The Table of Socion Dichotomy Traits in a Single Formula
Yu. P. Minaev Socionics, Mentology and Personality Psychology, No. 2, 2011, pp. 54–59 UDC 159.9.075 (Section: Mathematical Methods in Socionics)
Abstract
The mathematical properties of the table of socion dichotomy traits are examined for the case of the sign arrangement due to S. I. Churyumov. A formula is proposed for computing the signs of the Augustinavichiute–Reinin binary traits. It is shown that Churyumov’s sign arrangement, together with the choice of four basis binary traits, uniquely determines the ordering both in the list of types of information metabolism and in the list of socion dichotomy traits.
Keywords: socionics, system of types of information metabolism, system of socion dichotomy traits, mathematical methods in socionics.
Traditionally, in the table of Augustinavichiute–Reinin binary traits (ARTs), the accepted ordering is one in which the 8 individual traits come first, then the 4 dyadic ones, with the 3 quadra traits closing the list [2, p. 122]. Under this arrangement no simple rule can be discerned in the placement of the “pluses” and “minuses” that fill the table. And if a typo creeps in, it does not catch the eye. In particular, in the book by Grigory Romanovich Reinin that we have just cited, in at least one place a ”–” stands where a ”+” should be. Hence the desire arises either to make the table more orderly — so that even a cursory glance would suffice to spot unfortunate typos — or to try to find a mathematical formula by which anyone could correctly reconstruct all the signs.
The first path was taken by Semyon Ivanovich Churyumov. He showed that the ordering of TIMs (types of information metabolism) which he uses in his own works can be matched with an ordering of the binary traits such that the entire table, filled with “pluses” and “minuses,” is reconstructed from an elementary cell by “unfolding” according to a very simple rule [3, p. 143]. This rule can be depicted as follows:
[+] → [ + | + ] → [ + + | + + ] → ...
[ + | – ] [ + – | + – ]
[-----------]
[ + + | – – ]
[ + – | – + ]
Note that at each “unfolding” step the number of cells increases fourfold and equals 4ⁿ, where n is the number of the unfolding step. In brief, the proposed rule can also be depicted as:
After the fourth step one obtains a 16×16 table which, apart from the order of columns and rows, differs from the traditional table of socion dichotomy traits by the presence of a “zeroth” column filled entirely with “pluses.” It has been proposed to associate this “zeroth” column with the trait of existence (the Karpenko–Churyumov trait). The new trait is not differentiating (all TIMs receive a ”+” for the fact that they exist), but its introduction at least makes it possible to formulate a simple rule for reconstructing the whole table, provided the orderings of TIMs and ARTs are strictly coordinated.
This article will present a very simple formula by which all the signs of the socion dichotomy trait table, in the arrangement proposed by S. I. Churyumov, are easily reconstructed. We also intend to discuss some important properties of this arrangement to which, as far as we know, no attention has previously been paid. In particular, the meaning of the fundamental (or basis) dichotomies will become clear precisely in this representation of the table. It will be shown how the remaining binary traits are constructed on their basis and by what principle ordinal numbers are assigned to them. In addition, the advantages of writing the ordinal numbers of TIMs and ARTs in the binary number system will be demonstrated.
In what follows we shall assume that the TIMs and ARTs are given by their ordinal numbers in the binary number system from 0000 to 1111 (in decimal this corresponds to the numbers 00 through 15). A specific TIM, in accordance with its number, will be denoted by the symbol |t₁t₂t₃t₄⟩, and a specific ART by ⟨c₁c₂c₃c₄|. The table of socion dichotomy traits in S. I. Churyumov’s arrangement then takes the following form:
| TIM \ ART | 0000 | 0001 | 0010 | 0011 | 0100 | 0101 | 0110 | 0111 | 1000 | 1001 | 1010 | 1011 | 1100 | 1101 | 1110 | 1111 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 00 0000 | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| 01 0001 | + | – | + | – | + | – | + | – | + | – | + | – | + | – | + | – |
| 02 0010 | + | + | – | – | + | + | – | – | + | + | – | – | + | + | – | – |
| 03 0011 | + | – | – | + | + | – | – | + | + | – | – | + | + | – | – | + |
| 04 0100 | + | + | + | + | – | – | – | – | + | + | + | + | – | – | – | – |
| 05 0101 | + | – | + | – | – | + | – | + | + | – | + | – | – | + | – | + |
| 06 0110 | + | + | – | – | – | – | + | + | + | + | – | – | – | – | + | + |
| 07 0111 | + | – | – | + | – | + | + | – | + | – | – | + | – | + | + | – |
| 08 1000 | + | + | + | + | + | + | + | + | – | – | – | – | – | – | – | – |
| 09 1001 | + | – | + | – | + | – | + | – | – | + | – | + | – | + | – | + |
| 10 1010 | + | + | – | – | + | + | – | – | – | – | + | + | – | – | + | + |
| 11 1011 | + | – | – | + | + | – | – | + | – | + | + | – | – | + | + | – |
| 12 1100 | + | + | + | + | – | – | – | – | – | – | – | – | + | + | + | + |
| 13 1101 | + | – | + | – | – | + | – | + | – | + | – | + | + | – | + | – |
| 14 1110 | + | + | – | – | – | – | + | + | – | – | + | + | + | + | – | – |
| 15 1111 | + | – | – | + | – | + | + | – | – | + | + | – | + | – | – | + |
(Translator’s note: the sign pattern in the source PDF did not survive text extraction; the table above is reconstructed from the paper’s own formula, ⟨c|t⟩ = ”+” iff Σcᵢtᵢ is even, which the article states reproduces Churyumov’s arrangement exactly. Column order follows binary ordering 0000–1111 rather than the typographic order of the original.)
Attention should be paid to the columns numbered 1000, 0100, 0010 and 0001. It is precisely by the traits corresponding to these columns of the table that the socion is successively divided four times, so that at the end an ordered list of TIMs is obtained. Let us illustrate this with a scheme of the successive dichotomy of the socion (see next page).
The socion, consisting of 16 TIMs, is divided at the first step of the dichotomy into two groups of eight by the trait ⟨1000|. Those types which received a ”+” by this trait have “0” in the first position of their binary ordinal numbers, while those which received a ”–” have “1” in the first position. In the scheme, the TIMs falling into one group of eight are joined by a curly brace. It is easy to see that all types from the left-hand group of eight have “0” in the first position of their number, and those in the right-hand group have “1.”
At the second step, each group of eight is divided into two groups of four by the trait ⟨0100|. Note that the dichotomy trait is the same for both groups of eight. In the scheme, TIMs from one group of four are joined by a square bracket. The second digit in the type numbers is determined by the sign obtained precisely at this second step of dividing the socion: if ”+,” then “0”; if ”–,” then “1.” As we see, the principle has not changed. The same order is preserved at the subsequent steps of division. The pairs of TIMs obtained upon the dichotomy of the fours by the trait ⟨0010| are marked in the scheme by parentheses. The division of each pair by the trait ⟨0001|, uniform for all pairs, finally gives us a strictly ordered list of TIMs.
SOCION
+----------------+ dichotomy of the socion
+ – by the traits:
⟨1000| ... ⟨1000|
+ – + – ⟨0100|
... ... ... ... ⟨0010|
+ – + – + – + – ⟨0001|
TIMs: 0000 0001 0010 0011 | 0100 0101 0110 0111 || 1000 1001 1010 1011 | 1100 1101 1110 1111
From what has been said, the first convenience of numbers written in the binary system is already apparent. As for the ARTs we used in the successive dichotomy of the socion, even the “outward appearance” of their ordinal numbers (1000, 0100, 0010, 0001) underscores the distinguished role of these binary traits in obtaining the strict sequence of TIMs. Moreover, it is visible at which step of the division each played its part. The corresponding decimal numbers (08, 04, 02, 01) are clearly inferior in this kind of transparency.
As for the numbers of all TIMs without exception, they speak plainly of the signs of the poles of the binary traits used at the corresponding steps of the dichotomy. For example, take the type numbered 1010. The ones in the first and third positions mean that it received ”–” at the first step (by the trait ⟨1000|) and at the third step (by the trait ⟨0010|) of the socion’s dichotomy. And the zeros in the second and fourth positions testify that it has ”+” by the traits ⟨0100| and ⟨0001|.
Of course, some may find it inconvenient to associate ”+” with zero and ”–” with one — especially if accustomed to the coding by the Jungian traits. But this is the price to be paid for the fact that the “zeroth” TIM (number 0000) has all “pluses” in the table of socion dichotomy traits.
Speaking of the coding by Jungian traits: this coding cannot be tied to the ordinal numbers of TIMs without “tearing apart” the quadras — even if the zeros and ones are swapped. With these traits one can obtain ordered lists of TIMs in which representatives of other fours important in socionics are grouped together (by orientation toward kind of activity, by sociotemperaments, by styles of communicability, by stimuli to productive activity). But obtaining the quadras after the second step of the socion’s dichotomy, using only some two traits from the Jungian basis, is impossible in principle! And the fours of consecutively listed TIMs are obtained precisely at the second step, i.e. by means of just two traits: ⟨1000| and ⟨0100| (see the scheme). This means that if one does not wish to “tear apart” the quadras, then the candidates for these “basis” numbers can only be the quadra traits (merry / serious, judicious / decisive, democrats / aristocrats). As we see, the Jungian traits do not make this list.
Of course, the above remarks concerning the Jungian basis may seem strange, since there exist tables in which the TIMs are in the required order (by quadras) while only Jungian binary traits are used. Such a table exists, for example, in the relatively recently published book by Alexander Valentinovich Bukalov [1, p. 80]. And at first glance it may seem that such a table is easily converted into the socion dichotomy scheme we have just discussed. But this is not so. If at the first step the socion is divided, say, by the trait irrationality / rationality, then the first eight TIMs in the list must be irrational. But as is well known, the first two quadras do not consist entirely of irrationals. The Jungian traits are represented uniformly across the quadras. This fact is very well illustrated by the table from A. V. Bukalov’s book. But the question of why in the α and γ quadras two irrationals come first, while in β and δ two rationals do, remains unanswered. Or why in some irrational pairs a logician stands first, and in others an ethicist?
Thus, by successive division of the socion with a criterion uniform for all at each step (by one and the same Jungian binary trait), one cannot obtain an ordered list of TIMs in which the quadra grouping would be respected.
Let us return to the consideration of Churyumov’s sign arrangement in the Augustinavichiute–Reinin table of socion dichotomy traits, in order to work out how the signs in the remaining (non-basis) columns arise and by what principle they are connected with the ordinal numbers of the corresponding binary traits. If one looks at the signs in the column numbered 0011 and compares them with the signs in the columns numbered 0001 and 0010, it is easy to notice that when the basis columns have identical signs, the trait ⟨0011| yields a “plus,” and when they differ — a “minus.” And this testifies precisely to the fact that the trait ⟨0011| was generated by the two basis traits ⟨0001| and ⟨0010| according to the principle described by G. R. Reinin [2, p. 115]. As for the ordinal number of this new trait, it is obtained as the sum of the numbers of the corresponding basis traits into which the new trait can be “decomposed”: ⟨0011| = ⟨0010| + ⟨0001|. Let us generalize this “decomposition” principle to the case of an arbitrary trait:
What sign (”+” or ”–”) will a specific TIM |t₁t₂t₃t₄⟩ receive by the trait ⟨c₁c₂c₃c₄|? Recall that the poles of the fundamental dichotomies corresponding to a given TIM are encoded in its ordinal number by a simple rule: if at the i-th step of division (of which there are only four) the TIM received ”+,” then the i-th position of its ordinal number holds “0,” and if ”–,” then a one. The halving of the socion is carried out by the basis traits in the sequence ⟨1000|, ⟨0100|, ⟨0010|, ⟨0001|.
Clearly, the answer to the question posed must be fully determined by two “vectors”: the bra-vector of the ART, ⟨c₁c₂c₃c₄|, and the ket-vector of the TIM, |t₁t₂t₃t₄⟩ (the terminology is borrowed from quantum mechanics; from the English brackets). Let us introduce a kind of “scalar product” of these vectors:
The formula-definition just given answers the question posed: the signs in the Augustinavichiute–Reinin table of socion dichotomy traits in Churyumov’s arrangement are determined by the “scalar product” formula of the bra-vectors of the ARTs and the ket-vectors of the TIMs.
Suppose we need to find, for the TIM |0111⟩, the sign of the trait ⟨1011|. We compute: 1·0 + 0·1 + 1·1 + 1·1 = 2. The result is an even number. Hence the answer is the sign ”+.” Not difficult, is it?
For now let us regard the proposed formula as one arrived at by guessing, so as not to encumber the present article with reasoning concerning its formal-logical derivation. And that it yields the required distribution of “pluses” and “minuses,” everyone can verify directly for themselves.
Summary
Let us briefly sum up. After the works of A. Augustinavichiute and G. Reinin, all 16 TIMs and 15 ARTs were known, and for these the table of socion dichotomy traits was compiled. S. Churyumov, for the specific sequence of TIMs he uses in his works, showed how — by adding a “zeroth” column filled entirely with “pluses” and permuting the other columns — one can obtain a table of far more attractive “outward appearance.” The attraction consisted in the fact that the whole table was easily reconstructed by a very simple “unfolding” rule, starting from the cell in its upper-left corner. In doing so, of course, the order of the ARTs was rigidly fixed.
Our analysis of the resulting table showed that it is described by a formula which makes it possible, for any trait and any type, to easily “compute” the sign of the dichotomy pole, knowing only the ordinal numbers of the corresponding column and row. The result is obtained especially simply if the ordinal numbers of TIMs and ARTs are written in the binary number system. Moreover, this notation made it possible to indicate the numbers of those binary traits which it is advantageous to regard as the basis ones, since through them a rigid ordering is established without particular difficulty both for the TIMs and for the remaining (non-basis) ARTs.
However, we are compelled to state that Churyumov’s felicitous sign arrangement in itself — which allowed a simple formula to be found for its description — does not uniquely determine the substantive content of the basis traits ⟨1000|, ⟨0100|, ⟨0010|, ⟨0001|. And for any four independent ARTs that occupy the “basis” places in the ordering, there will be a corresponding order of TIMs. Let us illustrate this point with a concrete example.
Each of the three well-known socionists whose books we have cited prefers a particular order of TIMs in the socion list. These lists differ by permutations of TIMs within quadras, but the sequence of the quadras themselves is the same. This means that, after arranging the ARTs in such an order that the placement of “pluses” and “minuses” is à la Churyumov, in all three cases the trait merry / serious — separating the α and β quadras (the first eight TIMs) from γ and δ (the second eight) — will stand at number 1000 (or 08 in decimal), and at number 0100 (or 04 in decimal) the trait democrats / aristocrats, separating the quadras within the eights (α from β and γ from δ). But the substantive content of the traits ⟨0010| and ⟨0001| differs, and is as follows (in the same order): right / left and extraverts / introverts for A. V. Bukalov; right / left and positivists / negativists for G. R. Reinin; carefree / farsighted and extraverts / introverts for S. I. Churyumov.
It is easy to see that other variants are possible, even if one requires that the quadras not be “torn apart” and that the order of the quadras remain unchanged. And such variants do occur in the socionics literature. Thus, if there does exist a distinguished order of TIMs in the socion list, as S. I. Churyumov predicts, then one should seek far stricter constraints on this sequence than the sign arrangement he proposed for the table of socion dichotomy traits. And if such an order is found, it may turn out that the Churyumov arrangement discussed in this article — and with it our formula — will have more than merely mnemonic significance.
References
- Bukalov A. V. The Potential of Personality and the Riddles of Human Relations. — Moscow: Chornaya Belka, 2009. — 592 pp.
- Reinin G. R. Socionics: Typology. Small Groups. — St. Petersburg: “Obrazovanie — Kultura”, 2005. — 240 pp.
- Churyumov S. I. The Cheshire Cat’s Smile, or the Possible and the Impossible in Socionics: Problems, Hypotheses, Solutions. — Kyiv: Drohobych, “Vymir”, 2007. — 560 pp.
The article was received by the editors on 22.07.2010.