A Classification of the Directed Blocks of the Model A and the System of Formalised Designations of the Operators of the Classical Intertype Relations
Yu. P. Minaev Psychology and Socionics of Interpersonal Relations, No. 3–4, 2020, pp. 25–30 UDC 159.9.075 : 159.923 (Section: Mathematical Methods in Socionics)
Abstract
This article is the twenty-third in a series of publications devoted to the didactics of the mathematical methods of socionics. This time the notion of the directed blocks of the Model A is considered. The identification of each element of the set of directed blocks is carried out by means of four dichotomies. To these dichotomies are put into one-to-one correspondence dichotomies of the set of operators of the classical intertype relations. The connection of the indicated dichotomies with certain dichotomies of the socion is also traced.
Keywords: socionics, types of information metabolism, classical intertype relations, Model A, directed blocks, group of operators of the classical intertype relations, central dichotomies of socionic sets.
Translator’s note: operator notation as in the companion papers (1 identity, -1 super-ego, 1* kindred, -1* business, c extinguishment, -c duality, c* mirage, -c* semi-duality, m mirror, -m conflict, m* supervisee, -m* supervisor, cm quasi-identity, -cm activation, cm* benefactor, -cm* beneficiary). Figures typeset as images did not survive extraction; bracketed descriptions are supplied from the prose. TIM pictographs in running text are replaced by acronyms as determined by the prose.
Introduction
Examining the Model A of any individual TIM, one can discern in it the EGO blocks of all 16 types making up the socion. Of course, for this one must consider not only the horizontal blocks but the vertical ones as well. Moreover, each of the 8 undirected blocks must be “read” both from the odd-numbered functions toward the even-numbered ones and in the opposite direction. If one doubles the number of blocks by taking direction into account, and captions each directed block with the designation of the operator of transition from the initial type to the type whose EGO block was “discerned” in the Model A of the initial type, the resulting “picture” will not depend on which particular TIM’s model was examined.
In his time S. I. Churyumov, making use of this fact, proposed representing the Model A as a totality of sixteen radicals expressing the relations of the given socionic type to the others [3, p. 286]. The operator designations he used did not reflect the structure of their noncommutative group. Now, when the mathematical properties of this group have been studied and taken into account in the formalized designations of the operators [1], the representation of the Model A proposed by S. I. Churyumov can look as shown in Fig. 1 (the left-hand “picture”).
To the right of the “picture” representing the general Model A are placed “pictures” showing, by way of example, the models of two concrete TIMs: ИЛЭ (ILE) and ИЭЭ (IEE). Let us consider, as an illustration, the operator of transition to the beneficiary (-cm*). From the general Model A we see that this operator corresponds to the directed block (6, 7). This means that in the EGO block of the beneficiary the 1st (program) and 2nd (creative) functions will process the same IAs as the 6th (activation) and 7th (controlling) functions of the initial TIM, respectively. Thus the beneficiary of ILE is EIE, and the beneficiary of IEE is LIE. What has been said coincides with what can be “read off” the concretizations of the Model A for ILE and IEE. The corresponding formulas are written thus:
-cm*(ILE) = EIE; -cm*(IEE) = LIE.
In Fig. 1 it is not difficult to notice four central dichotomies of the set of operators of the classical intertype relations, which have found their reflection in their formalized designations. This set is divided exactly in half by the criteria of the absence / presence in the designations of the following four elements: the sign ”–,” the letter “c,” the letter “m,” the superscript ”.” As is seen from the example analyzed, the formalized designation of the operator of transition to the beneficiary (-cm) belongs in all four cases to the second pole (all elements present). The operator of identity (1), by contrast, belongs in all four cases to the first pole.
Fig. 1. The directed blocks in the Model A and in two of its concretizations.
[Three panels. Left: the general Model A as eight numbered cells (mental ring 1, 2 / 4, 3; vital ring 6, 5 / 7, 8), with each directed block captioned by its operator: (1,2) 1, (2,1) m, (4,3) -m, (3,4) -1, vertical blocks (1,4) 1*, (4,1) -m*, (2,3) -1*, (3,2) m*, (6,5) -c, (5,6) -cm, (7,8) c, (8,7) cm, (6,7) -cm*, (7,6) c*, (5,8) -c*, (8,5) cm*. Middle and right: the same diagram with the cells filled by the IA symbols of ILE and IEE respectively.]
The task of the present article is to show that to the indicated central dichotomies of the set of formalized designations of the operators of the classical IRs one can put into one-to-one correspondence quite meaningful central sections of the 16-element set of directed blocks of the Model A. We shall satisfy ourselves that to three of these four dichotomies one can also put into one-to-one correspondence bipolar traits of TIMs. These three traits constitute a system of generators for the common 8-element subgroup of the ART and JMT groups. In addition, it will be shown how easily the pole of the fourth dichotomy is determined once an ordered pair of TIMs is given and the sought IR operator effects the transition in this pair from the first TIM to the second.
Classification of the directed blocks of the Model A by four bipolar traits
As was already said, the number of directed blocks is twice the number of undirected ones, on account of the two possible directions between the two functions entering one and the same undirected block. For example, functions 6 and 7 form an undirected block uniting two directed ones: the block (6, 7) considered by us earlier, which we put into correspondence with the operator of transition to the beneficiary (-cm*), and the oppositely directed block (7, 6). As is seen from Fig. 1, to this second directed block (7, 6) corresponds the operator of mirage (c*).
Note that it was not at all necessary to look at Fig. 1. It sufficed to know that EGO blocks differing only in the order of the IA symbols belong to the TIMs linked by the operator m, i.e. the mirror partners. And the mirror partner of the beneficiary is the mirage partner. It is easy to satisfy oneself of this by multiplying the corresponding operators in the required order:
m ∙ (-cm*) = c*.
It is easy to see that a similar connection exists between any two operators corresponding to the two directed blocks entering one undirected block. Multiplying the operator m by one operator of such a pair, we must obtain the second. Let us illustrate this with one more example. Take the undirected block comprising the functions numbered 2 and 3. It consists of two directed blocks: (2, 3), to which corresponds the operator of transition to the supervisee (m*), and (3, 2), to which corresponds the operator of the business relations (-1*). As expected, we obtain:
m ∙ m* = -1* and m ∙ (-1*) = m*.
We have already mentioned the division of the undirected blocks into horizontal and vertical. These same words are quite suitable for the corresponding dichotomy of the set of directed blocks as well. Quite understandable, too, is the division of the undirected blocks into mental and vital, which can also be applied to the directed blocks.
Fig. 2 gives illustrative material for all four dichotomies we intended to consider. In the upper row are the “pictures” corresponding to the first pole of each division. In the lower row are the “pictures” representing the second pole of the corresponding dichotomies. As we see, the first pair of “pictures” from the left corresponds to the division of the blocks by the criterion horizontal / vertical.
Fig. 2. One-to-one correspondence between the dichotomies of the set of directed blocks of the Model A and of the set of operators of the classical IRs.
[Four pairs of panels on the numbered Model-A cells. Pair 1 (horizontal / vertical): top panel shows only the horizontal directed blocks with their operators (1, m, -m, -1, -c, -cm, cm, c); bottom panel only the vertical ones (1*, -m*, m*, -1*, c*, -cm*, cm*, -c*). Pair 2 (mental / vital): top panel the mental-ring blocks (no letter c in the operators); bottom the vital-ring blocks (all operators contain c). Pair 3 (odd / even): wide arrows run from odd- to even-numbered functions, thin arrows from even to odd; top panel gathers the blocks starting on odd functions (operators without m), bottom those starting on even functions (operators with m). Pair 4 (strong / weak): top panel gathers the blocks starting on the strong functions 1, 2, 7, 8 (operators without ”–”), bottom those starting on the weak functions 3, 4, 5, 6 (operators with ”–”).]
It is easy to see that to this division of the blocks corresponds, one-to-one, the absence / presence of the superscript ”*****” in the formalized designations of the operators. Let us note at once that this dichotomy is connected with the bipolar trait democrats / aristocrats, which halves the socion. Indeed, the IR operators corresponding to horizontal blocks, and only they, link TIMs belonging to one and the same pole of the named bipolar trait.
The second pair of “pictures” from the left corresponds to the division of the blocks by the criterion mental / vital. It is easy to see that to this division of the blocks corresponds, one-to-one, the absence / presence of the letter “c” in the formalized designations of the IR operators. Let us note here as well that to this dichotomy of the set of blocks corresponds the bipolar trait statics / dynamics, which is used for TIMs. The formalized designations of the operators of the classical IRs contain no letter “c” if, and only if, both types belong to one pole of this bipolar trait.
Examining the next pair of “pictures,” it is not hard to notice the dichotomy of the set of IR operators by the criterion of the absence / presence of the letter “m” in their formalized designations. To visualize the corresponding dichotomy of the set of directed blocks of the Model A, two shapes of arrows indicating the directions of the blocks are used. The wide arrows run from the odd-numbered functions toward the even-numbered ones, and the thin arrows from the even toward the odd. Depending on which function the arrow begins at, the directed blocks can be divided by the criterion odd / even. In the Model A the division into odd / even functions has a quite definite meaning. The membership of a TIM in the bipolar trait irrationals / rationals is determined by which IAs occupy the odd places. And this means that the letter “m” will be absent from the formalized designation of an IR operator if, and only if, both TIMs belong to one pole of the bipolar trait irrationals / rationals.
Finally, the last two “pictures” differ by the criterion of the absence / presence of the sign ”–” in the formalized designations of the operators. The directed blocks in this case must be distinguished according to whether they begin on the strong (1, 2, 7, 8) or the weak (3, 4, 5, 6) functions. In other words, the criterion strong / weak fixes the corresponding dichotomy of the set of directed blocks. Such a dichotomy has long been used for the horizontal blocks, but it can be extended to the vertical blocks by adopting the indicated convention.
Unlike the three preceding cases, the last dichotomy of the set of directed blocks of the Model A, and the corresponding dichotomy of the set of IR operators, cannot be tied to a single bipolar trait of TIMs. In this case there manifests itself that very effect of “splitting” which G. R. Reinin in his time encountered in attempting to analyze the classical IRs from the standpoint of the group of bipolar traits [2, p. 166]. Let us examine this effect with concrete examples.
Let the initial TIM be ИЛЭ (ILE). With this concretization, the last pair of “pictures” presented in Fig. 2 assumes the form shown in Fig. 3. Let us list all the TIMs to whose EGO blocks operators without the sign ”–” in their formalized designations are assigned in Fig. 3: ИЛЭ (ILE), ИЭЭ (IEE), ИЛИ (ILI), ИЭИ (IEI), ЛИЭ (LIE), ЛСЭ (LSE), ЛИИ (LII), ЛСИ (LSI). Is there a bipolar trait already known from the socionics literature such that all 8 named TIMs belong to one of its poles? There is such a trait — in the JMT group. The first function of all these TIMs processes a detached IA. Consequently, they all belong to the first pole of the trait detached / involved – 1. Note that the initial TIM (whose model is represented in Fig. 3) was ILE. But in the given case it was merely a representative of the democrats.
Fig. 3. The dichotomy of the set of directed blocks of the Model A by the criterion strong / weak, with the example of a democratic TIM.
[The ILE-concretized Model A shown twice: left, the directed blocks beginning on the strong functions with their minus-free operators (1, m, 1*, m*, c, cm, c*, cm*); right, those beginning on the weak functions with the minus-bearing operators (-m*, -1*, -m, -1, -c, -cm, -cm*, -c*).]
Let us now consider an example where the initial TIM is aristocratic. Let it be ИЭЭ (IEE). The corresponding “pictures” are presented in Fig. 4. Let us list all the TIMs to whose EGO blocks operators without the sign ”–” in their formalized designations are assigned in Fig. 4: ИЭЭ (IEE), ИЛЭ (ILE), ИЭИ (IEI), ИЛИ (ILI), ЭИЭ (EIE), ЭСЭ (ESE), ЭИИ (EII), ЭСИ (ESI). And is there in this case a known bipolar trait such that all 8 named TIMs belong to one of its poles? There is such a trait, again in the JMT group. The first function of all these TIMs processes an internal IA. Consequently, they all belong to the first pole of the trait internal / external – 1. Note once again that IEE in the given case was merely a representative of the aristocrats.
Fig. 4. The dichotomy of the set of directed blocks of the Model A by the criterion strong / weak, with the example of an aristocratic TIM.
[Same two-panel layout as Fig. 3, concretized with the IEE symbols.]
Having examined the two concrete examples, we have seen that the “splitting” of the IRs occurred along the line democrats / aristocrats. If the initial TIM on which the operator acts is a democrat, then in deciding whether the formalized designation contains the sign ”–” one must attend to the coincidence / difference in the pair of TIMs of the poles of the bipolar trait detached / involved – 1. This is connected with the fact that the democrats include the researchers [sayentists] (who are detached by functions 1, 2, 7, 8 — that is, by the strong ones) and the socials (whose strong functions are the involved ones). If the initial TIM is an aristocrat, one must attend to the trait internal / external – 1. Indeed, the aristocrats include the humanitarians (who are “doubly internal”) and the managers (whose strong functions process external IAs).
Conclusion
As we have ascertained, any of the 16 directed blocks of the Model A can be specified by the poles of four bipolar traits: horizontal / vertical, mental / vital, odd / even, strong / weak. Recall that the directed blocks in the model of a concrete initial TIM are representatives of the EGO blocks of all 16 socionic types. In determining the pole of the last two traits one must orient oneself by the 1st (program) functions of the resulting TIMs (by the “beginning” of the block).
To the indicated dichotomies of the set of directed blocks of the Model A we succeeded in putting into one-to-one correspondence the dichotomies of the set of operators of the classical IRs by the criteria of the absence / presence in the formalized designations of the following four elements: the superscript ”,” the letter “c,” the letter “m,” the sign ”–.” The stated orders of listing of the traits and of their poles are coordinated. For example, to the directed block characterized by the words vertical, vital, odd, strong will correspond the operator c. Indeed, the first two words gave us the information that in the formalized designation of the operator the superscript ”*” and the letter “c” will be present, and the last two words told us that the letter “m” and the sign ”–” will be absent from it.
We also succeeded in connecting the first three dichotomies of both sets with the coincidence / difference of the poles of bipolar traits in a pair of TIMs. At issue is membership in the poles of the following dichotomies of the socion: democrats / aristocrats, statics / dynamics, irrationals / rationals. If we again take the mirage operator as an example, then from its formalized designation (c*) one can understand that the TIMs coincide by the trait irrationals / rationals (as witnessed by the absence of the letter “m”), but do not coincide by the traits democrats / aristocrats (the superscript ”*****” is present) or statics / dynamics (the letter “c” is present).
What, then, does the absence of the sign ”–” in the designation of the mirage operator (c*) tell us, in the sense of the coincidence / difference of the poles of bipolar traits in a pair of TIMs? So far the answer to this question has been: if the initial TIM is a democrat, the absence of the sign ”–” tells us unambiguously that in the pair of TIMs the poles of the trait detached / involved – 1 coincide; if an aristocrat, then those of the trait internal / external – 1.
If it is known which TIMs make up the pair, determining their membership in the poles of the indicated traits presents no particular difficulty. Nevertheless, let us reformulate this rule in other terms, without resorting directly to the bipolar traits of TIMs. To this end, recall that the sign ”–” was absent from the designations of the operators taking the democrat ILE into ILE, IEE, ILI, IEI, LIE, LSE, LII, LSI, and the aristocrat IEE into IEE, ILE, IEI, ILI, EIE, ESE, EII, ESI.
The general rule in the language of the three-letter codes of TIMs is this: the sign ”–” is absent from the formalized designation of the operator if, and only if, the first letter of the code of the resulting TIM coincides with one of the first two letters of the code of the initial TIM. And in the language of the symbolic designations the same rule can be formulated thus: the sign ”–” is absent if, and only if, the shape of the first symbol of the resulting TIM coincides with the shape of one of the symbols of the initial TIM.
References
- Minaev Yu. P., Datsenko I. P., Pinda M. V. Formalized designations of the operators of intertype relations in place of the Augustinavichiute–Liashkiavichius table // Psychology and Socionics of Interpersonal Relations. — 2018. — No. 9–10. — pp. 41–46.
- Reinin G. Secrets of Type. Models. Groups. Traits. — Moscow: Chornaya Belka, 2010. — 296 pp.
- Churyumov S. I. The Cheshire Cat’s Smile, or the Possible and the Impossible in Socionics. 2nd ed., corrected and expanded. — Book 1. — Moscow: Chornaya Belka, 2011. — 408 pp.
The article was received by the editors on 15.05.2020.