Reinin Dichotomies
Reinin dichotomies or Reinin traits refer to a set of highertier dichotomies that were derived from first order Jungian dichotomies. Grigoriy Reinin, a mathematician and psychologist and one of the earliest socionists, mathematically proved the existence of these dichotomies, and their approximate content was elaborated by Aushra Augusta. Her work, The Theory of Reinin’s Traits, published in 1985, was intended to be an introduction to the 15 dichotomies  a draft of sorts  but further works have still not been written on the subject. The usefulness of most Reinin dichotomies is consistently questioned by a sort of Socionists.
In common use the term “Reinin dichotomies” often refers to the 11 nonJungian dichotomies.
Mathematics
Definition
Each half of a dichotomy is called a “trait”. Any pair of traits (such as E and P) may be combined to produce a third (static). These three traits form an interdependent triad, meaning that when two are held constant the third must too.
They are combined according to the relation *, defined as follows.
X*Y = (X & Y) or (~X & ~Y)
where ~X denotes the opposite of trait X (~E = I, etc.)
Thus,
static = E*P = E&P or I&J
In most cases, the “&” is understood.
static = EP or IJ
The relation * can be seen to be associative, meaning (X*Y)*Z = X*(Y*Z)
is always true.
Proof:
The relation may be restated more concisely as follows:
X*Y = (X == Y)
where == denotes logical equality  parentheses added for clarity.
We can use a truth table to show that the expressions are the same:
X 
Y 
Z 
(X==Y)==Z 
X==(Y==Z) 

T 
T 
T 
T 
T 
T 
T 
F 
F 
F 
T 
F 
T 
F 
F 
T 
F 
F 
T 
T 
F 
T 
T 
F 
F 
F 
T 
F 
T 
T 
F 
F 
T 
T 
T 
F 
F 
F 
F 
F 
Another property of ==, commutativity, transfers to *:
X*Y = Y*X
Since * is both associative and commutative, it forms an abelian group over four traits:
* 
1 
X 
Y 
Z 

1 
1 
X 
Y 
Z 
X 
X 
1 
Z 
Y 
Y 
Y 
Z 
1 
X 
Z 
Z 
Y 
X 
1 
Here 1 is the identity, which represents the trait that is true of all types. It might also be called the “nonnull” trait. Here X, Y, and Z are all interdependent.
Complete list
Given four initial independent dichotomies (that is, pairs of opposite traits), one can form new ones dependent on the original four. Here the “&”s may be understood, so as to simplify the generation of new dichotomies. Remember, * is associative and commutative, so the order doesn’t matter, and the *s can be left out. For example, E*N = N*E and (E*N)*T = E*(N*T). This means we can just write, in order, the Jungian dichotomies a derived dichotomy depends on. In other words, each new dichotomy can be uniquely specified in terms of dependence on each of the original four. In this case, each dichotomy can be replaced by one of its halves, the corresponding traits. For example, E/I can be replaced with just E or just I. The lack of a unifying name for most dichotomies makes this a convenient choice. ENTP is the preferred basis, for historical reasons. So, each new dichotomy is either dependent or independent of each original dichotomy. This later produces 2^4 = 16 dichotomies.
The first of these, the one dependent on no Jungian dichotomy, is the identity  not a dichotomy proper, as it does not split the socion. The remaining 15 are the type dichotomies as follow:

(null/nonnull)

E (extroversion/introversion)

N (intuitive/sensing)

T (logical/ethical)

P (irrational/rational)

EN (carefree/farsighted)

ET (obstinate/compliant)

EP (static/dynamic)

NT (aristocratic/democratic)

NP (tactical/strategic)

TP (constructivist/emotivist)

ENT (positivist/negativist)

ENP (reasonable/resolute)

ETP (subjectivist/objectivist)

NTP (process/result)

ENTP (questioner/declarer)
It is natural to classify these dichotomies based on how many of the Jungian foundation they depend on. The Jungian foundation is all the order1 traits, and null/nonnull are the only order0 traits.
Reinin dichotomies as combinations
From the above proof, we see that the Reinin dichotomies can be thought of as combinations of the original four.
C = n!/(k!(nk)!)
Where k is the number of elements in the subset and n is the number of elements in the set that is drawn from.
Here there are 4 elements in the original set (EI, NS, TF and PJ). The new sets will have between 2 to 4 parameters, so the number of Reinin dichotomies (not including the original four) is:
4!/(2!(42)!) + 4!/(3!(43)!) + 4!/(4!(44)!) = 6 + 4 + 1 = 11
It’s also easy to find out the combinations on pen and paper. Simply find all the ways you can combine four predefined letters (or any other things) into groups of two, three and four, where the order doesn’t matter.
Use of Reinin dichotomies
Empirical aspects
Above is the mathematical definition of the Reinin dichotomies. Whether they have empirical content, on the other hand, is a completely separate matter. Whether they are welldefined at all is one of the main criticisms of Reinin dichotomies.
The content of several of the derived dichotomies comes naturally from other standard parts of socionics theory. As the correspondence between the dichotomies and Model A becomes more complex, so does the dichotomy’s content become less obvious based on purely theoretical considerations.
Some of the simpler correspondences between the functional and dichotomous models:
Intuitive/sensing and logic/ethics determine a type’s strengths and weaknesses; Introversion/extroversion coincides with the orientation of the leading function (as well as all contact functions); Rationality/irrationality similarly coincides with the “rhythm” of the accepting functions; Static/dynamic determines the conscious (“mental”) elements in each types formula; Aristocratic/democratic determines which elements are blocked together; Reasonable/resolute and subjectivist/objectivist correspond to quadra values;
An interesting question is, can Reinin dichotomies be considered equal in their significance to the original Jungian dichotomies? After all, the derivation works in reverse too. Only speculation can provide an answer at this point. A humorous counterexample is the dichotomies of gender and blood type (reference). Obviously the conjunction of the two via * is totally meaningless. That said, the fact that Reinin dichotomies’ content can be elaborated at all is a testament to the highly general nature of socionic type, whose component dichotomies combine to form an integrated whole, represented mathematically as the nonnull trait: the human being.
Comparison with Model A
The central idea of socionics is that types are unbalanced entities. This understanding is built into Model A, but not into Reinin dichotomies. Therefore, Reinin dichotomies must be supplemented with other theoretical apparatus in order to explain things like relationships as fully as Model A. This could be a good thing in that Reinin dichotomies show the theoretical possibility of isolating the assumptions implicit in Model A and showing which are independent of which.
A few socionists, such as Mironov and others from St. Petersburg, consider the Reinin dichotomies to actually be information processing mechanisms. Three such dichotomy traits combine to form an information element. Most socionists consider the Reinin dichotomies to be divisions whose meaning can only be understood by analyzing their effect on Model A.
Criticism
Reinin Dichotomies has a problem which is that anything of substance is already explained with a thorough understanding of Model A, the Socionic functions, and information aspects, which further implies that it does not offer a whole of descriptive power beyond a narrow set of situations such as talking about ideas or giving evaluations when it comes to the existence of socion. Hence then, what other descriptions the dichotomies provide are vague and unsubstantiated enough to be rendered as useless.
Therefore, Reinin traits should be treated dichotomously instead of, or at least in addition to, in terms of information elements and Model A. The fundamental difference between rational and irrational information elements is not built into the Reinin model. Also, it has no concept of element dominance (i.e. functional ordering). However, it does necessarily render the tools used in Model A as useless, but it is important to keep in mind that the classical traits have a purely dichotomous definition, which further implies that they should also be understandable using dichotomies only. In fact, any regular categorization system will also be reflected into Model A in some more or less regular way  this observation doesn’t prove anything however.
Reinin Dichotomies had specific content intelligible to a majority of intellectual readers, as well as a satisfactory substantiation of the existence of the traits (a "mathematical" substantiation sounds convincing, but the existence of personality traits cannot be proven mathematically), they would have long ago been dismissed.
Superficially, the Reinin Dichotomies seem to fit into the overall theoretical framework of socionics. They show that the basic four Jungian dichotomies are just a subset of 15 hypothetical dichotomies that divide the 16 types into orthogonal halves. An attempt was made to provide tentative descriptions of the other 11 dichotomies. It seems so logical and yet available descriptions are inadequate, if not simply incorrect, when applied to real people. The vagueness of the descriptions makes it very hard to dismiss them as inaccurate, especially as we are used to fuzzy definitions in Socionics.
Augusta presented Reinin's Dichotomies as a hypothesis and stated that they needed further investigation. In contrast with her descriptions of the Jungian Dichomoties and the socionic functions, her descriptions of the Reinin Dichotomies are vague and generally muddled. Most Russian / Ukrainian socionists say that her descriptions weren't very good.
Possible dichotomous explanations
The most natural way to discover the meaning of the dichotomies is as a conflict or harmonizing of some sort between pairs of traits. For example, rationals want stability, but dynamics perceive change, creating a tendency for EJs to want to actively influence their environment to control that change  extroversion. Likewise, irrationals are more comfortable with fluid change, but statics see things as staying the same  which leads them to actively create the change in their environment (also extroversion).
An interesting area of research here would be interclub or intertemperament relationships. We would also need to establish descriptions for the other multipledichotomical categories, such as E/I combined with S/N, and so on.
Reference
[1] https://wikisocion.github.io/content/reinin_dich.html
[2] https://library.socionic.info/reinindichotomiesresearchresults
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