Preserving Visual Clarity in the Transition from Transformations of the Model A of Socionic Types to Transformations of Their Symbolic Designations
Yu. P. Minaev Psychology and Socionics of Interpersonal Relations, No. 1–2, 2021, pp. 40–46 UDC 159.9.075+159.923 (Section: Mathematical Methods in Socionics)
Abstract
This article is the twenty-fifth in a series of publications devoted to the didactics of the mathematical methods of socionics. This time, two representations of the operators of the classical intertype relations are compared. In one case the operators are represented by visual schemes of transformations of the Model A of the types of information metabolism. In the second case the formalized designations of the operators are used, and the types are represented by socionic symbols. It is shown that visual clarity is present in the second case as well. The transformations of the symbolic designations of the socionic types have found their reflection in the new operator designations.
Keywords: socionics, socionic types, classical intertype relations, Model A, group of operators, visual clarity.
Translator’s note: figures are typeset as images and did not survive text extraction; bracketed descriptions are supplied from the surrounding prose. Operator notation as in the 2016 paper: 1 identity, -1 super-ego, 1* kindred, -1* business, c extinguishment, -c duality, c* mirage, -c* semi-duality, m mirror, -m conflict, m* transition to supervisee, -m* transition to supervisor, cm quasi-identity, -cm activation, cm* transition to benefactor, -cm* transition to beneficiary.
Two representations of the “basis” operators of the classical intertype relations
By the classical intertype relations are understood (following M. M. Gut [2]) those introduced by the founder of socionics, Aushra Augustinavichiute [1], so as to distinguish them from the trait-based IRs, one variant of which was proposed by G. R. Reinin [7]. The operators of the classical IRs form a 16-element noncommutative group whose minimal set of generators consists of just three elements. In Fig. 1 one can see two representations of the triple of operators which transform the Model A of ИЛЭ (ILE) — or, correspondingly, the abbreviated symbolic designation of this TIM — into the Models A (or abbreviated symbolic designations) of the following TIMs: ИЛИ (ILI), ИЭЭ (IEE), ЛИИ (LII).
Fig. 1. Two representations of the set of generators of the group of operators of the classical intertype relations.
[Left: Model-A function stacks with arrows showing which cell contents are exchanged under each generator. Right: the chain of abbreviated symbolic designations — ILE transformed by c into ILI (vertical arrow), and by 1* into IEE and by m into LII (horizontal arrows).]
The representation of the operators used in transformations of the Models A practically coincides with that proposed by O. B. Karpenko [3]. The second variant of representation is oriented toward the transformation of the symbolic designations of TIMs and was proposed by us in article [4].
The connection of the first variant of operator representation with the classical theory of IRs is quite obvious, since in it the key premise of the theory is visually apparent: “Only identical elements transmit information to each other” [1, p. 128]. And this variant can be very useful not only in the expanded description of the intertype relations, but also at one’s first acquaintance with the mathematical properties of the group of operators of the classical IRs.
In the second variant of operator representation the connection with the classical theory of IRs is not so obvious, but the transformations of the symbolic designations of TIMs under the action of the operators are discernible quite well. The second variant of representation proves preferable in the solution of various problems concerning the given group.
Construction of the group of IR operators from a 3-element set of generators
The fact that the group of operators of the classical intertype relations can be built on the basis of just three operators, which are involutions (their own inverses), is well illustrated by the left-hand scheme presented in Fig. 2. It is a graph at whose vertices the abbreviated symbolic designations of the TIMs are placed. As we see, from any vertex of the graph one can reach any other vertex along undirected edges of just three kinds, corresponding to the following intertype relations: kindred (1*), extinguishment (c) and mirror (m).
Fig. 2. The minimal (in number of elements) and the extended set of generators of the group of operators of the classical IRs.
[Left: a cross-shaped (“Jung’s cross”) graph on the 16 TIM symbols with three edge types — solid (1*, kindred), dashed (c, extinguishment) and heavy/blue (m, mirror) — plus the formula {(1*)ᵖ · cˡ} · {(-1)ᵏ · m^q}, p, l, k, q ∈ {0; 1}, annotated “does not change the shape of the symbol” for the first factor and “changes the shape of the symbol” for the second. Right: the same graph redrawn so that the edges of the operator super-ego (-1) can also be shown; the extension of the generator set by this operator proves very convenient.]
Let us return to the left-hand scheme. Although the edges corresponding to the operator super-ego (-1) are not visible in it, it is easy to notice that the symbols of the TIMs linked by these relations are also placed symmetrically about the center of the cross — a cross reminiscent of Jung’s cross “with the axis of rationality running at right angles to the axis of irrationality” (as was said in his 1936 work “Psychological Typology,” which can be found in the “Appendix” to [8, p. 658]). Under the action of the operator super-ego (-1), the abbreviated symbolic designations of TIMs change their shape within one rationality pole (remaining on the same pole of the dichotomy irrationality / rationality). Neither the color of the symbol nor the sign changes in the process. The operator of extinguishment (c) changes only the color, and the operator of the kindred IRs (1*) leads only to a change of sign.
The operator of the mirror IRs (m), acting on the abbreviated symbolic designations of the socionic types, leads to a change of color, of sign, and of shape (with a change of the pole of the dichotomy irrationality / rationality). But what was said in the last sentence about the operator m still leaves two possibilities. For example, if ILE⁺ is taken as the example initial TIM, it remains to be determined which of the two variants of the resulting TIM should be chosen. For the correct choice one must in effect be able to determine, from the informational aspect of the first function and the sign, the IA of the second function. In the given case it is necessary to realize that the short symbol is an abbreviated record of the expanded designation, i.e. what is at issue is the intuitive-logical extravert (ИЛЭ, ILE). On the expanded designation of the TIM the operator m acts very simply: it changes the order of the symbols — the first and second functions swap places. Having examined attentively the schemes presented in Fig. 2, it is not difficult to grasp the rule by which the transition from the abbreviated symbolic designations of TIMs to the expanded ones, and back, takes place. It is easy to notice that for the right TIMs (those with the “plus” sign) the mirror partners can be found on both schemes by moving clockwise to the nearest symbol, while for the left ones (those with the “minus” sign) one must move in the opposite direction (counter-clockwise).
Let us show how a set consisting of just two elements can generate an 8-element subgroup of the group of operators of the classical IRs. In doing so we shall use, in parallel, the two representations of them discussed in the preceding part of the article. Let us turn to Fig. 3. In the upper row, the transition from ILE⁺ to LII⁻‘s kindred partner is effected in two steps: first from ILE⁺ to LII⁻ by means of the operator m, and then from LII⁻ to its kindred by means of the operator 1*.
Fig. 3. Introduction of a binary operation (multiplication) on the set of IR operators.
[Upper row: the Karpenko-style arrow schemes for m and 1* applied in succession to the Model-A cells, with the equality of the two-step result to a single scheme. Lower row: the scheme of the product m* = 1* · m, whose arrows are one-sided and form two closed rings.]
For the direct transition an operator m* is introduced as the product of the two already known: m* = 1* ∙ m (this is shown in the lower row of Fig. 3). Attention must be paid here to the order of the factors: the operator m acts first, and then the operator 1*, since m*(x) = 1*(m(x)). The operator m* is called the operator of transition to the supervisee (the supervised 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 the mirror partner.
The resulting operator of transition to the supervisee (m*) differs qualitatively from the operators of the mirror (m) and kindred (1*) IRs. This difference shows up well when the operators are represented as schemes (after Karpenko). In the schemes representing the operators m and 1*, two-sided arrows are used, which is evidence that these operators are their own inverses (are involutions): (m)⁻¹ = m; (1*)⁻¹ = 1*. The same fact can be written differently: m ∙ m = 1; 1* ∙ 1* = 1, where 1 is the operator of the identity IRs. Or thus: m² = (1*)² = 1. In the scheme representing the operator of transition to the supervisee (m*), by contrast, the arrows are one-sided and form two closed rings.
A natural question arises: “What operator results when the operator m* is multiplied by itself (squared)?” Fig. 4 shows the answer to this question. The resulting scheme corresponds to the operator super-ego, which we proposed to denote “-1.” As can be seen from the two-sidedness of the arrows, this operator is an involution and, correspondingly, satisfies the relation (-1)² = 1. The same figure shows, with a concrete example, how the operator super-ego acts on the abbreviated symbolic designation of a TIM (it changes the shape within one rationality pole without change of color or sign).
Fig. 4. Introduction of the operators super-ego (-1) and of transition to the supervisor (-m).*
[Upper row: m* · m* = -1, with the -1 scheme’s two-sided crossing arrows, and the action of -1 on a concrete short symbol. Lower row: two derivations of -m* — as the commuting product m* · (-1), and as m · 1* where the order of factors is essential.]
The lower row of Fig. 4 shows how the operator of transition to the supervisor (-m*) can be obtained. Two essentially different cases are considered. One can multiply the operators of transition to the supervisee (m*) and super-ego (-1). In this case the order of the factors does not matter! These operators are said to commute with each other (the product does not depend on the order of the factors). But in the second case considered, the order of the factors is essential. The supervisor is the mirror partner of the kindred (-m* = m ∙ 1*). Recall that the kindred of the mirror partner is the 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 can be written as (m*)⁻¹ = -m*. This means that these two operators are mutually inverse. In the scheme corresponding to the operator of identity (1) there will be no arrows at all.
Two more operators of the classical IRs are easily obtained: the conflict (-m = -1 ∙ m) and the business (-1* = -1 ∙ 1*) ones. The order of the factors in the products used does not affect the result. As it turned out, the operator super-ego (-1) commutes with absolutely all IR operators. Thus the set of two elements {1*, m}, which do not commute with each other, generated — by means of the binary operation of multiplication, natural for operators — the 8-element group {1, 1*, m, m*, -1, -1*, -m, -m*}. By multiplying elements of this group by one another in any order it is impossible to go beyond its limits. Just as is required of a group in the mathematical sense of the word!
Let us note at once that the TIMs linked to one another by operators from the constructed 8-element group belong to one 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 this trait. Indeed, the membership of a socionic type in a given 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, and the operator m puts into first place the IA that was 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 schemes (after Karpenko) visually shows that the arrows used in these schemes do not cross the line separating the mental ring from the vital one.
If to the pair of mutually non-commuting operators {1*, m} one adds the operator of extinguishment (c), one obtains a set of generators of the entire 16-element group of operators of the classical IRs. Fig. 5 shows an example of the action of this operator on the abbreviated symbolic designation and on the Model A of a concrete TIM, and also the derivation of the operator of transition to the benefactor (cm*) by multiplying the operators of extinguishment (c) and of transition to the supervisee (m*).
Fig. 5. Introduction of the operators of extinguishment (c) and of transition to the benefactor (cm).*
[The c scheme with its paired arrows applied to a Model A and to a short symbol; then the product c · m* = cm* with its one-sided arrows crossing at the center.]
Note that the operator of extinguishment (c) commutes with absolutely all IR operators. This makes it possible to obtain without difficulty all the remaining operators of the 16-element noncommutative group. Besides the operators already named, it came to include: quasi-identity (cm), mirage (c*), duality (-c), activation (-cm), semi-duality (-c*), transition to the beneficiary (-cm*). Two representations of this group are given in Fig. 6.
Fig. 6. The group of operators of the classical IRs.
[All 16 Karpenko-style schemes arranged in two rows: upper row 1, m, 1*, m*, c, cm, c*, cm*; lower row -1, -m, -1*, -m*, -c, -cm, -c*, -cm*. The 1 scheme carries the cell numbers 1, 2 / 4, 3 / 6, 5 / 7, 8 recalling the numbering of the Model-A cells.]
As we see from the one-sided arrows in four of the schemes, the operators m*, -m*, cm*, -cm* are not involutions. This four is one of the cosets with respect to the center of the group. The center of the group is the subgroup {1, -1, c, -c}, gathering all the operators that commute with all elements of the group. The two remaining cosets with respect to the center of the group are the fours {m, -m, cm, -cm} and {1*, -1*, c*, -c*}.
The visual clarity of the formalized designations of the IR operators
The operators that are not involutions are used in the case of the asymmetric intertype relations. As was already said, the representations of these operators as schemes (after Karpenko) are distinguished by the one-sidedness of the arrows used in the schemes. The corresponding formalized operator designations are easy to recognize by the simultaneous presence of the letter “m” and the “asterisk.”
It is easy to notice that the letter “m” occurs in the designations of exactly half of all the operators. In the other half it is absent. It is precisely this other half that is the 8-element commutative subgroup of the 16-element noncommutative group of the classical IRs which “works” in the half of the socion gathering the TIMs belonging to only one of the poles of the dichotomy irrationals / rationals. And can one spot some peculiarity in the schemes of the operators of the subgroup {1, -1, 1*, -1*, c, -c, c*, -c*}? There is such a peculiarity. In the corresponding schemes there are no arrows joining cells with numbers of different parity.
It is likewise easy to notice that in half of 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 only one of the poles of the dichotomy democrats / aristocrats. Is there any peculiarity of the schemes corresponding to the operators of the subgroup {1, -1, m, -m, c, -c, cm, -cm}? One can notice that the corresponding schemes have a vertical axis of symmetry — provided, of course, that the numerals recalling the cell numbers of the Model A are not included in the scheme of the identity operator.
Of the peculiarities of the representations of the operators of the 8-element subgroup {1, 1*, m, m*, -1, -1*, -m, -m*} it was already said earlier that the formalized designations do not contain the letter “c,” and the arrows in the schemes 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 only one of the poles of the dichotomy statics / dynamics. Fig. 7 may serve as an illustration of what was said in the last three paragraphs.
Fig. 7. Correspondence between three sections of the socion and of the set of IR operators.
[Left: a circle cut by the two coordinate axes into quadrants labeled ±1, ±1* (upper), ±c, ±c* (middle), ±cm, ±cm*, ±m, ±m* (lower) — the sections of the operator set. Right: the same circle with the TIM symbols in the corresponding sectors. Legend: I₁ = irrationals/rationals; I₂ = democrats/aristocrats; A₀ = statics/dynamics.]
By the three indicated sections the socion is partitioned into 8 super-ego pairs, for which special designations already exist. To these sections were put into one-to-one correspondence sections of the 16-element set of operators of the classical IRs. But to advance further in the construction of such a correspondence is impossible in principle. Here we run up against the mathematical fact, not recognized by everyone, that any group of bipolar traits is always commutative, whereas the 16-element group of operators of the classical IRs is noncommutative. Consequently, there exists no 16-element group of bipolar traits of TIMs that would be isomorphic to this group of operators.
What, then, does the presence or absence of the “minus” in the formalized designations of the operators of the classical IRs mean? In Fig. 6 the eight operators in whose designations there is no “minus” are well visible. They occupy the upper row ({1, c, m, cm, 1*, c*, m*, cm*}). But this eight 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 seen that the operators -m* and -cm*, which are the inverses of m* and cm* respectively, found no place in this eight. The operator super-ego (-1) does not lead to a change of pole of any of the triple of bipolar traits for which in Fig. 7 the designations I₁ (irrationals / rationals), I₂ (democrats / aristocrats), A₀ (statics / dynamics) are used [5]. This is a consequence of the fact that this operator is the square of other operators (m*, cm*, -m*, -cm*).
The answer to the question posed in the preceding paragraph can be found by noting where the arrows ending in the first cell in the schemes of Fig. 6 come from. In the cases of an empty first cell (i.e. in the cases of 1 and 1*) one must consider that the arrow begins and ends in that same cell, and for that reason is not displayed in the scheme. 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 designations of the operators of the classical IRs contain no “minus” (see the upper row in Fig. 6). Otherwise (see the lower row in Fig. 6) the “minus” will be present.
In the language of the symbolic designations of TIMs the same rule can be formulated thus: the “minus” sign in the formalized designation of an operator of the classical IRs is absent if, and only if, the shape of the IA symbol of the first function of the resulting TIM coincides with the shape of the IA symbol of one of the first two functions (i.e. of the functions of the Ego block) of the initial TIM [6].
Let us give an illustrating example of constructing the formalized designation of an operator of the classical IRs. Suppose we must find the operator that takes LSI⁺ into IEE⁻‘s symbol. One will need to “expand” the designation of the initial TIM to its two-function form. Since the IA symbol of the first function of the resulting TIM does not coincide in shape 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 designation of the conflict IRs.
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The article was received by the editors on 23.05.2021.