From Classical Intertype Relations to Bipolar Traits of Types and Back

Yu. P. Minaev

Socionics, Mentology and Personality Psychology (SMiPL), No. 6, 2019, pp. 8–18 Section: Research. UDC 159.9.075 : 159.923 Received by the editors 17.07.2019

Translator’s note. Translated from the Russian. Conventions as in the companion translations: “признак” = trait; ТИМ = TIM; ИО = IR (intertype relations); АРП = ART (Augustinavichiute–Reinin trait); ЮМП = JMT (Jung–Minaev trait). Relation names: тождество = identity, суперэго = superego, погашение = extinguishment, дуальные = dual, зеркальные = mirror, конфликтные = conflict, квазитождество = quasi-identity, активация = activation, родственные = kindred, деловые = business, миражные = mirage, полудуальные = semi-dual, ревизия = supervision, заказ = benefit. Tables 1–3 and all matrices appear as images in the scan; they have been reconstructed from the legible text and verified against the group-theoretic construction described in the body (the diagonal entries, vector-columns and matrix products all check out). In the graphic TIM symbols, filled shapes denote extraverted base functions and hollow shapes introverted ones: triangle = intuition, circle = sensing, square = logic, angle (∟) = ethics. In two places the TIM symbols were stripped from the scan’s text layer and could not be recovered; these are flagged below.

Abstract

A comparative analysis is carried out of two didactic directions for presenting the mathematical part of socionic theory. In one case, the classical model of intertype relations — resting on the representation of the types of information metabolism by their Models A — is considered first, and the bipolar traits of the types are introduced afterwards. In the other case, an attempt is made to obtain the classical table of intertype relations starting from 4-position codes in which the types’ membership in the poles of basis bipolar traits is fixed. The operators of intertype relations are then given by 4×4 matrices. The investigation of the second direction is continued here. New possibilities are shown for choosing a suitable basis in which the types are represented. The “semi-Jungian” basis of bipolar traits is analysed in detail.

Keywords: socionics, types of information metabolism (TIMs), intertype relations (IR), groups of intertype relation operators, symbolic calculus of intertype relations, bipolar traits, Augustinavichiute–Reinin traits (ARTs), Jung–Minaev traits (JMTs).

Introduction

The development of the mathematical apparatus of socionics makes it possible to return to old problems at a qualitatively new level. Such a return is useful from the standpoint of the didactics of socionics. The point is that the exposition of socionics teaching material usually operates under fairly rigid time constraints. A more developed mathematical apparatus allows the teaching material to be presented more compactly. At the same time, the use of an overly abstract mathematical formalism may become, for many students of socionics, an obstacle to understanding the connection between the initial postulates and the derived consequences. And the study of any discipline without understanding turns into primitive rote learning, which, as is well known, leads to sorry results.

Teachers of other subjects in which mathematical apparatus is actively used face similar contradictions. But socionics has its own peculiarities as regards the application of mathematical methods. For example, at the initial stages of a physics course it suffices to know the mathematics already included in the school curriculum (the elements of calculus and vector algebra). In studying socionics, however, the important branches of mathematics turn out to be ones not yet taught at school. It cannot be said that these branches are harder; they are simply not called for in a first acquaintance with physics — and it is physics, as a school subject, that most requires proper mathematical support. In the case of socionics, which is not yet a compulsory school subject, an acquaintance with elements of higher algebra — in particular, with elements of mathematical group theory — would do no harm.

Also important for the didactics of socionics is the circumstance that socionics as a scientific direction is quite young: its terminology has not settled, and its methods, including the mathematical ones, continue to develop actively. This development means that teaching materials prepared and used in instruction quite recently must be revised. As an example one may cite the series of instructional articles on mathematical methods in socionics whose publication began in 2016 in the journal Psychology and Socionics of Interpersonal Relations [20].

In parallel with the preparation of teaching materials, scientific work proper on developing the mathematical apparatus of socionics was under way. The scientific results obtained were translated into the language of teaching material, presented in the form of solved problems in mathematical socionics. By now about a hundred problems have been worked through. In the texts of the articles they are called exercises, though this word has proved not very apt, since by no means was it always a matter of perfecting particular skills. Often the assignments were of a problem-based character and aimed at the active mastery of unfamiliar material. Although the series of articles in question is still ongoing, it is already becoming clear in the light of new scientific results that the teaching material can be presented in a more compact form. Accordingly, it is time to begin work on an updated series of instructional articles devoted to mathematical methods in socionics.

The present article will consider a question that was not posed explicitly within the instructional series but may quite naturally arise for its readers. This is the question of the connection between the model of the TIM and the model of the IR. In the articles written jointly with G. R. Reinin [18; 19] in dialogue form, we distinguished the classical model of IR proposed by Aušra Augustinavichiute [2, pp. 109–181] from the “trait-based” model that Grigory Romanovich arrived at when he decided to analyse intertype relations from the standpoint of a group of bipolar traits [23, pp. 163–169]. The classical model of IR d oes not rest explicitly on the bipolar traits of TIMs. The “trait-based” model, by contrast, is founded on comparing, within a pair of TIMs, the poles of bipolar traits for coincidence. It turned out that the “trait-based” model of IR obtained by G. R. Reinin — “tied” to one quite definite group of bipolar traits, namely the ART group — leads to a table of IR that coincides only partially with the Augustinavichiute–Liashkiavichius table built on the classical model. The question arises: how can one achieve full coincidence with the table corresponding to the classical model of IR (in Augustinavichiute’s sense) while modelling the TIMs by fixing the poles of bipolar traits from a definite group?

Just such a question was posed by M. M. Gut in an article [5] published back in the now-distant year 2000. He managed to construct a “half-Reininian” basis of bipolar traits of TIMs suitable for representing the operators of the classical IR by 4×4 matrices. The models of the TIMs were four-position codes in which membership in the poles of the bipolar traits from the basis he built was fixed by the numbers “1” and “–1”. With the help of the matrices, the code of one TIM was transformed into the code of another in full accordance with the classical model of IR. Attempts to find a suitable basis in the only group of bipolar traits known at that time (the ART group) did not meet with success, and M. M. Gut even advanced the hypothesis that this is impossible in principle.

Fifteen years later it emerged that Gut’s “half-Reininian” basis is one of the 840 possible bases of the JMT group, which by then was already known. And since the connection between the ART and JMT groups had by that time been studied quite well [8], it proved possible without particular difficulty to find a suitable “Reininian” basis corresponding to the “half-Reininian” basis constructed by M. M. Gut [10]. However, the question remained open whether there exist in these two groups of bipolar traits other bases suitable for creating a matrix formalism of IR like the one created by M. M. Gut.

At present, now that the foundations of a symbolic calculus have been laid both for the classical and for the two “trait-based” models of IR [15; 16], it has become much easier to answer this question. The answer turned out to be positive. The search for suitable bases can now be conducted quite deliberately and without leaning on the “Gut basis”.

The direction from classical intertype relations to groups of bipolar traits

It was precisely this direction that was chosen as the introduction to mathematical socionics when the series of instructional articles mentioned above was being written. This choice was connected with the fact that the classical model of IR is unique of its kind, in contrast to the “trait-based” models, each of which is “tied” to “its own” group of central bipolar traits. By a central trait is meant one corresponding to a central section of the socion, i.e., a division of the 16-element set of TIMs exactly in half (8 TIMs belong to one pole and the remaining 8 to the other). The central sections of the socion number 6,435 [23, p. 234], and the number of 16-element groups of central sections is simply gigantic (64,864,800) [13].

How, out of this purely mathematically obtained enormous multitude, is one to select precisely those groups of central sections of the socion which it makes sense to study in more detail? The classical model of IR helped answer this question — the model on which the famous Augustinavichiute–Liashkiavichius table is built. This table, in various versions, can be found in practically any book on socionics. No few rapturous words have been said about its significance by leading socionists. This testifies to the consensus currently existing among socionists as to the recognition of the classical model of IR as the fundamental one.

The “trait-based” model of IR proposed by G. R. Reinin leads to a noticeably more ordered table of intertype relations [23, p. 234]. Nevertheless his table has not won such wide recognition. Note that other authors too had “trait-based” models [3; 24], though they apparently did not realise the effective coincidence of their models with G. R. Reinin’s model. Outwardly the approaches proposed by these authors differ, but they lead to one and the same commutative group of IR operators, which has a common 8-element subgroup with the non-commutative group of operators of the classical IR. One can also cite an example of an attempt to use in practice the “trait-based” model of IR “tied” to the ART group [1].

In the article [8] already mentioned, it was shown how to a given 16-element group of bipolar traits one can assign its own “trait-based” model of intertype relations. As an example, the model of IR “tied” to the JMT group was presented there. It was this possible multiplicity of “trait-based” models that gave pause. It was therefore decided to take as the point of departure the unique-of-its-kind group of operators of the classical IR, and then, from the standpoint of this distinguished group of operators, to look at the gigantic multitude of 16-element groups of bipolar traits, each of which is a subgroup of the 32,768-element group of all possible sections of the socion. Let us note in passing that relatively recently G. R. Reinin proposed calling the group of Reinin Traits (the RT group) precisely this 32,768-element group of all possible sections of the socion, and not one of the possible 16-element subgroups he had constructed in his time using the “Jung basis” as a system of generators [18].

How one can look at the bipolar traits of TIMs from the standpoint of the classical IR was first described in article [7]. The idea was that the socion was first divided into four parts equal in number of TIMs by means of the 4-element subgroups of the 16-element non-commutative group of operators of the classical IR, and then for each tetrachotomy one had to see which pairs of bipolar traits could be used to obtain the same division of the socion.

It turned out that for the 15 tetrachotomies of the socion (matching the number of those 4-element subgroups), 23 central bipolar traits are needed. By that time 15 of them were already known. It is these that, after the addition of the existence trait as the neutral element, form the 16-element ART group. For 8 central sections of the socion, new bipolar traits were needed. These 8 traits belong to the 16-element group that was later named the JMT group. The JMT group and the ART group have a common 8-element subgroup. Besides the existence trait, this common subgroup contains 7 bipolar traits, each of which divides the socion in half in exactly the same way as it can be divided by means of the corresponding 8-element subgroup of the group of operators of the classical IR [21].

The question naturally arises: are additional bipolar traits needed, going beyond the ART and JMT groups, which would be required to organise those octochotomies of the socion which can be realised by means of the 2-element subgroups of the group of operators of the classical IR? The answer to this question turned out to be negative [9]. And this means precisely that the view from the side of the classical IR singles out, in a first approximation, only the two named, partially overlapping 16-element groups out of the simply gigantic multitude of groups isomorphic to them.

The direction from groups of bipolar traits to classical intertype relations

Article [11] considered two approaches to constructing the socion and to examining the intertype relations within it. One of them was called “model-based”, because it rested on Model A for the TIMs. The other was called “trait-based”, since in that approach the TIMs were specified by fixing the poles of bipolar traits. It was said that the “model-based” approach, founded on Model A and the principle of information exchange between identical elements of the models in a pair of TIMs, leads to a non-commutative group of intertype relation operators. In this case, in some pairs of TIMs their order proves essential. The operator of the transition from the Model A of the first TIM to the Model A of the second may not coincide with the operator of the reverse transition. In such cases one speaks of asymmetric relations.

The “trait-based” approach, founded on a definite basis of bipolar traits, leads naturally to a theory of intertype relations determined by the spectrum of coincidences of the poles of the bipolar traits. In this case the group of IR operators comes out commutative, since when comparing poles it does not matter which TIM in the pair is counted first and which second. Accordingly, mathematically all the relations in this approach come out symmetric. Word-combinations of the type “rational control”, introduced by G. R. Reinin [23, p. 164], have masked this fact. Such word-combinations could appear in the theory only through comparison with the classical model of IR, in which the emergence of asymmetric relations is mathematically predetermined.

It is hard to disagree that, in the case where TIMs are specified by sets of poles of bipolar traits, the idea of comparing these poles in a pair of TIMs for coincidence seems entirely natural for building a theory of intertype relations. But what is one to do if one wants to specify the TIMs by traits and yet obtain the classical table of IR?

The “Gut matrices”, briefly described in the Introduction, satisfied this wish to a certain degree. Still, judging by article [5], M. M. Gut wanted to find a basis suitable for specifying the TIMs precisely in the ART group — and ideally, to fit 4×4 matrices for transforming TIM-codes written in the Jung basis. As already said, fifteen years after the appearance of Mikhail Mikhailovich’s work it emerged that the basis he constructed belongs to the JMT group, which is connected in a definite way with the ART group. The results of the study of this connection helped find the corresponding ART basis without particular difficulty. Let us show how one can obtain other “suitable” bases without leaning directly on the “Gut basis”.

Let us note once more that M. M. Gut, to fix a TIM’s membership in a particular pole of one or another bipolar trait, used the numbers “1” and “–1”, which recall the signs ”+” and ”–” figuring in the table “Traits of the dichotomy of the socion” compiled by G. R. Reinin [23, p. 155]. For the problem under discussion this proves more convenient than the frequently encountered coding of TIMs using the numbers “1” and “0” (see, e.g., [3; 25; 26]).

It must be understood that the 4×4 matrices for transforming TIM-codes will not in all cases be diagonal, since matrices in which only the elements on the main diagonal are non-zero commute with one another — while the group of operators of the classical IR is non-commutative. In addition, one must take into account that on multiplication by themselves the matrices representing the operators of symmetric IR must give the matrix of the identity operator, which is of course the unit matrix. The squares of the matrices of asymmetric IR must coincide with the matrix of the superego operator.

Recall that in the “trait-based” model of IR proposed by G. R. Reinin, 8 operators coincided with those obtained in the case of the classical model of IR. These were the identity operator and all the operators of relations within the quadra and with the TIMs of the opposite quadra [23, p. 163]. On the other hand, there exist 16 diagonal 4×4 matrices whose main diagonals hold only “ones” and “minus ones”. Each of these, on being squared, gives the unit matrix. All 16 of these matrices, as shown in [6], are used in the matrix formalism realising G. R. Reinin’s “trait-based” model. Thus 8 of these 16 diagonal matrices can serve for representing operators of the classical IR as well.

Note that in the case of the “trait-based” model of IR “tied” to the JMTs, the situation is entirely analogous. The difference consists only in certain changes in that eight of operators which coincide with the operators of the classical IR. In G. R. Reinin’s model, one obtained all the classical IR operators that do not lead to crossing the democrats/aristocrats boundary:

1 (identity), –1 (superego), c (extinguishment), –c (dual), m (mirror), –m (conflict), cm (quasi-identity), –cm (activation).

In the “trait-based” model of IR “tied” to the JMTs, one obtained all the classical IR operators that do not lead to crossing the irrationals/rationals boundary. Accordingly, the first four operators remained unchanged, while in place of the second four the following appeared [8]:

1* (kindred), –1* (business), c* (mirage), –c* (semi-dual).

The formalised designations of the operators of the classical IR were proposed after a study of the properties of their 16-element non-commutative group [12]. The first four operators {1, –1, c, –c} constitute, from the mathematical point of view, the centre of the group, since it comprises the operators that commute with the operators of the whole group. Let us write down the remaining four, in which all the operators of asymmetric IR — i.e., the transition operators — ended up:

m* (to the supervisee), –m* (to the supervisor), cm* (to the benefactor), –cm* (to the beneficiary).

This division of the 16-element non-commutative group into four fours is called the division into cosets with respect to the centre of the group. As we see, these cosets manifested themselves quite distinctly in the comparison of the classical model of IR with the two “trait-based” ones.

In the further reasoning we will also need the formalised designations for the ARTs and JMTs, formed with the use of Table 1, which was published relatively recently in article [17].

Table 1. Formalisation of the designations for ARTs and JMTs.

	I	A	B	C	AB	AC
0	S/∅ (existence)	stat/dyn	det/inv-1	judic/decis	α/γ-1	int/sens
1	irr/rat	extra/intro	intern/extern-1	carefree/farsighted	δ/β-1	tact/strat
2	dem/arist	quest/decl	det/inv-2	merry/serious	α/γ-2	log/eth
3	right/left	posit/negat	extern/intern-2	yield/obstin	β/δ-2	constr/emot

(Abbreviations: det/inv = detached/involved; judic/decis = judicious/decisive; int/sens = intuitives/sensorics; log/eth = logicians/ethicists; constr/emot = constructivists/emotivists.)

The bipolar traits recorded in columns “I” and “A” together constitute the common 8-element subgroup of the ART and JMT groups. Of the two groups under consideration, the traits from columns “B” and “AB” belong only to the JMT group, and the traits from columns “C” and “AC” only to the ART group.

Table 1 is composed in such a way that the binary operation introduced in the indicated groups of bipolar traits can be carried out easily. Attention should be paid to the fact that the union M = {I, A, B, AB} represents the factor group of the JMT group by the subgroup I, and the union R = {I, A, C, AC} the factor group of the ART group by the same subgroup. This helps one understand how to handle the letter part of the formalised designations. The groups M and R are Klein four-groups and are representable as the direct product of two cyclic subgroups of order 2. Isomorphic to the groups M and R is also the group of numerical indices {0, 1, 2, 3}, in which “0” plays the role of the neutral element of the group.

Suppose we need to find the product of two ARTs: positivists/negativists and constructivists/emotivists. Their formalised designations (see Table 1) are A₃ and (AC)₃ respectively. The simple multiplication rules give: A₃ ⊗ (AC)₃ = C₀ — that is, the sought product is the bipolar trait judicious/decisive. And here is an example with traits from the JMT group: intern/extern-1 ⊗ posit/negat = α/γ-2 (B₁ ⊗ A₃ = (AB)₂). More detailed explanations can be found in the corresponding article of the instructional series [22].

It is already well known that any operator of the classical IR can be expressed through three operators, each of which is its own inverse (is an involution) [14]. Let us consider a triple of operators suitable for this purpose: {c, 1*, m}. The operator c belongs to the centre of the group, and its action on any TIM leads to crossing neither the democrats/aristocrats boundary nor the irrationals/rationals boundary. Hence the corresponding matrix will be diagonal. The operator 1* does not lead to crossing the irrationals/rationals boundary, and the operator m does not lead to crossing the democrats/aristocrats boundary. What does this say about the form of the matrices representing them? If the TIMs are specified in a JMT basis, the matrix of the operator 1* will be diagonal, while the matrix of the operator m will not. In the case of an ART basis everything will be exactly the other way round.

Let us try to guess the possible form of the non-diagonal matrix of the operator m in a JMT basis. We need one and the same 4×4 matrix to “act” differently on the vector-columns of democratic and aristocratic TIMs in the sense of preserving/changing the poles of certain basis bipolar traits. How can this be arranged? To make the question easier to answer, let us compile Table 2, in which we mark the membership of the TIMs from two pairs of mirror partners in the poles of the JMTs. One of the pairs will be democratic (▲□ and □▲), and the other aristocratic (▲∟ and ∟▲).

Table 2. Membership of the TIMs from two pairs of mirror partners in the poles of the JMTs.

	I₀	I₁	I₂	I₃	A₀	A₁	A₂	A₃	B₀	B₁	B₂	B₃	AB₀	AB₁	AB₂	AB₃
▲□	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+
□▲	+	–	+	–	+	–	+	–	+	–	+	–	+	–	+	–
mᴰ	+	–	+	–	+	–	+	–	+	–	+	–	+	–	+	–
▲∟	+	+	–	–	+	+	–	–	+	+	–	–	+	+	–	–
∟▲	+	–	–	+	+	–	–	+	–	+	+	–	–	+	+	–
mᴬ	+	–	+	–	+	–	+	–	–	+	–	+	–	+	–	+

(Bold marks the cells shaded in the original — the columns B₀ and B₂, where the behaviour of democrats and aristocrats under m diverges.)

The order of the poles in the traits is chosen so that ▲□ belongs in all cases to the first pole. The signs +/– in the rows designated mᴰ and mᴬ signify the preservation/change of the poles of the traits under the action of the operator m in the case of democratic and aristocratic TIMs respectively. The existence trait I₀ is the neutral element of the group; all TIMs belong to its first pole.

In compiling Table 2, as in compiling Table 1, the four {I₁, I₂, A₀, B₀} was assumed as the basis of the JMT group. The ART basis {I₁, I₂, A₀, C₀} used in compiling Table 1 differs by only one element. The triple of coinciding elements {I₁, I₂, A₀} serves as a basis for the common 8-element subgroup of the ART and JMT groups.

After the columns corresponding to the four {I₁, I₂, A₀, B₀} have been filled in in Table 2, the rest of the table fills in very quickly if one takes account of the following relations, which became obvious after the introduction of the formalised designations: I₁ ⊗ I₂ = I₃; Iₙ ⊗ A₀ = Aₙ; Iₙ ⊗ B₀ = Bₙ; Aₙ ⊗ B₀ = (AB)ₙ, where n can take the values 0, 1, 2, 3.

Let us pay attention to the bipolar traits B₀ and B₂. The signs fixing the poles of these traits differ for any aristocrat and coincide for any democrat. This is explained by the formula I₂ ⊗ B₀ = B₂ and by the fact that on I₂ democrats have ”+” and aristocrats ”–“. Moreover, the action of the operator m on democratic TIMs does not lead to a change of poles on these traits, while on aristocratic ones it does. This is easy to understand by recalling the designations: B₀ is det/inv-1 and B₂ is det/inv-2 (see Table 1). Indeed, for democrats both information aspects in the Ego block are either detached (for sayentists) or involved (for socials). Hence for democrats, under the action of the operator m, which swaps the symbols of these aspects, no change of pole on traits B₀ and B₂ will occur. For aristocratic TIMs the situation is the opposite, since in their Ego block one IA is detached and the other involved. The indicated properties of the bipolar traits B₀ and B₂ can be exploited in creating the matrix formalism. Consider the following four equalities, in which one and the same 2×2 square matrix figures:

$\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\begin{array}{c}1\\ 1\end{array}\right)=\left(\begin{array}{c}1\\ 1\end{array}\right);\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\begin{array}{c}-1\\ -1\end{array}\right)=\left(\begin{array}{c}-1\\ -1\end{array}\right);$

$\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\begin{array}{c}1\\ -1\end{array}\right)=\left(\begin{array}{c}-1\\ 1\end{array}\right);\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\begin{array}{c}-1\\ 1\end{array}\right)=\left(\begin{array}{c}1\\ -1\end{array}\right).$

The “action” of this matrix on a vector-column amounts to swapping its elements. But in the cases written in the upper row, the “action” of the matrix does not lead to a change of the vector-column, whereas in the cases written in the lower row the elements undergo a change of sign. Let us imagine that the upper elements of these vector-columns indicate the pole of trait B₀ and the lower ones the pole of B₂. Then the 2×2 matrix figuring in all the equalities is precisely that submatrix of the sought 4×4 matrix which ensures the difference in the action of the operator m on democratic and on aristocratic TIMs.

The vector-column from the upper-left equality will enter into the 4-element vector-columns of the sayentists (detached in both the 1st and the 2nd function), and the one from the upper-right will belong to the socials (involved in both the 1st and the 2nd function). The initial vector-column from the lower-left equality will enter into the 4-element vector-columns of those aristocrats who are detached in the 1st function and involved in the 2nd function, and the initial vector-column from the lower-right will belong to the aristocrats who are involved in the 1st function and detached in the 2nd function.

Now let us pay attention to those four cells of Table 2 where the columns I₁ and A₁ intersect the rows mᴰ and mᴬ. The signs in these cells fix the fact that mirror partners always belong to different poles of the traits I₁ (irrationals/rationals) and A₁ (extraverts/introverts), irrespective of their membership in the poles of the trait I₂ (democrats/aristocrats). This means that if these two bipolar traits are included in the sought basis, the corresponding 2×2 submatrix of the 4×4 matrix will be diagonal, with the non-zero elements equal to “minus one”.

Using the formalised designations of the JMTs, it is easy to convince oneself that the traits entering into the chosen four {irr/rat, extra/intro, det/inv-1, det/inv-2} are independent — i.e., none of them can be expressed through the others. And for a basis of a 16-element group of bipolar traits it is precisely four independent elements that are required. Thus we have obtained the sought basis. In this basis the TIMs figuring in Table 2 will be represented by the following vector-columns:

$▲\square =\left(\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right),\square ▲=\left(\begin{array}{c}\\ -1\\ -1\\ 1\\ 1\end{array}\right),▲└=\left(\begin{array}{c}1\\ 1\\ 1\\ -1\end{array}\right),└▲=\left(\begin{array}{c}\\ -1\\ -1\\ -1\\ 1\end{array}\right).$

Using the considerations set out above together with Table 2, it is not difficult to find several more mathematically suitable JMT bases. The basis we have chosen as an example — which makes it possible to write down the matrices of the operators from the triple {c, 1*, m} — possesses an additional useful property. For any bipolar trait from this basis it is very easy to determine the pole to which a given TIM belongs. Let us recall once more that the vector-columns for the four TIMs that figured in Table 2 are written in the following basis:

{irr/rat (I₁), extra/intro (A₁), det/inv-1 (B₀), det/inv-2 (B₂)}.

This basis may be called semi-Jungian, since the first two dichotomies (I₁ and A₁) were already explicitly designated in Jung’s 8-element typology [27]. In socionics, by contrast, the 16-element set of TIMs is considered. With the help of the pair of dichotomies indicated, the socion is divided into 4 “temperamental” fours [4]. If to this pair of central sections of the socion we add a third, fixed by the bipolar trait detached/involved in the 1st function, we obtain an octochotomy of the socion into pairs of TIMs connected by kindred IR. Jung’s typology was 8-element because he oriented himself exclusively to the 1st function and accordingly, as S. I. Churyumov put it, “did not distinguish kindred types” [25, p. 386]. Hence for his typology the basis {I₁, A₁, B₀} would have been suitable. Let us note in passing that a basis for Jung’s typology cannot be found in the ART group.

To distinguish kindred partners, information about the 2nd function of the Ego block is necessary. For this it suffices to know to which of the poles of the dichotomy det/inv-2 (B₂) each of the TIMs entering into the given pair of kindred partners belongs. Anyone who wishes can convince themselves how simple it is, in this “semi-Jungian” basis we have proposed, to compose the vector-columns of the TIMs. The result obtained can be compared with Table 3.

Table 3. Membership of the TIMs in the poles of the traits of the “semi-Jungian” basis.

	▲⁺	▲⁻	●⁺	●⁻	△⁺	△⁻	○⁻	○⁺	■⁻	■⁺	◣⁺	◣⁻	□⁻	□⁺	∟⁺	∟⁻
irr/rat (I₁)	+	+	+	+	+	+	+	+	–	–	–	–	–	–	–	–
extra/intro (A₁)	+	+	+	+	–	–	–	–	+	+	+	+	–	–	–	–
det/inv-1 (B₀)	+	+	–	–	+	+	–	–	+	+	–	–	+	+	–	–
det/inv-2 (B₂)	+	–	+	–	+	–	+	–	+	–	+	–	+	–	+	–

(Column heads: the abbreviated graphic TIM symbols — filled/hollow triangle = extraverted/introverted intuition base, circle = sensing, square = logic, angle = ethics, with the +/– sign superscripts as in the original figure.)

The order of the TIMs in Table 3 is chosen so that it can be clearly seen how the socion is first divided in half by the bipolar trait irr/rat (I₁), then each half by the trait extra/intro (A₁), and so on. Let us now write out, in the “semi-Jungian” basis, the matrices of the operators of the classical IR {c (extinguishment), 1* (kindred), m (mirror)}, which constitute a set of generators of the group:

$\mathbf{c}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right);1^{\ast }=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right);\mathbf{m}=\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right).$

Compare the elements standing on the main diagonals, from top to bottom, in the matrices of the operators c and 1* with the elements of the columns ”△⁺” and ”▲⁻” respectively (see Table 3). This coincidence (up to the replacement of “1” by ”+” and “–1” by ”–”) is not accidental. The vector-columns of these TIMs must be obtained from the vector-column ▲⁺, containing exclusively “ones”, under the action of the matrices of precisely the operators c and 1*.

The non-diagonal 4×4 matrix of the operator m consists of four 2×2 submatrices, two of which are zero, while the purpose of the other two is easy to explain. The upper-left 2×2 submatrix answers for the change of the poles of the traits irr/rat (I₁) and extra/intro (A₁), which will occur absolutely identically for all TIMs under the action of the operator m (the mirror IR). The lower-right 2×2 submatrix contains the whole “trick” of choosing a basis suitable for the intended matrix formalism. This matrix does not change the poles of the bipolar traits det/inv-1 (B₀) and det/inv-2 (B₂) in the case of democratic TIMs, but does change them in the case of aristocratic ones.

Having at our disposal the matrices of the operators from the triple {c, 1*, m}, it is not difficult to obtain the matrices of all the remaining operators of the classical IR. Of course, to be able to use directly the canonical representation as a product of “basic” operators, one must obtain the matrix of the superego operator (–1), which is the square of any operator of asymmetric IR. Using the fact that m* = 1* • m (the supervisee is the kindred partner of the mirror partner), we find the operator of the transition to the supervisee:

${\mathbf{m}}^{\ast }=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right)\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right)=\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right).$

Multiplying the resulting matrix by itself (squaring it), we obtain the matrix of the superego operator (–1 = (m*)²):

$-1=\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right)\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right).$

Of course, this last matrix could have been obtained far more simply, by understanding that it must be diagonal. It was enough to look at the column ”●⁻” in Table 3. With the help of that same table one can immediately write down, in the “semi-Jungian” basis, all the diagonal matrices — i.e., the matrices of the operators that do not lead to crossing the irrationals/rationals boundary (for this is a basis of the JMT group). The matrix of the superego operator (–1) was obtained by a more complicated route in order to remind the reader that this operator was included in the “extended” system of generators of the group of operators of the classical IR only so that the canonical representation (–1)ᵏ • cˡ • (1*)ᵖ • m^q would be more convenient to use. After this extension one can confine oneself to just two values (0 and 1) of the exponents of the basis operators.

Obtaining the matrices of the remaining operators is not difficult once one has at one’s disposal the matrices of the operators m and m*, as well as of the operators that do not lead to crossing the irrationals/rationals boundary — i.e., those in whose canonical representation q = 0 (no operator m). One will need to multiply the known matrices, using the obvious formulas:

(–1) • m = –m; (–1) • m* = –m*; c • m = cm; c • m* = cm*; (–c) • m = –cm; (–c) • m* = –cm*.

There is hope that after so detailed an analysis of the construction of the matrix formalism on the example of the “semi-Jungian” basis, those who wish will be able to construct the matrices of the operators of the classical IR in other bases suitable for this purpose — not only of the JMT group but of the ART group as well. In the latter case one must not forget that the diagonal matrices will be those representing the operators that do not lead to crossing the democrats/aristocrats boundary. Accordingly, of the “basis” operators the non-diagonal matrix will be that of the operator 1* (the kindred IR), and not m (the mirror), as it was in the example analysed.

Of course, in the case of the ARTs, instead of Table 2 “Membership of the TIMs from two pairs of mirror partners in the poles of the JMTs” one will need to compile a table “Membership of the TIMs from two pairs of kindred partners in the poles of the ARTs”. The pairs taken as examples must belong to different poles of the trait irrationals/rationals. Suitable pairs could be {▲□-type pairs of kindred partners — the TIM symbols at this point are illegible in the source scan}. One variant of an ART basis is this:

{dem/arist (I₂), quest/decl (A₂), judic/decis (C₀), carefree/farsighted (C₁)}.

With this choice, the non-diagonal matrix of the operator 1* (the kindred IR) will be exactly the same as the matrix of the operator m (the mirror IR) was in the case of the “semi-Jungian” basis analysed in detail. Indeed, the operator 1* always changes the poles of the traits I₂ and A₂, while the poles of the traits C₀ and C₁ are preserved for irrationals but changed for rationals — and moreover the signs of these last two poles coincide for any individual irrational and differ for a rational.

What sequence of TIMs does it make sense to choose in compiling the table that would replace Table 3? If one wants the “body” of the table to remain the same, then in the case of the basis {I₂, A₂, C₀, C₁} the corresponding order must be chosen as follows — four fours of TIMs (the TIM symbols themselves are illegible in the source scan; only their +/– sign superscripts survive). This sequence is broken into four fours of TIMs so that it is easier to notice that the democrats are in the first half of the list; in each half the fours of questims come first; in each four the judicious mirror pair comes first; and in each mirror pair the carefree TIM is put in first place.

As we see, in this basis, in all pairs of mirror partners the difference lies only in the last trait. This means that the diagonal matrix of the operator m will coincide with the one that corresponded to the operator 1* in the case of the “semi-Jungian” basis. Recall that there, the difference within the pairs of (correspondingly) kindred partners was likewise only in the last trait of the basis {irr/rat (I₁), extra/intro (A₁), det/inv-1 (B₀), det/inv-2 (B₂)}.

And what will the matrix of the operator c (extinguishment) be in the ART basis under discussion? It will of course be diagonal, with the first (top) element equal to “one” and the remaining non-zero elements equal to “minus one”. Indeed, in any extinguishment pair the poles coincide only on the first of the four bipolar traits entering into the ART basis under consideration. Comparing with the case of the “semi-Jungian” basis, we see a difference consisting in the fact that the two lower diagonal elements have changed sign.

Conclusion

The investigation to which the present article is devoted was originally conceived as an investigation in the didactics of socionics. The intention was to compare two approaches to expounding the mathematical methods of socionics. However, in the course of analysing the approach in which the task is set of obtaining the table of the classical IR while representing the TIMs not by their Models A but by 4-element codes fixing membership in the poles of bipolar traits, new theoretical results were obtained concerning socionics proper rather than its didactics. It was shown that both in the JMT group and in the ART group there exist previously unknown bases which admit the construction of a group of 4×4 matrices transforming TIM-codes in accordance with the Augustinavichiute–Liashkiavichius table. Particular attention was paid to a convenient basis, named in the article “semi-Jungian”.

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