Chapter 4

Chapter 4. Classical Orbital Tetrachotomies and Verbalized Dichotomies of the Socion

History of the Question and Formulation of the Research Problem

The question of the principles of dividing the socion into four quartets of TIMs (types of information metabolism) is not new. After the construction of the group of bipolar traits now called the ART group (Augustinavichiute–Reinin traits), G.R. Reinin considered the possibility of organizing tetrachotomies of the socion by means of pairs of elements of that group. The number of such divisions turned out to equal 35 [12, p. 171]. It must be noted, however, that before this number was computed it was declared that the object of study would be “small groups differing in their spectrum of intertype relations” — while the question of which model of the intertype relations was thereby presupposed was not even discussed, although G.R. Reinin had in effect introduced his own “trait-based” model, rigidly tied to the ART group. That model led to a table of the IR [12, p. 168] which coincided with the then well-known Augustinavichiute–Liashkiavichius table in only half the cases. Without attending to which model of the IR is at issue, it is easy to become confused.

It turned out that the tetrachotomies of the socion obtained by means of pairs of dichotomies verbalized by ARTs differ substantially among themselves when looked at from the standpoint of the classical model of the IR proposed by Aušra Augustinavichiute [1]. In 15 of the 35 cases under discussion the quartets of TIMs were called by G.R. Reinin “homogeneous small groups,” and in the remaining 20 cases “non-homogeneous” [12, p. 172]. The division into “homogeneous” and “non-homogeneous” quartets of TIMs was originally described not very carefully, and in an unsuitable place, which later led to a certain confusion in the socionics literature. G.R. Reinin called homogeneous those groups “in which all the elements are in an identical intertype situation, represented by a set of three symmetric relations” — bearing in mind that the text quoted was written in a place where the “trait-based” model of the IR was in use.

Later, in the article [6], G.R. Reinin explained that the division into “homogeneous” and “non-homogeneous” quartets of TIMs can arise only from the standpoint of the classical model of the IR — and that asymmetric relations, too, can occur in “homogeneous” quartets. Here is a direct quotation from that article: “By homogeneous I mean groups in which all the types are in an identical intertype situation. Here I am, of course, speaking of the customary classical intertype relations and using the Augustinavichiute–Liashkiavichius table. For example, the request ring is a homogeneous group, because every type in it has a benefactor, has a receiver, and has a superego. Thus all the types in it are in an identical intertype situation.”

We draw the readers’ attention to the fact that the 15 cases, obtained by means of pairs of ARTs, of tetrachotomies of the socion into “homogeneous” quartets of TIMs divide into two classes. To the first, comprising 11 of the 15 tetrachotomies under discussion, belong those to which quite definite 4-element subgroups of the 16-element non-commutative group of operators of the classical intertype relations are put into one-to-one correspondence. To the second class, comprising the remaining 4 divisions of the socion, belong those in which the resulting quartets of TIMs are likewise “homogeneous,” but there is no common “formula” for all four quartets. By a “formula” is meant a system of three operators which, after the addition of the identity operator, turns into a 4-element subgroup of the group of operators of the classical IR. It is not hard to give an example of a tetrachotomy of the socion in which all the quartets of TIMs are “homogeneous” but the number of distinct “formulas” is not two but three [5].

Mention must also be made of the article by A. Kasiukov and S. Makarcheva [3]. That work was published not in a specialized journal but on the site “Socionics Through the Eyes of a Mathematician” [11]. In that publication, for the concepts we shall discuss in the present chapter, terms are used that are perhaps more comprehensible from the mathematicians’ standpoint. In particular, the authors speak of “orbital partitions” of the socion into quartets of TIMs. In the terminology more customary for us, we would speak of tetrachotomies of the socion into quartets of TIMs “homogeneous” from the standpoint of the classical IR. Moreover, in place of the long phrase “a single-formula tetrachotomy of the socion into quartets of TIMs homogeneous from the standpoint of the classical IR,” the word “factorization,” or the phrase “factorization tetrachotomy,” is used.

The authors of that article gave a diagram representing the classes of the partitions of the socion they considered. The number 15 figures in that diagram as the number of “factorizations by the action of the subgroups of order 4 in the group of relations.” And to obtain the 15 “orbital partitions” of the socion by pairs of ARTs, readers must themselves add two numbers we have already spoken of: the 11 factorizations that can be organized by means of pairs of ARTs, and the 4 tetrachotomies that are orbital partitions but not factorizations (being “two-formula” rather than “single-formula”). Separately indicated on the diagram are those 4 of the 15 factorizations that do not fall into the set of the 35 tetrachotomies obtained by pairs of ARTs.

We should like to draw the readers’ attention to the fact, fixed in the diagram under discussion, that not all factorizations can be obtained by pairs of ARTs. Unfortunately, even the authors of another interesting article [2] did not attend to this — it was through the reference list of that article that we came upon the publication of A. Kasiukov and S. Makarcheva. Although A.M. Banaru and D.A. Enina found all the elements of the set consisting of the 15 factorizations of the socion, they decided that this set is a subset of those 35 partitions of the socion which G.R. Reinin had indicated in his time.

At the end of the article by A. Kasiukov and S. Makarcheva the question is posed of the number of all possible orbital partitions of the socion into quartets of TIMs (i.e. into quartets “homogeneous” from the standpoint of the classical IR) — but the authors left it without an answer.

The 217 Orbital Tetrachotomies of the Socion

In the article [7] it was reported that, by means of an appropriate computer program, this number was found by us, and a catalogue was compiled of the partitions of the socion into those quartets of TIMs which G.R. Reinin called “homogeneous.” It turned out that of the 217-element set of orbital tetrachotomies of the socion, 15 are single-formula, 114 two-formula, 32 three-formula and 56 four-formula. The question arises: can these tetrachotomies be organized using dichotomies of the socion already verbalized in the socionics literature by the names of bipolar traits? The answer depends on precisely how these tetrachotomies are to be organized by means of the already verbalized dichotomies.

If one is guided by Reinin’s method (using pairs of orthogonal verbalized dichotomies), then only 23 of the existing 217 orbital tetrachotomies of the socion can be obtained [10] — this taking into account that at present one may count as verbalized not only those dichotomies fixed by the 15 Augustinavichiute–Reinin traits, but also the additional 8 that received names in the Jung–Minaev group [4]. Of these 23 orbital tetrachotomies of the socion, 15 are “single-formula” (factorizations) and the remaining 8 “two-formula.” The corresponding diagram is given in Fig. 4.1.

[Fig. 4.1. Organizing the orbital tetrachotomies of the socion by means of pairs of orthogonal dichotomies verbalized by ARTs and JMTs.]

One would like to widen the possibilities for obtaining orbital tetrachotomies of the socion without increasing the number of elements of the set of verbalized bipolar dichotomies used for the purpose. But for this one will have to renounce the principle of dividing the socion into four parts by pairs of orthogonal dichotomies specifically.

The 15 Factorization Tetrachotomies of the Socion

The factorization tetrachotomies of the socion were in effect spoken of in the article [4], though in different terminology. There the readers’ attention was drawn to the fact that the natural path to organizing such tetrachotomies lies in using the 4-element subgroups of the 16-element group of operators of the classical IR — and only after obtaining all 15 sought tetrachotomies (by the number of subgroups of order 4) should the question of verbalizing the corresponding dichotomies of the socion be posed. It was then presupposed that two orthogonal dichotomies of the socion generate a tetrachotomy into quartets of TIMs. On this path it became clear that the traits of the then-known ART group do not suffice. This fact was the impetus for building one more group of bipolar traits (the JMT group).

At present, for the traits of these two 16-element groups sharing a common 8-element subgroup, formalized notations convenient for mathematical computations have been proposed. Table 4.1, taken from the article [9], gives all 15 factorization tetrachotomies of the socion.

Table 4.1. The correspondence between triples of mutually dependent traits of TIMs and triples of IR operators.

№	Triple of bipolar traits	Triple of operators of the classical IR
1	I1, I2, I3 (irr/rat, dem/ar, right/left)	-1, c, -c
2	I1, A0, A1 (irr/rat, stat/dyn, extr/intr)	-1, 1*, -1*
3	I1, A2, A3 (irr/rat, quest/decl, posit/negat)	-1, c*, -c*
4	I2, A0, A2 (dem/ar, stat/dyn, quest/decl)	-1, m, -m
5	I2, A1, A3 (dem/ar, extr/intr, posit/negat)	-1, cm, -cm
6	I3, A0, A3 (right/left, stat/dyn, posit/negat)	-1, m*, -m*
7	I3, A1, A2 (right/left, extr/intr, quest/decl)	-1, cm*, -cm*
8	I1, B0, B1 (irr/rat, abstr/invol–1, intern/extern–1)	c, 1*, c*
9	I1, B2, B3 (irr/rat, abstr/invol–2, extern/intern–2)	c, -1*, -c*
10	I1, AB0, AB1 (irr/rat, α/γ–1, δ/β–1)	-c, 1*, -c*
11	I1, AB2, AB3 (irr/rat, α/γ–2, β/δ–2)	-c, -1*, c*
12	I2, C0, C2 (dem/ar, judic/decis, merry/serious)	-c, m, -cm
13	I2, C1, C3 (dem/ar, caref/farsight, yield/obstin)	-c, -m, cm
14	I2, AC0, AC2 (dem/ar, intuit/sens, log/eth)	c, m, cm
15	I2, AC1, AC3 (dem/ar, tact/strat, constr/emot)	c, -m, -cm

Each of the partitions of the socion listed in Table 4.1 can be organized both by means of the “formula” given in the right column and by means of any pair of the corresponding triple of mutually dependent bipolar traits from the left column. The table also gives the notations deriving directly from the corresponding socionic terms. We shall need this table again. But let us now turn to the following question: was there, after all, a possibility of organizing all 15 factorizations by means of bipolar traits while confining oneself to the ART group alone? For the 8 “new” traits not entering the ART group occur in Table 4.1 in only 4 rows of the 15 (numbers 8–11).

For the answer to become positive, one must renounce the division of the socion into 4 quartets of TIMs — traditional for socionics works — by exactly two orthogonal bipolar traits. Why not divide the socion in half by one trait, and in dividing each of the halves use two different bipolar traits? Let us demonstrate the application of this idea on a concrete example.

Suppose we must organize the factorization by the subgroup {1, c, 1*, c*} (see row 8 of Table 4.1). This can of course be done with the two traits I1 (irr/rat), which enters both the ART and the JMT group, and B0 (abstr/invol–1), which enters only the JMT group. But one can make do with the ART group alone, if one allows oneself to use three traits: keeping the trait I1 (irr/rat) for the first division of the socion, and then dividing the irrationals by the trait AC0 (intuit/sens) and the rationals by the trait AC2 (log/eth). The corresponding partition diagrams are presented in Fig. 4.2.

[Fig. 4.2. The factorization of the socion by the subgroup {1, c, 1*, c*}, carried out by means of three ARTs or two JMTs.]

One could proceed in an analogous manner in the remaining three cases, to which rows 9–11 of Table 4.1 correspond. A natural question arises: is there the possibility of organizing all 15 factorizations by means of bipolar traits while confining oneself to the JMT group alone? For the 8 traits belonging exclusively to the ART group likewise occur in only four rows of Table 4.1 (numbers 12–15).

The answer to this question is positive. From the standpoint of the classical model of the IR, these two 16-element groups of bipolar traits are mathematically of entirely equal standing. If for organizing all the factorization tetrachotomies of the socion one can make do exclusively with the ARTs, there must exist an analogous method allowing one not to step outside the JMT group. Of course, by means of pairs of JMTs one cannot obtain the tetrachotomies of the socion corresponding to rows 12–15 of Table 4.1 — but after lifting the restriction that only 2 bipolar traits must be used for dividing the socion into 4 quartets of TIMs, the possibility of accomplishing this task opens up. Let us demonstrate this on the example of the division of the socion into the clubs (Fig. 4.3).

[Fig. 4.3. The division of the socion into the clubs, carried out by means of three JMTs.]

One more celebrated division of the socion into 4 quartets of types of information metabolism, which cannot be obtained by means of two JMTs but can by means of three, is presented in Fig. 4.4. As we see, it is the division into quadras. If the partition of the socion into clubs is the factorization by the subgroup {1, c, m, cm}, then the partition into quadras is the factorization by the subgroup {1, -c, m, -cm}.

[Fig. 4.4. The division of the socion into the quadras, carried out by means of three JMTs.]

Of course, it is more customary to divide the socion into quadras and clubs by means of pairs of ARTs, but one may perfectly well confine oneself to bipolar traits from the JMT group. In that case, at the first step of the division one must use the trait I2 (dem/ar), and at the second step, for each half, use the two corresponding JMTs whose product equals I1 (irr/rat). The same rule works for the remaining two cases, fixed in rows 13 and 15 of Table 4.1.

Note that the order of the traits used at the second step of the division is essential. Let us change, in the example just considered, the order of the traits AB0 (α/γ–1) and AB1 (δ/β–1). As a result, instead of the division into quadras we obtain the factorization by the subgroup {1, -c, -m, cm}. The changes in the “formula” are especially well seen when the formalized notations for the IR operators are used: indeed, instead of {-c, m, -cm} we have obtained {-c, -m, cm}. In both cases the quartets of TIMs consist of dual pairs, but the two other operators “working” in the quartets have changed by a plainly visible rule — their sign has flipped. And a change of sign, with the formalized notations of the operators, means multiplication by the superego operator (-1). The corresponding illustrative diagram is given in Fig. 4.5; it is useful to compare it with the one in Fig. 4.4.

[Fig. 4.5. The factorization of the socion by the subgroup {1, -c, -m, cm}, carried out by means of the same three JMTs as the factorization by the subgroup {1, -c, m, -cm} (see Fig. 4.4).]

Let us note that if at the second step of the division corresponding to the “formula” {-c, m, -cm} ARTs had been used, the trait C0 (judic/decis) would have been required, while for the division corresponding to the “formula” {-c, -m, cm}, the trait C1 (caref/farsight). It should be clear that in the case of those 4 “two-formula” orbital tetrachotomies which cannot be organized by means of pairs of ARTs but can by means of pairs of JMTs, the situation is entirely analogous: there is the possibility of obtaining these tetrachotomies not with two but with three ARTs. At the first step of partitioning the socion one will have to use the bipolar trait I1 (irr/rat), and at the second step two ARTs will be needed whose product gives I2 (dem/ar).

The 114 “Two-Formula” Orbital Tetrachotomies of the Socion

After we had compiled the catalogue of the orbital tetrachotomies of the socion, it became clear that there exist 114 “two-formula” ones in all. But the question arose: can these tetrachotomies be obtained by means of bipolar traits, if the restriction is lifted that only a pair of bipolar traits may be used for organizing a tetrachotomy?

For the answer, Table 4.1 will again be useful. Begin with the analysis of a particular case. Let us divide the socion into democrats and aristocrats, and then divide each half of the socion in half once more, in such a way that two different “formulas” result for the democrats and the aristocrats. Table 4.1 gives all 15 possible “formulas” for “homogeneous” quartets of TIMs, but in only 7 cases does the “formula” correspond to a triple of bipolar traits including the trait I2 (dem/ar).

Suppose, for example, that for dividing the set of democrats the “formula” numbered 12 in Table 4.1 ({-c, m, -cm}) is chosen. Note that exactly the same division of the set of democrats would be obtained using the trait C0 (judic/decis) or the trait C2 (merry/serious) from the ART group — it is precisely these traits that, together with I2 (dem/ar), form the triple of mutually dependent bipolar traits put into correspondence with the “formula” under discussion in Table 4.1. To obtain a different “formula” for the aristocrats, only 6 variants remain for dividing their set into “homogeneous” quartets of TIMs, corresponding in Table 4.1 to the rows numbered 1, 4, 5, 13, 14, 15. Thus, after choosing one of the required divisions of the set of democrats, the set of aristocrats can be divided in 6 ways so that the “formulas” do not coincide. As an example let us take the “formula” numbered 14 ({c, m, cm}). The corresponding division diagram is given in Fig. 4.6.

[Fig. 4.6. An example of organizing a “two-formula” orbital tetrachotomy of the socion by means of three bipolar traits.]

The total number of “two-formula” tetrachotomies of the socion into “homogeneous” quartets of TIMs, when the division into democrats and aristocrats is performed first, turns out to equal 42 (the product of the numbers 7 and 6). The same number is obtained in the case where the socion is first divided by the trait I1 (irr/rat), since it too occurs in Table 4.1 exactly 7 times.

In the same Table 4.1, the following 5 traits occur exactly 3 times each: I3 (right/left), A0 (stat/dyn), A1 (extr/intr), A2 (quest/decl) and A3 (posit/negat). It should be clear that the total number of “two-formula” tetrachotomies into “homogeneous” quartets of TIMs, when the first division is performed by any of the 5 traits listed, turns out to equal 6 (the product of the numbers 3 and 2) — analogously to how we multiplied 7 by 6 in dealing with the traits I1 and I2, which occur in Table 4.1 seven times each. Thus, to compute the total number of “two-formula” orbital tetrachotomies of the socion it remains to perform the final arithmetic: 2 × 42 + 5 × 6 = 114.

Note the fact that, to obtain the sought “two-formula” tetrachotomies of the socion into “homogeneous” quartets of TIMs, we in each case used three different bipolar traits: with the first we divided the socion in half, and then for dividing each of the halves we used two further, different traits. On the other hand, we said that in 8 cases just two bipolar traits suffice for organizing the required tetrachotomies (see Fig. 4.1). There is no contradiction here. Consider, for example, the triple of bipolar traits {I2 (dem/ar), AC0 (intuit/sens), AC1 (tact/strat)}. In this case, for the quartets of democrats after their division by the trait AC0 (intuit/sens) the “formula” {c, m, cm} will work, and for the quartets of aristocrats after their division by the trait AC1 (tact/strat) the “formula” {c, -m, -cm}. Exactly the same division of the socion is obtained if at the second step of the division we use, in both cases, the trait B0 (abstr/invol–1) from the JMT group. Indeed, the pair of JMTs {I2 (dem/ar), B0 (abstr/invol–1)} generates a “two-formula” orbital tetrachotomy.

The 32 “Three-Formula” Orbital Tetrachotomies of the Socion

As we have said, the orbital tetrachotomies of the socion, when organized by means of pairs of orthogonal dichotomies, sometimes turn out “single-formula” (factorizations) and sometimes “two-formula.” This remarkable fact can be exploited to obtain the “three-formula” orbital tetrachotomies of the socion. The diagram proposed by Andrew Joynton and presented in Fig. 4.7 will help us in this.

[Fig. 4.7. The bipolar traits that will be used for organizing the “three-formula” orbital tetrachotomies of the socion.]

In this diagram the formalized notations of the bipolar traits are united into triples. If any pair of a triple makes it possible to organize a “single-formula” tetrachotomy of the socion, solid lines are used for uniting them into triples in the diagram. If, on the other hand, any pair of a triple defines a “two-formula” tetrachotomy, the symbols of the corresponding traits are united into triples by dashed lines. Let us show on a concrete example how this can be used.

Perform the first division of the socion by means of the trait B0 (abstr/invol–1), then divide one half by means of I1 (irr/rat) and the other by means of I2 (dem/ar). From the diagram presented in Fig. 4.7 one can see that in the half where the division is by the trait I1 one “formula” “works,” while in the half divided by the trait I2 two “formulas” will “work.” In this way one can obtain the two tetrachotomies of the socion presented in Fig. 4.8.

The bipolar trait taken for the first division of the socion enters the JMT group but not the ART group. That is why the “single-formula” half turned out to be the one in which the trait I1 (irr/rat) was used at the second step of the division: in the diagram of Fig. 4.7 the symbols B0 and I1 are joined by a solid line. In the half where the trait I2 (dem/ar) was used at the second step, two “formulas” will “work”: in the diagram just mentioned the symbols B0 and I2 are joined by a dashed line.

[Fig. 4.8. An example of organizing a “three-formula” orbital tetrachotomy of the socion by means of three bipolar traits.]

The bipolar traits entering only one of the two groups number 16 in all; in the diagram of Fig. 4.7 only two traits do not fall into their number (I1 and I2). And in each of the 16 cases we shall obtain two “three-formula” orbital tetrachotomies, by analogy with the two presented in Fig. 4.8. Thus the total number of tetrachotomies of this sort turns out to be 32.

As we see, the “three-formula” orbital tetrachotomies were obtained thanks to the fact that, after the first halving of the socion, one half was then divided so that in each of the resulting quarters of the socion one and the same “formula” turned out to “work,” while the other half was divided into quarters with different “formulas.” A natural question arises: can one organize two “formulas” in each of the two halves of the socion resulting from the first division?

The answer to this question turns out to be negative, if one stays within the already verbalized central sections of the socion. And this can be understood by turning once more to the diagram presented in Fig. 4.7. Each of the 16 bipolar traits that are elements of only one of the two groups of central sections of the socion verbalized at present enters just two triples suitable for organizing orbital tetrachotomies — and one of these triples always corresponds to a “single-formula” tetrachotomy (a factorization), the other to a “two-formula” one.

The 56 “Four-Formula” Orbital Tetrachotomies of the Socion

In the article [8] it was shown how to organize all 56 “four-formula” orbital tetrachotomies of the socion without resorting to a computer enumeration of variants. Use was made, admittedly, of a fact discovered on careful study of the resulting catalogue: in the tetrachotomies that now interest us, the superego operator (-1) enters the “formula” of every quartet of TIMs. In place of the 16-element socion one could consider the 8-element set of pairs of TIMs united into those pairs by the operator just named.

Thus, in place of the 16-element non-commutative group we obtained — by introducing the concept of inter-pair relations — an 8-element commutative one. This proved a substantial simplification of the task of finding the “four-formula” orbital tetrachotomies of the socion. The algorithm found was visual and simple in meaning, but required the building of the rather cumbersome table presented in Fig. 4.9.

[Fig. 4.9. A visual method for obtaining the “four-formula” orbital tetrachotomies of the socion.]

In the top row of Fig. 4.9 are shown all 7 factorizations of the socion by means of the superego-containing 4-element subgroups of the group of operators of the classical IR. To each of the seven “formulas” corresponds its own easily recognizable set of segments joining the cells with the superego pairs that form the quartets of TIMs of interest. The rest of the figure is occupied by 56 tables of size 4×2, demonstrating all the possible variants of uniting the superego pairs into quartets subject to the condition that the “formulas” of all four quartets of TIMs differ.

We shall now try to explain the idea of this algorithm without resorting to the building of such a table. Let us introduce a fairly obvious and at the same time useful one-to-one correspondence between the operators of the inter-pair relations and the pairs of TIMs united into those pairs by the superego operator (-1):

{Ne⁺,Se⁺} ↔ ±1; {Ni⁺,Si⁺} ↔ ±c; {Ne⁻,Se⁻} ↔ ±1*; {Ni⁻,Si⁻} ↔ ±c*; {Ti⁻,Fi⁻} ↔ ±m; {Te⁻,Fe⁻} ↔ ±cm; {Ti⁺,Fi⁺} ↔ ±m*; {Te⁺,Fe⁺} ↔ ±cm*.

Then replacements of the following sort become possible:

{{Ni⁻,Si⁻}, {Ti⁺,Fi⁺}} ↔ {±c*, ±m*}; {{Ne⁻,Se⁻}, {Ti⁻,Fi⁻}} ↔ {±1*, ±m}; etc.

Such replacements are very convenient: they allow one to work out quickly which “formula” corresponds to a quartet of TIMs uniting two superego pairs. For example, in the two cases just given we obtain {-1, cm, -cm} and {-1, m*, -m*} respectively — a consequence of the facts that (±c*) · (±m*) = (±cm) and (±1*) · (±m) = (±m*).

And so, we shall partition the socion into the quartets of TIMs of interest, taking care that the “formulas” do not repeat. The first quartet can be chosen to have the form {±1, ±x}, where ±x may be any of the following 7 elements: ±1*, ±c, ±c*, ±m, ±cm, ±m*, ±cm*. The “formula” for the first quartet will naturally be {-1, x, -x}. The second quartet of TIMs can be written {±y, ±z}, where ±y is the leftmost element of the 7-element list just given that differs from ±x. It is important to note that, after ±x has been chosen from the 7 variants, the element ±y is determined uniquely by the rule indicated. For ±z, however, there is a choice of 4 variants: indeed, ±z must differ from the 3 elements ±x, ±y and ±xy. The first two have already occurred, and ±xy is unsuitable because (±y) · (±xy) = (±x) — which means that in the quartet {±y, ±xy} the “formula” {-1, x, -x} would arise, and that cannot be allowed, since it occurred in the first quartet. In the second quartet the “formula” will be {-1, yz, -yz}, with yz not coinciding with x.

For the last two quartets only two variants remain: either {±xy, ±yz} and {±xz, ±xyz}, with “formulas” {-1, xz, -xz} and {-1, y, -y} respectively; or {±xy, ±xyz} and {±xz, ±yz}, with “formulas” {-1, z, -z} and {-1, xy, -xy} respectively. So we have two possible general forms of the “four-formula” orbital partitions of the socion into quartets:

1. {±1, ±x}, {±y, ±z}, {±xy, ±yz}, {±xz, ±xyz};
2. {±1, ±x}, {±y, ±z}, {±xy, ±xyz}, {±xz, ±yz}.

Each of them has 28 variants of concretization, since there were 7 variants for choosing ±x and 4 for choosing ±z (recall that the element ±y, once ±x was chosen, was determined uniquely). It is easy to see that in both general forms there is no repetition of TIMs; repetitions of “formulas,” too, we did not allow in the course of our reasoning. Here are these “formulas,” written out in the corresponding order for both kinds of tetrachotomy under discussion:

1. {-1, x, -x}, {-1, yz, -yz}, {-1, xz, -xz}, {-1, y, -y};
2. {-1, x, -x}, {-1, yz, -yz}, {-1, z, -z}, {-1, xy, -xy}.

Thus we have an algorithm for obtaining all 56 “four-formula” orbital tetrachotomies of the socion without recourse to the cumbersome table. Nevertheless, such tetrachotomies still cannot be obtained by means of the already known bipolar traits, as we managed to do in the cases of the partitions with a smaller number of distinct “formulas” in the quartets of TIMs.

Conclusion to Chapter 4

In the present chapter¹ we have analyzed the possibility of organizing the orbital tetrachotomies of the socion by means of the bipolar traits already known from the socionics literature, which verbalize 23 central sections of the socion. Those orbital tetrachotomies in which one and the same “formula,” consisting of three operators of the classical IR, “works” in every quartet of TIMs were called factorizations by the 4-element subgroups of the 16-element non-commutative group of operators of the classical IR. It was shown that if, in organizing the factorizations, one admits the possibility of using three bipolar traits (one at the first step of the division of the socion and two others at the second), then one can confine oneself to just one of the two known groups of bipolar traits (the Augustinavichiute–Reinin or the Jung–Minaev group).

The renunciation of the principle by which a tetrachotomy of the socion must be formed by exactly two orthogonal dichotomies made it possible to widen considerably the circle of those orbital tetrachotomies of the socion obtainable by means of the already verbalized central dichotomies of the socion. In this way one can obtain not 23 but 161 orbital tetrachotomies of the socion. The 56 “four-formula” orbital tetrachotomies of the socion cannot be obtained by means of the already known bipolar traits.

References for Chapter 4

1. Augustinavichiute A. Sotsionika [Socionics]. — Moscow: Chernaya belka, 2008. — 568 pp.
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Index of Illustrative Materials

• Fig. 1.1. Jung’s Compass — 10
• Fig. 1.2. The “folded” variant of the Socionic Compass — 13
• Fig. 1.3. The triple of operators {1*, c, m} of symmetric IR generates a 16-element non-commutative group — 15
• Fig. 1.4. The operators of the classical intertype relations on the “unfolded” variant of the Socionic Compass — 21
• Fig. 1.5. Cayley graphs for the three kinds of 8-element subgroups of the group of operators of the classical IR — 32
• Fig. 1.6. Cayley graphs for the two kinds of 4-element subgroups of the group of operators of the classical IR — 33
• Fig. 1.7. Using the Socionic Compass to introduce a generating system for the JMT group — 40
• Fig. 1.8. Using the Socionic Compass to introduce a generating system for the ART group — 46
• Fig. 1.9. Using the Socionic Compass to build the Model “A” of the SEI as a representative of the right democrats (for each TIM the numbering of the functions begins from the corresponding cell) — 55
• Fig. 1.10. Using the Socionic Compass to build the Model “A” of the ESE as a representative of the left democrats — 56
• Fig. 1.11. Using the Socionic Compass to build the Model “A” of the LSI as a representative of the right aristocrats — 57
• Fig. 1.12. Using the Socionic Compass to build the Model “A” of the SLE as a representative of the left aristocrats — 58
• Fig. 2.1. Two representations of the generating set of the group of operators of the classical intertype relations — 64
• Fig. 2.2. Introducing a binary operation (multiplication) on the set of IR operators — 66
• Fig. 2.3. Introducing the operators of superego (-1) and of passage to the supervisor (-m*) — 67
• Fig. 2.4. Introducing the operators of extinguishment (c) and of passage to the benefactor (cm*) — 69
• Fig. 2.5. The group of operators of the classical IR — 69
• Fig. 2.6. The symmetry of the tetragonal prism as an image of the group of the classical IR (from the article by A.M. Banaru [3]) — 73
• Fig. 2.7. The correspondence between the elements of the extended generating sets of the group of operators of the classical IR and of the group of self-superposition transformations of the tetragonal prism — 75
• Fig. 2.8. The isomorphism between the symmetry group of the square and that 8-element subgroup of the group of operators of the classical IR which is realized in each of the halves of the socion divided by the trait statics / dynamics — 77
• Fig. 2.9. The passage to the symbolic notations of the TIMs as their “folded” Models “A” — 79
• Fig. 2.10. Expressing the operators of superego (-1) and of passage to the supervisee (m*) through the operators of the kindred (1*) and mirror (m) relations — 81
• Fig. 2.11. Uniting the TIMs into the socion by means of the generating system {c, 1*, m} of the group of operators of the classical intertype relations — 82
• Fig. 2.12. Four obvious dichotomies of the 16-element set of formalized notations of the operators of the classical IR — 84
• Fig. 2.13. A visual demonstration of the difference in the results of the action of the operators of the classical IR on the abbreviated symbolic notations of democratic and aristocratic TIMs — 89
• Fig. 3.1. A visual representation of the composition of the four existing 16-element subgroups (without the common 4-element subgroup “I”) of the 32-element group of sections of the socion with generating system {I1, I2, A0, B0, C0} — 116
• Fig. 3.2. The generating system of the common order-8 subgroup of the ART and JMT groups as a basis of the typology of superego pairs (-1), and the Cayley graph of the group of inter-pair relations for such pairs — 120
• Fig. 3.3. Four tetrachotomies of the socion into homogeneous quartets of TIMs, organized by means of the “formulas”: {c, 1*, c*}, {c, -1*, -c*}, {c, m, cm}, {c, -m, -cm} — 124
• Fig. 3.4. Four tetrachotomies of the socion into homogeneous quartets of TIMs, organized by means of the “formulas”: {-c, 1*, -c*}, {-c, -1*, c*}, {-c, m, -cm}, {-c, -m, cm} — 127
• Fig. 4.1. Organizing the orbital tetrachotomies of the socion by means of pairs of orthogonal dichotomies verbalized by ARTs and JMTs — 139
• Fig. 4.2. The factorization of the socion by the subgroup {1, c, 1*, c*}, carried out by means of three ARTs or two JMTs — 142
• Fig. 4.3. The division of the socion into the clubs, carried out by means of three JMTs — 143
• Fig. 4.4. The division of the socion into the quadras, carried out by means of three JMTs — 143
• Fig. 4.5. The factorization of the socion by the subgroup {1, -c, -m, cm}, carried out by means of the same three JMTs as the factorization by the subgroup {1, -c, m, -cm} (see Fig. 4.4) — 144
• Fig. 4.6. An example of organizing a “two-formula” orbital tetrachotomy of the socion by means of three bipolar traits — 146
• Fig. 4.7. The bipolar traits that will be used for organizing the “three-formula” orbital tetrachotomies of the socion — 148
• Fig. 4.8. An example of organizing a “three-formula” orbital tetrachotomy of the socion by means of three bipolar traits — 149
• Fig. 4.9. A visual method for obtaining the “four-formula” orbital tetrachotomies of the socion — 151
• Table 1.1. The group of bipolar traits of the information aspects used here — 36
• Table 1.2. The distribution of the dual pairs over the quadras, taking into account membership in the poles of the bipolar trait right / left — 51
• Table 3.1. Formalization of the notations for the JMTs — 110
• Table 3.2. Formalization of the notations for the ARTs — 111
• Table 4.1. The correspondence between triples of mutually dependent traits of TIMs and triples of IR operators — 141

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¹ Translator’s note: the original reads “In the present article” — a trace of the chapter’s origin in the article series [7]–[10].

Translator’s note on the superego-pair correspondence table: reconstructed from the reference-pair convention of Fig. 3.2 ({Ne⁺, Se⁺} ↔ ±1) by applying each operator; the two replacement examples then follow uniquely from their stated products (±cm and ±m*). Section headings for the three- and four-formula parts do not appear in the printed table of contents but are present in the body text and are kept here.