain superposition of ‘pure’ types, a mixing of their characteristics. For such an understanding of type it is shown that a description by means of n pure types is equivalent to a description by means of (n − 1) properties.”

Commentary. It is precisely this reference that contains the answer to the question of the existence, in socionics with its 16 types, of 15 DTs (differentiating traits) — i.e., 11 more than Jung’s dichotomies provided. This, properly speaking, is the key assertion for socionics about the number of DTs. Given this reference, one must admit that priority here belongs to Ekman. However, Ekman — and Reinin after him — did not understand that a type is not an extensive but an intensive, i.e., synthetic, set of poles, and only in the case of a single property do the extensive and the intensive partially coincide — and even then shades of meaning are possible. Note that Ekman defines a type as a certain pole of a measuring axis — a “pure” type. This can be understood such that in this case the type coincides with an individual DT and we are dealing with a monotype, which in no way affects the theoretical conclusions connected with the concept of type. If we add that, per Ekman, “concrete objects are described by means of a certain superposition of ‘pure’ types,” and “concrete objects” are obviously types resting on several DTs — i.e., the talk is of poly-trait types — then all the differences between Ekman’s and Reinin’s approaches reduce to definitions and formulations. Unfortunately, neither author carries out such clarifications.

Reinin: “The action of this law is considered (in Ekman) on the example of E. Kretschmer’s pure types (pyknic, athletic, leptosome). The author (i.e., Ekman) holds that the system of types is constant, whereas the system of properties is always open and any number of dimensions can be added to it or removed from it.”

Commentary. Note that in Kretschmer, owing to the underdevelopment of theory, the types in fact coincide with the traits. But whereas before Kretschmer psychiatry was a purely clinical discipline distinguishing mental disorders by empirical symptomatology, Kretschmer makes a serious theoretical generalization, coming to the conclusion that a mental disorder differs from the norm only in the degree of expression of some invariant property. Such properties — like “cyclo,” “epi,” “schizo” — Kretschmer called psychic radicals. These radicals are, properly speaking, identical to traits matched with types: one trait — one type. As for the openness of the system of properties: for an empirical set, because we do not know everything about an object in advance, this is partially fair; but for a deductive set it is not. Ekman, because positivism lay at the base of his methodology, had only empirical sets in mind. And Ekman’s work should be considered an important step in the development of the concept of psychological typology. This matters within the methodological principle of complementarity, which presupposes the continuity of theoretical generalizations and the cumulativeness of scientific knowledge. By cumulativeness is understood here the effect of the accumulation of scientific information. For the philosophy of science this concept has exceptional importance, since on its acceptance or rejection depends the interpretation of historical facts in the development both of individual scientific directions and of science as a whole.

Reinin: “A. Anastasi reformulated J. Ekman’s proposition about n types and (n − 1) properties as follows: ‘Any set of data that can be adequately described by means of n types can be described equally well by means of (n − 1) properties.‘”

Commentary. This phrase, in light of the preceding, contains a logical error. For it was said there — presupposed — that the set of traits is open and may contain any number of “dimensions,” i.e., traits — and hence, also none at all! This was neither specified nor refuted! But this likewise means that any data set adequately describable by means of n types can be described equally well by means of any number of properties — one need only qualify the evaluative expression “equally well,” replacing it with “somehow, or with a definite degree of accuracy,” for if the traits are removed altogether, a description can still somehow be made, as, for example, is done by that same Kretschmer. Despite the strangeness of such an assertion, it follows logically from the set of premises contained in the reasoning of Ekman and Anastasi. But as we now understand it, the basis traits suffice to describe the variation in any set. This is possible because the remaining traits depend on them linearly. Nonetheless, turning to them may prove valuable, because it reveals shades that might not have been obvious — but this too must be proven, not accepted by default.

Reinin: “As will be shown below, this proposition, proven only for a particular case, is valid also for the notions of typology, its volume and content, adopted here.”

Commentary. The author seems to think that Ekman proved his assertion only for a particular case — but how can one speak of a particular case when Ekman’s assertion involves an arbitrary number of types n? Apparently Reinin believes that his proposed definition of typology (which, as shown above, is far from perfect) fundamentally changes the essence of the matter, since Ekman’s understanding of typology differs from Reinin’s. Apparently Reinin believes he has discovered DTs connected with arbitrary sections of any sets whatsoever — but their existence, as was shown in the analysis carried out, was precisely in the general case not proven. Obviously the differences between Reinin’s and Ekman’s approaches are not essential for solving the problem of the relation of types and traits and their numbers. It is fully solved by Ekman’s formula. However, Ekman’s empiricism did not allow him to arrive at the theoretical conclusion of the isomorphism — in the general case, homeomorphism — of these sets. As a result he misses the possibility of the existence of a non-distinguishing trait, whose addition leads to the conclusion that N types can adequately be described by n traits.

Further, Reinin — insisting on his introduced definitions of the content and volume of a typology which, as shown above, are erroneous — proceeds to consider the number of traits in an arbitrary typology, so concretely determined in Ekman.

Reinin: “The problem of determining the number of traits corresponding to a given typology reduces, as already said, to the problem of determining the cardinality of the set of intertype differences.”

Commentary. Note that in the given case the concepts “number” and “cardinality” are synonymous, the latter term being standard in set theory.

One can understand the author when, formulating the initial conditions of his further reasoning, he says: “Suppose there is a typology distinguishing n types on the set of objects under study: T₁, T₂, …, Tₙ.” But doubt arises when, following this, he stipulates two seemingly quite natural conditions. One he calls the requirement of completeness and writes as a formula meaning that the union of all types equals their initial set: ⋃ⁿᵢ₌₁ Tᵢ = S. Obviously in this formula S represents the previously mentioned typology. However, this requirement seems so identical to the initial enumeration of all the types of the given typology that, in essence, it adds nothing to the author’s reasoning beyond mathematical dress. It may seem to raise the rigor of the reasoning, but in fact it complicates it without necessity, only hindering understanding of the text, and looks like coquetry with mathematical turns of phrase.

And when the author speaks of the “mutual non-intersection of types” and writes the corresponding mathematical expression Tᵢ ∩ Tⱼ = Ø, where i, j = 1…n and i ≠ j, doubt arises about the applicability of the intersection operation to individually taken types, since the author has not proven that types constitute sets. All the more so since, when it turns out that a type is formed from an extensive collection of properties — i.e., a set of traits — such intersections will, first, make sense and, second, not necessarily be empty; and in the case of socionic types they are always nonempty.

Reinin: “Let Jₙ be the set of types:

Jₙ = {T₁, T₂, …, Tₙ} (7)

Then to any trait (a, ā) from A there corresponds a certain partition of the set Jₙ into two subsets, which can be written in the form:

X = ⟨x, x̄⟩ (8)

where x is the set of types belonging to one pole of the trait — a; and x̄ the set of types belonging to the other pole — ā.”

Commentary. Note that the author has, for no evident reason, changed the notation: instead of the previously used S he denotes the set of types Jₙ; X = (Sₐ, S_ā) he has changed to X = ⟨x, x̄⟩; and he repeats what was already said — see (3). From the standpoint of scientific editing, such things need to be standardized; otherwise the text is complicated without need and its understanding hindered.

Reinin: “The number of ways of distinguishing on the set Jₙ some subset — of possible combinations of n taken i at a time — will equal:

Σⁿᵢ₌₀ Cₙⁱ = 2ⁿ (9)”

Commentary. This is a perfectly correct formula — only it is entirely unclear what relation it bears to the DTs. Dividing the set into more than two subsets makes no sense in this context, since the DTs, per the author, are bipolar. Moreover, judging by everything, the author supposes that any partition of a set from any typology into any two subsets automatically points to a DT by which these two subsets can be opposed to each other. In some empirical cases this is probably possible. For example, someone may ask how these 7 people differ from those 19. And the answer will come: “These seven are bandits, and those 19 peasants; these are all young, under 30, those over 40; these all tall — above average — and those all of average height; these all Americans, and those Mexicans…“. But what happens if one bandit and one peasant are swapped? Will there now be even one essential DT by which the two new groups can be opposed? G. Kelly, in his repertory-grid method, supposes that a subject will always find a trait by which two people known to him differ from a third. For example, to the question of how you differ simultaneously from your father and your mother, one person may point to age, another to worldview, and a third to the difference in films watched. If one is now asked how you and your father together differ from your mother, one person will stress sex, while another will point out that he was in Alaska with his father while his mother was not. But here the sets are extremely small and concretely selected for the goals of the testing procedure — what about arbitrary empirical situations? Suppose, say, you have 277 rock samples in your laboratory. Will you be able to run sequentially through all 2ⁿ sections and establish the corresponding DTs? Perhaps in some cases these traits are obvious, but are there even any logical grounds for asserting that such a trait exists in every case? And what is needed is not just any old trait but an essential one. If you take, say, 10 billiard balls, then obviously any sections of this set will be semantically identical, and no essential traits can be found here. One could probably find more or less random differences in the molecular structures of the material the balls are made of, but these differences bear no relation to billiard semantics.

Matters stand analogously in deductive sets, to which the socion belongs. Can one, for example, find a DT for the section opposing the single intuitive-logical extravert type to all the rest at once? Not the concrete people belonging to this type, but the theoretical constructs which represent the types studied in socionics, and whose properties are potentially given by the algorithm of TIM construction? A positive answer to this question is problematic — though, of course, research is needed here. In any case, without answering the question of the existence of DTs for arbitrary sections of any (!) typologies, it is senseless to reason about the set of DTs in the arbitrary case and to find an expression for this number through binomial coefficients. Therefore the author’s claim to have solved this question in the general case, in contrast to J. Ekman, remains unjustified, and the latter’s assertion of the existence of n−1 DTs for n types retains the character of precisely the general, not a particular, case. So for the 16 socionic types there are, precisely per Ekman, 15 DTs.

Note also that Reinin’s basic assertion — “to arbitrary sections of any typologies DTs may be put in correspondence, matching the semantics of the corresponding typology” — belongs in logic to the category of universal-affirmatives, and to refute such assertions a single counterexample suffices. The examples with the bandits and peasants, and with the opposition of the ILE to all the other TIMs, quite suffice to refute this assertion. On this, properly speaking, the analysis of Reinin’s ideas could be concluded — ideas that are, incidentally, very witty; it is even a pity they turned out to be erroneous.

However, for methodological reasons, let us consider the further reasoning of the author of the student work.

Note also that the author perhaps did not notice that the partitions of an arbitrary set corresponding to the binomial coefficients will only in particular cases divide the set in half, as the polarity of the trait declared by the author requires. This too is an argument against the “binomial” approach.

Reinin: “The quantities Cₙⁱ — the binomial coefficients — are conveniently represented in the form of Pascal’s triangle (Fig. 1).

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1

Fig. Re.01. Pascal’s triangle

In the n-th row of Pascal’s triangle stand the quantities:

Cₙ⁰ = 0; Cₙ¹; Cₙ²; … Cₙⁿ⁻¹; Cₙⁿ = 1. (10)”

Commentary. Since the question of the set of arbitrary sections of any typology was resolved above in the negative sense, and all these sections turned out to be a flight of intuitive fancy, Pascal’s triangle here is likewise beside the point — but let us follow the author’s thought to the end. Perhaps it is not of great importance, but against the background of the carelessness discovered in the presentation and elaboration of the theoretical side of the text under consideration, let us note all the same that the number of combinations of n taken i at a time is written Cᵢₙ — that is, the total number goes below and the variable above. C⁰ₙ — or, in the author’s notation, Cₙ⁰ — equals 1, not 0 (as in (10)); and further on he gives the correct relation Cₙ⁰ = Cₙⁿ = 1, so that in (10) he commits either an error or a slip. An error is when he actually thinks so; a slip is when he knows it is not so but, out of absent-mindedness or carelessness, writes it anyway.

Reinin: “Since one section distinguishes at once two sets, of k and (n − k) elements, then, taking into account that

Cₙᵏ = Cₙᵏ⁻¹ (11)

it is not hard to determine the number L of possible sections of the set Jₙ:

L = ½ Σⁿᵢ₌₀ Cₙⁱ = 2ⁿ⁻¹ (12)

Note here that to the combinations Cₙ⁰ = Cₙⁿ = 1 there corresponds a certain identity section. It can be regarded as the limiting case of the partition of the set of types — the identity operation.

E = ⟨{T₁, T₂, …, Tₙ}, Ø⟩ = ⟨Jₙ, Ø⟩ (13)

Thus the maximum number of possible intertype differences (N) for a set of n types will be:

N = L − 1 = 2ⁿ⁻¹ − 1 (14)”

Commentary. 1) In formula (11) an error is made: instead of k−1 one must write n−k.

2. Equality (11) speaks only of the number of the one and the other sections and of their cardinality, but the sets themselves are different and, in the general case, they should correspond to different DTs — provided, of course, that the existence of the corresponding DTs is proven. Therefore (9) should be kept; but let us not forget that Cₙⁱ ought to correspond to the number of DTs = Aᵢ, and from it one should remove, or subtract, the unit relating to the non-distinguishing trait, whence we obtain: Σⁿᵢ₌₁ Aᵢ = 2ⁿ − 1. This would be the maximum number of possible intertype differences, if each section really did correspond to a concrete DT.

For example, for A = 3 this will be 7: C¹₃ = 3; C²₃ = 3; C³₃ = 1, i.e., 3+3+1 = 7. Here C¹₃ = C²₃ = 3, but these are, of course, different sets, and different DTs should correspond to them (if the author’s idea were correct): the first set is three sets of 1 element each, and the second three sets of two elements each. Hence the halving in (12) was pointless. But since the existence of the corresponding DTs was nowhere proven, all the reasoning about arbitrary sections supposedly generating DTs loses its meaning as an intuitive error.

3. Also inexact is the reasoning that “to the combinations Cₙ⁰ = Cₙⁿ = 1 there corresponds a certain identity section.” First of all, there should be two of them, and consequently two sections should be put in correspondence with them, not one. It is likewise clear that, from group considerations, there can be only one identity section — so what, then, is the set-theoretic sense of each of these sections? Is each of them an identity, and on what grounds is such a name assigned to them? The point is that although these combinations are matched with one and the same quantity, their sense differs. To the combination Cₙⁿ (read: of n elements take n, i.e., take them all at once) corresponds the whole set, while to Cₙ⁰ (read: of n elements take 0, i.e., take none) corresponds the empty set. One might try to unite these two coefficients and put in correspondence with them a single section, one part of which is the initial set and the other the empty set. But for such an identification there must be axiomatic grounds, and no such grounds have been proposed in set theory. Nonetheless, to the coefficient Cₙⁿ one can put in correspondence the identity section, which leaves the initial set untouched. The coefficient Cₙ⁰ remains without a section, but one operation, albeit empty, corresponds to it. It should therefore not be included in the total number of sections — which the author does, but from other considerations: in formula (14) one ought to subtract 2, but the author “erroneously” subtracted 1, proceeding from the equality Cₙ⁰ = Cₙⁿ = 1 — yet Cₙ⁰ + Cₙⁿ = 2, and by the author’s own logic one should subtract 2.

Of course, one must understand that we are dealing with mathematical abstractions, and the term section is used in a generalized sense as the operation of partitioning a set into two subsets, whatever meaning is put into the concept. A particular case of such a partition is the partition into two subsets, one of which coincides with the initial set while the other is empty. Note that the indefinite pronoun “a certain” is here used incorrectly by the author, for the identity section is one, and there is no choosing among several. If, moreover, it is proven that the set of sections of any set is a multiplicative group, then the section corresponding to the coefficient Cₙⁿ will be the identity. Since we are dealing here with a universal-affirmative assertion, a single counterexample suffices for its refutation. Such an example might be the set {1, 3, 7}, which is not a group — neither additive nor multiplicative. Therefore, to develop his rather beautiful idea, the author from the very start ought to have spoken not of any sets whatsoever but only of those which are additive or multiplicative groups.

4. It is also unclear why the author in (14) again changed the notation — what is N doing here, when for the set of traits he used the symbol A (essential traits), for the set of sections the symbol R was used, and now — L? But it turns out the result will in any case be different: not 2ⁿ⁻¹ − 1 but 2ⁿ − 1. Let us check. Count for 3 → 2ⁿ⁻¹ − 1 = 3, whereas 2ⁿ − 1 = 7 (C⁰₃=1, C¹₃=3, C²₃=3, C³₃=1 → 8, but the first case must be discarded as having null sense — that is, this case enters the count of sections but its sense is absent, and it a priori cannot generate a DT, by analogy with division by zero, which is inadmissible).

Reinin: “Record (8) can be regarded simultaneously both as a section of the set of types and as a description of a certain trait by means of a set of n types.”

Commentary. What does (8) have to do with it? The author draws the conclusion from (14), and since he keeps changing — or possibly simply confusing — the notation, everything begins to unravel…

Reinin: “In the latter case the result obtained can be formulated as follows: by means of n types one can describe no more than 2ⁿ⁻¹ − 1 different traits.”

Commentary. This is a trifle, but the variable designation has changed yet again — in formula (2) the number of possible intertype differences is denoted by the letter m, and in this paragraph by n.

When the author writes that “by means of n types one can describe no more than…”, this is of course correct, but the corresponding formula indicates the exact number, which means not only no more but also no less — that is, it is an equality, not an inequality. The author’s qualification may therefore mislead the reader, in the sense that there cannot be more but there can be fewer. Well, since the author does strive for mathematical correctness. Of course, someone may content himself with describing a smaller number of traits, or refuse them altogether, but the formula takes into account not the subjective attitudes of the operator but the exact quantitative measure.

Reinin: “So, the two quantities — the number of types and the maximum number of intertype differences — turn out to be rigidly connected with each other by relation (14).”

Commentary. In this phrase the word “rigidly” sounds too rigid. It would be far better to say “unambiguously,” since the existence of thematically significant meanings that could be understood as DTs of combinatorial types connected with arbitrary sections of arbitrary sets is not only unproven but, in the general case, not even realized. Using socionics as an example, one can say that typological meanings, even in theoretical combinatorial typologies, can be found only in “regular” subsets, where “regular” presupposes the preservation in the subsets of the necessary symmetries. In socionics such subsets are the 15 regular octaves of Augustinavichiute, the 35×4 = 140 regular tetrads (quadruples) of Reinin, and the 120 regular dyads (pairs) of Churyumov.

Reinin: “The set of classifying traits A may not include all the intertype differences. The number of traits in it may be nearer to or farther from the maximum. For example, to obtain 16 types four independent traits suffice, whereas the number of possible differences between these 16 types will be almost 33 thousand. Let us introduce the notation Rₙ for the set containing all the variants of intertype differences for a typology of n types:

Rₙ = {E, X₁, X₂, X₃, … X_N} (15)

R is the set of all intertype differences, which, by definition, corresponds to the content of the typology of n types. The relation between the volume and the content of the typology is presented in Table 1.”

Commentary. 1) How can it be that “the set of classifying traits A may not include all the intertype differences”? What is the author talking about? Quite plausibly he has additional considerations on this score, or he has simply forgotten what was under discussion.

Table of the relation of volume (VT = n) and content (CT = 2ⁿ⁻¹ − 1) of a typology — Table Re.01

VT	1	2	3	4	5	…	14	15	16
CT	0	1	3	7	15	…	8191	16383	32767

For the classifying traits and the intertype differences are one and the same, and the author has introduced no corresponding distinction.

Reinin: “From the table it is evident that a typology’s capacity to assess the properties of objects, measured by the number of intertype differences, is small at N < 10, but already at N > 10 their number passes a thousand, and at N > 16 it already exceeds the average volume of the written-speech vocabulary of the principal languages.”

Commentary. This reasoning contains two errors which devalue it. The number of differentiating traits must be computed by Ekman’s formula. And then table Re.01 will have to be rewritten as Ekman’s table (Re.02).

Table of the relation of the number of types (T = n) and traits (DT = n − 1) of a typology — Table Re.02

TIM	1	2	3	4	5	…	14	15	16
DT	0	1	2	3	4	…	13	14	15

It is highly doubtful that psychologically colored concepts in language have a theoretical rather than an empirical origin. But additional semantic research is needed here. An analogous error was in his time committed by E. Shepetko, though his reasoning is constructed differently. He proceeded from the notion that DTs are generated by arbitrary octaves of the socion. Their number equals 16×15×14×13×12×11×10×9/8! = 12,870. This number is somewhat smaller than what Reinin proposed (32,767), and it is close to the psychological word-stock of developed languages. However, it is almost obvious — and at least the contrary is unproven — that no DTs correspond to the non-dichotomous octaves, and Shepetko’s hypothesis therefore loses its grounds.

Taking into account the remarks made earlier, one must acknowledge that, since the sets of DTs and types are isomorphic, one and the same archetypes stand behind their semantics, of which they are projections. Therefore, in the author’s terms, either of these sets may claim the role of content, and the second set may be used for assessing its volume. Obviously in that case such a division into volume and content loses its meaning. With what was said earlier in mind, the table should assume a completely trivial form, making its use superfluous.

Table of the relation of the number of types (T = n) and traits (DT = n) of a typology — Table Re.03

TIM	1	2	3	4	5	…	14	15	16
DT	1	2	3	4	5	…	14	15	16

Reinin: “Let us return, however, to typological descriptions. If each trait Aᵢ from A is regarded as a certain logical condition determining an object’s belonging either to the set Sₐ or to S_ā, then the typology corresponding to the set of traits constitutes, as already said, the totality of their logically consistent realizations.”

Commentary. The logical condition presupposed here could be explicitly formulated as follows: “If a type Tⱼ possesses the trait Aᵢ, the type falls into the section Sₗ, where l takes one of two values: a or ā.” As applied to the socion this reasoning is, to say the least, strange. The point is that all socionic types possess all 15 DTs — though such a use of words needs special stipulation, since a trait is a bipolar construct and a type possesses not the whole trait but only one of its poles. Furthermore, the attempt to describe the situation with types and traits in the terms used above can only with a stretch be called logical, since the concept of belonging pertains to set theory and is used in logic as a borrowing. The form of implication used above is not at all obligatory here and can be replaced by the set-theoretic formula of belonging: “Tⱼ(Aᵢ) ∈ Sₗ,” which makes the implicative form redundant, since it reduces to belonging and forms no logical system.

Reinin: “If all the traits X₁, X₂, X₃, …, X_N were independent of one another, the number of types would prove equal to 2^N. However, since already at n > 3 the quantity 2^N is much greater than n, it is clear that we are dealing with a system of traits possessing a developed structure of internal connections.”

Commentary. It is strange that the author loses sight of the fact that, per Ekman’s postulate, the number of traits is always one less than the number of types, while per his own derivations, with 16 types, as in the socion, no more than four traits can be independent. The second part of the paragraph rests on the author’s false conclusions, commented on above.

Reinin: “Taking into account the complexity of the object of analysis, let us consider the structure of the set Rₙ in the following aspects: Depth of content along the line ‘essence–appearance.’ Composition of the set of traits. Structure of connections. Possibility of manifestation for the observer.”

Commentary. Here we enter the sphere of psychological metaphysics, at whose base lie the semantic errors committed earlier, while the author’s reasoning lies in the plane of psychology. In themselves the constructs the author considers are of definite interest, even if further analysis should find semantic defects in them. In any case these reasonings are deep, and they could have become the basis of a great typological discovery, had it turned out that they contained no errors.

Reinin: “Let us consider the sections composing Rₙ from the standpoint of the traits corresponding to them. In the general case, behind each partition X from Rₙ stands a certain set of traits from A. This set is an equivalence class containing all the traits that generate one and the same partition of the set S.”

Commentary. It was shown above that, in the general case, the assertion that a DT stands “behind each partition X from Rₙ” is not true. In the case of empirical typologies, several objects falling into one section may possess the same essential properties, but this by no means signifies that each of these properties uniquely determines that section, since objects belonging to other sections may possess each such property, while the given section is determined by the totality of the selected properties. In the case where several properties, each taken separately, correspond to one and the same section, the talk can only be of inessential traits, having no principled significance for the concrete typology. Thus a warehouse may store vegetables, fruit, dishware, and clothing, and within each category individual items may be of one color: green apples, green squash, green pots, and green socks. Green will here define its own section, but it is hard to imagine a natural situation in which this would be essential. And even if the color of these objects has some significance for a consumer — say he collects only green things — then, in the general-scientific sense, such a situation may be considered random. Nonetheless, green here did define its own section. One can artificially add any number of inessential traits that will fall only into this section. One can, for example, put a yellow sticker on all the green items in the warehouse and wrap them all in cellophane. Clearly such cases have no universal significance. An essential trait distinguishes its own section, and such a section is elementary, in the sense that no other essential traits cover it. And if a section uniquely determines a trait, that trait can only be unique, by definition — that is, one elementary section corresponds to at most one trait. The intersection of sections, however, may point to any number of traits. Consequently, there can be no talk of any equivalence classes of the set of traits generating one and the same partition of the set S — for otherwise we run into the paradox of the simultaneous presence of the essentially incompatible.

Reinin: “This class may include: Traits lying on the line ‘essence–appearance.’ For example, atomic weight is appearance; the number of nucleons in the nucleus is essence. Some authors, studying this aspect, introduce a division of traits into ‘trait-causes’ and ‘trait-manifestations’ [60, p. 25]. Projections onto different areas of manifestation of one and the same property, not directly connected with one another by a cause-and-effect relation. One and the same property may be manifested in entirely different spheres of human existence. As a rule, the situation realized is one in which not essential but observable traits are distinguished, i.e., those lying below some level along the line ‘essence–appearance’ (fig. 12). 3. A situation is also possible in which different causes generate identical effects.”

Commentary. 1) Obviously, as soon as we oppose essence and appearance, we thereby separate the essential from the inessential. And though we can designate the essential by means of the inessential, this changes nothing. Of course, an appearance may be essential for us, but only because an essence stands behind it, possibly one not yet known to us. Yet as soon as we cognize the essence, we understand the relativity of the appearance. Moreover, it was long ago proven that a dogmatic opposition of essence and appearance is unproductive, and we can control an appearance knowing its essence. The division of traits into “trait-causes” and “trait-manifestations” does not look apt, for the concept of a trait presupposes from the start its external status, and a trait always remains at the level of appearance, though it may not catch the eye.

Consider the following situation. The destruction of a building, occurring as the result of a hurricane gust of wind, led to the death of terrorists, as a result of which a supermarket containing a large number of people was not blown up. The cause of the hurricane was the difference in atmospheric pressures between two geographic zones. And this difference, as a sign of the coming hurricane, could be measured directly on the ground and displayed on synoptic maps, and a person able to read such maps can, from the signs present there, conclude that a hurricane is approaching. But can such a sign be considered a “trait-cause”? Obviously the marks on the map are not the cause of the hurricane, but point to it. The readings of the instruments measuring atmospheric pressure are likewise not such a cause, but point to it. We learn of a cause by its signs. In the example under consideration we are dealing with a chain of causes and effects. The cause of the supermarket’s not being blown up is the death of the terrorists. The cause of the terrorists’ death is the destruction of the house they were in. The cause of the house’s destruction was the hurricane wind. The cause of the wind was the drop in atmospheric pressure. The cause of the drop in atmospheric pressure was complex processes developing in the atmosphere under the influence of many causes, among them the activity of the Sun. Each of these causes we recognize by its signs. On this simple example one can see how each cause itself turns out to be an effect; but in every case we recognize causes by their signs.

However, this example illustrates the possibility of the coexistence of phenomena of different energo-informational levels, whose content is neither reducible to nor derivable from one another. There is no sense in classifying terrorists by the numerical marks on meteorological maps.

Reinin: “The aspect ‘essence–appearance’ requires, however, a separate, rather philosophical analysis, which already goes beyond the bounds of the topic under consideration. Essential for us will be only the statement of the presence of a certain essential dimension on the set of traits — depth.”

Commentary. The essential dimension of a trait is its essentiality, and within the theory of typology the question is thereby exhausted.

Reinin: “From these positions it is interesting to consider the meaning of the identity section E. Behind it are: Traits that are generic for the set of objects under study. In other words, general traits of the substrate of the typology, i.e., relating to the entire set of objects under study as a whole. All traits not relating to the content of the given typology. For example, the trait of sexual dimorphism is not essential for a typology of temperaments.”

Commentary. Indeed, what is the meaning of the identity section, and is this question philosophical? The concept of a section belongs to set theory, while the semantic filling of the sections of concrete sets depends on the subject area and on the concrete tasks the user is solving. However, there also exist general set-theoretic considerations that must be taken into account when discussing the semantics of sections.

Note that a section can be called an identity only if it is proven that the set of types constitutes a multiplicative group — which means a multiplication operation has been introduced on the set of types, and its result belongs to that same set. Consequently, in the case of a typology, the result of multiplying one type by another must be a type. One could probably invent some typology of numbers whose objects can be multiplied, with the result of the multiplication belonging to that same typology. But can one assert that all empirical and deductive typologies without exception are groups — which the author assumes a priori?

But let us return to the concept of the identity section. Note, to begin with, that with equal right — but perhaps more productively for typological sets — it could be called the zero section. In that case, however, the talk must be not of a multiplicative but of an additive group, i.e., a set in which the addition operation is realized, and the addition of the zero element does not change the value of the sum. The difference between these two varieties of sets is easy to understand in connection with their numerical origin. Thus the set of natural numbers, into which zero does not enter, is a multiplicative but not an additive group — easily understood even without special mathematical training, for any natural numbers can always be multiplied and the result will necessarily be a whole number. In this group, one naturally plays the role of the identity element, since multiplication by it does not change the result. If zero is added to the set of natural numbers, it can now be regarded as an additive group, in which the role of the group identity is performed by zero.

Although in the abstract-mathematical sense the multiplicative identity and the additive zero perform one and the same function in their groups, their original semantics differ and must be taken into account when discussing the semantics of a set and its elements. Thus, to call a set-theoretic partition an identity before it is proven that the corresponding set is a group is incorrect.

As for the semantics of the identity section, apparently it cannot be assigned arbitrarily and linked to everything that did not enter into the content of the traits. Obviously its level of concretization or abstraction must correspond to those indicators of the trait array — in other words, it too must be a trait, but a non-differentiating one. Such a trait, common to all typologies, is the existence of types.

Reinin: “In the most general case, behind E stand all the traits complementing the set {X₁, X₂, …, Xₙ} to the universe of constructs. Thus the set Rₙ reflects, in folded form, the entire universe of constructs possible in language.”

Commentary. Both of these judgments may be assigned to the category of strong ones, but they cannot be accepted without proof. A critique of the first judgment was given in the previous commentary. Complementing the set of sections to a universe of constructs is an entirely incomprehensible action. For the talk should be of the semantics of the sections, not of the sections themselves. The universe of sections is exhausted by the sections themselves. The second judgment looks like a conclusion from the first, but there the talk is of sections and here of traits. In any case, the second judgment has remained unproven, and, on the contrary, it is quite plausible that no positive content stands behind it.

In the section “2.5. Analysis of the structure of the set of intertype differences…” there are several errors in the mathematical derivations: there is no correspondence between fig. 13 and expressions (18), (19), and (20); in expression (22) the variables are confused, and as a result the group identity E (the set-theoretic zero) is obtained, whereas the group variable Z should have resulted. At first I even thought Reinin was no mathematician at all, and that someone had invented all this for him; but in personal conversation I understood that Reinin grasps everything at half a word, that he is a normal mathematician, and that the errors are the result of oversight — it happens to anyone — but in a book they are nonetheless out of place.

Figure Re.02 shows part of a “bad” page from Reinin’s Socionics with the commentator’s pencil marks.

[Fig. Re.02. A “bad” page from Reinin’s “Socionics” with my marks — reproduces the two quadrant diagrams for traits X, Y and the derived trait Z, together with formulas (17)–(20): J₄ = {T₁, T₂, T₃, T₄}; X = ⟨{T₁, T₃}, {T₂, T₄}⟩ = ⟨x, x̄⟩; Y = ⟨{T₁, T₂}, {T₃, T₄}⟩ = ⟨y, ȳ⟩; Z = ⟨{T₁, T₄}, {T₂, T₃}⟩ = ⟨Z, Z̄⟩; Z = X ⊕ Y = ⟨xy ∪ x̄ȳ, x̄y ∪ xȳ⟩ = ⟨{T₁, T₄}, {T₂, T₃}⟩ = ⟨Z, Z̄⟩.]

Reinin: “Considering the set R, we are dealing with a system possessing a developed structure of internal connections.”

Commentary. Since the “binomial hypothesis” has turned out to be false, of the “enormous” set of traits R only n−1 remain, and hence there is no sense in speaking of the structure of connections between traits as being so highly developed. In the changed context such words are perceived as redundant and “grandiloquent.” Together with “Pascal’s triangle,” the talk of central and peripheral traits has lost its meaning — only Ekman’s traits remain. As for the structure of the set of traits, from the remaining text it is difficult to draw any far-reaching conclusions.

Reinin: “Let us begin the analysis of the structure of the set Rₙ of traits with the simplest, two-dimensional case.”

Commentary. Still, apparently, the talk in the given case should be not of the structure of the set but simply of the set — if only because the concept of structure, as applied to the case under consideration, should first be defined. Structure presupposes at least some topology, and without that it is not correct to speak of it.

Reinin: “Consider two bipolar traits:

X = ⟨x, x̄⟩ and Y = ⟨y, ȳ⟩. (16)

These traits divide the set of objects under study S into 4 types: T₁, T₂, T₃, T₄.”

Commentary. Of course, traits may be designated by coordinate symbols, but in doing so it must be stipulated that we are dealing with discrete mathematics, that the axes here have no continual sense and are not graduated, but are used merely to designate polarity. Even after that, the direction of the arrows creates an illusion of the trait’s orientedness, while the minus sign imposes a semantics of negativity which traits do not have. Of course we mathematicians understand symbolic conventions, but since the reasoning here stands at the very sources of the topic, it must be maximally correct. However, the above almost elementary reasoning contains something of a trick, connected with the coordinate representation of polarity instead of, say, a combinatorial one. In the latter case the talk would be of a pair of binary traits, of which six pairwise combinations are possible, which would give three pairs of bipolar traits — and this corresponds in the given case to three types. All the more so since one has to deal with combinations of coordinates anyway — but what we need is not coordinates but types and traits. In the coordinate variant, the presence of two more traits besides the “obvious” ones looks like a miracle. They appear like a rabbit from a magician’s hat — though, of course, the coordinate method in itself is perfectly good, but, applied to polar qualities, it introduces the semantics of negativity for half the poles. Naturally this effect can be weakened if all the reservations are made in advance.

Reinin: “All three traits X, Y, Z belong to R₄, since they are sections of the set J₄. However, as is evident from the construction, the partition of the set of objects under study by these three traits does not lead to the appearance of new types. Such traits we will call mutually dependent… The mathematical reflection of this dependence is the binary operation of the product of sections…”

Commentary. R₄ is a set of four traits. The author shows only three. Clearly the fourth trait is the identity. However, earlier — for example in formula (15): Rₙ = {E, X₁, X₂, X₃, … X_N} — E is apparently not included in the count, since it stands without an index and is, as it were, not considered a trait, the author having several times stressed that the number of possible intertype differences (N) for a set of n types will be N = L − 1 = 2ⁿ⁻¹ − 1 (14). But, per the author, n is the set of types, not of traits. Yet in formula (15) this index stands at the symbol for the set of traits, of which, per (14), for n = 4 there should be 7, not 3 and not 4 — which can easily mislead the reader.

As for the mutual dependence of the traits, it flows naturally from the combinatorics of types, when one and the same type enters different sections defining different traits, and they thereby turn out to be connected by the semantics of the common type.

Reinin: “From the mathematical point of view all three sections, as will be shown below, are identical. This fact suggests the presence of a substantive interpretation for the third bipolar axis as well.”

Commentary. Note that the author speaks of the identity of the sections. Obviously identity is here an unfortunate term, since the sections are different. Apparently the author wants to indicate that none of these sections has advantages over the others. From which, incidentally, follows the absence of hierarchy — and hence of any complex structure whatsoever, which was mentioned above. This is likewise an additional argument against dividing traits into central and peripheral.

As for the presence of a substantive interpretation for the third bipolar axis, in the general case this problem has no solution. Which means that a combined trait can formally be designated, but a substantive interpretation for it may turn out not to exist. As an example, consider the case of a set consisting of hens and ducks. The substantiveness of the species and sex differences here is obvious, which gives us four sets: hens, roosters, ducks, and drakes. But what trait corresponds to the section “hens and drakes” versus “roosters and ducks”? This fact suggests that in an enormous number of empirical sets the derivative traits will turn out to be, in practical respects, meaningless.

Reinin: “The set of traits of the typology Jₙ constitutes an abelian group.”

Commentary. Despite the large number of critical remarks made above in connection with the concept of typology, this assertion can be substantiated — but for that, all the fictitious traits must be removed from the concept of typology, since for them the proof will be fictitious. Then, in a purely mathematical sense, this will be a not-bad result — though from the standpoint of semantics it gives the researcher in a concrete subject area nothing. On the contrary, it generates many questions. What matters to a concrete researcher is knowing of the existence of traits dependent on the basis ones — but this follows not from group properties, rather from typological combinatorics, from which Ekman’s result (theorem) is obtained. The connection of a typology’s group foundation with its subject content can be compared metaphorically to the way a television set’s construction affects the programs shown on it. Yet without the television there would be no programs — whereas the group organization of a typology appears not before but after the elaboration of the typology, or simultaneously with it.

Among the questions arising in connection with the group status of a typology is, for instance, this one: “How does the group operation performed on the typology’s traits as elements of the group relate to the real values of the traits — do they, too, get added or multiplied?” What result will there be in the case of multiplying the traits of hen-ness and duck-ness from the example above? No, it would be absurd to deny the importance of mathematical concepts, but their elaboration and introduction must, without question, be correct.

REININ’S PASSIONS OVER SOCIONICS

G. R. Reinin gave, at the 25th socionics conference, a talk that was, in my view, very strange, later published in SMiPL No. 6 for 2008 [1]. I take up this text in the hope of receiving a valuable message from an intelligent man and an experienced mathematician. I read into every line, trying to find that message. But, alas, I find there some kind of intuitive-emotional sententiae with vague hints at portentous mysteries of being. This, incidentally, is quite characteristic of bearers of the ILI TIM, which the author is.

Author: Let us conduct the following thought experiment: let us remove from socionics Model Yu, remove Kalinauskas’s Helm, Shulman’s Periodic System, Gulenko’s energy model, Talanov’s Model, the lesser-known models of Alekseev and Medvedev, remove the Reinin Traits, remove, finally, Model A, remove the elements of informational metabolism… And, in the end, let us remove Jung’s traits. Will socionics remain? It certainly will. Because the socion will remain. And precisely this set of 16 types of people — the key position of socionics — will remain all the same. And this is the point on which all researchers, the authors of all the models, converge. And in general everyone who is engaged in this.

Editor: A somewhat strange thought experiment… Of course socionics contains much that is superfluous, and one can, without any complications whatever, remove from it Kalinauskas’s Helm, Shulman’s Periodic System, Gulenko’s energy model, Talanov’s Model, and even the lesser-known models of Alekseev, Medvedev, and others… But Model “A” and Model “Yu” can by no means be removed. Jung is the forerunner of socionics, but it was created by Aushra Augustinavichiute, and before her this science did not exist. And of course Jung had no Model “Yu” — Aushra reconstructed it and named it after Jung. In fact it is Model “A-Yu” — Aushra–Jung — as it should be called after Aushra’s death. And if these two models are removed from socionics, there will be no 16 types, and hence no socion and no socionics — for the 16 types are the 16 admissible variations of these models.

And it is entirely unclear where the author got the idea that “this is the point on which all researchers, the authors of all the models, converge.” Types and their models are inseparable from one another — they are one and the same. Two sides of one coin — the formal and the substantive. A model is the formal representation of a TIM, and a TIM is the content of a model.

Author: The human psyche constitutes a certain quality, the set of whose concrete realizations is the socion as the totality of 16 types of psyche.

Editor: Of course, it is hard to believe that the author is here giving his own working definition of the psyche. It is so — a free flight of intuition; but then, some subconscious task may stand behind it. I, for example, consider the psyche an energo-informational complex organizing the behavior of a living being, and not simply “a certain quality.” Dozens of other definitions can be proposed, but under the definition “it is a certain quality” anything whatsoever in this world fits; and if you listen closely to the second part of this definition of the psyche, you will understand that here the psyche is defined through itself: the psyche is 16 realizations of the psyche. But surely some deep thought, or at least some subconscious task, must stand behind this.

Author: The socion can be regarded as a hologram. What is principal is the wholeness of each type, described by various informational models.

Editor: Of course, if the author gives no definition of a hologram, he may put anything whatsoever into the concept. Let us nonetheless try to assess the author’s claim closer to scientific concepts rather than at the level of instantaneous intuitive insights. By a hologram, modern science understands both the material carrier on which an image of an object is recorded in a special way, and the image itself, which under certain conditions can be observed directly by looking at the hologram’s material carrier. So with which does the author connect his notion of the socion as a hologram — with an image of something, or with the carrier of that image? It is also interesting why, or to what end, he resorts to this image. Obviously the socion cannot be the material carrier of anything at all, simply because it is an ideal construction describing the virtual structure of society. There remains the possibility that the socion is some image — but, alas, images in our world do not exist without a material carrier. Consequently, the author has irresponsibly followed the lead of his imagination and created a beautiful combination of words unbacked by real meaning. On the template the author used, one can create many such beautiful, intuitively attractive absurdities: “the psyche is a hologram,” “human thinking is a hologram,” “an emotional reaction is a hologram,” “human relations are a hologram”… If an intuitive strains his imagination, he will manage not only to fill each such formula with content understandable to him personally, but also to deploy arguments in defense of the image he has created.

For the author, the socion is a primary essence which precedes the psyche. The socion, in the author’s opinion, will remain if, in accordance with the author’s thought experiment, everything is removed. Perhaps it is precisely for this reason that he is prepared to regard it as a hologram. A hologram, though ephemeral, is — in the variant of a sticker on a document — quite tangible. And when we try this image on the socion, the socion becomes fixed in consciousness not only as a certain quality but even as a certain essence, which can be discerned in the shimmering light of intuition’s holograms.

The second phrase in this quotation, though, looks more enigmatic. The concept of wholeness is used here. The author gives no explicit definition of it, and consequently it is presupposed that, by default, everyone knows what it is. Yet in the second part of this assertion there is a hint at the constructive character of the concept of wholeness, which is described “by various informational models.” Only it remains unclear whether, in the author’s view, all the models describe it differently, or identically.

Author: Actually, I want to recall that socionics began precisely with descriptions of types. And if you take Aushra’s first type descriptions — they were not based on a functional model, though models had already appeared by then. They were simply unstructured descriptions. The description of Don Quixote she wrote from herself and from several acquaintances of that type, seeing certain common structures behind them. The description of Gabin, for example, she wrote from her niece Egle, and so on. But Aushra’s genius consisted in seeing and distinguishing certain common moments in the process.

Editor: I do not believe that socionics began with descriptions of types. Types can begin to be described only when there is a model. And it was precisely with Model “A-Yu” that socionics began. This model not only set the analytical formula of the TIM but made it possible to model the intertype relations.

How is one to understand the author’s assertion that “Aushra’s first type descriptions were not based on a functional model”? This appears to be the author’s subjective opinion, bordering on a failure to understand Aushra’s creative process. The model, and the semantics of its functions, were absolutely necessary for Aushra in order to form a notion of the set of types, to draw at least primary conclusions about the peculiarities of each of them, and hence to have the possibility of finding a prototype and giving the type a name. Only on this basis could the difficult work be launched of forming the skills of socionic diagnostics, of learning to detect the typologically essential in the behavior of concrete people. Moreover, one can speak of the existence of a model in Jung, though he did not use the concept. The combinatorial algorithm of the psychological type itself — four primary psychic functions, ordered even if weakly, admitting four combinations, and two primary attitudes, extraversion and introversion, bringing the number of combinations to eight — this is a model, and only on its basis could Jung give his primary descriptions.

Incidentally, in the first part of the author’s book Socionics not only is the alien Model “K” used, but an intelligible mechanism for interpreting the model in forming a type description is absent. Possibly this became the subconscious ground of the belief that socionics was possible without a model.

Author: At present, the existence … of the set of 16 types … can be considered a “medical fact,” from my point of view.

Editor: What did the author want to say with the words “consider it a medical fact”? Empirically reliable, is it? And why only “at present”? Have we grown used to it, or what? The problem of the TIM’s existence is considerably more complex than a “medical” statement of subjective confidence. The TIM is, above all, a theoretical construct, and one must try very hard to establish its correspondence to empirically observed facts [see 2].

Author: 16 types result from the orthogonalization of most of the most popular personality questionnaires. 16 types result from the unification of the best-known clinical typologies. Most of the most developed theoretical typologies contain 16 types. Moreover, such ancient sources as Ayurveda and the Tarot point to 16 types.

Editor: In this quotation we have four assertions identical in semantic thrust. The author appears to be trying to explain the existence of the 16 types by their occurrence in various sources and by many researchers arriving at this number. Well, plus the authority of ancient sources…

Of course, the 16 types do not exist for that reason. On the one hand, 16 is a power of two, which appears naturally in any processes presupposing dichotomies, and so there is nothing surprising in the creators of Ayurveda, the Tarot, and other sources presupposing a typology having arrived at this number. On the other hand, it is the evolutionary answer of the species Homo sapiens to the action of its habitat. And also because psychic plasticity admitted such a possibility. But it would be unreasonable to consider identical any and all constructs that happen to be formatted by this number.

Author: At the same time, we cannot at present say why there are exactly 16 of them. Maybe someone here present already can, but I cannot yet. I do not know. Possibly it is determined by the human genome. Possibly by some other factors.

Editor: Here the author honestly admits he does not know why there are 16 socionic types. Well, he doesn’t know, so he doesn’t know — plenty of people don’t know plenty of things, and where there’s nothing, there’s nothing to claim… Though it is strange to hear such a statement from a professional mathematician and credentialed psychologist, for such a man’s opinion carries a certain weight and may influence people with unstable psyches.

Author: So, browsing around the Internet, I came across this little picture. Botanists, you see, have found that there exist 16 isomers of the leaf. That is, there exist 16 types of linden leaves.

Editor: Here the author again seeks the solution to the problem of the existence of the 16 types in references to other researchers’ searches and, with a certain pomp of mysteriousness, presents to the reader what is apparently the key argument: a picture with 16 linden leaves arranged in a 4×4 matrix. But in essence the talk here is of the dependence of 16 types on four binary parameters — that is, it is no more than the same old number 2⁴, and wherever such parameters interact, 16 types necessarily appear. Naturally, with a different number of parameters the number of types will be different too. The content of the typology will depend on how essential the parameters turned out to be. Socionics was lucky in this sense — its basic parameters proved to be ultimate ones, and the typology lying at its foundation proved fundamental.

Author: Maybe in the human being, too, all this is likewise determined by the genome. There is no such work as yet. I hope it will appear some day. Clearly such a proof will go far beyond the bounds of socionics itself.

Editor: The author continues to build up the air of mystery around the theme of the number 16. He even tries to refer the reader to the genome. But this is the well-known scientific move of readdressing causality. Thus there are biologists who believe that life was brought to Earth from other planets, not understanding that this does not solve the riddle of the origin of life on Earth but merely readdresses it to another and, on the whole, indeterminate place.

Author: Historically, the socion was discovered by means of a model. And the names of the types bear the stamp of a definite model. But the socion is not a consequence of that model. And the descriptions obtained by means of the model are far poorer than the real descriptions of the types.

Editor: The intuitive’s train of thought is curious. At the beginning of the article he tried to disavow the significance of the model in socionics — which, he asserted, “began with descriptions of types,” descriptions that “were not based on a functional model.” Here he apparently wants to underscore the model’s sole meaning, connected with its having helped to discover the socion. And that is all. And then the author continues the chain of his strange and enigmatic reasonings, again disavowing the model, asserting that “the socion is not a consequence of this model.” This reasoning deserves closer examination.

Let us pose the question thus: “Can the socion be a consequence of Model ‘A’?” If the socion is a virtual essence connected with the appearance in society of structural organizednesses with definite parameters, then the model can only be a mental construct reflecting the existence of such an essence. And no essence can be a consequence of its own model. If, on the other hand, someone does not recognize the socion as an objective essence, then it is entirely unclear what is being talked about. And if someone considers the socion no more than a mental construct — that is, the image of some model — he will have to arrive at the conclusion that it is that model’s intellectual consequence. This reasoning permits the conclusion that the author must consider the socion a real essence. But all this needs to be explicitly formulated, not turned into enigmatic hints.

Author: A type has far more traits than a model can describe. Any model. Models illuminate some subject sphere, but they do not generate it.

Editor: Here the author commits an elementary error. He speaks of the type as a real thing, and not as an intellectual construct. An intellectual construct possesses properties completely determined by the algorithm of its construction. From the author’s reasoning it follows that the type exists as an independent thing, not all of whose properties are known in advance — and hence attempts to build a model of this thing will always be incomplete.

Author: In socionics, from the very beginning, functional models were used, based on the idea of the superposition of semantic fields. What is this? It is Model Yu, it is Model A, it is Kalinauskas’s helm — well, most of them — and Alekseev’s model, for whoever knows it. They all consisted of two parts: the first, the static part — a definite functional structure with a given semantics of the functions; and the second, the dynamic part — the elements of informational metabolism, likewise with a given semantics.

Editor: This is a banal disquisition on what a scientist ought to know from the very start. But Jung did not know it. He did not use the concept of a model, though implicitly one can be ascribed to him. Model “Yu” was invented by Aushra, not by Jung. But one can say that Jung discovered several semantic fields and superimposed them on one another, as a result of which he obtained eight extensive sets, whose combinatorial synthesis allowed him to arrive at typologically concretized images, whose intuitive descriptions he offered to humanity as the final product of his typology. However, the author introduces into his reasoning, as though they had equal standing, the derivative and defective models of Kalinauskas and Alekseev, which makes the reasoning somehow non-binding. The impression is that the author does not see the main thing in socionics and wanders in the dark of enigmatic searches for the “mystery of the number 16.” Yet there is no mystery at all.

But possibly the author’s subconscious is trying to construct some argumentation justifying his errors connected with the adoption of “Kalinauskas’s Helm” as the basic and deeper model of the psyche. This is worthy of regret.

Author: And so, depending on which places of the functional structure this or that element occupies, the authors and users of this or that model obtain a definite text of type descriptions, intertype relations, explanations of all manner of interactions, small groups, and so on. To obtain these texts the user needs great experience and skill in the work. And, as we have already seen here at the conference, there are people who wield the model virtuosically — for example, in Galina Chikirisova’s talk, and in other talks, it is visible that this is turning — indeed has in reality already turned — into an art.

Editor: Here the author shares his observations on the “textual” possibilities of socionics. This is the standard possibility of any language — to generate textual coherences depending on the combinatorics of its elements. But this should not be marveled at; it should be accepted as a given, after which the work should be done of revealing the text-generating rules. Yet the author himself did not do this work in his book. His readings of Model “K” do not look convincing. They are, rather, intuitive improvisations on the possible symbolic meanings of the model’s elements. I have partially done this work in chapter 19 of UChK, but so far have not met with understanding even among the most conscientious representatives of the socionics community.

Author: In antiquity such systems were often used. In astrology, for instance. There, too, there is a static part — the system of houses — and a dynamic part — the planets. The planets, each with a definite semantics, in the context of the semantic field of this or that house, generate a definite text. A skilled astrologer reads a natal chart as a coherent, well-structured text.

Editor: Where is the author heading with this? Is he seeking an analogy between socionics and astrology? A structural analogy is indeed present. The planets are analogous — but merely analogous — to the socionic functions, and the zodiacal fields are analogous to the IAs. The difference between these typologies, however, is fundamental. Astrology rests on the hard-to-verify assumption of the stars’ influence on human Fate, whereas at the base of socionics lie fundamental parameters of human character, whose presence quite admits of empirical verification.

Author: Take Feng Shui, for example. There is a static part there — the Ba-Gua square. And there are objects of household use which, landing in different places of this square, generate new semantics. Or, say, the Tarot. There is the spread — that is, a certain static part, and many variants of the spread are known — and there are 78 cards which have a definite semantics and, landing in different places of the static part, generate a definite text. There exists a multitude of specialists who, virtuosically manipulating these semantic fields, on the basis of a concrete spread produce a text: “what was, what will be, and how the heart will be consoled.” There are a great many of these spreads; we can see them in the various descriptions of the deck, abundantly represented on esoteric stalls.

Editor: The author runs through the ancient symbolic systems, discerning in them the common capacity to generate texts — but this is something obvious. These are special languages, and they, just like verbal language, are designed to generate texts. So what is there to marvel at? One simply needs to sit down and work, rather than pose one another riddles about the incomprehensible “16.”

Author: In socionics, too, there are many spreads. Many models. And to speak of their truth, to argue about which model is true and which is not, seems to me absurd. One can speak only of their usefulness, their convenience, their descriptive power, and one’s ability to work with them.

Editor: At last the author has formulated an important methodological point. He considers it absurd to compare models from the standpoint of their truth. He considers only one thing important — their usefulness. And here we return to the old philosophical problem of practice as the criterion of truth. The answer has long been known: practice answers the question of a theory’s usefulness, but not of its truth. Moreover, I am certain that with these reasonings the author is masking the error he himself committed by using, in place of the true Model “A,” the deficient Model “K” (Kalinauskas’s).

Author: Here one can hold competitions, compare by various parameters. And there is no need to reduce the models to one another — each of them has its own value in itself. The popularity of one or another is determined to a large degree by the number of people using it, by habits, and by skill in working with it.

Editor: Again and again the relativist motif surfaces in the author’s reasoning: all models have their own value; there is no need to compare them. But then in what does the work of a scientist consist? Why does the possibility of comparing models and posing the question of their truth so strain the author? It seems that somewhere at the subconscious level the author understands that in his book he committed an error, leading the reader down the false trail of the deficient Model “K.”

Author: And so, we are dealing with a multiple, spatially unattached object called the “socion.” I see this object as a kind of hologram. Here it may be apt to recall the parable — it has already been recalled here — of how five sages felt an elephant in a dark room, and each described certain of its properties. At the same time they all agreed that the elephant exists, but described it differently.

Editor: The hologram again. But what does the author actually see? It turns out the author sees an elephant in a dark room where blind men are feeling it. The allegory is obvious. The elephant is the socion, and the blind men are the socionists. They are in principle incapable of seeing the whole, and are therefore doomed to fragmentary impressions. Nonetheless the Truth-Elephant exists, but the socionists are in principle incapable of discovering it, owing to their intellectual limitations. Here, however, the author clearly acknowledges the socion as a real object.

Author: In our socionics it is the same — there is a position with which absolutely no one argues, but the different models look at it differently and are sometimes entirely irreducible to one another. In effect, Jung opened a little window into this dark room and shone a flashlight in, revealing certain qualities of this object. Myers and Briggs, shall we say, increased the flashlight’s brightness.

Editor: There is no longer any “our socionics”; there are two and a half dozen socionic schools, which insist on the truth of their own understanding of socionics and not only do not wish to enter into academic dialogue with one another but are incapable of it. The metaphor of the dark room and the flashlights merely equates the different searches in socionics with one another — whereas what is far more important now is to demarcate, so that it becomes clear on what basis unification is possible. As for the Myers–Briggs variant, it only showed how a heavily distorted typology, resting on false notions of typological truth, can nonetheless become a commercial product and enjoy social demand — though, as we now know, owing to its serious defects, its use differed little from gypsy hypnosis, against whose background a psychologist, drawing on his everyday experience, can give quite useful recommendations which in no way follow from his theories.

Author: Aushra Augustinavichiute created the functional model and described the intertype relations. This proved to be a powerful searchlight already. Then other functional models arose, built on the same principle. Then new dichotomy traits appeared — that is, a structured phenomenological description of the socion. The notion of small groups appeared, groups possessing various properties differing from one another depending on the spectrum of intertype relations within the group. And here is what I think: however disunited the socionists of different cities may be, this is all the same a collective engaged in studying the set of 16 psychological types of people. In their own way, each differently, but they are all engaged with this object. And at present there is altogether rapid growth; many new directions and applications are being born in socionics.

Editor: The author persistently pushes the metaphor of the flashlight and the searchlight in the dark. Well, light is a metaphor of knowledge and of truth. Here one can agree that Aushra discovered a certain truth — that is precisely what people followed her for. Then models appeared at whose base lay other principles — otherwise why would they have been needed? But closer analysis shows that the principle laid, for example, at the base of Model “K” was alien to the structure of socionics. The MQS was created by Kalinauskas for an entirely different occasion and was purely artificially “dressed,” “stretched,” onto socionics. Kalinauskas masterfully wields the method he developed, and the impression arises that he is capable of applying it to anything whatsoever — but socionics already possesses a model adequate to its task, so why, one asks, substitute the artificial for the natural? Why, instead of developing methods for the adequate interpretation of Model “A,” replace it with a model developed for the solution of entirely different tasks — namely, the metaphysical mastering of arbitrary informational arrays in which at least some coherence is presupposed? Is the whole point really the positivist principle of the possibility of arbitrary descriptions of an object? But why, all the same, replace an adequate model with a surrogate one?

Author: I would not separate socionics from psychology. Psychology is engaged with the very same object; it simply reflects on this poorly. And in this sense socionics is located — I believe — inside psychology, and it can become for psychology the same kind of instrument that mathematics became for physics. The object is the very same. If the entire psychometric, psychological toolkit were structured in such a way that first the general-psychological type were determined, and then came tests adapted and standardized for each type — then, probably, it would all work better.

Editor: Of course socionics should not be opposed to psychology, but their specificity and their working zones of applicability should be clearly distinguished. At the base of socionics lies the structuring of the psyche by means of psychic functions of a high level of generalization. In this sense socionics is a special part of psychology. Psychology, for its part, is engaged with everything that concerns the manifestations of the human psyche, all levels of psychic phenomenology. However, socionics offers a basic model of the structure of the psyche and a method for its deductive construction, which can be laid at the foundation of the construction of psychology as a whole — and this is what will make socionics for psychology what mathematics became for physics.

Author: I think the most important thing is that socionics introduces into culture the notion of 16 types of people, the notion of intertype relations, the notion that people are not simply different, but at times differ from one another systemically. I very much liked the title of Livia Mikelsone’s book: “Don’t Change — Just Know Who You Are.” This is the notion socionics introduces into culture, and I believe that is exactly what is happening now.

Editor: In general the author’s position is understandable. But, alas, it contains principled inaccuracies which lay the groundwork for systematic errors in the further development of socionics. Do not Jung’s typology and its simplified-and-distorted Myers–Briggs variant introduce the notion of 16 types? But there this is done on a defective basis, without a basic model of the psyche. And in this sense these are notions which differ from one another systemically. And one can by no means agree with the author that the position of socionics reduces to Livia Mikelsone’s aphorism “Don’t change — just know who you are.” Knowing who you are is the foundation of psychology; but knowing it, a person gains the possibility of choosing the direction of his development and of changing accordingly.

Author: Aushra Augustinavichiute defined socionics as the science of the socion. In other words, there is a certain quality — the human psyche — and there is a certain set of its concrete realizations, which is called the socion. The set consists of 16 elements.

Editor: The author has already said everything, but apparently in conclusion he wishes to underscore something most important. But what is it? Aushra’s definition is important, but it is not the only one and admits variants pointing to other essential sides of socionics. For example, that socionics investigates the generalized forms of IM in the structures of the psyche. In that case the socion turns out to be a consequence of the division and grouping of psychic structures in maximally generalized processes of IM.

Author: At present, the existence of this set can be considered a “medical fact,” from my point of view. So if you walk through a forest where lindens grow and collect little leaves, we will fill 16 little bags (Fig. Re.03). And that’s no fake. [Untranslatable pun: “lipa” means both “linden” and “a fake.“] There they distinguish precisely 4 independent dichotomy traits, by which the 16 types of leaf are determined.

[Fig. Re.03. The complete set of variations of the linden leaf’s form — a 4×4 matrix of 16 leaf drawings.]

Maybe such is the structure of the noosphere? I don’t know.

Editor: Again this “I don’t know.” Given the author’s general intonation, this can be understood as a hint at the unknowability of the world. But why not a call to know it — for then, after all, the question of truth will have to be resolved, at least in its typological variant. As for collecting little linden leaves into little bags in the forest — that is a sweet and sentimental description of an empirical situation. And indeed one can collect not only linden — there are many other plants, and they have not only leaves but many other parts. And then, possibly, one will have to speak of many other regularities. But of what these are the consequence — the structures of the noosphere, or the structures of the whole world — remains to be sorted out.

Reinin’s article was received by the editors on 21.10.2008.

CONCLUSIONS:

1. G. R. Reinin has carried out a large and interesting work on the analysis of the concepts “classification” and “typology.” For the first time in the world literature, relying on the mathematical apparatus of set theory, he proved in general form J. Ekman’s theorem on the sufficiency of n−1 traits for describing a set of N elements.
2. His work contains a small number of formal inaccuracies, which can be explained by a lack of time for proofreading the work and by editorial oversight.
3. His reasonings on the semantic groundedness of any sections of any sets whatsoever are extremely interesting, but bear the romantic character of intuitive imagination unsupported by strict logical arguments.
4. The replacement in his work of Model “A” by “Kalinauskas’s Helm” is unjustified and introduces systematic distortions into the interpretation of the TIM.

Literature

1. Reinin G. R., Socionics, St. Petersburg, “Obrazovanie-kultura,” 2005 — 238 pp.
2. Churyumov S. I., The Smile of the Cheshire Cat, Drogobych, “Promin,” 2007 — 560 pp.
3. For the references within the Reinin quotations, see [1].

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Translator’s notes. DP (дифференциальные/дифференцирующие признаки) is rendered as DTs (differential/differentiating traits) and АРП as ARTs (Augustinavichiute–Reinin traits); ИО = IRs (intertype relations); ИА = IAs (information aspects); ИМ = IM (informational metabolism); МКС = MQS (method of qualitative structures); МНС = PLR (Place of Least Resistance); УЧК = UChK (“The Smile of the Cheshire Cat”). “Vertness” translates вертность (the extraverted/introverted orientation of a function or aspect). Figures Re.02 and Re.03 are page images in the source and are described in bracketed captions at their positions.