Gut Matrices and the Jung–Minaev Bipolar Traits

Yu. P. Minaev Socionics, Mentology and Personality Psychology, No. 1, 2015, pp. 5–16 UDC 159.923.2 (Section: Research)

Abstract

It is shown that the basis of bipolar traits of TIMs which M. M. Gut in fact used in constructing his matrix formalism of the classical intertype relations (per Augustinavichiute) is one of the possible bases of the Jung–Minaev group of traits. A comparative analysis of the 4×4 Gut matrices and the 3×3 Minaev matrices is carried out. A basis of the Augustinavichiute–Reinin group of traits is found which is suitable for constructing matrices of the classical IR operators by analogy with the Gut matrices.

Keywords: socionics, intertype relations, Augustinavichiute–Reinin group of traits, Jung–Minaev group of traits, Gut matrices, Minaev matrices, matrix formalism of intertype relations.

Translator’s note: the matrix figures in the original are typeset as images and did not survive text extraction; they are indicated below by bracketed descriptions. IR abbreviations follow the same key as in the 2016 paper: ТО identity, СЭ super-ego, РО kindred, ДЕ business, ПО extinguishment, ДУ duality, МИ mirage, ПД semi-duality, ЗЕ mirror, КФ conflict, Р–/Р+ supervision, КВ quasi-identity, АК activation, З+/З– benefit.

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Introduction

More than a year and a half after the publication of articles [3; 4], devoted to the matrix formalism of intertype relations that I had proposed, Olga Bogdanovna Karpenko drew my attention to the work of M. M. Gut [2], which, in her opinion, had not received a proper appraisal from socionists. That work considered a mathematical representation of the classical IRs by means of 4×4 matrices. By “classical” were meant the intertype relations per Augustinavichiute, as opposed to the IRs per Reinin.

Recall that in the theory of intertype relations developed by Aushra Augustinavichiute [1, pp. 109–181], the basic postulate was that “only identical elements transmit information to each other.” Therefore the models of IRs were diagrams in which line segments joined the symbols of identical informational aspects in the side-by-side graphical models of two TIMs. For G. R. Reinin, by contrast, the models of IRs were in fact “coincidence spectra of the bipolar traits” in the pairs of TIMs under consideration. Moreover, these traits were taken from a quite definite group, called for short the group of ARTs (Augustinavichiute–Reinin traits) [11, pp. 163–169].

In my article [4] I compared precisely the Augustinavichiute–Liashkiavichius and Reinin IR tables on the basis of the matrix formalism. In the first case I used 3×3 matrices for the IR operators, and the TIM was given by the three-letter coding adopted in socionics. The resulting matrix formalism corresponded to the IRs per Augustinavichiute. In the second case I had 4×4 matrices for the IR operators, and for the TIMs the four-letter Myers–Briggs coding was used. In this latter case the matrices turned out to be diagonal and in fact reflected shortened “coincidence spectra” per Reinin. Therefore the fact that 4×4 matrices were used by M. M. Gut for the mathematical representation of precisely the classical IRs (per Augustinavichiute) could not fail to interest me. A more detailed acquaintance with work [2] revealed several more interesting points, which are the subject of this article.

M. M. Gut’s numerical coding of TIMs and its slight modification

M. M. Gut first introduced a coding for the informational aspects (IAs), using the following basis: object / relation between objects (i.e. body/field, or black/white); statics/dynamics; internal/external. The first pole was assigned the number “–1” and the second “1.” By way of example, the code of black intuition came out as (–1; –1; –1). The numerical coding of a TIM could be represented “in the form of a matrix, where the upper row corresponds to the 1st info-aspect and the lower row corresponds to the 2nd info-aspect” [2].

Having noticed that “the value of the 1st position of the 2nd info-aspect is always opposite to the corresponding value of the 1st info-aspect, and the value of the 2nd position of the coding of the 2nd info-aspect always coincides with the corresponding value of the 1st info-aspect,” M. M. Gut proposed “to represent the coding of a TIM in the form of a row vector, where the 1st, 2nd and 3rd positions correspond to the coding of the 1st info-aspect, and the 4th position corresponds to the 3rd position of the 2nd info-aspect” [2].

One more important quotation from M. M. Gut’s work, connected with the coding of TIMs: “As is known, by means of a four-position code a TIM can be specified in any of the Reinin bases. The question then arises whether the coding method used by the author is one of the Reinin bases. If one examines all the positions of the TIM coding used, it turns out that the 1st position corresponds to the trait extraversion–introversion, the 2nd position corresponds to the trait statics–dynamics, whereas the 3rd and 4th positions correspond to none of the known Reinin traits, being, nonetheless, individual dualizing traits! Yet the product of these two last traits gives the quadral trait aristocrats–democrats. That is, the basis used is only half Reininian” [2].

It must be that in those days, when M. M. Gut voiced the thought of a “half-Reininian” basis, it seemed so heretical that it could not be taken seriously by the socionics community. Even more than 10 years later, when I arrived at this same thought from entirely different considerations and voiced it not only in the concluding part of article [4] but also in a talk at a socionics conference, no one told me that this had long been known to everyone. On the contrary, people began assuring me that I was mistaken. The greatest activity in this was shown by Semyon Ivanovich Churyumov, with whom I subsequently discussed more than once, in private conversations and correspondence, the question of the possible existence of a group of traits for TIMs that only partially coincides with the group of ARTs.

In Semyon Ivanovich I was dealing with a well-known socionist who purposefully tracks all the novelties of socionic thought, so it gradually dawned on me that the possibility of the existence of non-Reininian traits is far from obvious to everyone, and that I needed to present explicitly a group alternative to the ARTs and carry out their comparative analysis. The results of the corresponding investigations were reflected in a series of my articles published in the journals of the International Institute of Socionics. And whereas in the first articles a rather long phrase of the type “a group of bipolar traits alternative to the ARTs, corresponding more closely to Jung’s typology” was used when mentioning this new group, I later began using the abbreviated variant that arose in the heat of the polemic with S. I. Churyumov. In this article too we shall compare the ARTs with the JMTs (Jung–Minaev traits). In the group of JMTs one can choose a four-element basis in such a way that, after discarding one bipolar trait, one obtains a basis that could be used for K. G. Jung’s 8-element typology. With the group of ARTs this cannot be done [6].

It turned out that M. M. Gut had in fact been using one of the variants of such a basis in the group of JMTs. The third and fourth positions in his coding of TIMs cannot be matched with ARTs, but with JMTs they can: the third with internal/external-1, and the fourth with internal/external-2. One of my articles even contains a figure with the caption: “Generation of a known bipolar trait by ‘multiplying’ two dichotomies that do not belong to the group of ARTs” [7]. The point was precisely how the well-known trait aristocrats/democrats is obtained by multiplying the just-named JMTs by each other.

If M. M. Gut had done with the basis used in his work [2] what G. R. Reinin in his time did with the Jung basis (which it would be more accurate to call the Jung–Augustinavichiute–Reinin basis), he would automatically have arrived at the group we now call the group of JMTs. It would not contain those 8 bipolar traits which are among the ARTs: intuitives/sensorics; logicians/ethicists; merry/serious; judicious/decisive; constructivists/emotivists; tacticians/strategists; yielding/obstinate; carefree/farsighted. They would be replaced by a different eight. For the new traits I gave formalized names so that it would be immediately clear what is being discussed. Besides the two actually used by M. M. Gut, among the JMTs there are 6 more traits with the following formalized names: detached/involved-1, detached/involved-2, α/γ-1, α/γ-2, β/δ-1, β/δ-2.

Fig. 1 presents a diagram in which the common 8-element subgroup of the two 16-element groups of bipolar traits (Augustinavichiute–Reinin and Jung–Minaev) is clearly visible. It should be said that the idea of this diagram belongs to S. I. Churyumov, who by means of such a diagram tried to convince me of the impossibility of the existence of two groups of bipolar traits with a common subgroup.

Fig. 1. Two groups of bipolar traits of TIMs (Augustinavichiute–Reinin and Jung–Minaev) with a common 8-element subgroup.

[Figure: two boxes feeding into a common lower box. Left box, “Group of ARTs”: constructivists/emotivists; yielding/obstinate; intuitives/sensorics; judicious/decisive; tacticians/strategists; carefree/farsighted; logicians/ethicists; merry/serious (bold = basis trait). Right box, “Group of JMTs”: α/γ-1; detached/involved-1; β/δ-2; external/internal-2; α/γ-2; detached/involved-2; δ/β-1; internal/external-1 (bold = basis trait). Common lower box (shared 8-element subgroup): extraverts/introverts; irrationals/rationals; questims/declatims; democrats/aristocrats; positivists/negativists; right/left; statics/dynamics; the existence trait of Karpenko–Churyumov.]

Incidentally, as generating elements of this 8-element subgroup of each of the compared groups (ARTs and JMTs) one can take the following triple of traits: statics/dynamics, right/left, democrats/aristocrats. If to this triple one adds, say, the trait merry/serious, one obtains one of the possible bases for the ARTs; if, say, internal/external-1 — then for the JMTs.

The ARTs and JMTs taken as basis traits are set in bold in Fig. 1. The order of the poles is chosen so that the first corresponds to ILE. The order of the traits themselves is also not accidental. At the very bottom stands the neutral element of all possible groups of bipolar traits (the existence trait of Karpenko–Churyumov). From the first two (lowest in the diagram) basis traits (statics/dynamics and right/left), multiplication (as introduced by G. R. Reinin [11]) yields the trait written above them, positivists/negativists. Higher up comes one more trait, which in this case is included in the basis (democrats/aristocrats). The next 3 traits, written still higher, are obtained by multiplying the trait democrats/aristocrats by the preceding 3 traits (those lower in the diagram).

That is the whole 8-element group. Any product of traits belonging to this group will also belong to this group (in the mathematical sense). Multiplication of any trait by itself (squaring) gives the neutral element of the group (the existence trait of Karpenko–Churyumov).

This 8-element group is a subgroup both of the group of ARTs and of the JMTs. The remaining ARTs are obtained by successively multiplying the traits belonging to the 8-element subgroup by one more basis bipolar trait, now purely ARTian (merry/serious). For example, the trait logicians/ethicists is obtained as the product of this fourth basis trait of the ART group and the trait statics/dynamics.

All the remaining JMTs are obtained analogously. But as the fourth basis trait we took the purely JMTian trait internal/external-1, which is closer in spirit to K. G. Jung’s typology [12]. Then, for example, multiplying this trait by the trait statics/dynamics, we obtain the trait δ/β-1, which is not among the ARTs. It is a purely JMTian trait. Its pole δ-1 belongs to those TIMs whose 1st function processes one of the aspects of the δ-values.

Of course, one must be aware that some of the JMTs, like some of the ARTs, are called traits (in the sense of typological characteristics of individuals) only by a considerable stretch. Rather, what is at issue is formal divisions of the socion into two equal parts of 8 TIMs each. For at least some of them to become reliably registrable differentiating traits, a great deal of work by a large number of researchers (both theorists and experimenters) is still required.

Let us return to the analyzed work of M. M. Gut, where there is a table titled “Numerical coding of TIMs via the system-forming basis of info-aspects.” In its place, in the present article we give Table 1.

Table 1. Coding of TIMs in the JMT basis used by M. M. Gut.

JMTs \ TIMs	IEE	SLI	ILE	SEI	LII	ESE	LSI	EIE	SLE	IEI	SEE	ILI	ESI	LIE	EII	LSE
extrav/introv	+	–	+	–	–	+	–	+	+	–	+	–	–	+	–	+
statics/dynam	+	–	+	–	+	–	+	–	+	–	+	–	+	–	+	–
intern/extern-1	+	–	+	–	–	+	–	+	–	+	–	+	+	–	+	–
intern/extern-2	+	–	–	+	+	–	–	+	–	+	+	–	–	+	+	–

What, then, is the slight modification of the TIM coding compared with M. M. Gut’s? First, the TIMs are represented not by row vectors but by column vectors. Second, ”+” was assigned to the TIM corresponding to the first pole of the bipolar trait, and ”–” to the one corresponding to the second pole. Third, only the signs were recorded; the digit “1” was not written but understood. Note also that Table 1 uses the “square-wise” order of TIMs, which for the case of the JMTs is more natural than the “quadra-wise” one [9].

The possibility of using non-diagonal 4×4 matrices to represent the IR operators per Augustinavichiute in Gut’s approach

As already said, in the matrix formalism I proposed, 4×4 matrices arose when the TIMs were given by column vectors whose elements were the letters of the Myers–Briggs code. Let us give an example of how one and the same matrix “acts” on two different column vectors:

[Figure: a 4×4 matrix with entries –, +, –, + on the diagonal acting on the column (E, N, T, P)ᵀ to give (I, N, F, P)ᵀ, labelled ξ(ILE) = IEI; and the same matrix acting on (I, N, F, J)ᵀ to give (E, N, T, J)ᵀ, labelled ξ(EII) = LSE.]

The “coincidence spectrum” of the ART poles (the IR per Reinin) is the same in these two cases. But on the irrational TIMs ξ “acts” like the operator МИ, and on the rational ones like ПД.

G. R. Reinin ran into similar unpleasantnesses when he attempted to carry out an analysis of intertype relations “from the standpoint of the group of bipolar traits” (for more detail see [7]).

But here it must be noted that when the Myers–Briggs coding is used for writing the column vectors, the matrices are necessarily diagonal, since replacements (upon multiplication by ”–”) can occur only according to the scheme E ↔ I, N ↔ S, F ↔ T, P ↔ J, and the order of the bipolar traits in the code is rigidly fixed. If, however, one uses not a letter coding but a numerical one, there is no such rigid requirement of diagonality of the matrices. It is this circumstance that allowed M. M. Gut to select matrices that would correspond to the operators of the classical IRs (per Augustinavichiute). Had he confined himself exclusively to diagonal matrices, he would have arrived at “coincidence spectra” — but of the JMTs, not the ARTs. Let us give the corresponding example:

[Figure: one and the same diagonal 4×4 matrix acting on the column vector of ILE (+,+,+,–)ᵀ to give (–,+,–,+)ᵀ, labelled ν(ILE) = LII; and on the column vector of EII to give the vector of SEE (per the coding of Table 1), labelled ν(ILE-partner example) — i.e. on democratic vs aristocratic columns the same matrix realizes different classical IRs.]

The column vectors in this example are taken in the same coding as recorded in Table 1. And the matrix is such as if we had deliberately restricted ourselves to the requirement of using only diagonal matrices. We repeat that when the Myers–Briggs coding is used for the TIMs, the diagonality of the matrices arises automatically. One more reminder: by “plus” one must understand the number “1,” by “minus” the number “–1,” and empty cells in the matrix must be taken as the number “0.”

What, then, does the matrix that is the same in both cases of the example just given correspond to? Clearly, among the IRs per Augustinavichiute we shall find no correspondence, since in the first case this matrix “ought” to represent the operator ЗЕ, and in the second — КФ. Nor shall we find a correspondence among the “coincidence spectra” per Reinin, because there the ARTs are used, and in that case the mirror and conflict IRs do not “split” (in G. R. Reinin’s terminology [11, p. 166]) and quite definite (different!) “coincidence spectra” correspond to them. But in the case of the JMTs, one and the same 4×4 matrix, which we denoted by the Greek letter ν, acts on the democratic column vectors like the operator ЗЕ, and on the aristocratic ones like КФ. More detail can be found in [7]. One need only bear in mind that in my article analyzing the problem of the “splitting” of IRs by “coincidence spectra,” the following basis is used in the JMT case: 1) extraverts/introverts; 2) internal/external-1; 3) internal/external-2; 4) irrationals/rationals. As we see, it differs somewhat from the basis proposed in the analyzed work of M. M. Gut. Readers who decide to verify everything for themselves will therefore have to do a little work.

We shall orient ourselves by the basis proposed by M. M. Gut, since it is recorded in our Table 1 and there is no need to consult other works. The same table contains the corresponding column vectors for all the TIMs. Lifting the requirement of diagonality of the matrices, we can obtain, for the same TIMs that figured in the last example, the following:

[Figure: two non-diagonal 4×4 matrices; the first realizes ЗЕ(ILE) = LII, the second КФ(ILE) = SEE — now as fixed operators.]

On what grounds can one assert that these new non-diagonal matrices will correspond to quite definite IR operators per Augustinavichiute for all TIMs, if these TIMs are represented by column vectors in Gut’s JMT basis? Let us interchange the two TIMs, leaving the matrices in place:

[Figure: the same two matrices acting on the interchanged column vectors, yielding ЗЕ(EII) = ESI-partner and КФ(ILE-swapped) = SEE-partner — the results again come out correctly.]

As we see, everything came out right. This, of course, is not yet a proof, but for the JMT basis the check on interchanging TIMs differing by the trait aristocrats/democrats is encouraging. In article [7] it is shown that when the ARTs are used, the “splitting” of IRs, when considered “from the standpoint of the bipolar traits,” occurs along the line rationals/irrationals (this fact was noticed already by G. R. Reinin [11, p. 166]), while when the JMTs are used — along the line democrats/aristocrats.

The last of the facts cited gives grounds to expect non-diagonality of the matrices, when the JMT basis is used, for the following IR operators per Augustinavichiute: Р+, Р–, З+, З–, ЗЕ, АК, КВ, КФ — since it is precisely in these cases that “splitting” along the line democrats/aristocrats occurs if one uses “coincidence spectra.” In other words, if the named IR operators “act” on democratic TIMs, the “coincidence spectra” differ from those which will occur under action on aristocratic TIMs (see Table 4 in [7]). If, however, we are dealing with the operators of these IRs: ТО, ДУ, СЭ, ПО, ДЕ, МИ, РО, ПД — then their “coincidence spectra” in the JMT basis will not depend on which TIM they “act” upon. Recall that G. R. Reinin in his time discovered that in the case of the ARTs the “coincidence spectrum” can be unambiguously tied to the following classical IRs (per Augustinavichiute): ТО, ДУ, СЭ, ПО, ЗЕ, АК, КВ, КФ, while for the remaining relations “splitting” occurs.

Comparison of the 4×4 Gut matrices and the 3×3 Minaev matrices

In article [8] attention was already drawn to the fact that the whole 16-element group of classical IRs (per Augustinavichiute) can be reconstructed from three generating elements. The choice of a system of generators is not unique. One may take the triple ДУ, ЗЕ, РО. It is easy to see that by means of ЗЕ and РО one can obtain the operators of all the IRs linking TIMs belonging to one pole of the dichotomy statics/dynamics. Indeed, Р– = РО·ЗЕ (that is, the supervisee is the kindred of the mirror partner), ДЕ = ЗЕ·(Р–), СЭ = РО·ДЕ, КФ = ЗЕ·СЭ, Р+ = РО·КФ, ТО = РО·РО.

All the remaining IR operators are obtained from those already listed by multiplication by the operator ДУ: АК = ДУ·ЗЕ, З– = ДУ·(Р–), МИ = ДУ·ДЕ, ПО = ДУ·СЭ, КВ = ДУ·КФ, З+ = ДУ·(Р+), ПД = ДУ·РО.

Thus, if the matrices for the generating elements are written out explicitly, all the remaining matrices can be obtained by the matrix multiplication rules well known from linear algebra. Let us compare the 3×3 matrices that I introduced in work [3] and used in [4] with the 4×4 Gut matrices.

Note that in article [3] the 3×3 matrices went as the “second variant.” In the “first variant” of the matrix formalism I proposed, 2×2 matrices were used, and the TIMs were given not by three-letter codes but by a two-element sequence of the IA symbols of their EGO blocks. In that variant, admittedly, one had to introduce the “numbers” c and f, responsible for the change of color and shape of the IA symbols. A change of shape (internal ↔ external) did not alter the IA’s membership in a given pole of the dichotomy rational/irrational, while a change of color took place within the macro-aspects (intuition, sensorics, logic, ethics).

When comparing the IRs per Augustinavichiute and per Reinin in article [4], it was more convenient to use the “second variant” with the 3×3 matrices. It will be more convenient for us now as well. It should be said that, for ease of comparison of my 3×3 matrices with the 4×4 matrices proposed by M. M. Gut, I have somewhat modified the latter. Instead of “zeros” there will be empty places, instead of “1” a ”+,” and of “–1” only the ”–” will remain. In the last two examples we analyzed, such modified Gut matrices “representing” the operators ЗЕ and КФ were already used. To make it still easier to compare the 3×3 Minaev matrices and the 4×4 Gut matrices, the blocks of the matrices important for the comparison are set off with bold lines. Then for the generating elements of the group of IR operators per Augustinavichiute the following comparative table results (Table 2).

Table 2. The generating elements of the group of IR operators per Augustinavichiute, represented by 3×3 Minaev matrices and 4×4 Gut matrices.

[Figure: three columns headed ДУ, ЗЕ, РО. Top row: the 3×3 Minaev matrices; bottom row: the 4×4 Gut matrices. In each pair, a 2×2 block and a corner cell are set off with bold lines; the signs in the corresponding 2×2 blocks and corner cells of each Minaev/Gut pair coincide.]

The marked 2×2 blocks in the compared matrices (corresponding to one and the same IR operator) coincide. The signs in the corner cells set off by bold lines also coincide. The latter fact has the simplest explanation. Both in the Minaev matrices and in the Gut matrices the signs in the corresponding cells are determined by the coincidence (”+”) or non-coincidence (”–”) of the poles of the bipolar trait extraverts/introverts in the pair of TIMs linked by the IR operator. The compared matrices are either both diagonal or both not. Diagonal are the matrices linking the column vectors of TIMs both of which belong to one pole of the dichotomy rationals/irrationals.

The important fact is that the element of the Gut matrix for which no correspondence was found in the Minaev matrix is easily “computed” if one knows the element responsible for the change of version (extra/intro), and also whether the rationality pole changes (whether or not the matrix is diagonal). The element located in the Gut matrix at the intersection of the second row and second column is responsible for the change of pole of the dichotomy statics/dynamics. And, as is known, this bipolar trait can be viewed as the “product” of the traits extraverts/introverts and irrationals/rationals. This means that in diagonal Gut matrices (“representing” those IR operators that do not change rationality) the signs of the elements responsible for statics/dynamics and for extraversion/introversion must necessarily be the same, while in non-diagonal matrices they must be different.

Having noted this fact, one could easily discover an unfortunate typo in the analyzed article of M. M. Gut. The operator ЗЕ changes rationality, hence the matrix must be non-diagonal, and the signs of the two upper nonzero elements must consequently differ. In the article, however, “–1” stands in both cases, although the matrix is, as it should be, non-diagonal. This value is correct for the element responsible for the change of version, but incorrect for the element responsible for statics/dynamics. The operator ЗЕ does not change the pole of this dichotomy!

Why, all the same, did the 2×2 blocks turn out identical? Let us first consider the case of diagonal matrices with the concrete example of the operator ПД = ДУ · РО “acting,” say, on LIE. The ПД matrix in both variants is easily obtained from the РО matrix, known from Table 2, by a change of signs. Indeed, multiplying the operator РО by ДУ amounts to replacing all signs by their opposites in the matrices “representing” the operator РО (both in Minaev’s and in Gut’s).

[Figure: the Minaev 3×3 matrix realizing ПД(LIE) = SLI on the letter column (Л, И, Э)ᵀ → (Э, И, И)ᵀ; and the Gut 4×4 matrix realizing the same ПД(LIE) = SLI on the sign columns.]

What do the signs in the 2×2 blocks mean? In the Minaev matrix: the ”–” led to the replacement Л → Э for the first function, while the ”+” left the intuition in 2nd place of the EGO block unchanged. In the Gut matrix: the ”–” led to a change of pole of the trait internal/external-1, while the ”+” left the pole of the trait internal/external-2 unchanged. But this is one and the same thing, only in different words! Taking into account, of course, the change of version and the invariance of rationality.

Let us consider the case of non-diagonal matrices with the example of the operator Р– = РО·ЗЕ, which will “act” on IEI. First we find the Minaev and Gut matrices “representing” this operator by multiplying the matrices already known from Table 2:

[Figure: multiplication of the РО and ЗЕ matrices in both formalisms, yielding the Р– matrices: РО · ЗЕ = Р–.]

And now let us see how the formula (Р–)(IEI) = ESE looks in matrix form for the two compared approaches:

[Figure: the Minaev 3×3 matrix acting on (И, Э, И)ᵀ to give (Э, С, Э)ᵀ; the Gut 4×4 matrix acting on the sign column of IEI to give that of ESE.]

In both cases the change of version is indicated by the ”–” standing in the corner cell set off by bold lines. And what do the signs in the 2×2 blocks mean?

The Minaev-matrix case: the letter “Э” moved from the second position to the first, since the ”+” stands in the first row and second column; while the intuition “turned into” sensorics (both IAs of the same rationality) and in that form moved to 2nd place because of the “minus” in the second row and first column.

The Gut-matrix case: in the “resulting” TIM (ESE) the 1st function deals with an IA belonging to the same pole of the trait internal/external as the IA of the 2nd function of the “initial” TIM (IEI) (as witnessed by the ”+” in the 2×2 block); while the 2nd function of ESE and the 1st function of IEI process IAs of different poles of this same trait (as indicated by the ”–” placed in the corresponding spot). Note that what was just discussed was a bipolar trait of the informational aspects, not of the TIMs.

The question may arise: if the 1st function of the “initial” TIM (IEI) dealt with white intuition, why must the 2nd function of the “resulting” TIM process precisely white sensorics, when three other IAs also belong to the pole of the trait internal/external opposite to white intuition? Obtaining the answer to this question is not so difficult if one takes into account that the operator (Р–) changes both version (as witnessed by the ”–” in the corner element set off by bold lines) and rationality (which manifests itself in the non-diagonality of the matrix). One may, of course, also note the invariance of the pole of the trait statics/dynamics.

As we see, in the case of non-diagonal matrices too (under which rationality changes), the interpretations of the Minaev and Gut matrices essentially coincide, but in my variant, it seems to me, explanations are easier to give and shorter. What is this connected with?

As was said, in the Gut matrices the information about the change of pole of the trait statics/dynamics is duplicated. It can be “computed” without invoking the element directly responsible for it. In the Minaev matrices there is no such duplication; they are simpler to use. Moreover, the matrices I proposed are immediately and unambiguously determined once the “initial” and “resulting” TIMs are known. Any matrix with the structure of the 3×3 Minaev matrices corresponds to a definite IR operator per Augustinavichiute. It is easy to see that there can be only 16 matrices of such structure. But there can be twice as many matrices with the structure of the 4×4 Gut matrices. For from every “correct” Gut matrix one can make an “incorrect” one (corresponding to no IR per Augustinavichiute) simply by changing the sign in the cell responsible for statics/dynamics. We have in fact already discussed this question when speaking of the typo in M. M. Gut’s article, which is hard to notice unless one knows that the sign in the just-mentioned cell is unambiguously determined by the arrangement of signs in the remaining cells of the matrix.

One may say that I was lucky with the 3×3 matrices. It turned out that the three-letter coding of TIMs adequately reflects the structure of their graphical models, which Aushra Augustinavichiute used in constructing her theory of IRs. And the 3×3 matrices I proposed were able to replace adequately the diagrams in which line segments joined the symbols of identical IAs in pairs of graphical models of TIMs. In doing so, there were no “superfluous” (“incorrect”) matrices. As for what considerations guided M. M. Gut in selecting the “correct” matrices, he did not share them with his readers in his article. The matrices were simply presented ready-made. In the present article I have given only my hypothesis as to what those considerations might have been.

Nevertheless, the “half-Reininian” basis was proposed by M. M. Gut; matrices consistent with this basis, capable of “representing” the IR operators per Augustinavichiute, were presented. Had this work received further development in its time, socionics might now look somewhat different.

Can matrices of the classical intertype relations be constructed in any of the Reinin bases?

Precisely this question arose for M. M. Gut after he had constructed the 4×4 matrices of the IR operators per Augustinavichiute using the “half-Reininian” basis he had proposed. His attempts to construct matrices of the classical IRs for several “arbitrarily taken” bases of the ART group met with no success, which gave him grounds to suppose that it cannot be done. But it was said that “this still requires a rigorous mathematical proof.”

I, however, had grounds for the opposite answer to the question under discussion. What are these grounds? As already said in the present article, I had had to construct the group of JMTs explicitly and to carry out a comparative analysis with the group of ARTs. By formal criteria these groups are not inferior to each other. In article [9] I gave a large number of diagrams and tables “demonstrating the symmetry and complementarity of the views of the ring structure of the socion from the standpoints of the two groups of bipolar traits.” In articles [5; 10] I showed that the subgroups of the group of IR operators per Augustinavichiute can be found on the basis of the matrix formalism I developed, without invoking any group of bipolar traits of TIMs. To these subgroups of the group of classical IR operators one can put into correspondence certain “good” divisions of the socion into parts in each of which one and the same set of IRs is “realized.”

And if one wants to obtain these same divisions of the socion by means of bipolar traits, both the ARTs and the JMTs will be required in equal measure. Therefore, if M. M. Gut succeeded in finding a JMT basis for constructing his system of 4×4 matrices which can be put into correspondence with the classical IRs, then a corresponding ART basis must also exist. But how to find it?

Unlike M. M. Gut, I did not have to take all possible ART bases at random and try to select suitable matrices. I could use M. M. Gut’s JMT basis and the results of my previous investigations on the comparative analysis of the ART and JMT groups. It sufficed to look at the numbers under which the traits that were in Gut’s basis (extraverts/introverts, statics/dynamics, internal/external-1, internal/external-2) appear in my tables for the JMTs, and then to write out from the tables for the ARTs the traits appearing under the same numbers (see the tables in [9]). Here is the ART basis that resulted: questims/declatims, statics/dynamics, intuitives/sensorics, tacticians/strategists. Now instead of Table 1 we obtain Table 3.

Table 3. Coding of TIMs in the ART basis corresponding to Gut’s JMT basis.

ARTs \ TIMs	ILE	SEI	LII	ESE	LSI	EIE	SLE	IEI	SEE	ILI	ESI	LIE	EII	LSE	IEE	SLI
quest/declat	+	–	+	–	–	+	–	+	+	–	+	–	–	+	–	+
statics/dynam	+	–	+	–	+	–	+	–	+	–	+	–	+	–	+	–
intuit/sensor	+	–	+	–	–	+	–	+	–	+	–	+	+	–	+	–
tact/strateg	+	–	–	+	+	–	–	+	–	+	+	–	–	+	+	–

In this case the “quadra-wise” order of TIMs is used, as more natural for the ARTs. The attentive reader will notice, comparing Tables 1 and 3, that the “square-wise” (Table 1) and “quadra-wise” (Table 3) orders are obtained from one and the same “ring” of the socion (for more detail see [9]). Let us write out explicitly the 4×4 matrices “representing” the generating elements of the group of IR operators per Augustinavichiute for the resulting ART basis (compare them with the corresponding Gut matrices for the JMT basis!).

[Figure: three 4×4 matrices labelled ДУ, ЗЕ, РО in the new ART basis.]

How to obtain the remaining matrices? One can, of course, reconstruct the whole group using the matrices of the generating elements. For example, understanding that Р– = РО · ЗЕ, we obtain

[Figure: the matrix product РО · ЗЕ = Р– written out in the ART basis.]

But one can proceed differently. One need only realize that diagonal must now be the matrices corresponding to those IR operators per Augustinavichiute which do not change the pole of the dichotomy democrats/aristocrats (ТО, ДУ, ЗЕ, АК, СЭ, ПО, КФ, КВ). If we wished to obtain the matrix of the operator (Р–) without multiplying the matrices (РО · ЗЕ), we could act as follows. First one must understand that the matrix will be non-diagonal (the operator (Р–) changes the pole of the trait democrats/aristocrats). One would also need to know that (Р–)(ILE) = LSI (the supervisee of ILE is LSI). And now, from the codes of ILE and LSI from Table 3, the non-diagonal matrix of the known structure is reconstructed quickly and unambiguously:

[Figure: the reconstructed (Р–) matrix acting on the column of ILE (+,+,+,+)ᵀ to give the column of LSI (–,+,–,+)ᵀ; labelled (Р–)(ILE) = LSI.]

Note that this is now written not in Gut’s JMT basis but in the corresponding ART basis. It is, of course, very inconvenient to have to ask every time in which basis the TIM column vectors and IR matrices are written. For by their outward appearance this cannot be determined, if one uses the “trait-based” matrix formalism that Gut proposed for a specific JMT basis, or the one that can be constructed by analogy for a specific ART basis (as demonstrated in this work). In my “model-based” matrix formalism of the classical IRs, this problem is absent.

Conclusion

One of the possible variants of a matrix formalism of the classical IRs (per Augustinavichiute) was proposed by M. M. Gut more than a decade before I developed my own variant. In my case the TIMs were given by the customary three-letter codes, while M. M. Gut’s code consisted of 4 digits and was based on a “half-Reininian” basis. As it turned out, this Gut basis is one of the possible bases of the Jung–Minaev group of bipolar traits, which I constructed in my time as a complement to the well-known Augustinavichiute–Reinin group of traits, guided by didactic considerations of a consistent transition from Jung’s 8-element typology to the 16-element socion.

The comparative analysis of the 4×4 Gut matrices and the 3×3 Minaev matrices showed that the larger size of the Gut matrices is connected with the duplication of the information about the change of pole of the trait statics/dynamics in a pair of TIMs.

Not agreeing with M. M. Gut’s hypothesis about the impossibility of using an ART basis for constructing a “trait-based” matrix formalism, I found such a basis, relying on the JMT basis proposed by M. M. Gut and on the results of the comparative analysis of the ART and JMT groups that I had carried out earlier. Nevertheless, I consider more convenient the “model-based” matrix formalism of IRs proposed by me, in which the TIMs are given by the customary three-letter coding.

References

1. Augustinavichiute A. Socionics. — Moscow: Chornaya Belka, 2008. — 568 pp.
2. Gut M. M. A mathematical representation of intertype relations // Socionics, Mentology and Personality Psychology (SMiPL). — 2000. — No. 1. — pp. 60–69.
3. Minaev Yu. P. A matrix formalism of intertype relations // SMiPL. — 2012. — No. 5. — pp. 56–61.
4. Minaev Yu. P. A comparative analysis of the Augustinavichiute–Liashkiavichius and Reinin intertype relations tables on the basis of the matrix formalism // SMiPL. — 2012. — No. 6. — pp. 35–42.
5. Minaev Yu. P. 15 small groups of intertype relations which generate 23 bipolar traits for the types of information metabolism // SMiPL. — 2014. — No. 1. — pp. 11–18.
6. Minaev Yu. P. The didactic aspect of the question of choosing bases of bipolar traits // SMiPL. — 2014. — No. 2. — pp. 32–40.
7. Minaev Yu. P. The problem of the “splitting” of intertype relations by coincidence spectra of the poles of bipolar traits // SMiPL. — 2014. — No. 3. — pp. 30–37.
8. Minaev Yu. P. A comparative analysis of variants of ordering the set of socionic types // SMiPL. — 2014. — No. 4.
9. Minaev Yu. P. Arguments for the polemic on the two groups of bipolar traits // SMiPL. — 2014. — No. 5.
10. Minaev Yu. P. From intertype relations to two groups of bipolar traits // SMiPL. — 2014. — No. 6.
11. Reinin G. Secrets of Type. Models. Groups. Traits. — Moscow: Chornaya Belka, 2009. — 304 pp.
12. Jung C. G. Psychological Types / trans. from the German. — Moscow: Universitetskaya Kniga / AST, 1998. — 720 pp.

The article was received by the editors on 20.08.2014.