Chapter 2. Operators of Intertype Relations as Operators of Transformations of the Models “A” of Types of Information Metabolism

The Group of Operators of the Classical IR as a Group of Operators of Transformations of the 2D Models “A” of TIMs

The operators of the classical IR form a 16-element non-commutative group whose minimal set of generators consists of just three elements. In Fig. 2.1 one can see two representations of the triple of operators that transform the 2D Model “A” of the ILE (or, correspondingly, the abbreviated symbolic notation of that TIM) into the Models “A” (or abbreviated symbolic notations) of the following TIMs: ILI, IEE, LII.

The representation of the operators used in transforming the 2D Models “A” practically coincides with the one proposed by O.B. Karpenko [7]. The operators are in this case represented by 4×2 tables with arrows. The second variant of representation (in the form of the formalized notations) is oriented to transforming the symbolic notations of the TIMs and was proposed by us in the article [9].

[Fig. 2.1. Two representations of the generating set of the group of operators of the classical intertype relations.]

The connection of the first variant of representation with the classical theory of the IR is entirely obvious, since the key message of the theory is plainly visible in it: “Information is transmitted to each other only by identical elements” [1, p. 128]. And this variant can be very useful not only in the detailed description of the intertype relations but also in one’s first acquaintance with the mathematical properties of the group of operators of the classical IR.

In the second variant of representation the connection with the classical theory of the IR is not so obvious, but the transformations of the symbolic notations of the TIMs under the action of the operators are seen quite well. The second variant proves preferable in solving various problems concerning the group in question.

In Chapter 1 it was shown that the group of operators of transformations of the abbreviated symbolic notations of the TIMs can be built on the basis of just three operators, which are involutions (their own inverses). This proposition was illustrated by the diagram presented in Fig. 1.3. It is in effect a graph at whose vertices the abbreviated symbolic notations of the TIMs are placed. It was plainly visible that from any vertex of the graph one can reach any other along undirected edges of just three kinds, corresponding to the following intertype relations: kindred (1*), extinguishment (c) and mirror (m).

The extinguishment operator (c) changes only the colour of a TIM’s abbreviated symbolic notation, and the operator of the kindred IR (1*) leads only to a change of sign. The operator of the mirror IR (m), acting on the abbreviated symbolic notations of the socionic types, leads to a change of colour, of sign and of form (with a change of pole of the dichotomy irrationality / rationality). But what was said of the operator m in the last sentence still leaves two possibilities. If, for example, we take Ne⁺ as the initial TIM, it remains to determine which of the two candidate results (Ti⁻ or Fi⁻) is to be chosen. For the correct choice one must, in effect, be able to determine the IA of the second function from the information aspect of the first function and the sign.

In the present case it is necessary to realize that Ne⁺ is the abbreviated record of the unfolded notation (Ne, Ti) — that is, we are speaking of the intuitive-logical extravert (ILE). On the unfolded notation of a TIM the operator m acts very simply: it interchanges the order of the symbols, i.e. from (Ne, Ti) one obtains (Ti, Ne) — that is, Ti⁻.

Let us first show how a set consisting of just two elements can generate an 8-element subgroup of the group of operators of the classical IR. In doing so we shall use, in parallel, the two representations spoken of. Turn to Fig. 2.2. In the upper row, the passage from Ne⁺ to Ti⁺ is carried out in two “steps”: first from Ne⁺ to Ti⁻ by means of the operator m, and then from Ti⁻ to Ti⁺ by means of the operator 1*.

For the direct passage from Ne⁺ to Ti⁺ the operator m* is introduced as the product of the two already known: m* = 1* · m (this is shown in the lower row of Fig. 2.2). Here attention must be paid to the order of the factors: first the operator m acts, and then the operator 1*, since m*(Ne⁺) = 1*(m(Ne⁺)) = 1*(Ti⁻) = Ti⁺. The operator m* is called the operator of passage to one’s supervisee (the controlled one). Note that the order of the operators in the formula m* = 1* · m coincides with the order of the corresponding words in the phrase: the supervisee is the kindred of one’s mirror partner.

[Fig. 2.2. Introducing a binary operation (multiplication) on the set of IR operators.]

The resulting operator of passage to the supervisee (m*) differs qualitatively from the operators of the mirror (m) and kindred (1*) IR. This difference shows up clearly when the operators are represented as diagrams (after Karpenko). In the diagrams representing the operators m and 1*, two-way arrows are used — evidence that these operators are their own inverses (involutions): (m)⁻¹ = m; (1*)⁻¹ = 1*. In the diagram representing the operator of passage to the supervisee (m*), by contrast, the arrows are one-way and form two closed rings. The question arises: “What operator results from multiplying the operator m* by itself?”

The resulting diagram (Fig. 2.3) corresponds to the superego operator (-1). As the two-way arrows show, this operator is an involution and accordingly satisfies the relation (-1)² = 1. The same figure shows that the superego operator changes the form of a TIM’s abbreviated symbolic notation (within a single -nality, without change of colour or sign).

The lower row of Fig. 2.3 shows how the operator of passage to the supervisor (-m*) can be obtained. Two substantially different cases are considered. One can multiply the operators of passage to the supervisee (m*) and of superego (-1) — in this case the order of the factors does not matter! One says that these operators commute with each other (the product does not depend on the order of the factors). In the second case considered, however, the order of the factors is essential. The supervisor is the mirror partner of one’s kindred (-m* = m · 1*). Recall that the kindred of one’s mirror partner is one’s supervisee (m* = 1* · m). Thus the operators 1* and m do not commute with each other. Anyone can verify without difficulty that m* · (-m*) = 1, which may be written as (m*)⁻¹ = -m*. This means that these two operators are mutually inverse. In the diagram corresponding to the identity operator (1) there will be no arrows at all.

[Fig. 2.3. Introducing the operators of superego (-1) and of passage to the supervisor (-m*).]

Two more operators of the classical IR are easily obtained: the conflict operator (-m = -1 · m) and the business operator (-1* = -1 · 1*). The order of the factors in the products used does not affect the result. As it turned out, the superego operator (-1) commutes with absolutely all the IR operators. Thus the set of two mutually non-commuting elements {1*, m} has generated, by means of the multiplication operation natural for operators, the 8-element group {1, 1*, m, m*, -1, -1*, -m, -m*}. By multiplying the elements of this group by one another in any order it is impossible to step outside it — exactly as is required of a group in the mathematical sense of the word!

Let us note at once that TIMs linked by operators of the 8-element group just built belong to one and the same pole of the bipolar trait statics / dynamics. This follows automatically from the fact that neither operator of the generating set {1*, m} leads to a change of pole of that trait. Indeed, a socionic type’s membership in a particular pole of this trait is fully determined by the IA of the first function. The operator 1* does not change the first function at all, while the operator m puts in first place the IA that stood in second. And the structure of the Model “A” is such that the first four functions (the whole mental ring) contain either only static elements or only dynamic ones. The representation of these operators as 4×2 tables with arrows (after Karpenko) shows visually that the arrows used in those diagrams do not cross the line separating the mental ring from the vital one.

If to the pair of mutually non-commuting operators {1*, m} we add the extinguishment operator (c), we obtain a generating set of the whole 16-element group of operators of the classical IR. Fig. 2.4 shows an example of the action of this operator on the abbreviated symbolic notation and on the Model “A” of a specific TIM, as well as the derivation of the operator of passage to the benefactor (cm*) by multiplying the operators of extinguishment (c) and of passage to the supervisee (m*).

[Fig. 2.4. Introducing the operators of extinguishment (c) and of passage to the benefactor (cm*).]

Note that the extinguishment operator (c) commutes with absolutely all the IR operators. This makes it possible to obtain all the remaining operators of the 16-element non-commutative group without difficulty. Besides the operators already named, it includes the following: quasi-identity (cm), mirage (c*), duality (-c), activation (-cm), semi-duality (-c*), passage to the beneficiary (-cm*).

In Fig. 2.5 we have two representations of the group of operators of the classical intertype relations (by means of the 4×2 tables and of the formalized notations).

[Fig. 2.5. The group of operators of the classical IR.]

As the one-way arrows on four of the diagrams show, the following operators are not involutions: m*, -m*, cm*, -cm*. The corresponding formalized notations of the operators are easily recognized by the simultaneous presence of the letter “m” and the “asterisk.” This quadruple is one of the cosets with respect to the centre of the group. The centre of the group is the subgroup {1, -1, c, -c}, gathering all the operators that commute with all the elements of the group. The two remaining cosets with respect to the centre are the quadruples {m, -m, cm, -cm} and {1*, -1*, c*, -c*}.

It is easy to notice that the letter “m” occurs in the notations of exactly half of all the operators; in the other half it is absent. It is precisely this other half that constitutes the 8-element commutative subgroup of the 16-element non-commutative group of the classical IR which “works” in the half of the socion gathering the TIMs belonging to just one of the poles of the dichotomy I1 (irrationals / rationals). And can one spot some peculiarity of the diagram representations of the operators of the subgroup {1, -1, 1*, -1*, c, -c, c*, -c*}? There is such a peculiarity: in the corresponding diagrams there are no arrows connecting cells whose numbers have different parity.

It is likewise easy to notice that in half the cases the “asterisk” is absent. This half is the 8-element commutative subgroup that “works” in the half of the socion gathering the TIMs belonging to just one of the poles of the dichotomy I2 (democrats / aristocrats). Is there any peculiarity of the diagrams corresponding to the operators of the subgroup {1, -1, m, -m, c, -c, cm, -cm}? One may notice that the corresponding diagrams have a vertical axis of symmetry — provided, of course, that the diagram for the identity operator does not include the digits recalling the numbers of the cells in the Model “A.”

Of the peculiarities of the representations of the operators of the 8-element subgroup {1, 1*, m, m*, -1, -1*, -m, -m*} it has already been said that the formalized notations do not contain the letter “c,” and the arrows in the diagrams do not cross the horizontal line separating the mental ring from the vital one. Recall that this subgroup is not commutative and “works” in the half of the socion gathering the TIMs belonging to just one of the poles of the dichotomy A0 (statics / dynamics).

What, then, does the presence or absence of the “minus” in the formalized notations of the operators of the classical IR mean? In Fig. 2.5 the octet of operators whose notations contain no “minus” is plainly visible: it occupies the upper row ({1, c, m, cm, 1*, c*, m*, cm*}). But this octet of operators is not a group in the mathematical sense of the word: any group must contain, together with each element, its inverse. It is immediately apparent that the operators -m* and -cm*, which are the inverses of m* and cm* respectively, found no place in this octet.

The answer to the question posed in the preceding paragraph can be found by attending to where the arrows ending in the first cell come from in the diagrams of Fig. 2.5. (In the cases of an empty first cell — i.e. for 1 and 1* — one must reckon that the arrow begins and ends in that same cell and is therefore not shown in the diagram.) If the arrows ending in the first cell come from the cells numbered 1, 2, 7 and 8 (i.e. from the strong blocks Ego and Id), then the formalized notations of the operators of the classical IR contain no “minus” (see the upper row of Fig. 2.5). Otherwise (see the lower row of Fig. 2.5) the “minus” will be present.

In the language of the symbolic notations of the TIMs this same rule can be formulated thus: the “minus” sign is absent from the formalized notation of an operator of the classical IR if and only if the form of the IA symbol of the first function of the resulting TIM coincides with the form of the IA symbol of one of the first two functions (i.e. the Ego-block functions) of the initial TIM [16].

Let us give an illustrative example of building the formalized notation of an operator of the classical IR. Suppose we must find the operator that carries Ne⁻ into Ti⁺.¹ We shall need to “unfold” the notation of the initial TIM to (Ne, Fi). Since the IA symbol of the first function of the resulting TIM does not coincide in form with the IA symbols of the Ego block of the initial TIM, a “minus” must be placed. There will be no letter “c,” since both TIMs are statics. The letter “m” must be placed, since the TIMs belong to different poles of the trait irrationals / rationals. Finally, there will be no “asterisk,” since both TIMs are aristocrats. In the end we obtain “-m” — and that is the notation of the conflict IR.

The Group of Operators of the Classical IR as a Group of Operators of Transformations of the 3D Models “A” of TIMs

In the preceding section the isomorphism was established between the group of operators of transformations of the TIMs’ abbreviated symbolic notations and the group of operators of transformations of their 2D Models “A.” The elements of the latter group, which directly reflects the basic idea of the classical IR, were represented as 4×2 tables with arrows. Such a representation, as we saw, is quite visual. Indeed, if the TIMs too are represented by 4×2 tables whose cells are appropriately filled with IA symbols, it becomes clear how the operators transform the 2D Model “A” of one TIM into the model of another. It is not hard to understand, either, how the multiplication of two operators yields a third, and one can verify for oneself that the result of multiplication often depends on the order of the factors. It is likewise obvious how, from the table representing a particular element of this group, to build the table representing the element inverse to it.

Such visual clarity of the representation of the group of operators of the classical IR as a group of operators of transformations of the 2D Models “A” is a great help at the initial stage of acquaintance with the group of operators of the classical IR that interests us. But this representation turns out not to be particularly convenient in working with the group when “computations” must be performed quickly and in one’s head. It must also be noted that the group of operators of transformations of the 2D Models “A” of TIMs cannot be familiar to people who have not studied socionics. It would therefore be good to find a group that is isomorphic to the group of interest and yet familiar to us from other fields of knowledge.

The authors of the work [2], by their education and principal professional activity, were well acquainted with the crystallographic groups. This allowed them, when they became interested in socionics, to “recognize” in the group of operators of the classical IR a group familiar to them — the symmetry group of the tetragonal prism, i.e. a right prism with a square base.

Fig. 2.6 shows one possible way of putting the elements of the group of IR operators into one-to-one correspondence with the symmetry group of the tetragonal prism. The illustration is taken from an article by A.M. Banaru [3]. As we see, here the TIMs are represented by points located on the edges of the prism and labelled with the three-letter codes.

[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]).]

Which operator of the classical IR corresponds, in this figure, to reflection in the horizontal plane through the centre of the prism, say? That is quite a simple question. Points lying on one vertical are labelled with codes differing only in the last letter; hence the operator in question is extinguishment (c). It is likewise easy to see that rotation through 180° about the vertical axis through the centre of the prism corresponds to the superego operator (-1): in this case the last letter of the code stays unchanged while the first two change, but within a single -nality (S↔I, L↔E). Gradually one can work out the remaining correspondences as well. But under this approach the key idea of the classical theory of the IR — “Information is transmitted to each other only by identical elements” [1, p. 128] — is not particularly traceable.

I proposed using the tetragonal prism not as in Fig. 2.6, but as a frame for a 3D variant of the Model “A.” It is important that the height of such a prism does not coincide in length with the side of the base — i.e. it is not a cube, though its base is a square. The point is that the cube has more symmetry elements than the group of operators of the classical IR.

This idea proved promising. At the initial stage of acquaintance with the properties of the 16-element non-commutative group of operators of the classical IR, it is useful, for representing the TIMs, to become acquainted with the 3D variant of the Model “A” in the form of a tetragonal prism with suitably numbered vertices [11]. The models of the individual TIMs will differ in which IA symbols are “assigned” to each of the prism’s 8 vertices.

Using the self-superposition (symmetry) transformations of the tetragonal prism — of which there are 15, not counting the identity transformation — we can obtain from the model of one TIM the models of all the rest. Fig. 2.7 shows how, under the action of the operators of the extended generating set of this symmetry group of the tetragonal prism, from the 3D model of the ILE (Ne⁺) are obtained the 3D models of the ILI (Ni⁺), the IEE (Ne⁻), the LII (Ti⁻) and the SEE (Se⁺). In the same figure the geometric transformations are put into one-to-one correspondence with the formalized notations of the generators of the group of operators of the classical IR.

[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.]

It is not hard to work out which geometric transformations correspond to these operators. To the operator c (extinguishment) corresponds reflection in the horizontal plane through the centre of the prism. To the operator 1* (kindred IR) corresponds reflection in the plane passing through the vertices numbered 1, 3, 5, 7 — whereupon the IA symbols with odd numbers stay in place, and those with even numbers exchange places by the rule 2 ↔ 4, 6 ↔ 8. To the operator m (mirror) corresponds reflection in the vertical plane through the centre of the prism parallel to the faces 1467 and 2358; from Fig. 2.7 it is easy to see that the IA symbols exchange places by the rule 1 ↔ 2, 3 ↔ 4, 5 ↔ 6, 7 ↔ 8. Finally, to the superego operator (-1) corresponds rotation through 180° about the vertical axis through the centre of the prism.

Note that the arrows in Fig. 2.7, labelled with the formalized notations of the operators of the extended generating set, are two-way. This means that on repeated application of these operators we return to the initial TIM. In socionic language one says these are operators of symmetric relations; in the language of mathematical group theory, that they are involutions. One can also say that the squares of these operators equal the operator of the so-called identity transformation (1), under which the symbols of all the IA stay in place. This fact is recorded thus:

c² = (1*)² = m² = (-1)² = 1.

It is important to realize that the generating set can be confined to three elements only: {c, 1*, m}. Rotation through 180° about the vertical axis through the centre of the prism can be expressed through the transformations corresponding to the operators 1* and m:

(1* · m)² = (m · 1*)² = -1.

It is interesting that the “product” 1* · m is not equal to the “product” m · 1*. Geometrically, both expressions mean rotations through 90° about the vertical axis through the centre of the prism; both rotations replace the successive reflection in two planes. But the directions of the rotations are different! The direction depends on the order in which the reflections in the planes (corresponding to the operators 1* and m) are performed. And this is what it means for the operators 1* and m not to commute: their product depends on the order of the factors. The operators c and -1, by contrast, commute not only with each other but with the operators 1* and m as well. Anyone can verify the truth of this for themselves by picturing how, under the action of the operators in question, the model of one TIM turns into the model of another.

Using such 3D models of the TIMs, one can easily show that both of the isomorphic groups under consideration can be represented as the direct product of the dihedral group of order 8 and the cyclic group of order 2 (D₈×C2). In the case of the operators of the classical IR the cyclic group is the group {1, c}; in the case of the symmetry group of the prism it is the group consisting of the identity transformation and reflection in the plane through the centre of the prism parallel to its square bases. As for the dihedral group of order 8, Fig. 2.8 makes it clear that this is both the symmetry group of the square and the group {1, 1*, m, m*, -1, -1*, -m, -m*} of IR operators.

[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.]

The squares in Fig. 2.8 are to be understood as the lower or upper faces of the 3D Models “A” of the TIMs — i.e. as the mental or vital rings. It is not hard to work out which self-superposition transformations of the square are put into correspondence in this figure with the operators of the classical IR. The transformations of the square are easy to follow by keeping an eye on the arrow.

For example, let us take the ILE as the initial TIM and understand the square as the mental ring, the arrow running in the direction from the first function to the second. Accordingly, at the start of the arrow one should picture the symbol Ne, and at its end Ti. Then the remaining 7 squares will symbolize the mental rings of the remaining static TIMs — provided one reckons that under the transformations the function numbers stay in place and only the IA symbols “attached” to the vertices of the square move. In other words, everything is analogous to what we had in Fig. 2.7 for the self-superposition transformations of the tetragonal prism.

Clearly, from the position of the arrow (i.e., in our case, from the positions of the symbols Ne and Ti) the positions of the symbols of the two other IA entering the mental rings of the statics (Se and Fi) are easily reconstructed. The vital rings of these TIMs are found automatically, once one recalls the structure of the proposed 3D variant of the Model “A.”

One must, of course, understand that to the 8-element group of self-superpositions of the square corresponds an 8-element subgroup of the 16-element group of self-superpositions of the tetragonal prism. In this subgroup, reflections in the axes of symmetry of the square correspond to reflections in the vertical planes of symmetry of the prism, and rotations about the centre of the square to rotations about the vertical axis of symmetry of the prism. The remaining 8 transformations of the tetragonal prism are obtained as compositions (as products) of the elements of this 8-element subgroup with reflection in the horizontal plane through the centre of the prism. The remaining 8 elements of the group of IR operators, correspondingly, are obtained by multiplying the elements of the subgroup {1, 1*, m, m*, -1, -1*, -m, -m*} by the operator c (extinguishment).

The symmetry group of the tetragonal prism is non-commutative and contains four transformations that are not involutions. But how a non-involution is obtained from two involutions can be grasped already on the example of the symmetry group of the square. Indeed, two reflections carried out in turn in axes of symmetry of the square that form an angle of 45° can be replaced by a single rotation about the centre through 90°. But the direction of the rotation depends on the order in which the reflections in the indicated axes of symmetry are performed.

For all the visual clarity possessed by the explanation of the structure of the group of operators of the intertype relations in the language of geometric transformations of 3D models of the TIMs, it requires much time for preparing the illustrative material. Moreover, mental operations involving transformations of three-dimensional figures come easily to far from everyone.

Let us recall, for comparison, the interpretation that can be given to the elements of the extended generating set of the group of operators of the classical IR using far more “folded” models of the TIMs — which is what their symbolic notations effectively are. In the left diagram of Fig. 2.9 the TIMs are represented by their Ego blocks, while in the right diagram the abbreviated symbolic notations of the TIMs are used outright.

[Fig. 2.9. The passage to the symbolic notations of the TIMs as their “folded” Models “A”.]

Comparing these two diagrams, even those unfamiliar with the abbreviated symbolic notations of the TIMs can try to grasp the principle of the abbreviation. If intuition and logic are united in the Ego block (as with the ILE, ILI and LII present in the diagrams), then the intuition symbol receives the sign “plus” (+) as superscript and the logic symbol the sign “minus” (−) — and this does not depend on which poles of the traits irrationals / rationals (I1) and statics / dynamics (A0) the given TIM belongs to. What matters is the union of intuition and logic in one block. Once this fact has been noticed, placing the signs in the remaining cases presents no difficulty whatever.

Now let us recall how the operators of the 4-element extended generating set transform the abbreviated symbolic notations of the TIMs. The extinguishment operator (c) simply changes the colour of the symbol (recall that “colour” is the English word). The operator of the kindred IR (1*) merely “toggles” the superscript. The superego operator (-1) changes only the form of the symbol, within a single -nality (i.e. without changing the pole of the trait irrationals / rationals (I1)). The action of the mirror operator (m) is easy to explain when the TIM is given by its Ego block (as in the left diagram of Fig. 2.9²): one need simply exchange the IA symbols. To carry out in one’s head the passages from the abbreviated symbolic notation to the Ego block and back, a little practice is needed. The very notation of the operator comes from the English word “mirror.”

Using only two operators (1* and m) of the four considered, and starting from the 3D Model “A” of, say, the ILE (Ne⁺), we can obtain the 3D Models “A” of all the statics. But it is simpler to transform the abbreviated symbolic notations of the TIMs by the simple rules just formulated. The result is presented in Fig. 2.10 (left diagram). Using this diagram, it is easy to verify the formula given earlier expressing the superego operator (-1) through the operators 1* and m. Guided by the same diagram, we can write:

(1* · m)(Ne⁺) = 1*(m(Ne⁺)) = 1*(Ti⁻) = Ti⁺ and (m · 1*)(Ne⁺) = m(1*(Ne⁺)) = m(Ne⁻) = Fi⁺.

If we now denote the product 1* · m as m*, and the product -1 · m* as -m*, then, bearing in mind that -1(Ti⁺) = Fi⁺ (see the left diagram), we obtain: m · 1* = -m*. As we see, the interchange of the factors led to the appearance of the “minus” (-). The right diagram shows the two so-called supervision rings (or control rings). The direction of the arrows runs from the supervisor (or controller) to the supervisee (or the controlled one). It therefore makes sense to call the operator m* the operator of passage to the supervisee (the controlled one), and its inverse (-m*) the operator of passage to the supervisor (the controller). The squares of these operators, as we established earlier, equal the superego operator (-1), and their fourth powers the identity operator (1). These facts are easily established from the right diagram of Fig. 2.10 as well.

[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.]

It should be clear that had we started not from the ILE (Ne⁺) but from the ILI (Ni⁺), we would have obtained diagrams differing from those of Fig. 2.10 only in the colour of the symbols. And if the abbreviated symbolic notations of TIMs differ only in colour, this means, as we already know, that such TIMs are linked by the extinguishment relations (c). In this way it is visually demonstrated (see Fig. 2.11) that all 16 TIMs making up the socion can be linked by just three operators that are their own inverses, i.e. involutions: {c, 1*, m}.

[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.]

In Fig. 2.11 it is plainly visible that from any TIM one can “reach” any other along a “trajectory” made up of line segments of three kinds: single solid (1*), double solid (m), dashed (c). The same figure shows two partitions of the socion into four quartets of TIMs. One tetrachotomy of the socion corresponds to the 4-element subgroup with generators {c, 1*}, and the other to that with generators {c, m}. The last possible pair ({1*, m}) of the triple of operators under consideration is a generating system not of a 4-element but of an 8-element subgroup, by means of which the socion can be divided in exactly the same way as by the trait A0 (statics / dynamics). And this dichotomy of the socion, too, is clearly visible in Fig. 2.11.

And although the minimal generating system of the group of operators of the classical IR consists of just 3 operators, after its extension to 4 elements through inclusion of the superego operator (-1) it becomes possible to write any element of this group in a form convenient for use, which has been called canonical [10]:

(-1)ᵏ · cˡ · (1*)ᵖ · mᑫ,

where k, l, p and q can take only the values 0 or 1. In this record, what is essential is the order of the two mutually non-commuting operators — i.e. that the operator 1* stands precisely to the left of the operator m. One must also not forget the convention of denoting the product 1* · m (with precisely that order of factors!) as m*. If in the canonical representation of an IR operator the condition p = q = 1 holds, that operator will not be its own inverse — i.e. one of the four operators of asymmetric IR results: {m*, -m*, cm*, -cm*}.

If p = 0, only operators of the 8-element subgroup with generating system {-1, c, m} result. By means of this subgroup the socion can be divided in exactly the same way as by the bipolar trait I2 (democrats / aristocrats). If q = 0, only operators of the 8-element subgroup with generating system {-1, c, 1*} result — i.e. of the one dividing the socion in the same way as the bipolar trait I1 (irrationals / rationals).

Under the condition p = q = 0 only the quadruple {1, -1, c, -c} remains. It comprises all the operators that commute not only among themselves but with all the other operators of the classical IR. This quadruple is called the centre of the group.

The Group of Operators of the Classical IR as a Group of Operators of Transformations of the One-Dimensional Models “A”

Let us now turn to Fig. 2.12, which presents all 16 transformations of the one-dimensional variant of the Model “A” of the initial TIM of an ordered pair into the model of the resulting one, under the action of the corresponding operator of the classical intertype relations. This illustration of the group of the classical IR was proposed by Andrew Joynton after his acquaintance with the system of formalized notations of its elements [14]. He made use of a remarkable property of the Model “A”: from the IA symbol occupying one of the odd places in the model of a specific TIM, all the other symbols standing in odd places can be reconstructed. The analogous assertion holds for the IA symbols occupying the even places. As a result, the number of diagrams was halved, since each half of each diagram was used twice (for two operators).

[Fig. 2.12. Four obvious dichotomies of the 16-element set of formalized notations of the operators of the classical IR.]

Note that for the initial TIMs of a pair (those on which the IR operator acts) the upper part of the one-dimensional Model “A” is always allotted to the odd-numbered elements. For the diagrams presented in the left half of the figure, the same can be said of the models of the resulting TIMs; in the right half of the figure, the upper parts of the Models “A” of the resulting TIMs are occupied by the even-numbered elements. How is this difference reflected in the formalized notations of the IR operators? Is it connected with the initial and resulting TIMs belonging, in the one case, to the same pole of some bipolar trait, and in the other to different poles?

It is easy to notice that the notations of the operators of the left half of Fig. 2.12 lack the letter m, while those of the right half have it. To answer the second question, one must recall that in the Model “A” all the odd-numbered elements “deal” either only with irrational IA or only with rational ones — from which the analogous assertion about the even-numbered elements should be clear.

As we see, in all the diagrams of the left half of Fig. 2.12 the arrows go from odd numbers to odd and from even to even. This means that the initial and resulting TIMs belong to one and the same pole of the bipolar trait irrationals / rationals (I1). In all the diagrams of the right half, as is easily noticed, the situation is the reverse. We thus arrive at the following simple rule: the letter m is present in the formalized notation of an IR operator if and only if the initial and resulting TIMs belong to different poles of the trait I1.

Let us now compare the upper and lower halves of Fig. 2.12. The difference in the formalized notations of the IR operators is easily seen: in the upper half the operator notations contain no letter c, in the lower half they do. Can this be tied to the coincidence / difference of the poles of some bipolar trait for the initial and resulting TIMs? To answer, one must recall one more property of the Model “A”: the mental ring (elements numbered 1, 2, 3, 4) can contain the symbols either only of static IA or only of dynamic ones.

As we see, in all the diagrams of the upper half of Fig. 2.12 the arrows go from the numbers 1–4 to numbers of the same quartet. This means that the initial and resulting TIMs belong to one and the same pole of the bipolar trait statics / dynamics (A0). In the diagrams of the lower half the situation is the reverse. We thus arrive at another simple rule: the letter c is present in the formalized notation of an IR operator if and only if the initial and resulting TIMs belong to different poles of the trait statics / dynamics (A0).

The third clearly visible central dichotomy of the set of formalized notations of the IR operators is the absence / presence of the superscript “asterisk” (*). Once again we ask: can this be tied to the coincidence / difference of the poles of some bipolar trait of the initial and resulting TIMs?

To answer, let us recall the structure of the Model “A” in its 2D variant, in which one distinguishes the so-called horizontal blocks ({1, 2}, {3, 4}, {5, 6}, {7, 8}) and vertical blocks ({1, 4}, {2, 3}, {5, 8}, {6, 7}). Among the democrats, intuition and logic — and likewise sensorics and ethics — are united in the horizontal blocks. Among the aristocrats, only intuition with ethics, and sensorics with logic, can be together in a horizontal block.

Let us trace where the IA symbols that stood in the horizontal block {1, 2} of the initial TIM’s model end up. Analysis of Fig. 2.12 shows that they land in a horizontal block of the resulting TIM’s model exclusively in those cases where the formalized notation of the operator contains no “asterisk.” And passage from horizontal block to horizontal block means that both TIMs belong to one and the same pole of the trait democrats / aristocrats (I2).

We thus arrive at the following simple rule: the “asterisk” (*) is present in the formalized notation of an IR operator if and only if the initial and resulting TIMs belong to different poles of the trait democrats / aristocrats (I2).

Finally, the fourth clearly visible central dichotomy of the set of formalized notations of the IR operators is the absence / presence of the “minus” sign (-). And again one wants to ask: can this dichotomy, too, be tied to the coincidence / difference of the poles of some bipolar trait of the initial and resulting TIMs? Unlike the previous three cases, one cannot hope for an equally simple answer here.

Indeed, if we managed to put this dichotomy of the set of operators of the classical IR, as well, into one-to-one correspondence with a bipolar trait of TIMs that was independent of the already considered traits I1, I2 and A0 and orthogonal to them, then the group of IR operators would turn out to be isomorphic to a 16-element commutative group of bipolar traits — and that is not the case for the classical model of the intertype relations.

From the mathematical standpoint, the classical model of the IR turns out to be somewhat more complex than the “trait-based” models, which are built on assigning IR operators to the spectra of coinciding traits in a pair of communicants. In the “trait-based” models the operator of passage from one TIM to the other always coincides with the operator of the reverse passage; such models do not lead to the so-called asymmetric intertype relations. Terms like the phrase “irrational request” [18, p. 164] could only have arisen from comparing the results of a “trait-based” model of the IR with the previously created classical model. If a “trait-based” model of the IR is tied not to the AR-trait group but to the JMT group, one can with equal success speak, for example, of a “democratic request” [8]. We have named only two 16-element groups of central (8:8) sections of the socion out of their simply gigantic number [17]. These two groups turned out to be singled out precisely from the standpoint of the unique classical model of the IR [12].

When G.R. Reinin analyzed the intertype relations “from the position of the group of bipolar traits” (at that time only the AR-trait group was at issue), he had to state that not only the asymmetric relations “split,” but also the following symmetric ones: 1* (kindred), -1* (business), c* (mirage), -c* (semi-dual). To these relations no definite spectrum of coincidences of AR-trait poles could be assigned: for each of the operators listed, two spectra of pole coincidences could be indicated — one corresponding to the irrational pairs and the other to the rational ones. Note that the action of the operators listed does not lead to a change of pole of the bipolar trait irrationals / rationals (I1).

Carrying out the analogous analysis not from the position of the AR-trait group but from that of the JMT group, one finds that of the symmetric IR it is the mirror (m), conflict (-m), quasi-identity (cm) and activation (-cm) relations that undergo “splitting.” Note that the action of the operators listed does not lead to a change of pole of the bipolar trait democrats / aristocrats (I2). Correspondingly, for each of the operators listed two spectra of pole coincidences can be indicated — one corresponding to the democratic pairs and the other to the aristocratic ones.

Recall that in Fig. 2.12 we could clearly see four dichotomies of the set of formalized notations of the operators of the classical IR, to three of which we at once assigned dichotomies of the socion, but not to the fourth:

absence / presence of the letter “m” ↔ (+/− I1);
absence / presence of the superscript ”*” ↔ (+/− I2);
absence / presence of the letter “c” ↔ (+/− A0);
absence / presence of the sign ”-” ↔ (???).

The fourth central dichotomy of the set of formalized notations of the operators is just as obvious as the previous three. But a corresponding dichotomy of the socion simply cannot exist. What is to be done?

We must take into account that the formalized notations for the operators of the classical IR were created with a view to their being convenient when the TIMs are represented by their abbreviated symbolic notations, in which the IA of precisely the first function is explicitly fixed. This means we must attend to that group of bipolar traits which is, in a certain sense, closer to Jung’s 8-element typology — a typology that effectively takes into account the IA of the first function only, as a result of which it did not distinguish kindred types.

The group in question is the 16-element JMT group, and one should expect a “splitting” along the line democrats / aristocrats. Let us look separately at how the operators of the classical IR act on democratic and aristocratic types. As representatives of the democrats and the aristocrats we take the ILE and the IEE respectively, which are kindred.

As we know, neither the operator 1* nor the operator c changes the form of the IA symbol of the first function of any TIM, regardless of whether it is a democrat or an aristocrat. Hence the form will not change under the action of the operator c* either, which is the product 1* · c. Nor does the operator 1 change the form — it changes nothing at all. Let us now unite into quartets the operators whose formalized notations differ by the traits of absence / presence of the superscript ”*” and absence / presence of the letter “c,” but coincide by the traits of absence / presence of the letter “m” and absence / presence of the sign ”-“. One then obtains the very telling picture presented in Fig. 2.13.

[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.]

As expected, the “splitting” occurs along the line democrats / aristocrats. Note the quartet of operators {m, cm, m*, cm*}, obtained by multiplying the operator m by the operators of the subgroup {1, c, 1*, c*}, the operator m being multiplied on the right. For example, c* · m = cm*. Correspondingly,

cm*(Ne⁺) = c(1*(m(Ne⁺))) = c(1*(Ti⁻)) = c(Ti⁺) = Te⁺; cm*(Ne⁻) = c(1*(m(Ne⁻))) = c(1*(Fi⁺)) = c(Fi⁻) = Fe⁻.

The form of the symbol N changed differently under the action of the operator m for the TIMs Ne⁺ and Ne⁻, which belong to different poles of the trait I2 (democrats / aristocrats). The question arises: “The poles of which bipolar traits answering for the form of the symbols were preserved in the process?” The TIMs Ne⁺ and Ti⁻, linked by the mirror relations, belong to one and the same pole of the trait B0 (abstract / involved — 1). On the other hand, the TIMs Ne⁻ and Fi⁺, which likewise form a mirror pair, belong to one and the same pole of the trait B1 (internal / external — 1).

Having considered this particular example, we should understand that the property of preserving the pole of the trait B0 under the action of the operator m is possessed not by Ne⁺ alone: the same property is characteristic of all the democrats. Indeed, those democrats belonging to the Club of Scientists have two abstract IA in the Ego block, and those belonging to the Club of Socials two involved ones. Under the action of the operator m, the symbol of the aspect that stood in second place in the initial TIM’s Ego block moves into first place. And since both aspects are either abstract or involved, such a transformation cannot lead to a change of pole of the trait B0.

An entirely analogous situation obtains with Ne⁻. It simply acted as representative of the aristocrats, who under the action of the operator m do not change the pole of the trait B1. Bearing in mind the rules for filling the Ego block with IA symbols, the Managers can be called doubly external and the Humanitarians doubly internal — just as the Scientists could be called doubly abstract and the Socials doubly involved.

Thus the following “splitting” resulted: if the initial TIM is a democrat, it must be compared with the resulting TIM by the bipolar trait B0, and if an aristocrat, by the trait B1. The sign ”-” enters the formalized notation of an operator of the classical IR in that case, and only in that case, when under such a comparison the poles of the corresponding trait for the initial and resulting TIMs do not coincide. This is the last rule, and it has turned out somewhat more complex than the previous three. In the case where the question was settled of the absence / presence of the sign ”-”, additional preliminary information was required about the initial TIM’s membership in a particular pole of the trait I2 (democrats / aristocrats). Herein showed itself that difference between the model of the classical IR and the “trait-based” models of which we have already spoken: in the “trait-based” models a pair of TIMs figures as unordered, whereas in the case of the classical IR the division of the TIMs of a pair into initial and resulting has a definite meaning.

Now that we have dealt with all four dichotomies of the set of formalized notations of the IR operators, an algorithm can be proposed for composing such notations when the initial TIM (the one on which the IR operator acts) and the resulting TIM are given. The user must be able to distinguish the notations of the TIMs by the following five bipolar traits:

I1 (irrationals / rationals),
I2 (democrats / aristocrats),
A0 (statics / dynamics),
B0 (abstract / involved — 1),
B1 (internal / external — 1).

From the information on the membership of each TIM of the ordered pair under consideration in the poles of these traits, decisions will have to be made on including in the formalized notation the following sequence of elements: 1) the sign ”-”, 2) the letter c, 3) the letter m, 4) the superscript ”*“. This is precisely the order in which the decisions must be made if the formalized notation of the operator is written from left to right.

The division of the TIMs of the pair into initial and resulting is taken into account at the first step of the algorithm; at the remaining three steps one need only track the coincidence / difference of the poles of the corresponding bipolar traits, regardless of the order of the TIMs in the pair. Here is this simple algorithm.

Place the sign ”-” if one of the following conditions holds: a) the initial TIM is a democrat, but the poles of the TIMs do not coincide by the trait B0 (abstract / involved — 1); b) the initial TIM is an aristocrat, but the poles of the TIMs do not coincide by the trait B1 (internal / external — 1).
Place the letter c if the poles of the TIMs do not coincide by the trait A0 (statics / dynamics).
Place the letter m if the poles of the TIMs do not coincide by the trait I1 (irrationals / rationals). If neither c nor m has been placed, then 1 must be placed.
Place the superscript ”*” if the poles of the TIMs do not coincide by the trait I2 (democrats / aristocrats).

¹ Translator’s note: the specific conflict pair in this worked example is not recoverable from the text layer; the constraints stated (conflict relation, both statics, both aristocrats) admit two pairs — {IEE, LSI} and {EII, SLE} — and the IEE→LSI direction is used here. Every step of the reasoning is identical for the other choice.

² The original reads “Fig. 2.8” here; from context the reference is to the left diagram of Fig. 2.9.

The Two Languages of Thinking in Vekker’s Theory and the Rules for Combining Functions of Information Metabolism into the Blocks of Model “A”

The formation of the blocks in the Model “A” was discussed in the work of the founder of socionics, Aušra Augustinavichiute, “The Socion, or the Foundations of Socionics” [1, p. 82]. She wrote that elements of statics are “blocked” only with elements of statics, and elements of dynamics only with elements of dynamics. The rule was illustrated with examples from which, by the symbols of the information aspects processed by a single block, it was evident that these aspects necessarily belong to different poles of the bipolar traits irrationality / rationality and extraversion / introversion.

In the work of Aušra Augustinavichiute cited, the terms extraversion / introversion, which figured explicitly also in C.G. Jung’s Psychological Types [19], were in many cases replaced, as applied to the “elements of information metabolism,” by the terms body / field and black / white. As for the bipolar trait irrationality / rationality, it was effectively not used in introducing the information aspects, although 4 of the 8 aspects were nevertheless called rational “following C.G. Jung.”

Jung, however, tied “pure perception” to irrationality, which was opposed to rationality [19, p. 552]. To catch the difference between the irrational and rational IA from their descriptions in the work of Aušra Augustinavichiute under consideration is difficult.

Of course, let us not forget that the founder of socionics herself wrote plainly that her work could not be considered finished — that it still contained many imprecisions and contradictions. This was explained by the fact that “one had to feel one’s way, both because of the novelty of the object and because of the author’s lack of special training.” She was fully aware that some of her propositions might seem “not fully proven, as it were hanging in the air,” and even gave examples of such propositions (“why there are exactly eight aspects of the world, why blocks are formed from them, why precisely these and not some others, and so on” [1, p. 63]).

It is now already possible to attempt answers to these “whys” of Aušra Augustinavichiute, which she herself posed in connection with her own Model “A,” whose sixteen varieties make up the socion. In my view, help here can come from the unified theory of psychic processes of Lev Vekker, who wrote explicitly of the need to take “informational regularities” into account in considering psychic processes [5, p. 48]. We shall pay special attention to how thinking is defined in Vekker’s theory, and try to relate that definition to Jung’s “psychological functions” and to the principle of “blocking” of the elements of Model “A” proposed by Aušra Augustinavichiute.

To introduce the eight IA by means of bipolar traits, the number of such traits must be at least three. It might seem that, in addition to the bipolar constructs body / field (extraversion / introversion) and statics / dynamics, one could take the bipolar trait internal / external. It is precisely this triple of traits that some socionists consider the “basis” (generating system) for building the whole group that “voices” the corresponding group of central sections of the 8-element set of information aspects.

Aušra Augustinavichiute explained how the trait internal / external found its reflection in the graphic symbols of the IA. Extraverted sensorics was denoted by a circle (●) — “a figure that creates the impression of the fullest contact with the whole external world.” Intuition is denoted by a triangle (▲), “which fits perfectly into the circle.” A square with one corner removed symbolizes ethics, which is the internal side of logic — for whose designation the square (■) was chosen as “a symbol of strictness.” The corresponding introverted elements were assigned the same symbols, only white in colour [1, p. 14].

It should be clear that even on formal grounds one could not take, as the third element in place of the trait internal / external, the trait irrationality / rationality — for it is the product of the bipolar traits body / field (extraversion / introversion) and statics / dynamics. That would give a triple of mutually dependent traits: from the known poles of two traits of the triple, the pole of the third would be determined automatically. Accordingly, it would be impossible to distinguish, for example, Se (●) from Ne (▲), since both IA are characterized by the terms: body, irrationality, statics.

With the triple of bipolar traits {internal / external, statics / dynamics, body / field} there are no such problems: it is a triple of independent traits. Specifying their poles, we arrive unambiguously at a single IA. But this “basis” has another problem. Take, for example, Ne (▲) and Ni (△), which both belong to the macroaspect intuition. To distinguish these two IA it suffices to add to the word intuition the appropriate adjective (say, extraverted or introverted); this division corresponds to the bipolar trait body / field included in the “basis” under discussion.

One might expect the macroaspect intuition itself to be determined, in that case, by the corresponding poles of the two remaining traits. But this hypothesis — natural, in my view — is not borne out. The only common pole is the pole of the trait internal / external. In general, in the triple of bipolar traits {internal / external, statics / dynamics, body / field} one cannot choose two such that specifying their poles fixes a specific macroaspect of the information flow.

Thus, as we see, when the “basis” under discussion is used, the terms denoting the macroaspects are in effect left without definitions through the poles of bipolar traits. Nevertheless these terms are used in the two-word names of the IA: the noun fixes the macroaspect, and the adjective specifies which aspect is meant within the macroaspect named by the noun. A natural question arises, which we shall consider on concrete examples.

For extraverted intuition (▲) and introverted intuition (△) the same form of graphic symbol (the triangle) is used, and the names of these IA share the generic word intuition, the specific differences being fixed by the appropriate adjectives. Yet these IA share the pole of only one of the three “basis” traits. At the same time, in the case of Ne and Fi the poles of two “basis” traits coincide — yet graphic symbols of different forms and two different nouns (intuition and ethics) are used. What deep meaning is there in this? In my view, questions of this kind should push us toward choosing a more felicitous “basis.”

Recall, moreover, how the dichotomy internal / external arose, and its reflection in the graphic symbols. By this trait, after all, the irrational part of the information flow was divided into its intuitive and sensory components, and the rational part, correspondingly, into the ethical and logical ones. This means that, for example, the term intuition replaces the phrase “internal irrationality.” It is on the next step of the division of the information flow (when the macroaspects are divided into IA) that the specifying adjectives are used (e.g. extraverted intuition and introverted intuition).

Building a hierarchical sequence of bipolar traits (for example, irrationality / rationality → internal / external → extraversion / introversion), we automatically obtain a triple of independent traits (no one of them expressible through the other two). On the other hand, the important socionic concepts corresponding to the level of the macroaspects (intuition, sensorics, ethics, logic) will not be left “hanging in the air.”

It is another matter that for the passage from one conceptual level to another it may prove convenient to use other traits. For example, after dividing the information flow into its irrational and rational parts, there is in principle the possibility of using, for the passage to the level of the macroaspects, the dichotomy abstract / involved. Then the dichotomy internal / external, which is reflected in the form of the macroaspect symbols, would figure as a derived one, obtained by multiplying the traits irrationality / rationality and abstract / involved. And in passing from the level of the macroaspects to the level of the IA one might try using the bipolar trait statics / dynamics. Under such a passage to the IA level, the graphic symbol ▲ would correspond to the phrase static intuition, and the symbol △ to the phrase dynamic intuition. Admittedly, one would have to explain that the colour of the symbols is tied not to the dichotomy statics / dynamics but to the dichotomy obtained as the product of the dichotomies irrationality / rationality and statics / dynamics — that is, to extraversion / introversion.

It is important to understand that any 4-element set has only three central sections. This means that the central bipolar traits of the macroaspects already mentioned, entering the triple {irrationality / rationality, internal / external, abstract / involved}, represent all the possible ways of dividing this set strictly in half. Other words may be chosen to designate the corresponding central sections — but no more than that. These three central sections are, of course, mutually dependent. For example, to the pole of the abstract belong “internal irrationality” (intuition) and “external rationality” (logic), and to the pole of the involved, correspondingly, “internal rationality” (ethics) and “external irrationality” (sensorics). On the same principle the bipolar trait internal / external can be represented through the traits irrationality / rationality and abstract / involved.

Of course, the trait irrationality / rationality too can be represented as the product of the traits internal / external and abstract / involved. This may push us toward the conclusion that all three traits are of equal standing. But we should be restrained by semantic considerations connected with our notion of the hierarchy of concepts. Obtaining logic and ethics as the result of a dichotomy of the rational part of the information flow, and intuition and sensorics as species of the irrational part, we thereby single out the trait irrationality / rationality against the background of the other two. Are there weighty grounds for this, apart from the reference to Jung?

Let us give additional arguments in favour of the view that, in introducing the IA by means of bipolar traits “voicing” the successive division of the information flow into two, four and then eight parts, one must begin with the dichotomy irrationality / rationality. These arguments were found in L.M. Vekker’s unified theory of psychic processes — namely, in that part of it directly concerned with the organization of thought processes. Moreover, turning to this psychological theory will, as already said, allow us to come closer to understanding the principles of block formation in the Model “A” proposed by Aušra Augustinavichiute.

It must be said at once that the term “thinking” in L.M. Vekker’s theory was subjected to careful analysis and received precisely the interpretation to which I should like to draw the readers’ attention. Above all, this interpretation does not coincide with Jung’s, who understood thinking “as one of the four basic psychological functions (…) which, following its own laws, brings given contents of representations into conceptual connection” [19, p. 538]. The founder of socionics, Aušra Augustinavichiute, as is well known, also declined Jung’s understanding of the term “thinking.” Here is a rather long quotation from her work “On the Dual Nature of Man”:

“…when C.G. Jung opposes the thinking types to the emotional ones, the impression is created that the former ‘think’ while the latter ‘live by emotions.’ In actual fact everyone thinks — only they think about different things: the former about the objective, the latter about the subjective in the surrounding world. Therefore, if we say of these latter, ‘ethical thinking predominates in them,’ this will not sound such an absurdity as if we said, ‘emotional thinking predominates in these people.’ People with ethical thinking have an inclination to assess and judge their surroundings from the standpoint of the ethicality or unethicality of relations and actions. People with logical thinking are mainly interested in the logicality or alogicality of relations and actions” [1, p. 9].

As we see, Aušra Augustinavichiute wished to underline that all people think, and not only the representatives of the “thinking types.” Let us note in passing that Jung opposed the thinking types to the feeling (emotional) ones within the rational types, while within the irrational types the opposition ran between the intuitive and the sensing types. One and the same type could not be simultaneously, say, intuitive and rational. In socionics such a combination became possible, since the through-going dichotomies intuitives / sensorics and logicians / ethicists were introduced. By means of these dichotomies the whole set of types was divided in half, and not one of its halves (the irrational or the rational respectively). In such divisions the orientation was not only to the principal (first) function, as with Jung, but to the auxiliary (second) as well. In socionics “rational intuitives” are possible — those for whom intuition is the second (auxiliary) rather than the first (principal) function. More on this can be read in the article [13].

As for the difference between the understanding of the term “thinking” in L.M. Vekker’s theory and its understanding in other psychological theories, special attention is paid to this question in his monograph, which even has a special section, “The Incompleteness of the Traditional Definitions of Thinking” [5, p. 172]. Summing up his critical analysis of the definitions of thinking, L.M. Vekker writes: “Be that as it may, between the fact of thinking’s monopoly belonging to man alone (clearly indicating the social-labour determination of its genesis and structure and tracing a definite boundary between thought processes and pre-thought processes) and the excessive generality of the criteria considered (which give no points of support for drawing such a boundary line) there is an evident mismatch and even contradiction” [5, p. 177].

Then, after a careful examination of the generic and specific criteria of thinking, he formulates a hypothesis “concerning the specific informational-psychological principle of the organization of the thought process, allowing a clear structural boundary to be traced between image and thought” [5, p. 271]. Here is the formulation of this hypothesis: “Thinking as a process is a continuous reversible translation of information from the language of simultaneous-spatial object gestalts, represented by images of various levels of generality, into the symbolic-operator language represented by the one-dimensional successive structures of speech signals. An individual thought, as a structural unit and result of the thought process, in its psychological specificity represents a psychically reflected relation as the invariant of the reversible translation from one language into the other” [5, p. 274].

Of course, such a formulation of the hypothesis presupposes a reader prepared for its reception by the whole preceding text of the monograph. I therefore urge those readers unacquainted with L.M. Vekker’s theory to make its acquaintance without fail. In my view, the informational approach of this theory’s author to psychic processes is very close to socionics. I draw attention to the fact that Vekker not only advanced, on good grounds, the hypothesis of the existence of two complementary languages of thinking, but also explained on its basis many experimental facts — in particular, the phenomenon of understanding.

What relation does Vekker’s hypothesis of the bilingualism of thinking have to the structure of the Model “A”? As it seems to me, the language of simultaneous-spatial object gestalts spoken of in Vekker’s theory is represented in socionics by the irrational information aspects, and the symbolic-operator language by the rational ones. For thinking through any question (problem situation) both languages are necessary. Let us turn to the text of his monograph:

“Incomprehensibility creates the motivational spring and the subjective starting signal of the dynamics of the thought process, while the concluding comprehensibility, or comprehendedness, of the corresponding sought relation constitutes the objective basis and the subjective finishing signal of the given stretch of the process, which concludes with a thought-solution.

Theoretical analysis of this process has led to the conclusion that the initial deficit of comprehensibility of the problem situation standing at its start — embodied chiefly in the incomprehensibility of the sought relation — and then the whole subsequent dynamics of the intermediate phases of understanding in the course of the search for an answer, flow from the organization of the thought process as a reversible interlingual translation. The initial incomprehensibility is then an expression of the mismatch of the languages, and the subsequent dynamics of understanding is determined by their growing coordination, which at each stage is determined by the measure of reversible translatability. Behind the latter, in turn, lies the invariance of the corresponding sought relation, constituting its objective basis. Thus reversibility contains within itself not only the indicator and measure of the invariance of the translation, but together with that — and precisely thereby — it also contains the measure of the comprehensibility of the corresponding relation, disclosed in the course of the thought operations of the search for an answer” [5, p. 275].

If one agrees with Vekker’s hypothesis of the bilingualism of thinking, and with my supposition of the correspondence of these languages to the two parts of the information flow (the irrational and the rational), then the thought becomes entirely natural that in the model of a TIM the elements will be united into blocks of two, one of which “answers” for an irrational IA and the other for a rational one. And if one attends to the fact that the object of thought is either in statics or in dynamics, it becomes clear that both elements of one block must be “occupied” by information aspects belonging to one and the same pole of the dichotomy statics / dynamics. These considerations already suffice to understand that, for example, Ne can form a block only with Ti or with Fi. Accordingly, if for any of the 8 IA placed first in a block there are only 2 candidates for second place, the total number of “permitted” variants of two-element blocks is 16. Here, of course, it is understood that the block, say, (Ne, Ti) differs from (Ti, Ne). In Aušra Augustinavichiute’s work “The Socion, or the Foundations of Socionics” a division of the elements of one and the same block into the acceptive (first) and the productive (second) is introduced [1, p. 83].

How can Vekker’s hypothesis of the bilingualism of thinking help one grasp the rules for filling the “cells” of the Model “A” with IA symbols? Speaking of “cells,” we shall orient ourselves to the 2D variant of the model under discussion (as a 4×2 table).

For the coordination of the two languages of thinking and the achievement of the reversibility of translation of which Vekker speaks, there must be the possibility of carrying out definite transformations within at least one language. Some people find it easier to work with images (simultaneous-spatial object gestalts), while others find it easier to transform one-dimensional successive structures. In mathematics, for example, one speaks of geometric and algebraic thinking. In socionics, TIMs are divided into irrational and rational depending on which IA the acceptive elements in their Models “A” “deal” with.

The Id block is coordinated with the Ego block. To perform the work for which, say, the Scientists are “intended,” one must have “multi-dimensional” (in the sense introduced by A.V. Bukalov [4, p. 373]) intuition and logic, in both the static and the dynamic variant. Thinking over the possible structure of some system (statics), one should not forget to ask oneself how that system will work (dynamics). It is another matter that some people find it easier to speak of static aspects (form, structure and the other relatively constant characteristics of individual objects and their systems), and others of dynamic aspects (processes and the characteristics connected with them). The former are called statics and the latter dynamics. In the former, not only the Ego block but the Superego block as well is static; these two blocks form the so-called mental ring. In the latter, it is the blocks of the vital ring (Id and Superid) that are static.

From which IA “occupy” the multi-dimensional blocks Ego and Id, it follows automatically which IA will “occupy” the low-dimensional blocks Superego and Superid. Consequently, among the Scientists it is sensorics with ethics that are united in the low-dimensional blocks. If sensorics and ethics are represented by the multi-dimensional blocks (and intuition and logic, correspondingly, by the low-dimensional ones), we obtain the TIMs united in the Club of Socials. If sensorics is united with logic in the multi-dimensional blocks (and intuition with ethics, correspondingly, in the low-dimensional ones), we obtain the Club of Managers. And among the Humanitarians, sensorics with logic are in the low-dimensional blocks.

The coordination of the blocks just considered within a single TIM leads to the specific variant of the Model “A” being fully determined by the specification of the Ego block. And this automatically leads to the number of variants of the Model “A” coinciding with the number of possible variants of the two-element blocks — that is, to 16.

As has already been noted, in the specific variant of the Model “A” corresponding to a particular TIM, the acceptive elements of all four blocks are “occupied” by information aspects belonging to one and the same pole of the bipolar trait irrationality / rationality. This allows one to speak not only of irrational and rational information macroaspects and aspects, but of irrational and rational TIMs.

All the blocks of a specific variant of the Model “A” are united not only by the membership of their acceptive elements in one pole of irrationality / rationality. At the level of the blocks there appears a fundamentally new dichotomy, to one pole of which belong all four horizontal blocks of the model of a single TIM: one may speak of democratic and aristocratic blocks. The Models “A” of the democrats (Scientists and Socials) consist only of democratic blocks, in which intuition is united with logic and sensorics with ethics. Among the aristocrats (Humanitarians and Managers) all the horizontal blocks are aristocratic (intuition with ethics, and sensorics with logic).

Note that the dichotomies statics / dynamics and extraversion / introversion do not exist at the level of the macroaspects of the information flow. They appear at the level of the IA and are carried over quite naturally to the level of the blocks. With statics / dynamics everything is entirely simple, since, for the reason already indicated, a block unites elements “occupied” by information aspects belonging to one pole of that dichotomy. The division of blocks into extraverted and introverted can be effected on the same principle as the division into irrational and rational (with orientation to the acceptive element). The carrying-over of the terms statics / dynamics to the level of the TIMs proceeds with orientation to the mental ring, which includes the blocks Ego and Superego.

For example, in the extraverted statics (ILE, IEE, SLE, SEE) both blocks of the mental ring are static and extraverted in the sense indicated above. Recall that we call extraverted those blocks whose acceptive element is extraverted. In these same TIMs, both blocks of the vital ring are dynamic and introverted.

Let us note once more that in any of the 16 variants of the Model “A” all four horizontal blocks (Ego, Superego, Superid, Id) belong to one pole of irrationality / rationality, and likewise to one pole of democracy / aristocracy. At the level of the TIMs, too, the socion is divided strictly in half by the trait irrationals / rationals, and likewise by the trait democrats / aristocrats. The product of these traits is the trait right / left — though some socionists use other words to verbalize the resulting section of the socion (process / result types, evolutors / involutors).

As should already be clear, dividing the socion by means of two bipolar traits from the triple {I1 (irr / rat), I2 (dem / arist), I3 (right / left)} leads to quartets of TIMs whose Models “A” consist of the very same horizontal blocks. The socion can be divided into these same quartets of TIMs on the grounds that within them the operators of the intertype relations of duality (-c), superego (-1) and extinguishment (c) are “realized.” This tetrachotomy of the socion played an important role in the introduction of the formalized notations for the operators of the IR [9] and for the elements of the two groups of bipolar traits of TIMs [15].

In the present section an attempt has been made to explain the principles of the construction of the Model “A” by relying on the bilingual structure of thinking proposed by L.M. Vekker in his unified theory of psychic processes. The hypothesis was advanced that the irrational information aspects may be put into correspondence with the language of simultaneous-spatial gestalts, and the rational ones with the language of one-dimensional successive speech structures. Such a hypothesis explains the requirement to unite in one block of the Model “A” two elements differing by the trait irrationality / rationality. The requirement to unite in one block two elements belonging to one pole of the trait statics / dynamics is explained by the fact that the object of thought at a given moment is either the structure of some system (statics) or a process in which it participates (dynamics). This section also advances the idea of a hierarchical principle for building the “basis” (generating system) of the group, used in socionics, of central bipolar traits of the set of information aspects.

Conclusion to Chapter 2

This chapter has been devoted to the classical model of the intertype relations (IR), which is based on the proposition advanced by Aušra Augustinavichiute: “Information is transmitted to each other only by identical elements” [1, p. 128] — meaning the elements of the communicants’ models.

As the models of the communicants, their Models “A” were considered. Each element of such models, called a function of information metabolism (FIM), “processes” one of the 8 information aspects (IA). The structure of the Model “A” admits 16 distinct types of information metabolism (TIMs). The classical model of the IR presupposes the existence of a 16-element group of operators whose elements allow one, from the Model “A” of one TIM, to obtain the remaining 15 models, which together with the initial one form the socion. The sixteenth element is the identity operator, which is the identity (neutral) element of the group.

Chapter 2 has considered the operators of transformation of three variants of representation of the Model “A” of the TIMs. In each of these cases the connection with the basic idea of the classical theory of the IR was entirely transparent. These operators are attractive for their visual clarity, but are not suited to the quick and mental performance of the required mathematical actions. For that, the group of operators of transformations of the abbreviated symbolic notations of the TIMs proves more suitable — no less visual, though its connection with the basic idea that information is transmitted only by identical elements is not so transparent. That is precisely why it had to be proved that it is isomorphic to the group of operators of the classical IR.

We have not considered the earlier attempts at a mathematical representation of the classical IR involving the use of substitutions (permutations) and matrices (see e.g. [6]). In their time they played a definite role in the study of the group of operators of the classical IR, but there now exists the more convenient mathematical apparatus of a distinctive symbolic calculus, to the explanation of whose advantages the present monograph is devoted. Having mastered this apparatus, one can arrive quite independently at results whose derivation cost socionists a good deal of time owing to the imperfection of the mathematical instruments they used.

In the last section of Chapter 2 an attempt was made to give a psychological grounding to the structure of the Model “A,” relying on Lev Markovich Vekker’s hypothesis of the bilingualism of thinking, advanced in his “Unified Theory of Psychic Processes.” Vekker’s taking account of “informational regularities” in the consideration of psychic processes is, in my view, very close in spirit to the scientific direction we call socionics.

References for Chapter 2

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Banaru A.M., Enina D.A. A geometric representation of the group of intertype relations // SMiPL. — 2014. — №2. — pp. 25–31.
Banaru A.M. From the structural formula of the socion to the deactualization of the asymmetric intertype relations // SMiPL. — 2015. — №5. — pp. 50–54.
Bukalov A.V. Potentsial lichnosti i zagadki chelovecheskikh otnosheniy [The Potential of the Personality and the Riddles of Human Relations]. — Moscow: Chernaya belka, 2009. — 592 pp.
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Minaev Yu.P. The problem of the “splitting” of intertype relations by spectra of coincidences of the poles of bipolar traits // SMiPL. — 2014. — №3. — pp. 30–37.
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Minaev Yu.P., Joynton A. A representation of the classical intertype relations by means of four user-friendly dichotomies // SMiPL. — 2020. — №3–6. — pp. 29–40.
Minaev Yu.P., Datsenko I.P., Pinda M.V. Simple rules for multiplying bipolar traits // PiSMO. — 2017. — №11–12. — pp. 25–35.
Minaev Yu.P. A classification of the directed blocks of Model “A” and the system of formalized notations of the operators of the classical intertype relations // PiSMO. — 2020. — №3–4. — pp. 25–30.
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Reinin G. Tayny tipa. Modeli. Gruppy. Priznaki [Secrets of Type. Models. Groups. Traits]. — Moscow: Chernaya belka, 2010. — 296 pp.
Jung C.G. Psikhologicheskie tipy [Psychological Types] / Trans. from the German. — Moscow: Universitetskaya kniga; AST, 1998. — 720 pp.

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