Encyclopedia of Model G and Model A Dichotomy Systems (contains proofs of new mathematical discoveries)

Introduction

Very recently, I discovered a new system of "functional dichotomies" (dichotomies which describe the positions within a Socionics model so they should arguably be called "positional dichotomies"), called the Gulenko-Newman Dichotomies. They uniquely define the workings of Model G. This system of functional dichotomies is distinct and independent from the 7 functional dichotomies which describe the positions of Model A! This new system of dichotomies can also describe a distinct, independent set of 15 type dichotomies (independent from the 15 Reinin dichotomies) which work better in Model G than in Model A (whereas the Reinin dichotomies work better in Model A, though technically both sets of dichotomies can be made to work in both models, just without symmetry in one of the two). This newer set of 15 type dichotomies was previously discovered by the Socionists and mathematicians Ibrahim Tencer and Yuri Minaev (both likely LSI-N in Model G) using group theory and lattice theory, but here I present a new derivation of the Tencer-Minaev dichotomies and structural implications of these dichotomies which Tencer and Minaev did not discover: they lead to an independent system of 35 small group systems, just as the Reinin dichotomies do, but the Tencer-Minaev small groups largely describe the structure of Model G. Probably they did not discover these implications of their work because they are not familiar with Model G. I also discovered a new system of "elemental dichotomies" (which divide the 8 elements of Socionics, Ne, Ti, etc., into pairs). This system describes the horizontal and vertical blocks of Model G, showing that the way Model G divides the elements of Socionics is equally mathematically valid as the way Model A divides them.

The rest of the post will be the explanation and elaboration of all that, so if that was confusing and/or made you curious, that's okay. If you are interested in a thorough understanding of Socionics dichotomy systems and the proofs for the dichotomy systems of Model A and Model G, then you'll want to read this small encyclopedia thoroughly. It will also serve as a good introduction to the concept for someone who wants a deeper understanding of Socionics theory. As this is an encyclopedia, it's pretty long (about 87 pages in Microsoft Word with a normal font size), so if you aren't interested in a thorough understanding of the mathematics of Socionics dichotomy systems and all of the details they generate, then you probably won't be interested in reading this from beginning to end. It's not as long as it seems though; a lot of it is reference information that hardly anyone will read in its entirety at any one time, so the text itself is not especially long. If you are only interested in the results of my recent mathematical discoveries, then you can find them in the much shorter linked article here (https://varlawend.blogspot.com/2023/06/mathematical-discovery-model-g-has-its.html).

I'll include a more advanced table of contents in the future, but for now I'll just include the title of each section so you can find the specific content you need using your search function (ctrl-F):

Table of Contents

Introduction
Table of Contents
Important History of Socionics Mathematics
Model A and interpreting the Reinin Dichotomies
Information/Elemental Dichotomies
Information/Elemental Sets of Blockings
Functional/Positional Dichotomies
Functional/Positional Sets of Blockings
The Reinin Dichotomies
Towards A More Mathematically Complete Socionics Than Model A
All 35 Partitions of 8 Functions/Positions
Criteria of any structurally valid system of Functional/Positional Dichotomies in Socionics
Applying Criterion 3 to all 35 partitions of 8 functions/positions
Deriving and Validating the Gulenko-Newman Dichotomies
Gulenko-Newman Functional/Positional Dichotomies
Gulenko-Newman Functional/Positional Sets of Blockings
Gulenko/Newman Elemental Dichotomies
Gulenko/Newman Elemental Sets of Blockings

Types as Isomorphisms and more General Model A and G Proofs

Set of Bijective Mappings Between Model A Elemental and Functional/Positional Blocking Sets

Set of Bijective Mappings Between Model G Elemental and Functional/Positional Blocking Sets

Proof of Sociotypes as Isomorphisms
Deriving the Tencer-Minaev Dichotomies for Model G
Tencer-Minaev Type Dichotomies
Tencer-Minaev (Model G) Dichotomies
Small Groups for the Reinin (Model A) dichotomies and Tencer-Minaev (Model G) dichotomies
Orbital Small Group Systems (Reinin + Tencer-Minaev)
Mixed Small Group Systems (Reinin)
Mixed Small Group Systems (Tencer-Minaev)
Pure Reinin Small Group Systems
Pure Tencer-Minaev Small Group Systems
Conclusion and Limitations on the Mathematics

____________

Important History of Socionics Mathematics

A mathematically complete description of Socionics has been interesting to many in the Socionics community for a long time. One manifestation of this quest includes the discovery of the 15 Reinin dichotomies, which were discovered by the mathematician and Socionist Grigoriy Reinin in 1985. This was critical to demarcating Socionics from other forms of Jungian typology, and making it a much more complex and thorough theory. Originally, Jungian typologies such as the MBTI and Jungian Cognitive Function theories based on four-letter codes used only 4 dichotomies:

    1. Extroverted/Introverted

    2. Intuitive/Sensing

    3. Thinking/Feeling (in Socionics, we call this Logical/Ethical for a more precise name)

    4. Perceiving/Judging (in Socionics, we call this Irrational/Rational for a more Jungian name)

These 4 dichotomies are enough to produce the familiar type codes (ENTP, INTJ, ENTp, INTj, ILE, LII, etc.). In Socionics, these dichotomies are called the Jungian dichotomies since they are derived directly from classical Jungian typology.

- Extroverted/Introverted refers to whether the Program (sometimes known as Base or Dominant) function of a person is extroverted or introverted.

- Intuitive/Sensing refers to whether the strongest "irrational functions" (Ne/Se/Ni/Si) of a person are intuitive or sensing.

- Logical/Ethical refers to whether the strongest "rational functions" (Ti/Fi/Te/Fe) of a person are logical or ethical.

- Irrational/Rational refers to whether the Program function of a person is irrational or rational.

All of these dichotomies have additional implications which we will explore later, but technically they are almost all we need to make an initial derivation of the Reinin dichotomies and appreciate much of the discovery of Reinin. Let's abbreviate them as follows:

    E: Extroverted/Introverted

    N: Intuitive/Sensing

    T: Logical/Ethical

    P: Irrational/Rational

That's not sufficient to describe a dichotomy system though, because dichotomies can be combined! For example, we could just evaluate whether a person is Extroverted or Introverted, and that's a dichotomy. We could represent Extroverted as "0", and Introverted as "1".

- Extroverted ~= 0

- Introverted ~= 1

*n.b. the symbol "~=" denotes "equality under our definitions", which is similar to “corresponds to”

We could also evaluate both "whether a person is Extroverted or Introverted" and "whether a person is Intuitive or Sensing". And these are two different dichotomies which can be joined together in a binary (Boolean) code, where "Extroverted" people have a "0" on the first element, "Introverted" people have a "1" on the first element, "Intuitive" people have a "0" on the second element, and "Sensing" people have a "1" on the second element:

- Extroverted + Intuitive ~= 00

- Extroverted + Sensing ~= 01

- Introverted + Intuitive ~= 10

- Introverted + Sensing ~= 11

Those are two separate dichotomies, but if we look closely, they also define a third (distinct) dichotomy! That's because we can perform binary addition (also known as modulo 2 addition on the integers) on the elements of each of our codes, and we'll get either a "1" or "0" as the result for each one:

- Extroverted + Intuitive ~= 00 ~= 0 + 0 ~= 0

- Extroverted + Sensing ~= 01 ~= 0 + 1 ~= 1

- Introverted + Intuitive ~= 10 ~= 1 + 0 ~= 1

- Introverted + Sensing ~= 11 ~= 1 + 1 ~= 0

Based on the above calculations, we can see that:

- Extroverted + Intuitive ~= 0 ~= Introverted + Sensing

- Extroverted + Sensing ~= 1 ~= Introverted + Intuitive

So, we have another dichotomy where "Extroverted + Intuitive" and "Introverted + Sensing" are joined on the "0" side, with something in common between these opposite extremes, and "Extroverted + Sensing" and "Introverted + Intuitive" are joined in the same way on the "1" side of the dichotomy. In classical Socionics, the “0” side of this new dichotomy is called “Carefree” whereas the “1” side is called “Farsighted”. This operation of dichotomy addition has previously been defined as interleaving, and we can perform it on any two dichotomies (or on any combination of dichotomies with any other combination of dichotomies, since a combination of dichotomies simply reduces to another singular dichotomy, which is a property called Closure).

In order to derive all of the possible dichotomies which could describe the types in a given dichotomy system, we need a basis (https://en.m.wikipedia.org/wiki/Basis_(linear_algebra)) of dichotomies which are sufficient to single out all of the types. We need the basis because the system of dichotomies can also be understood as a vector space (of sets of binary numbers), and linear combinations of the elements of a basis set are the smallest set sufficient to combinatorially describe all elements of the vector space. The basis for a system of Socionics types has to be exactly 4 dichotomies because there are 16 types, dichotomies split the number of types in half, and:

16 = 2*2*2*2 = 2^4

Thus, 4 is the minimum number of dichotomies we need to describe all the types, which means 4 must be the size of the basis set. Conveniently, as we know from the popular type codes (ENTp, INTj, etc.), the 4 Jungian dichotomies are already sufficient to precisely identify each type and distinguish them from one another (singling them out as we required), so our basis can be exactly ENTP, as we abbreviated previously (we only need to use one basis since it’s sufficient to describe all elements of the system, even though other bases exist). For the sake of clarity, let's show how we can convert this orthographic basis ENTP into a Boolean form that we can use for our interleaving operation, where each type code is equivalent to one unique permutation of 4 Boolean digits:

- ENTp ~= 0000

- ENTj ~= 0001

- INTj ~= 1001

- INFj ~= 1011

- ISFj ~= 1111

n.b. that the Boolean digits are either 0 or 1 precisely depending on the dichotomy in the same position of the type code

Our Jungian basis of abbreviated dichotomies (ENTP from earlier) and the interleaving operation are the only two components we need to define a whole system of dichotomies. Remember that in each code, E represents the first Boolean digit, N represents the second, and so on. Under this interpretation, each dichotomy in our system is defined by a combination of bolded letters, where the letters included signify the specific digits which must add to the same Boolean sum in order to be on the same side of that dichotomy. Notice also that singling out which positions in the type code need to add to the same Boolean sum is exactly what defines our vector space of dichotomies, described by all combinations of the basis set of Jungian dichotomies that make up the type code. For the sake of clarity, let's give two examples:

Example 1: Dichotomy E: Extroverted/Introverted (this only depends on the Boolean sum of the 1st digit, and all other digits are disregarded), which can be defined as the vector (1,0,0,0), which happens to be an element of our basis set

- Extroverted types (sum 0): 0000, 0100, 0010, 0001, 0110, 0101, 0011, 0111

- Introverted types (sum 1): 1000, 1100, 1010, 1001, 1110, 1101, 1011, 1111

Example 2: Dichotomy ENT: Positivist/Negativist (this depends on the Boolean sum of the 1st, 2nd and 3rd digits, and the 4th digit is disregarded), which as you can see is a combination of three elements of our basis set, representable as (1,1,1,0)

- Positivist types (sum 0): 0000, 1100, 1010, 0110, 0001, 1101, 1011, 0111

- Negativist types (sum 1): 1000, 0100, 0010, 1110, 1001, 0101, 0011, 1111

Let's categorize the entire Boolean dichotomy system of Reinin dichotomies as follows (with their most canonical names in Socionics discourse):

- E: Extroversion/Introverted

- N: Intuitive/Sensing

- T: Logical/Ethical

- P: Irrational/Rational

- EN: Carefree/Farsighted

- ET: Yielding/Obstinate

- EP: Static/Dynamic

- NT: Democratic/Aristocratic

- NP: Tactical/Strategic

- TP: Constructivist/Emotivist

- ENT: Positivist/Negativist

- ENP: Judicious/Decisive

- ETP: Merry/Serious

- NTP: Process/Result

- ENTP: Asking/Declaring

We've just successfully defined 11 new dichotomies, which are often referred to as the non-Jungian dichotomies or Reinin dichotomies, using nothing except for our Jungian basis and interleaving operation. The Reinin dichotomies can also refer to the entire system of 15 dichotomies. The side of the dichotomy listed before each “/“ defines 8 types which have a Boolean sum of “0” on the subset of basis vectors represented by the bolded letters, and the side listed after the “/“ represents the other 8 types which have a Boolean sum of “1” for the same subset of basis vectors.

All these dichotomies, differentiations and distinctions make for a potentially much richer theory than any simple Jungian typology that excludes all this logical subtlety. The obstacle to using these non-Jungian dichotomies in a more practical context is that they are only useful if they are empirically interpreted in a way that is semantically (not just mathematically) consistent with the Jungian dichotomies and the rest of the structure of Socionics.

Although the syntactical structure of the Reinin dichotomies should be clear enough now, we haven't explained anything about the names of these dichotomies, their semantic interpretations, or how they relate to the other structural facets of Socionics. For this, we will need to get into the details of Model A (the most popular model of Socionics which was created by Ausra Augustinaviciute, the founder of Socionics).

____________

Model A and interpreting the Reinin Dichotomies

The first structural facets one needs to know about Model A are the elements that it’s filled with: information elements (also sometimes called functions). They correspond with the 8 classical Jungian cognitive functions, but are often defined with a lot more specificity in Socionics. When conceived of as “information elements”, they are subjective properties of the psyche which process different types of information about reality (the type of information is called an information aspect and it has the same name and symbol as the corresponding information element that processes it, which is why information aspects and elements are often confused):

- Ne (Extroverted Intuition, Opportunity Intuition, Black Intuition)

- Se (Extroverted Sensing, Power Sensing, Black Sensing)

- Ti (Introverted Logic, Structural Logic, White Logic)

- Fi (Introverted Ethics, Relation Ethics, White Ethics)

- Ni (Introverted Intuition, Temporal Intuition, White Intuition)

- Si (Introverted Sensing, Comfort Sensing, White Sensing)

- Te (Extroverted Logic, Business Logic, Black Logic)

- Fe (Extroverted Ethics, Emotion Ethics, Black Ethics)

In diagnosing type in Socionics, we need various methods to recognize these information elements in a person and determine their position in a model (like Model A or Model G). In some schools of Socionics the emphasis of this recognition is on tracking certain holistic behaviors associated with the element (Holism), and in other schools the emphasis is on tracking certain structural properties from which the holistic element can be derived (Reductionism). For our purposes, what's important is that these information elements or functions HAVE a specific internal structure which can be described by a set of 7 dichotomies (which we could call elemental dichotomies or information dichotomies) that are important to the structure of all Socionics:

Information/Elemental Dichotomies:

Static/Dynamic:

- Static: Ne, Se, Ti, Fi

- Dynamic: Ni, Si, Te, Fe

Irrational/Rational:

- Irrational: Ne, Se, Ni, Si

- Rational: Ti, Fi, Te, Fe

Extroverted/Introverted:

- Extroverted: Ne, Se, Te, Fe

- Introverted: Ni, Si, Ti, Fi

Detached/Involved:

- Detached: Ne, Ti, Ni, Te

- Involved: Se, Fi, Si, Fe

Implicit/Explicit:

- Implicit: Ne, Fi, Ni, Fe

- Explicit: Se, Ti, Si, Te

Alpha/Gamma:

- Alpha: Ne, Ti, Si, Fe

- Gamma: Se, Fi, Ni, Te

Delta/Beta:

- Delta: Ne, Fi, Si, Te

- Beta: Se, Ti, Ni, Fe

... ... ... ...

One thing we can immediately do with these dichotomies is show how they form an elegant, inter-connected system of 7 sets of blockings. These pair together the elements of Socionics in different ways to get different types of abstract relationships which we see enacted in mature Socionics models. We can show:

- how each of the 7 blocks is formed out of the intersection of 3 of the above Information/Elemental dichotomies (which exactly replicates the structure of the interleaving operation for the Reinin dichotomies, in which two blockings that are opposite in two dichotomies of the same blocking set must be the same in the third dichotomy in the set due to adding to the same Boolean sum, e.g. 00 ~= 0 + 0 ~= 0 ~= 1 + 1 ~= 11, since the sum represents the third dichotomy)

- the specific set of 4 blockings that results from intersecting all logically consistent combinations of these dichotomies

Information/Elemental Sets of Blockings:

Temperament Blocking Set:

- Static/Dynamic

- Irrational/Rational

- Extroverted/Introverted

- Ne+Se: Static, Irrational, Extroverted

- Ti+Fi: Static, Rational, Introverted

- Ni+Si: Dynamic, Irrational, Introverted

- Te+Fe: Dynamic, Rational, Extroverted

Democratic Revision Blocking Set:

- Static/Dynamic

- Detached/Involved

- Alpha/Gamma

- Ne+Ti: Static, Detached, Alpha

- Se+Fi: Static, Involved, Gamma

- Ni+Te: Dynamic, Detached, Gamma

- Si+Fe: Dynamic, Involved, Alpha

Aristocratic Revision Blocking Set:

- Static/Dynamic

- Implicit/Explicit

- Delta/Beta

- Ne+Fi: Static, Implicit, Delta

- Se+Ti: Static, Explicit, Beta

- Ni+Fe: Dynamic, Implicit, Beta

- Si+Te: Dynamic, Explicit, Delta

Extinguishment Blocking Set:

- Irrational/Rational

- Detached/Involved

- Implicit/Explicit

- Ne+Ni: Irrational, Detached, Implicit

- Se+Si: Irrational, Involved, Explicit

- Te+Ti: Rational, Detached, Explicit

- Fe+Fi: Rational, Involved, Implicit

Duality Blocking Set:

- Irrational/Rational

- Alpha/Gamma

- Delta/Beta

- Ne+Si: Irrational, Alpha, Delta

- Se+Ni: Irrational, Gamma, Beta

- Fe+Ti: Rational, Alpha, Beta

- Te+Fi: Rational, Gamma, Delta

Democratic Order Blocking Set:

- Extroverted/Introverted

- Detached/Involved

- Delta/Beta

- Ne+Te: Extroverted, Detached, Delta

- Se+ Fe: Extroverted, Involved, Beta

- Ni+Ti: Introverted, Detached, Beta

- Si+Fi: Introverted, Involved, Delta

Aristocratic Order Blocking Set:

- Extroverted/Introverted

- Implicit/Explicit

- Alpha/Gamma

- Ne+Fe: Extroverted, Implicit, Alpha

- Se+Te: Extroverted, Explicit, Gamma

- Ni+Fi: Introverted, Implicit, Gamma

- Si+Ti: Introverted, Explicit, Alpha

... ... ... ...

There are two major points I want to make here before we move on from elemental dichotomies, for the sake of maximum clarity:

- You might be asking "How do we know these are all the blocks we could produce with these dichotomies?". We know because we've listed all logically consistent combinations of the dichotomies. Let's use Alpha/Gamma as an example. In the Democratic Revision Blocking Set, we combined it with Static/Dynamic and Detached/Involved. In the Duality Blocking Set, we combined it with Irrational/Rational and Delta/Beta.  In the Aristocratic Order Blocking Set, we combined it with Extroverted/Introverted and Implicit/Explicit. Thus we've uncovered the logical relationships to Alpha/Gamma with all 6 of those dichotomies. Combined with Alpha/Gamma itself, and that's all 7 elemental dichotomies, thus all possible logical combinations. You might ask: "Why can't we combine Alpha/Gamma with Extroverted/Introverted and Delta/Beta?" We can't do that because these dichotomies aren't structurally related in this way: if we have Alpha and Extroverted functions, we have Ne+Fe, for example. This implies nothing about Beta and Delta functions, and neither does any other combination of Alpha/Gamma with Extroversion/Introversion. Those two dichotomies only structurally imply something about Implicit/Explicit functions, as we showed in the Aristocratic Order Blocking Set (in a word, Implicit/Explicit is the dichotomy which structurally relates Alpha/Gamma and Extroversion/Introversion, and no other dichotomy fulfills this structural role). So these are all possible blockings of the Informational/Elemental dichotomies, because of the limits of how these dichotomies structurally relate in their system.

- You might notice that I underlined three of the above dichotomies: Static/Dynamic, Irrational/Rational, and Extroverted/Introverted. The reason I did that is to highlight the structure that these dichotomies create, since we will see them again later on and they are especially important to the structure of Socionics models. We can call them the "Bridge Dichotomies", or, if you prefer a more familiar name, the "Temperament Dichotomies". They are part of a larger group of dichotomies called the Orbital Dichotomies (of which there are 7) which describe a lot of necessary structure in Socionics (rather than the content which fills it) and are necessary to understand Socionics from the perspective of Group Theory. This is not important for us, as long as we understand that the 3 orbital dichotomies I underlined are the only one's that show up at the level of the underlying models in Socionics based on 7 dichotomies, and the other 4 (Process/Result, Aristocratic/Democratic, Positivist/Negativist, Asking/Declaring) only show up at the level of systems of 15 type dichotomies (as we already saw with the Reinin dichotomies).

Now that we have all these elements of Socionics, and we've described them to a considerable logical depth while showing their interconnections, we need some way to organize them within a type structure. This is where Model A comes in! (we could use other models like Model G, but we'll get into this later) Model A is a model which describes 8 different positions that the 8 different elements can be slotted into (with some common names in parentheses):

- Position 1 (Program, Base)

- Position 2 (Creative, Instrumental)

- Position 3 (Role, Regulatory)

- Position 4 (Vulnerable, Painful)

- Position 5 (Suggestive, Seeking)

- Position 6 (Activating, Mobilizing)

- Position 7 (Ignoring, Limiting)

- Position 8 (Demonstrative, Background)

Just as the 7 Elemental Dichotomies describe the internal structure of the 8 elements of Socionics, the 7 functional dichotomies (which also could be called the positional dichotomies) describe the internal structure of the 8 positions of Model A. These are all the ways we can group the positions together and distinguish them from one another in a symmetric and binary way within Model A:

Functional/Positional Dichotomies:

Mental/Vital:

- Mental: 1, 2, 3, 4

- Vital: 5, 6, 7, 8

Accepting/Producing:

- Accepting: 1, 3, 5, 7

- Producing: 2, 4, 6, 8

Bold/Cautious:

- Bold: 1, 3, 6, 8

- Cautious: 2, 4, 5, 7

Strong/Weak:

- Strong: 1, 2, 7, 8

- Weak: 3, 4, 5, 6

Inert/Contact:

- Inert: 1, 4, 6, 7

- Contact: 2, 3, 5, 8

Valued/Unvalued:

- Valued: 1, 2, 5, 6

- Unvalued: 3, 4, 7, 8

Evaluatory/Situational:

- Evaluatory: 1, 4, 5, 8

- Situational: 2, 3, 6, 7

The numbers are somewhat arbitrary; they are just used to describe certain positions in Model A in a logically consistent way, via the conventional number associated with each position.

... ... ... ...

From these functional/positional dichotomies, we can generate 7 blocking sets with precisely the same abstract structure as the 7 blocking sets of the elemental dichotomies:

Functional/Positional Sets of Blockings:

Temperament Blocking Set:

- Mental/Vital

- Accepting/Producing

- Bold/Cautious

- 1+3: Mental, Accepting, Bold

- 2+4: Mental, Producing, Cautious

- 5+7: Vital, Accepting, Cautious

- 6+8: Vital, Producing, Bold

Freudian Blocking Set:

- Mental/Vital

- Strong/Weak

- Valued/Unvalued

- 1+2 (Ego): Mental, Strong, Valued

- 3+4 (Superego): Mental, Weak, Unvalued

- 7+8 (Id): Vital, Strong, Unvalued

- 5+6 (Superid): Vital, Weak, Valued

Vertical Blocking Set:

- Mental/Vital

- Inert/Contact

- Evaluative/Situational

- 1+4: Mental, Inert, Evaluative

- 2+3: Mental, Contact, Situational

- 6+7: Vital, Inert, Situational

- 5+8: Vital, Contact, Evaluative

Extinguishment Blocking Set:

- Accepting/Producing

- Strong/Weak

- Inert/Contact

- 1+7: Accepting, Strong, Inert

- 3+5: Accepting, Weak, Contact

- 2+8: Producing, Strong, Contact

- 4+6: Producing, Weak, Inert

Duality Blocking Set:

- Accepting/Producing

- Valued/Unvalued

- Evaluative/Situational

- 1+5: Accepting, Valued, Evaluative

- 3+7: Accepting, Unvalued, Situational

- 2+6: Producing, Valued, Situational

- 4+8: Producing, Unvalued, Evaluative

Dimensional Blocking Set:

- Bold/Cautious

- Strong/Weak

- Evaluative/Situational

- 1+8 (4D): Bold, Strong, Evaluative

- 3+6 (2D): Bold, Weak, Situational

- 2+7 (3D): Cautious, Strong, Situational

- 4+5 (1D): Cautious, Weak, Evaluative

Confidence Value Blocking Set:

- Bold/Cautious

- Inert/Contact

- Valued/Unvalued

- 1+6: Bold, Inert, Valued

- 3+8: Bold, Contact, Unvalued

- 4+7: Cautious, Inert, Unvalued

- 2+5: Cautious, Contact, Valued

... ... ... ...

- We know this system of blocking sets is complete for the same reasons we knew it for the elemental dichotomies; it's literally the same structure with different names.

- As with the elemental dichotomies, you can see that I have underlined three of the functional/position dichotomies in the blockings above. You can also see that they form the same structure that we saw in the elemental dichotomies (all 3 together in one blocking set, then each of them in two other separate blocking sets). The reason for this is that they correspond exactly to the three underlined dichotomies from the elemental dichotomies, which will be very important when combining the elements with the numbered positions to form a complete working model of Sociotype. Mental/Vital corresponds to Static/Dynamic; if Mental functions are Static, Vital functions are Dynamic, and if Mental functions are Dynamic, then Vital functions are Static. Accepting/Producing corresponds to Irrational/Rational in the same way. Bold/Cautious corresponds to Extroverted/Introverted in the same way. We will see all 6 of these dichotomies again when we expand the dichotomies even more, so don't forget them.

We can also show exactly how the sets of functional/positional blockings of Model A are filled by the sets of elemental blockings for any given type. Let's use ILE as an example:

- the Temperament Blocking Set of the elemental dichotomies corresponds exactly to the Temperament Blocking Set of the functional/positional dichotomies

- the Democratic Revision Blocking Set of the elemental dichotomies corresponds exactly to the Freudian Blocking Set of the functional/positional dichotomies

- the Aristocratic Revision Blocking Set of the elemental dichotomies corresponds exactly to the Vertical Blocking Set of the functional/positional dichotomies

- the Extinguishment Blocking Set of the elemental dichotomies corresponds exactly to the Extinguishment Blocking Set of the functional/positional dichotomies

- the Duality Blocking Set of the elemental dichotomies corresponds exactly to the Duality Blocking Set of the functional/positional dichotomies

- the Democratic Order Blocking Set of the elemental dichotomies corresponds exactly to the Dimensionality Blocking Set of the functional/positional dichotomies

- the Aristocratic Order Blocking Set of the elemental dichotomies corresponds exactly to the Confidence Value Blocking Set of the functional/positional dichotomies

It's notable that there are three separate Blocking Sets here which are exactly the same between models, which gives us two interesting insights:

- the Temperament Blockings Sets are the same, which will be very important later when we start building new systems of dichotomies (this similarity is a necessity for the compatibility and synchronized functioning of any system of elemental dichotomies with any system of functional/positional dichotomies)

- the Extinguishment Blocking Sets and Duality Blocking Sets are both the same between systems, which is curious because they share a corresponding dichotomy: Irrational/Rational (for the elements) and Accepting/Producing (for the positions). This corresponds to the fact that Irrational/Rational is a more special dichotomy than even the other temperament dichotomies and most of the other orbital dichotomies; it is called a central dichotomy because of some facets of group theory and the central role it plays in organizing many aspects of Socionics theory (we can see one manifestation of that centrality here due to the unusual similarity of these blocking sets between systems). The other central dichotomies are Democratic/Aristocratic and Process/Result. We can see Democratic/Aristocratic playing a special role here in how it organizes the Order and Revision Blocking Sets into two different sets each (to see this, just look at the titles of the blocking sets and see which function pairs they correspond to). The special role of Process/Result is harder to see here, but it happens to be generated exactly out of the dichotomies of Irrational/Rational + Democratic/Aristocratic = Process/Result, which we will see later (and in time we will explore this dichotomy much more). For now, just understand that these dichotomies have a special importance, and we will focus on this higher mathematics later.

... ... ... ...

The Reinin Dichotomies

Now we have a positional model (the functional/positional dichotomies of Model A) and something to fill it with (the elements described by the elemental dichotomies). From this, we can go full circle to generate a more complicated dichotomy system: the Reinin dichotomies (dichotomies at the level of the type itself) that we already generated using our Jungian basis combined with Boolean logic. Now we can also generate them from elemental and functional/positional dichotomies to connect all these parts into a seamless whole:

E: Extroversion/Introverted

- Extroverted: Bold positions are filled with Extroverted elements, Cautious positions are filled with Introverted elements

- Introverted: Bold positions are filled with Introverted elements, Cautious positions are filled with Extroverted elements

N: Intuitive/Sensing

- Intuitive: half of the Strong positions are filled with Detached+Implicit (Irrational) elements, half of the Weak positions are filled with Involved+Explicit (Irrational) elements

- Sensing: half of the Strong positions are filled with Involved+Explicit (Irrational) elements, half of the Weak positions are filled with Detached+Implicit (Irrational) elements

T: Logical/Ethical

- Logical: half of the Strong positions are filled with Detached+Explicit (Rational) elements, half of the Weak positions are filled with Involved+Implicit (Rational) elements

- Ethical: half of the Strong positions are filled with Involved+Implicit (Rational) elements, half of the Weak positions are filled with Detached+Explicit (Rational) elements

P: Irrational/Rational

- Irrational: Accepting positions are filled with Irrational elements, Producing positions are filled with Rational elements

- Rational: Accepting positions are filled with Rational elements, Producing positions are filled with Irrational elements

EN: Carefree/Farsighted

- Carefree: half of the Evaluative positions are filled with Alpha+Delta (Irrational) elements, half of the Situational positions are filled with Gamma+Beta (Irrational) elements

- Farsighted: half of the Evaluative positions are filled with Gamma+Beta (Irrational) elements, half of the Situational positions are filled with Alpha+Delta (Irrational) elements

ET: Yielding/Obstinate

- Yielding: half of the Evaluative positions are filled with Delta+Gamma (Rational) elements, half of the Situational positions are filled with Alpha+Beta (Rational) elements

- Obstinate: half of the Evaluative positions are filled with Alpha+Beta (Rational) elements, half of the Situational positions are filled with Delta+Gamma (Rational) elements

EP: Static/Dynamic

- Static: Mental positions are filled with Static elements, Vital positions are filled with Dynamic elements

- Dynamic: Mental positions are filled with Dynamic elements, Vital positions are filled with Static elements

NT: Democratic/Aristocratic

- Democratic: Freudian Blocks are filled with Democratic Revision Blockings, Vertical Blocks are filled with Aristocratic Revision Blocks, Dimensional Blocks are filled with Democratic Order Blockings, and Confidence Value Blocks are filled with Aristocratic Order Blockings

- Aristocratic: Freudian Blocks are filled with Aristocratic Revision Blockings, Vertical Blocks are filled with Democratic Revision Blocks, Dimensional Blocks are filled with Aristocratic Order Blockings, and Confidence Value Blocks are filled with Democratic Order Blockings

NP: Tactical/Strategic

- Tactical: half of the Inert positions are filled with Detached+Implicit (Irrational) elements, half of the Contact positions are filled with Involved+Explicit (Irrational) elements

- Strategic: half of the Inert positions are filled with Involved+Explicit (Irrational) elements, half of the Contact positions are filled with Detached+Implicit (Irrational) elements

TP: Constructivist/Emotivist

- Constructivist: half of the Inert positions are filled with Involved+Implicit (Rational) elements, half of the Contact positions are filled with Detached+Explicit (Rational) elements

- Emotivist: half of the Inert positions are filled with Detached+Explicit (Rational) elements, half of the Contact positions are filled with Involed+Implicit (Rational) elements

ENT: Positivist/Negativist

- Positivist: the combined traits of Extroverted+Democratic, or the combined traits of Introverted+Aristocratic (there is more than one way to form this but it's not straightforward in Model A)

- Negativist: the combined traits of Extroverted+Aristocratic, or the combined traits of Introverted+Democratic (there is more than one way to form this but it's not straightforward in Model A)

ENP: Judicious/Decisive

- Judicious: half of the Valued positions are filled with Alpha+Delta (Irrational) elements, half of the Unvalued positions are filled with Gamma+Beta (Irrational) elements

- Decisive: half of the Valued positions are filled with Gamma+Beta (Irrational) elements, half of the Unvalued positions are filled with Alpha+Delta (Irrational) elements

ETP: Merry/Serious

- Merry: half of the Valued positions are filled with Alpha+Beta (Rational) elements, half of the Unvalued positions are filled with Delta+Gamma (Rational) elements

- Serious: half of the Valued positions are filled with Delta+Gamma (Rational) elements, half of the Unvalued positions are filled with Alpha+Beta (Rational) elements

NTP: Process/Result

- Process: the combined traits of Irrational+Democratic, or the combined traits of Rational+Aristocratic (there is more than one way to form this but it's not straightforward in Model A)

- Result: the combined traits of Irrational+Aristocratic, or the combined traits of Rational+Democratic (there is more than one way to form this but it's not straightforward in Model A)

ENTP: Asking/Declaring

- Asking: the combined traits of Static+Democratic, or the combined traits of Dynamic+Aristocratic (there is more than one way to form this but it's not straightforward in Model A)

- Declaring: the combined traits of Static+Aristocratic, or the combined traits of Dynamic+Democratic (there is more than one way to form this but it's not straightforward in Model A)

... ... ... ...

For the sake of clarity, let's also fill out the Reinin dichotomies of the type ILE in Model A using functional and elemental dichotomies, as an example.

n.b. the link below contains a table with a key to which symbols correspond to which information elements to use if you aren't aware of how they correspond (https://en.wikipedia.org/wiki/Socionics#Information_metabolism_elements_(often_confused_with_memetics)).

E: Extroversion

- We can see that ILE is Extroverted because all of its Extroverted (black) elements are in Bold positions (1, 3, 6, 8).

N: Intuitive

- We can see that ILE is Intuitive because half of its Strong positions (1, 2, 7, 8) are filled with Intuitive (triangular) elements, in positions 1 and 7

T: Logical

- We can see that ILE is Logical because half of its Strong positions (1, 2, 7, 8) are filled with Logical (square) elements, in positions 2 and 8

P: Irrational

- We can see that ILE is Irrational because all of its Irrational (triangular and circular) elements are in Accepting positions (1, 3, 5, 7)

EN: Carefree

- We can see that ILE is Carefree because half of its Evaluative positions (1, 4, 5, 8) are filled with Alpha, Delta and Irrational (Ne and Si) elements, in positions 1 and 5

ET: Yielding

- We can see that ILE is Yielding because half of its Evaluative positions (1, 4, 5, 8) are filled with Delta, Gamma and Rational (Fi and Te) elements, in positions 4 and 8

EP: Static

- We can see that ILE is Static because all of its Static (Ne, Ti, Se and Fi) elements are in Mental positions (1, 2, 3, 4)

NT: Democratic

- We can see that ILE is Democratic because its Freudian blocks (1+2, 3+4, 5+6, 7+8) are filled with Democratic Revision Blockings (1+2~=Ne+Ti, 3+4~=Se+Fi, 5+6~=Si+Fe, 7+8~=Ni+Te)

NP: Tactical

- We can see that ILE is Tactical because half of its Inert positions (1, 4, 6, 7) are filled with Intuitive (triangular) elements, in positions 1 and 7

TP: Constructivist

- We can see that ILE is Constructivist because half of its Inert positions (1, 4, 6, 7) are filled with Ethical (L-shaped) elements, in positions 4 and 6

ENT: Positivist

- We can see that ILE is Positivist since we can already see that it's Extroverted and Democratic

ENP: Judicious

- We can see that ILE is Judicious since half of its Valued positions (1, 2, 5, 6) are filled with Alpha, Delta and Irrational (Ne and Si) elements, in positions 1 and 5

ETP: Merry

- We can see that ILE is Merry since half of its Valued positions (1, 2, 5, 6) are filled with Alpha, Beta and Rational (Ti and Fe) elements, in positions 2 and 6

NTP: Process

- We can see that ILE is Process since we can already see that it's Irrational and Democratic

ENTP: Asking

- We can see that ILE is Asking since we can already see that it's Static and Democratic

It is also interesting that some of these dichotomies (Positivist/Negativist, Process/Result, Asking/Declaring, all of which happen to be Orbital Dichotomies) aren't entirely straightforward using the classical Model A approach, which is an insight we may come back to later. Nonetheless, they are still defined in a sufficiently clear and consistent way, and we've now successfully connected three systems of dichotomies (the Reinin type dichotomies, the information/elemental dichotomies and the functional/positional dichotomies) into a seamlessly synchronized whole, derivable from multiple directions. This is one of the impressive things about Socionics that makes it so much more elegant, deep and thorough than almost any other theory of personality typology in existence. What's even more amazing is that we can go even further than this, creating a deeper structure in which both Model A and Model G (the other most popular model in Socionics created by the 2nd most cited Socionist, Victor Gulenko) can co-exist in a more harmonious whole.

____________

Towards A More Mathematically Complete Socionics Than Model A

It's important to mention that Socionics is not limited to these mathematical, structural, syntactic issues; it also includes the problem of semantically interpreting all of these dichotomies, elements and models. Because of this, it has become a point of pride and intellectual thoroughness for Socionists to semantically define, or at least experimentally research, as many of the Reinin dichotomies as possible, and a number of them have already made important attempts:

- Victor Gulenko (School of Humanitarian Socionics)

- Vladimir Mironov (School of Dynamic Socionics)

- Sergey Kutuzov (School of Imperative Socionics)

- Ruslan Stepanov (Psi-Mechanics)

- Vera Stratievskaya (Socionics from Stratievskaya)

- Tatyana Prokofyeva (Socionics Research Institute)

- Vladimir Vincent (World Socionics Society)

- Kimani White (Model L)

- Jack Oliver Aaron (World Socionics Society)

While this might have been difficult and taken a lot of effort, in addition to having led to some useful insights into people, it is not sufficient for a mathematically complete and justified Socionics! That's because Victor Gulenko, Ibrahim Tencer, Yuri Minaev and I have discovered even more dichotomy systems which follow from the fundamental mathematical structure of Socionics, and they are mathematically independent, sound and complete in their own right. Thus, until these mathematical structures are interpreted, or until they are falsified empirically, our work in understanding and defining Socionics will never be complete with the Reinin dichotomies. Ultimately, both Model A and Model G seem to be connected in a greater harmonious whole, so it behooves us even more to work together to uncover the secrets of Socionics and aid in the understanding and development of humanity that Socionics can provide, instead of engaging in counterproductive squabbling amongst Socionists and typologists.

One step we need to take towards this is to seek out and acknowledge any unquestioned assumptions or arbitrary constructions in our model, relaxing these inappropriate assumptions and finding logically solid intellectual ground before moving forward again. One large example of that in the mathematical work that we've done is in generating the functional/positional dichotomies which serve as the foundation for the Model A blocks. They are 7 ways of splitting the functions which happen to give us a structurally sound model:

- Mental: 1, 2, 3, 4, Vital: 5, 6, 7, 8

- Accepting: 1, 3, 5, 7, Producing: 2, 4, 6, 8

- Bold: 1, 3, 6, 8, Cautious: 2, 4, 5, 7

- Strong: 1, 2, 7, 8, Weak: 3, 4, 5, 6

- Inert: 1, 4, 6, 7, Contact: 2, 3, 5, 8

- Valued: 1, 2, 5, 6, Unvalued: 3, 4, 7, 8

- Evaluatory: 1, 4, 5, 8, Situational: 2, 3, 6, 7

These dichotomies are partitions of the set of numbers from 1 to 8 into 2 equal-sized groups. However, there are obviously many more ways to partition these numbers than the 7 we chose. In a set of 8 elements, we partition them into 2 groups by choosing 4 of them (mathematically, this is 8 choose 4). It's a little more complicated than that because the partition {1, 2, 3, 4} is the same as {5, 6, 7, 8}, since they partition the set into the same two groups. Thus, the true number of dichotomous partitions of an 8-function model is:

8C4 / 2 = 70 / 2 = 35

That's a lot of dichotomies, but not such an overwhelming amount that we couldn't investigate them by hand. Prima facie, the other 28 dichotomies on this list could be equally as valid as the 7 dichotomies which serve as a somewhat logically arbitrary foundation for Model A. We can represent all 35 of these dichotomous partitions in an abbreviated form by including the side of each partition which happens to include position 1 (because every partition has at least one half that includes 1 by definition, and it gives us a standardized way of writing them to include only the half that includes 1 each time):

All 35 Partitions of 8 Functions/Positions

1234: Mental/Vital

1235

1236

1237

1238

1245

1246

1247

1248

1256: Valued/Unvalued

1257

1258

1267

1268

1278: Strong/Weak

1345

1346

1347

1348

1356

1357: Accepting/Producing

1358
1367

1368: Bold/Cautious

1378
1456

1457

1458: Evaluative/Situational

1467: Inert/Contact

1468

1478

1567

1568

1578

1678

... ... ... ...

We can see the 7 original functional/positional dichotomies of Model A marked among the 35. Although some of the others might also be meaningful or structurally valid partitions, we would need some criteria to decide which one's; it would be just as indiscriminate as our original Model A assumptions to assume that all of them are valid. The key to knowing which partitions are valid is to remember that these functional/positional dichotomies don't operate alone; they generate a model which has to be filled by (and thus must be structurally compatible with) the 8 elements of Socionics, and thus a set of 8 elemental dichotomies which describe them. That's convenient, because we already noticed a key structural similarity between the elemental dichotomies and the functional/positional dichotomies; there is a precise correspondence between the 3 underlined dichotomies in the sets of blockings of each of them, which we named temperament dichotomies or bridge dichotomies. More precisely, we discovered 3 sets of functional partitions which exactly correspond to the 3 temperament dichotomies: Mental/Vital (which corresponds to Static/Dynamic), Accepting/Producing (which corresponds to Irrational/Rational), and Bold/Cautious (which corresponds to Extroverted/Introverted). Based on the logic of how these dichotomies must combine with other temperament dichotomies and how they must relate to the non-temperament dichotomies in any dichotomy system, we can produce three minimal criteria for valid functional/positional partitions:

Criteria of any structurally valid system of Functional/Positional Dichotomies in Socionics

Criterion 1: The system must include Mental/Vital, Accepting/Producing and Bold/Cautious (although they could have different names, and importantly, by their interpretation in Socionics, these dichotomies are necessarily included in the same blocking set)

- Justification: The elements are a fundamental aspect of Socionics, without which it's hard to say that we are doing Socionics anymore. And the elements are already divided according to the temperament dichotomies: Static/Dynamic, Irrational/Rational, Extroverted/Introverted; in fact, every elemental blocking set includes at least one of these dichotomies. Since a set of elemental blockings with this structure needs to fill the positions of any Socionics model we create, the positional blocking sets must be wholly structurally compatible with those 3 temperament dichotomies. The structure of the temperament dichotomies is that they form a Temperament Blocking Set with exactly themselves, and this Temperament Blocking Set needs to be able to fill a precisely compatible Temperament Blocking Set in the functional/positional dichotomy system (which corresponds exactly to the blocking of Mental/Vital, Accepting/Producing, and Bold/Cautious). For purpose of contradiction, let's assume that a structurally different set of dichotomies could produce exactly the same blocking (1+3, 2+4, 5+7, 6+8, n.b. technically the numbers are arbitrary up to logical consistency and we are only using these by Model A convention, but the point is that the two positions included in each blocking must have elements of the same temperament). Let's call the first dichotomy x1/x2, the second dichotomy y1/y2, and the third dichotomy z1/z2. Without loss of generality, we could define 1+3 as being formed by {x1, y1, z1}. This implies that 2+4 is formed by {x1, y2, z2} (we could also say it's formed by {x2, y1, z2} or {x2, y2, z1}, but this doesn't make any difference to the proof since the order of dichotomies is arbitrary, thus it would just mean that whatever dichotomy we originally assigned to x1/x2 would apply to y1/y2 or z1/z2 instead, respectively). Accordingly, 5+7 is formed by {x2, y1, z2}, and 6+8 is formed by {x2, y2, z1}. But, this is necessarily exactly the same as Mental/Vital standing in for x1/x2, Accepting/Producing standing in for y1/y2 and Bold/Cautious standing in for z1/z2 (this completes the proof by contradiction). Thus, although these dichotomies might have different names or number designations, we have proved that they would always have to be structurally identical with {Mental/Vital, Accepting/Producing, Bold/Cautious}. This also proves that any blocking sets are produced by a structurally unique set of three dichotomies (since the proof trivially generalizes to any blocking set). Thus, since the Temperament Blocking Set in the elemental dichotomies must map to a Blocking Set in the functional/positional dichotomies, and any three functional/positional dichotomies that the Temperament Blocking Set maps to are by necessity structurally identical to {Mental/Vital, Accepting/Producing, Bold/Cautious}, Criterion 1 follows.

Criterion 2: Each of the 4 non-temperament dichotomies in a system of functional/positional dichotomies needs to be included in 3 unique blocking sets, each with one other non-temperament dichotomy and one temperament dichotomy.

- Justification: Let's take a given non-temperament dichotomy D. In the functional/positional systems of dichotomies that includes D, there are necessarily 6 more dichotomies which can be labeled N1, N2 and N3 for the other 3 non-temperament dichotomies, and T1, T2 and T3 for the 3 temperament dichotomies. There is no loss of generality from our labeling since the order of the dichotomies doesn't affect the logic, as long as we respect any necessary logical structure. We know by our proof in Criterion 1 that there is a Temperament Blocking Set, so we already have one blocking set {T1, T2, T3} for certain. Since we know {T1, T2, T3} is a blocking set, we know that no temperament dichotomy can appear with any other temperament dichotomy in any other blocking set, since if we had, for example, {T1, T2, ?}, then we would know that ?=T3 due to the unique logical relationship of these dichotomies (and order clearly doesn't matter here, since we could have chosen any of the temperament dichotomies as "?" and it follows in the same way). Since T1 can't pair with T2 or T3 anymore, there are 4 more dichotomies it can possibly pair with, and it could do so as follows: {T1, D, N1}, {T1, N2, N3}. For T2 we would have {T2, D, N2}, {T2, N1, N3}. For T3 we would have {T3, D, N3}, {T3, N1, N2}. There are no more possible blockings, since the 7 we have produced include each of the dichotomies with each of the other 6 exactly one time each. Thus, we've proven that every non-temperament dichotomy is included in 3 unique blocking sets, each with one other non-temperament dichotomy and one temperament dichotomy, so Criterion 2 follows.

Criterion 3: Both sides of all functional/positional dichotomies in the system must respect the symmetry of Mental/Vital, Accepting/Producing and Bold/Cautious.

- Justification: This is trivially true for the 3 temperament dichotomies since together they form a Blocking Set; this implies that each half of each of the dichotomies can be split into two equal parts by each of the other two dichotomies (for examples of that, just see the Blocking Sets we've already exhaustively listed). To see why each of the non-temperament dichotomies must respect the symmetry of all temperament dichotomies, it's enough to understand what it means for two dichotomies to be in the same blocking set. Let's say we have a given functional/positional dichotomy "x" with positions {x1, x2, x3, x4} on one side, and positions {x5, x6, x7, x8} on the other side. If I interleave another distinct dichotomy "y" with x, y partitions each side of x based on which positions are consistent with the two sides of y (and y partitions the same positions that x does, x1 through x8, since all functional/positional dichotomies are ways of partitioning the same positions). There are two ways to partition one side of dichotomy x with dichotomy y; we could split three into one subpart, and one into the other (for example, {x1, x2, x3 / x4} on one side and {x5 / x6, x7, x8} on the other, so the two sides of x do not respect the symmetry of y), or y could split each side of x into two equal halves with two positions each (for example, {x1, x2 / x3, x4} on one side and {x5, x6 / x7, x8} on the other, so the two sides of x properly respect the symmetry of y). It's not possible for y to split any half of x into 4 positions on one side and 0 on the other because the only dichotomy that could do that would be x itself (which contradicts the distinctness of x and y). If we split 3 on one side and 1 on the other (and thus both sides of x do not respect the symmetry of y), then we generate a degenerate blocking set. An adequate blocking set (i.e. not degenerate) contains exactly 4 sets of 2 positions each, since they need to be filled with 4 pairings of element blocks that come from element blocking sets). Thus, for an adequate blocking set that contains dichotomy x and dichotomy y, both sides of x must respect the symmetry of y. Since each of the non-temperament dichotomies is in a blocking set with each of the other temperament dichotomies (as we saw from Criterion 2), it follows that each of the non-temperament dichotomies must respect the symmetry of each of the temperament dichotomies. Criterion 3 also follows. [It also follows from this criterion that the non-temperament dichotomies must respect the symmetries of the other non-temperament dichotomies in the same system, but this is not very significant because by respecting the symmetry of the temperament dichotomies, another 3 symmetrical non-temperament dichotomies would necessarily be generated by the combination of one symmetry respecting non-temperament dichotomy with each of the other 3 temperament dichotomies.]

... ... ... ...

We can now apply this set of criteria for a structurally valid system of functional/positional dichotomies to our list of all 35 partitions of 8 functions/positions. This will allow us to discern which partitions could be included in a structurally valid system of functional/positional dichotomies, and which cannot. The easiest criterion to start with is Criterion 3, since it is easy to check whether a given partition respects the symmetry of Mental/Vital, Accepting/Producing, and Bold/Cautious.

Applying Criterion 3 to all 35 partitions of 8 functions/positions

n.b. we are using Model A conventions in names and position numbers, which is useful for familiarity but technically unnecessary, so don't mistake it for being anything special about Model A since Model G conventions give the same results)

1234: Mental/Vital

1235: doesn't respect the symmetry of Mental/Vital (123 are the same in Static/Dynamic)

1236: doesn't respect the symmetry of Mental/Vital (123 are the same in Static/Dynamic)

1237: doesn't respect the symmetry of Mental/Vital (123 are the same in Static/Dynamic)

1238: doesn't respect the symmetry of Mental/Vital (123 are the same in Static/Dynamic)

1245: doesn't respect the symmetry of Mental/Vital (124 are the same in Static/Dynamic)

1246: doesn't respect the symmetry of Mental/Vital (124 are the same in Static/Dynamic)

1247: doesn't respect the symmetry of Mental/Vital (124 are the same in Static/Dynamic)

1248: doesn't respect the symmetry of Mental/Vital (124 are the same in Static/Dynamic)

1256: Valued/Unvalued

1257: doesn't respect the symmetry of Bold/Cautious (257 are the same in Extroverted/Introverted)

1258: New Valid Dichotomy!

1267: New Valid Dichotomy!

1268: doesn't respect the symmetry of Bold/Cautious (168 are the same in Extroverted/Introverted)

1278: Strong/Weak

1345: doesn't respect the symmetry of Mental/Vital (134 are the same in Static/Dynamic)

1346: doesn't respect the symmetry of Mental/Vital (134 are the same in Static/Dynamic)

1347: doesn't respect the symmetry of Mental/Vital (134 are the same in Static/Dynamic)

1348: doesn't respect the symmetry of Mental/Vital (134 are the same in Static/Dynamic)

1356: doesn't respect the symmetry of Bold/Cautious (136 are the same in Extroverted/Introverted)

1357: Accepting/Producing

1358: doesn't respect the symmetry of Bold/Cautious (138 are the same in Extroverted/Introverted)
1367: doesn't respect the symmetry of Bold/Cautious (136 are the same in Extroverted/Introverted)

1368: Bold/Cautious

1378: doesn't respect the symmetry of Bold/Cautious (138 are the same in Extroverted/Introverted)
1456: New Valid Dichotomy!

1457: doesn't respect the symmetry of Bold/Cautious (457 are the same in Extroverted/Introverted)

1458: Evaluative/Situational

1467: Inert/Contact

1468: doesn't respect the symmetry of Bold/Cautious (168 are the same in Extroverted/Introverted)

1478: New Valid Dichotomy!

1567: doesn't respect the symmetry of Mental/Vital (567 are the same in Static/Dynamic)

1568: doesn't respect the symmetry of Mental/Vital (568 are the same in Static/Dynamic)

1578: doesn't respect the symmetry of Mental/Vital (578 are the same in Static/Dynamic)

1678: doesn't respect the symmetry of Mental/Vital (678 are the same in Static/Dynamic)

n.b. We only wrote one side of each partition of the functions, but if that side respects the symmetry of the temperament dichotomies, the other side must also, since a 2/2 split on one side and 3/1 split on the other side is impossible (it would imply that the temperament dichotomies have 5 positions on one side and 3 on the other, which we know is false)

... ... ... ...

It turns out that most partitions of 8 functions/positions cannot be included in a structurally valid dichotomy system. Nonetheless, it's very interesting to note that we newly discovered 4 partitions that meet a major criterion (Criterion 3) to be included in a structurally valid dichotomy system. Since we already had 7 (all of the existing functional/positional dichotomies of Model A), we now have a total of 11 potentially valid functional/positional dichotomies. It follows obviously that the 4 we discovered are not included in the functional/positional dichotomies of Model A, so we need to investigate them further to figure out what they might mean and apply Criterion 1 and Criterion 2 to them (to ultimately confirm their validity).

It's already possible for me to say a little about what they mean, since I am versed in more models in Socionics than Model A.

1258: In Model A, this refers to the Program function, the Creative function, the Suggestive function and the Demonstrative function.

We can see immediately that this dichotomy is not very symmetric in Model A, nor is it obviously meaningful. But what about in Model G? The functions it refers to for Model G are the Program function (1), Implementation function (2), Demonstrative function (5) and Dual function (6).

Not only is this dichotomy perfectly symmetrical in Model G, but it exactly represents the High Energy functions on one side (the one's we listed), and the Low Energy functions on the other (a well-known Model G dichotomy)! We can confirm that the other 3 structurally valid partitions are also symmetrical and meaningful in Model G (while being asymmetric and less obviously meaningful in Model A). [more pictures may be added later]

1267: In Model A, this refers to the Program, Creative, Mobilizing and Ignoring functions, which are not symmetric in the model. In Model G, this refers to the Program (1), Launcher (4), Demonstrative (5), and Control (8) functions. Not only are these functions perfectly symmetric in the model, but they also correspond to a well-known Model G dichotomy of Low Brakes functions on one side (the one we listed), and High Brakes functions on the other!

1456: In Model A, this refers to the Program, Vulnerable, Suggestive and Mobilizing functions, which are not symmetric in the model. In Model G, this refers to the Program (1), Launcher (4), Dual (6) and Brake (7) functions. Not only are these functions perfectly symmetric in the model (the two sides form a U-shape and N-shape [in the sense of a logical intersection symbol, as opposed to union] which can be rotated by 180 degrees for a symmetry), but they have a speculative interpretation in Model G where the functions on the U-shape (which we listed) correspond to more Impressionable (Intuitive) functions, whereas functions on the N-shape correspond to relatively Unimpressionable (Deliberative) functions (though this interpretation is still being actively considered).

1478: In Model A, this refers to the Program, Vulnerable, Ignoring and Demonstrative functions, which are not symmetric in the model. In Model G, this refers to the Program (1), Implementation (2), Brake (7) and Control (8) functions. Not only are these functions perfectly symmetric in the model, but they also correspond to a well-known Model G dichotomy of Tensioned blocks on one side (the ones that we listed which includes the Social Mission and Inflation blocks), and Relaxed blocks on the other side (which includes the Self-Affirmation and Social Adaptation blocks which we did not list).

Now that we know how meaningful and symmetric these functional/positional partitions are in Model G, it is even more interesting and important to confirm their validity according to Criteria 1 and 2 for structurally valid functional/positional dichotomies. If we confirm the validity of these dichotomies, then we have not only discovered a new system of functional/positional dichotomies unique to Model G, but we have also proven Model G to be as mathematically valid, symmetric, elegant and complete as Model A (which means Socionics that takes mathematics seriously can no longer legitimately ignore it).

____________

Deriving and Validating the Gulenko-Newman Dichotomies (a functional/positional dichotomy system for Model G)

It might not be obvious where to start in building this new system of functional/positional dichotomies, but it's more obvious than it may appear because of Criterion 1. Since Criterion 1 dictates that we must include the temperament dichotomies (a.k.a. the bridge dichotomies) in our system, they are the first three that we will include {Mental/Vital, Accepting/Producing, Bold/Cautious}! This has the nice consequence that these Model G positional dichotomies are already consistent by Criteria 1 and 3, so we just need to check Criterion 2. In order to check Criterion 2, we'll need 4 more dichotomies to serve as our non-temperament dichotomies, so that we can check that (once we combine them into a system with the 3 temperament dichotomies) each of them is included in 3 unique blocking sets (each with one other non-temperament dichotomy and one temperament dichotomy).

Yet again, at first it may not be obvious which 4 dichotomies we should choose, but yet again it's more obvious that it may appear because we discovered precisely 4 new potentially valid functional/positional partitions, which also happen to be symmetrical in Model G (as opposed to the 4 non-temperament dichotomies which we used for Model A, which are symmetrical in that model but which aren't symmetric in Model G thus less likely to work well with it in practical terms). Since there are only 7 dichotomies in a functional/positional system, we can only choose 4 more and the 4 we discovered are clearly the most natural choice to start from to satisfy our criteria and build a system of functional/positional dichotomies based on Model G. Let's list our prospective system in the conventions of Model A at first (then later I'll show how to transform it into Model G names and positions, and this won't have any effect on anything we prove structurally since the names and numbers we use are merely conventions and not a genuine part of the mathematics save for using them in a logically consistent way):

Gulenko-Newman Functional/Positional Dichotomies

Mental/Vital:

- Mental: 1, 2, 3, 4

- Vital: 5, 6, 7, 8

Accepting/Producing:

- Accepting: 1, 3, 5, 7

- Producing: 2, 4, 6, 8

Bold/Cautious:

- Bold: 1, 3, 6, 8

- Cautious: 2, 4, 5, 7

Energetic/Informational (partition 1258 from the previous section):

- Energetic: 1, 2, 5, 8

- Informational: 3, 4, 6, 7

Excitable/Inhibitable (partition 1267 from the previous section):

- Excitable: 1, 2, 6, 7

- Inhibitable: 3, 4, 5, 8

Impressionable/Unimpressionable (partition 1456 from the previous section):

- Impressionable: 1, 4, 5, 6

- Unimpressionable: 2, 3, 7, 8

Tensioned/Relaxed (partition 1478 from the previous section):

- Tensioned: 1, 4, 7, 8

- Relaxed: 2, 3, 5, 6

The reason I call them the Gulenko-Newman dichotomies is because Victor Gulenko (in discovering and using Model G for many years) had already discovered 3 of these dichotomies, yet I discovered the last one and figured out that they connect in an independent, mathematically sound set of dichotomy systems of the sort used in Model A (which we are in the process of proving).

... ... ... ...

In order to check Criterion 2, we just need to generate the blocking sets of these dichotomies (which are independently interesting anyways):

Gulenko-Newman Functional/Positional Sets of Blockings

Temperament Blocking Set (you'll notice that this is exactly the same as in Model A, which is as it should be due to Criterion 1):

- Mental/Vital

- Accepting/Producing

- Bold/Cautious

- 1+3: Mental, Accepting, Bold

- 2+4: Mental, Producing, Cautious

- 5+7: Vital, Accepting, Cautious

- 6+8: Vital, Producing, Bold

Revision Blocking Set:

- Mental/Vital

- Energetic/Informational

- Excitable/Inhibitable

- 1+2: Mental, Energetic, Excitable

- 3+4: Mental, Informational, Inhibitable

- 5+8: Vital, Energetic, Inhibitable

- 6+7: Vital, Informational, Excitable

Cross-Revision Blocking Set:

- Mental/Vital

- Tensioned/Relaxed

- Impressionable/Unimpressionable

- 1+4: Mental, Tensioned, Impressionable

- 2+3: Mental, Relaxed, Unimpressionable

- 7+8: Vital, Tensioned, Unimpressionable

- 5+6: Vital, Relaxed, Impressionable

Diagonal Blocking Set:

- Accepting/Producing

- Energetic/Informational

- Impressionable/Unimpressionable

- 1+5: Accepting, Energetic, Impressionable

- 3+7: Accepting, Informational, Unimpressionable

- 2+8: Producing, Energetic, Unimpressionable

- 4+6: Producing, Informational, Impressionable

Cross-Diagonal Blocking Set:

- Accepting/Producing

- Excitable/Inhibitable

- Tensioned/Relaxed

- 1+7: Accepting, Excitable, Tensioned

- 3+5: Accepting, Inhibitable, Relaxed

- 2+6: Producing, Excitable, Relaxed

- 4+8: Producing, Inhibitable, Tensioned

Order Blocking Set:

- Bold/Cautious

- Energetic/Informational

- Tensioned/Relaxed

- 1+8: Bold, Energetic, Tensioned

- 3+6: Bold, Informational, Relaxed

- 2+5: Cautious, Energetic, Relaxed

- 4+7: Cautious, Informational, Tensioned

Cross-Order Blocking Set:

- Bold/Cautious

- Excitable/Inhibitable

- Impressionable/Unimpressionable

- 1+6: Bold, Excitable, Impressionable

- 3+8: Bold, Inhibitable, Unimpressionable

- 2+7: Cautious, Excitable, Unimpressionable

- 4+5: Cautious, Inhibitable, Impressionable

... ... ... ...

Now we can easily check if our 4 non-temperament dichotomies (in the context of the Gulenko-Newman system) satisfy Criterion 2:

-Energetic/Informational is in the Revision Blocking Set, the Diagonal Blocking Set, and the Order Blocking Set, and all of those have exactly one temperament dichotomy

-Excitable/Inhibitable is in the Revision Blocking Set, the Cross-Diagonal Blocking Set, and the Cross-Order Blocking Set, and all of those have exactly one temperament dichotomy

-Impressionable/Unimpressionable is in the Cross-Revision Blocking Set, the Diagonal Blocking Set, and the Cross-Order Blocking Set, and all of those have exactly one temperament dichotomy

-Tensioned/Relaxed is in the Cross-Revision Blocking Set, the Cross-Diagonal Blocking Set, and the Order Blocking Set, and all of those have exactly one temperament dichotomy

And that's it; we've just satisfied Criterion 2 and proved that the Gulenko-Newman functional/positional dichotomies form a structurally valid system of functional/positional dichotomies for Model G, in a way that is completely independent from Model A yet also shares a number of important structural features that connect the two systems in a greater whole. It is no longer possible to dismiss Model G on mathematical or logical grounds; the only possible route to verify or falsify Model G is empirical and semantic, which is where the especially challenging and substantive parts of typology come in (this is challenging because the structure of reality is even more complicated than the math we are doing here).

Since the existence of this new system of functional/positional dichotomies might be a surprise, you might be wondering if there are any more systems of functional/positional dichotomies which obey the same 3 criteria we previously set out. It is actually very easy to prove that there are no more systems of functional/positional dichotomies which satisfy these criteria. Thus, the classical Model A functional/positional dichotomies and the Gulenko-Newman (Model G) functional/positional dichotomies are the only two such systems.

- Justification: Suppose, for the sake of contradiction, that there was another system of functional/positional dichotomies, different from the two we have already shown. It would have to include the 3 temperament dichotomies by Criterion 1. For its 4 non-temperament dichotomies, it would have to choose from the 8 we have already used, since they are the only remaining valid partitions. It couldn't use the same 4 as Model A or Model G (since then it would not be distinct), so it would have to mix Model A dichotomies and Model G dichotomies. This doesn't work, because no combination of a Model A non-temperament dichotomy with a Model G non-temperament dichotomy could be blocked with a temperament dichotomy. This is because the Model A blocking sets already show all possible ways that the Model A non-temperament dichotomies can be blocked with temperament dichotomies, since each of them are blocked with all 3 of the temperament dichotomies exactly one time. Combining a Model A non-temperament dichotomy with a Model G non-temperament dichotomy will thus always result in something different from a temperament dichotomy. And since a Boolean algebra is distributive, we have to combine every Model A non-temperament dichotomy we used with every Model G non-temperament dichotomy we used, and it will always result in an illegitimate blocking set that doesn’t include a temperament dichotomy. Thus, we know there are no other systems of functional/positional dichotomies by inherent contradiction.

Up until now, we’ve presented all these Gulenko-Newman functional/positional dichotomies and blocking sets under the conventions of Model A, since it would serve as an easier to understand bridge for those familiar with Model A. However, these dichotomies really work better and are more symmetric in Model G, so we should present them in that model for greater ease of use (and from here on out, we will use Model G conventions unless we indicate otherwise).

Gulenko-Newman Functional/Positional Dichotomies (Under Model G names and numbers)

Opening/Closing:

- Opening: 1, 3, 5, 7

- Closing: 2, 4, 6, 8

Stable/Unstable:

- Stable: 1, 3, 6, 8

- Unstable: 2, 4, 5, 7

Externality/Internality:

- Externality: 1, 2, 3, 4

- Internality: 5, 6, 7, 8

Energetic/Informational:

- Energetic: 1, 2, 5, 6

- Informational: 3, 4, 7, 8

Excitable/Inhibitable:

- Excitable: 1, 4, 5, 8

- Inhibitable: 2, 3, 6, 7

Impressionable/Unimpressionable:

- Impressionable: 1, 4, 6, 7

- Unimpressionable: 2, 3, 5, 8

Tensioned/Relaxed:

- Tensioned: 1, 2, 7, 8

- Relaxed: 3, 4, 5, 6

… … … …

Gulenko-Newman Functional/Positional Sets of Blockings (Under Model G names and position numbers)

Temperament Blocking Set:

- Opening/Closing

- Stable/Unstable

- Externality/Internality

- 1+3: Opening, Stable, Externality

- 5+7: Opening, Unstable, Internality

- 6+8: Closing, Stable, Internality

- 2+4: Closing, Unstable, Externality

Revision Blocking Set (these are the 4 vertical dimension blocks of Model G):

- Opening/Closing

- Energetic/Informational

- Excitable/Inhibitable

- 1+5: Opening, Energetic, Excitable

- 3+7: Opening, Informational, Inhibitable

- 2+6: Closing, Energetic, Inhibitable

- 4+8: Closing, Informational, Excitable

Cross-Revision Blocking Set:

- Opening/Closing

- Tensioned/Relaxed

- Impressionable/Unimpressionable

- 1+7: Opening, Tensioned, Impressionable

- 3+5: Opening, Relaxed, Unimpressionable

- 2+8: Closing, Tensioned, Unimpressionable

- 4+6: Closing, Relaxed, Impressionable

Diagonal Blocking Set:

- Stable/Unstable

- Energetic/Informational

- Impressionable/Unimpressionable

- 1+6: Stable, Energetic, Impressionable

- 3+8: Stable, Informational, Unimpressionable

- 2+5: Unstable, Energetic, Unimpressionable

- 4+7: Unstable, Informational, Impressionable

Cross-Diagonal Blocking Set:

- Stable/Unstable

- Excitable/Inhibitable

- Tensioned/Relaxed

- 1+8: Stable, Excitable, Tensioned

- 3+6: Stable, Inhibitable, Relaxed

- 4+5: Unstable, Excitable, Relaxed

- 2+7: Unstable, Inhibitable, Tensioned

Order Blocking Set (these are blocks like Social Mission/Self-Affirmation/Social Adaptation/Inflation):

- Externality/Internality

- Energetic/Informational

- Tensioned/Relaxed

- 1+2: Externality, Energetic, Tensioned

- 3+4: Externality, Informational, Relaxed

- 5+6: Internality, Energetic, Relaxed

- 7+8: Internality, Informational, Tensioned

Cross-Order Blocking Set:

- Externality/Internality

- Excitable/Inhibitable

- Impressionable/Unimpressionable

- 1+4: Externality, Excitable, Impressionable

- 2+3: Externality, Inhibitable, Unimpressionable

- 5+8: Internality, Excitable, Unimpressionable

- 6+7: Internality, Inhibitable, Impressionable

... ... ... ...

As we did for Model A, one of the most useful things we can do to show how this model works is to show how the elemental blockings fill the functional/positional blockings for a given type. However, our functional/positional blockings changed this time, so we need to check whether the elemental blockings can properly fill them. In fact, we can easily see that they can't!

- One blocking set that we have from the elemental blockings is {Ne+Si, Se+Ni, Fe+Ti, Te+Fi}, the Duality Blocking Set. Unfortunately, this fits nowhere in the Gulenko-Newman blocking sets since they pair everything except the temperaments differently than Model A. Two of the duality blocks in Model G are in the Diagonal Blocking Set (1+6, 3+8), but the other two blocks in that set are extinguishment blocks! And there is another split between duality and extinguishment blocks in the Cross-Diagonal Blocking Set. The fact that the elemental blocking sets don't fit into the functional/positional blocking sets of Model G might seem like it could be problem, but it isn't. The reason for this is that Model G has its own set of elemental dichotomies as well!

In order to find new dichotomies to generate a new set of elemental dichotomies, we would need to find new sets of partitions from the set of all 35 dichotomous partitions of 8 elements, just as we did from the set of all 35 dichotomous partitions of 8 functions/positions (the fact that these sets have the same number of partitions is obvious since they are both sets of all possible dichotomous partitions of 8 elements). If we define the same interleaving operation on those two sets with the same structure (the structure is the set of 3 criteria that these dichotomy systems must meet combined with the Boolean algebra performed by the interleaving operation), then the two sets of partitions are isomorphic to one another, since all we have done is change the names of the 8 objects we are using and apply the same structure to them. In fact, since we could establish a trivial bijective mapping from the set of Socionics elements to the set of numbers 1 through 8 (which is exactly the set used to represent the functions/positions in Socionics), the isomorphism from the set of all 35 dichotomous partitions of 8 elements to the set of all 35 dichotomous partitions of 8 functions/positions (which are just the numbers 1 through 8) is isomorphic to an automorphism from the set of all 35 dichotomous partitions of the numbers 1 through 8 to itself.

Thus, as with the functional/positional partitions, we can assert that there are exactly 4 elemental partitions that we have not yet discovered (which respect the symmetries of the temperament dichotomies) besides the 7 already included in the Model A elemental dichotomies. Instead of going through the long process of deriving them, I can just tell you what these 4 dichotomies are (and you can easily verify that they respect the symmetries of the temperament dichotomies, and know that no more partitions can possibly exist due this due to the isomorphism between these systems, in addition to the fact that these partitions won't be able to mix with the Model A one's, just like the functional/positional dichotomies for Models A and G don't mix):

LII+EIE High Energy Functions/ESI+LSE High Energy Functions:

- LII+EIE High Energy Functions: NeNiFeTi

- ESI+LSE High Energy Functions: SeSiTeFi

LIE+EII High Energy Functions/ESE+LSI High Energy Functions:

- LIE+EII High Energy Functions: NeNiTeFi

- ESE+LSI High Energy Functions: SeSiFeTi

ILE+SLI High Energy Functions/SEE+IEI High Energy Functions:

- ILE+SLI High Energy Functions: NeSiTeTi

- SEE+IEI High Energy Functions: SeNiFeFi

IEE+SEI High Energy Functions/SLE+ILI High Energy Functions:

- IEE+SEI High Energy Functions: NeSiFeFi

- SLE+ILI High Energy Functions: SeNiTeTi

n.b. These names are rather long and unwieldly for dichotomies, so let's abbreviate "High Energy Functions" to "HEF" going forward. Perhaps we can come up with more descriptive names in the future; these are just chosen for clear reference. You can also see how these dichotomies "cross" the duality blocks and extinguishment blocks, which already gives a hint as to how they would be compatible with the Gulenko-Newman blocking sets

... ... ... ...

Now we can present what is, without a doubt, the full set of 7 Model G elemental dichotomies:

Gulenko-Newman Elemental Dichotomies

n.b. it should be obvious why we are including the three temperament dichotomies

Static/Dynamic:

- Static: Ne, Se, Ti, Fi

- Dynamic: Ni, Si, Te, Fe

Irrational/Rational:

- Irrational: Ne, Se, Ni, Si

- Rational: Ti, Fi, Te, Fe

Extroverted/Introverted:

- Extroverted: Ne, Se, Te, Fe

- Introverted: Ni, Si, Ti, Fi

LII+EIE HEF/ESI+LSE HEF:

- LII+EIE HEF: Ne, Ni, Fe, Ti

- ESI+LSE HEF: Se, Si, Te, Fi

LIE+EII HEF/ESE+LSI HEF:

- LIE+EII HEF: Ne, Ni, Te, Fi

- ESE+LSI HEF: Se, Si, Fe, Ti

ILE+SLI HEF/SEE+IEI HEF:

- ILE+SLI HEF: Ne, Si, Te, Ti

- SEE+IEI HEF: Se, Ni, Fe, Fi

IEE+SEI HEF/SLE+ILI HEF:

- IEE+SEI HEF: Ne, Si, Fe, Fi

- SLE+ILI HEF: Se, Ni, Te, Ti

... ... ... ...

Now let's show the blocking sets (which as you will see, will follow all the familiar structure organized by the three temperament dichotomies):

Gulenko-Newman Elemental Sets of Blockings

Temperament Blocking Set:

- Static/Dynamic

- Irrational/Rational

- Extroverted/Introverted

- Ne+Se: Static, Irrational, Extroverted

- Ti+Fi: Static, Rational, Introverted

- Ni+Si: Dynamic, Irrational, Introverted

- Te+Fe: Dynamic, Rational, Extroverted

Asking Revision Blocking Set:

- Static/Dynamic

- LII+EIE HEF/ESI+LSE HEF

- ILE+SLI HEF/SEE+IEI HEF

- Ne+Ti: Static, LII+EIE HEF, ILE+SLI HEF

- Se+Fi: Static, ESI+LSE HEF, SEE+IEI HEF

- Ni+Fe: Dynamic, LII+EIE HEF, SEE+IEI HEF

- Si+Te: Dynamic, ESI+LSE HEF, ILE+SLI HEF

Declaring Revision Blocking Set:

- Static/Dynamic

- LIE+EII HEF/ESE+LSI HEF

- IEE+SEI HEF/SLE+ILI HEF

- Ne+Fi: Static, LIE+EII HEF, IEE+SEI HEF

- Se+Ti: Static, ESE+LSI HEF, SLE+ILI HEF

- Ni+Te: Dynamic, LIE+EII HEF, SLE+ILI HEF

- Si+Fe: Dynamic, ESE+LSI HEF, IEE+SEI HEF

Irrational Extinguishment Rational Duality Diagonal Blocking Set:

- Irrational/Rational

- LII+EIE HEF/ESI+LSE HEF

- LIE+EII HEF/ESE+LSI HEF

- Ne+Ni: Irrational, LII+EIE HEF, LIE+EII HEF

- Se+Si: Irrational, ESI+LSE HEF, ESE+LSI HEF

- Fe+Ti: Rational, LII+EIE HEF, ESE+LSI HEF

- Te+Fi: Rational, ESI+LSE HEF, LIE+EII HEF

Irrational Duality Rational Extinguishment Diagonal Blocking Set:

- Irrational/Rational

- ILE+SLI HEF/SEE+IEI HEF

- IEE+SEI HEF/SLE+ILI HEF

- Ne+Si: Irrational, ILE+SLI HEF, IEE+SEI HEF

- Se+Ni: Irrational, SEE+IEI HEF, SLE+ILI HEF

- Te+Ti: Rational, ILE+SLI HEF, SLE+ILI HEF

- Fe+Fi: Rational, SEE+IEI HEF, IEE+SEI HEF

Negativist Order Blocking Set:

- Extroverted/Introverted

- LII+EIE HEF/ESI+LSE HEF

- IEE+SEI HEF/SLE+ILI HEF

- Ne+Fe: Extroverted, LII+EIE HEF, IEE+SEI HEF

- Se+Te: Extroverted, ESI+LSE HEF, SLE+ILI HEF

- Ni+Ti: Introverted, LII+EIE HEF, SLE+ILI HEF

- Si+Fi: Introverted, ESI+LSE HEF, IEE+SEI HEF

Positivist Order Blocking Set:

- Extroverted/Introverted

- LIE+EII HEF/ESE+LSI HEF

- ILE+SLI HEF/SEE+IEI HEF

- Ne+Te: Extroverted, LIE+EII HEF, ILE+SLI HEF

- Se+Fe: Extroverted, ESE+LSI HEF, SEE+IEI HEF

- Ni+Fi: Introverted, LIE+EII HEF, SEE+IEI HEF

- Si+Ti: Introverted, ESE+LSI HEF, ILE+SLI HEF

n.b. We see two more Orbital Dichotomies make an appearance here! Asking/Declaring and Positivist/Negativist dichotomies were not straightforward to derive from the structure of Model A, but they are straightforward in Model G, as we will see. And it should be obvious at this point why this system of dichotomies is structurally valid, based on how this blocking set meets the criteria (in addition to the isomorphism and automorphism we pointed out).

... ... ... ...

Now, as we did with Model A, we can show how these Model G elemental dichotomies and functional/positional dichotomies combine perfectly in a given type (let's use ILE again):

- the Temperament Blocking Set of the GN (Gulenko-Newman) elemental dichotomies corresponds exactly to the Temperament Blocking Set of the GN functional/positional dichotomies

- the Asking Revision Blocking Set of the GN elemental dichotomies corresponds exactly to the Revision Blocking Set of the GN functional/positional dichotomies

- the Declaring Revision Blocking Set of the GN elemental dichotomies corresponds exactly to the Cross-Revision Blocking Set of the GN functional/positional dichotomies

- the Irrational Extinguishment Rational Duality Diagonal Blocking Set of the GN elemental dichotomies corresponds exactly to the Cross-Diagonal Blocking Set of the GN functional/positional dichotomies

- the Irrational Duality Rational Extinguishment Diagonal Blocking Set of the GN elemental dichotomies corresponds exactly to the Diagonal Blocking Set of the GN functional/positional dichotomies

- the Negativist Order Blocking Set of the GN elemental dichotomies corresponds exactly to the Cross-Order Blocking Set of the GN functional/positional dichotomies

- the Positivist Order Blocking Set of the GN elemental dichotomies corresponds exactly to the Order Blocking Set of the GN functional/positional dichotomies

____________

Types as Isomorphisms and more General Model A and G Proofs

Previously, we've claimed that the dichotomy systems of Model A (the elemental and functional/positional dichotomies) are structurally compatible but not compatible with the dichotomy systems of Model G (and vice versa, that the Gulenko-Newman Model G dichotomy systems are compatible with one another but not with Model A dichotomy systems). Here we're going to provide a more rigorous proof of compatibility. We can prove structural compatibility by providing a bijective mapping between compatible systems (or an account of all possible bijective mappings between them). We can then use this bijective mapping to prove that the types themselves can be understood as isomorphisms and automorphisms, in a sense.

First let's prove that the Model A elemental dichotomies are compatible with the Model A functional/positional dichotomies by showing all possible bijective maps that could occur between their Blocking Sets (which preserve the structure of each dichotomy system since they are built from their 3 structural criteria). Indeed, for every Sociotype, such a bijective map between Blocking Sets occurs, which can be described by some features of the Sociotype. Since the temperament dichotomies in one system always map to the temperament dichotomies in the other by definition, we know for sure that a blocking set from one system can only map to a blocking set in the other system if it shares the compatible temperament dichotomy(ies). We can thus specify the rest of the information in the mapping as follows:

Set of Bijective Mappings Between Model A Elemental and Functional/Positional Blocking Sets

- Since they both contain all three temperament dichotomies, the Temperament Blocking Set from the elemental dichotomies must always map to the Temperament Blocking Set from the functional/positional dichotomies (remember that the temperament dichotomies necessarily map to one another)

- Since two blockings in each Model A dichotomy system contain Static/Dynamic (in elemental dichotomies) or Mental/Vital (in functional/positional blockings) and no other temperament dichotomies, we have the following two possible cases of two mappings between them, one of which will always be chosen in each type based on the Democratic/Aristocratic dichotomy:

1. In a Democratic type, the Democratic Revision Blocking Set from the elemental dichotomies must always map to the Freudian Blocking Set from the functional/positional dichotomies, and the Aristocratic Revision Blocking Set from the elemental dichotomies must always map to the Vertical Blocking Set from the functional/positional dichotomies
2. In an Aristocratic type, the Aristocratic Revision Blocking Set from the elemental dichotomies must always map to the Freudian Blocking Set from the functional positional dichotomies, and the Democratic Revision Blocking Set from the elemental dichotomies must always map to the Vertical Blocking Set from the functional/positional dichotomies

- Since two blockings in each Model A dichotomy system contain Irrational/Rational (in elemental dichotomies) or Accepting/Producing (in functional/positional blockings) and no other temperament dichotomies, we have the following two necessary mappings between them:

1. The Duality Blocking Set from the elemental dichotomies must always map to the Duality Blocking Set from the functional/positional dichotomies
2. The Extinguishment Blocking Set from the elemental dichotomies must always map to the Extinguishment Blocking Set from the functional/positional dichotomies

- Since two blockings in each Model A dichotomy system contain Extroverted/Introverted (in elemental dichotomies) or Bold/Cautious (in functional/positional blockings) and no other temperament dichotomies, we have the following two possible cases of two mappings between them, one of which will always be chosen in each type based on the Democratic/Aristocratic dichotomy:

1. In a Democratic type, the Democratic Order Blocking Set from the elemental dichotomies must always map to the Dimensional Blocking Set from the functional/positional dichotomies, and the Aristocratic Order Blocking Set from the elemental dichotomies must always map to the Confidence Value Blocking Set from the functional/positional dichotomies
2. In an Aristocratic type, the Aristocratic Order Blocking Set from the elemental dichotomies must always map to the Dimensional Blocking Set from the functional/positional dichotomies, and the Democratic Order Blocking Set from the elemental dichotomies must always map to the Confidence Value Blocking Set from the functional/positional dichotomies

... ... ... ...

Now, we can also specify the set of bijective mappings between Model G elemental and functional/positional Blocking Sets using the Gulenko-Newman dichotomies in a similar manner:

Set of Bijective Mappings Between Model G Elemental and Functional/Positional Blocking Sets

- Since they both contain all three temperament dichotomies, the Temperament Blocking Set from the elemental dichotomies must always map to the Temperament Blocking Set from the functional/positional dichotomies

- Since two blockings in each Model G dichotomy system contain Static/Dynamic (in elemental dichotomies) or Opening/Closing (in functional/positional blockings) and no other temperament dichotomies, we have the following two possible cases of two mappings between them, one of which will always be chosen in each type based on the Asking/Declaring dichotomy:

1. In an Asking type, the Asking Revision Blocking Set from the GN elemental dichotomies must always map to the Revision Blocking Set from the GN functional/positional dichotomies, and the Declaring Revision Blocking Set from the GN elemental dichotomies must always map the Cross-Revision Blocking Set from the GN functional/positional dichotomies
2. In a Declaring type, the Declaring Revision Blocking Set from the GN elemental dichotomies must always map to the Revision Blocking Set from the GN functional/positional dichotomies, and the Asking Revision Blocking Set from the GN elemental dichotomies must always map the Cross-Revision Blocking Set from the GN functional/positional dichotomies

- Since two blockings in each Model G dichotomy system contain Irrational/Rational (in elemental dichotomies) or Stable/Unstable (in functional/positional blockings) and no other temperament dichotomies, we have the following two possible cases of two mappings between them, one of which will always be chosen in each type based on the Irrational/Rational dichotomy:

1. In an Irrational type, the Irrational Duality Rational Extinguishment Blocking Set from the GN elemental dichotomies must always map to the Diagonal Blocking Set from the GN functional/positional dichotomies, and the Irrational Extinguishment Rational Duality Blocking Set from the GN elemental dichotomies must always map the Cross-Diagonal Blocking Set from the GN functional/positional dichotomies
2. In a Rational type, the Irrational Extinguishment Rational Duality Blocking Set from the GN elemental dichotomies must always map to the Diagonal Blocking Set from the GN functional/positional dichotomies, and the Irrational Duality Rational Extinguishment Blocking Set from the GN elemental dichotomies must always map the Cross-Diagonal Blocking Set from the GN functional/positional dichotomies

- Since two blockings in each Model G dichotomy system contain Extroverted/Introverted (in elemental dichotomies) or Externality/Internality (in functional/positional blockings) and no other temperament dichotomies, we have the following two possible cases of two mappings between them, one of which will always be chosen in each type based on the Positivist/Negativist dichotomy:

1. In a Positivist type, the Positivist Order Blocking Set from the GN elemental dichotomies must always map to the Order Blocking Set from the GN functional/positional dichotomies, and the Negativist Order Blocking Set from the GN elemental dichotomies must always map the Cross-Order Blocking Set from the GN functional/positional dichotomies
2. In a Negativist type, the Negativist Order Blocking Set from the GN elemental dichotomies must always map to the Order Blocking Set from the GN functional/positional dichotomies, and the Positivist Order Blocking Set from the GN elemental dichotomies must always map the Cross-Order Blocking Set from the GN functional/positional dichotomies

... ... ... ...

Proof of Sociotypes as Isomorphisms

We can show that the 16 Sociotypes themselves are (abstractly) equivalent to three mappings:

- An Isomorphism between the Model A elemental dichotomy system and the Model A functional/positional dichotomy system

- An Isomorphism between the Gulenko-Newman (Model G) elemental dichotomy system and the Gulenko-Newman (Model G) functional/positional dichotomy system

- An Automorphism from the set of all 35 dichotomous partitions of the numbers 1 through 8 to itself, in such a way that the elemental dichotomy systems from both Model A and G collectively bijectively map to the functional/positional dichotomy systems from both Model A and G, the most mathematically complete description of Sociotype to date

An isomorphism is a bijective homomorphism, i.e. a one-to-one mapping between two mathematical structures which preserves the structure on both sides of the mapping. Thus, in order to prove that the Sociotypes are the isomorphisms I claimed, we need to prove two things:

- We need to show a one-to-one mapping from one structure to the other. The reason for the one-to-one mapping (one part of the structure to another) is that it is bijective and thus can map in either direction (i.e. in this case it would start from elemental dichotomies and go to functional/positional dichotomies, or start from functional/positional dichotomies and go to elemental, and it doesn't matter since the mapping is one-to-one and thus could be trivially inverted), which is needed to preserve all structure from one structure to another.

- We need to prove that our one-to-one mapping is "structure-preserving". This requires proving 4 statements:

1. the temperament dichotomies in one structure must each map to their corresponding temperament dichotomy in the other structure (this will preserve Criterion 1)
2. each non-temperament dichotomy in one structure must map to a non-temperament dichotomy in the other, and when a non-temperament is mapped to the other structure, it must be included in a blocking set with exactly one temperament dichotomy and one other non-temperament dichotomy (this will preserve Criterion 2)
3. the dichotomies we use must be valid partitions which respect symmetry of each of the temperament dichotomies (this will preserve Criterion 3)
4. the interleaving operation ("+") must be preserved, i.e. if I have dichotomies E1, E2 and E3 in one structure where E1 + E2 = E3, then for the map "t" to be isomorphic it must be the case that t(E1 + E2) = t(E1) + t(E2)

Proof:

Let's start by listing all of the types as exactly the one-to-one (bijective) mappings that I describe in this article (https://varlawend.blogspot.com/2023/06/sociotypes-as-isomorphisms-between.html), as it will be useful for reference if you want to check my claims. This covers one-to-one mappings for Model A system isomorphisms, Model G system isomorphisms and the automorphisms on partitions which include both Model A and Model G systems. Now we just need to show that they will all be structure-preserving:

Proof of Statement 1: You can simply check the mappings in my linked article to confirm that Static/Dynamic is always mapped to Opening/Closing (or its Model A correlate Mental/Vital), Irrational/Rational is always mapped to Stable/Unstable (or its Model A correlate Accepting/Producing), and Extroverted/Introverted is always mapped to Externality/Internality (or its Model A correlate Bold/Cautious). Thus, our mappings preserve the structure of Criterion 1.

Proof of Statement 3: You can simply check the mappings in my linked article to confirm that I only used dichotomies in my mappings which were confirmed to be valid partitions that respect the symmetry of all three temperament dichotomies. Thus, our mappings preserve the structure of Criterion 3.

The proofs of statements 1 and 3 together imply that non-temperament dichotomies are all mapped to other non-temperament dichotomies, since valid partitions are either temperament dichotomies or non-temperament dichotomies, and a non-temperament dichotomy mapping to a temperament dichotomy would contradict our proof of statement 1 (since the mappings are one-to-one, we can only have exactly 1 dichotomy mapping to another).

Another important fact about the mappings in my linked article is that, in both Model A and Model G, the positions of each element are uniquely determined. Thus, if we changed any of the mappings, it would change the positions of the elements and no longer preserve the structure of the type. This shows that the mapping for each type is unique; there is no other way to map the dichotomies from one system in the model to the other.

The uniqueness of type mappings is important because in two sections above (Set of Bijective Mappings Between Model A Elemental and Functional/Positional Blocking Sets, Set of Bijective Mappings Between Model G Elemental and Functional/Positional Blocking Sets), we showed rules that allow us to produce a unique bijective mapping between the elemental blocking sets and functional/positional blocking sets of each type, for models A and G (that's 32 total mappings of blocking sets, since while some types map the sets in the same way, they do it via a different map of elements, as we will see next). To map one set to another, the elements of those sets must have a one-to-one correspondence. However, the elements of those sets are the same dichotomies used in the mappings in my linked article, which, as we just proved, have a unique one-to-one correspondence for every type. Therefore, the only one-to-one correspondence we can use in our bijective mappings of blocking sets for each type are the mappings from the linked article. This proves that the mappings in the linked article respect the structure of the blocking sets, since they are directly used to generate the bijective maps between the blocking sets. To be more specific:

- Suppose that D1 + D2 = D3 where D1, D2 and D3 are dichotomies in one of our dichotomy systems, i.e. {D1, D2, D3} are a blocking set. Since we proved that, for each type, the bijective map "t" of our dichotomies generates a bijective map of blocking sets in which every blocking set is always uniquely mapped to a blocking set in the other dichotomy system, if we apply our mapping "t" to the blocking set {D1, D2, D3} such that t(D1) = S1, t(D2) = S2, and t(D3) = S3, then we know that {S1, S2, S3} is a blocking set in the other dichotomy system.

Proof of Statements 2 and 4: We've already proven that every non-temperament dichotomy maps to another non-temperament dichotomy in the other system. For any given dichotomy E1, it's part of a blocking set {E1, E2, E3} of one of our dichotomy systems, where E2 and E3 are two other dichotomies in that system. We know that for each type, the bijective map of dichotomies from the linked article generates a bijective map of blocking sets, so when we apply this map "t" such that t(E1) = P1, t(E2) = P2, t(E3) = P3, we know that {P1, P2, P3} is a blocking set for the dichotomy system on the other side of the mapping. However, we've already proven that all of the dichotomy systems we use obey Criterion 2 individually, so {P1, P2, P3} contains one temperament dichotomy and two non-temperament dichotomies. Thus, our mappings preserve the structure of Criterion 2. We also know that E1 + E2 = E3 and P1 + P2 = P3, due to these dichotomies being in the same blocking sets respectively. Thus, t(E1 + E2) = t(E3) = P3 = P1 + P2 = t(E1) + t(E2). Hence, t(E1 + E2) = t(E1) + t(E2), so we've also proven Statement 4 and proved that our mappings all preserve the interleaving operation.

Via this proof of bijectivity and structure preservation, we've shown that all of the mappings in the linked article are either isomorphisms or automorphisms (depending on what was claimed). Thus, the Model A elemental dichotomies are isomorphic to the Model A functional/positional dichotomies, the Gulenko-Newman elemental dichotomies are isomorphic to the Gulenko-Newman functional/position dichotomies, and Model A and Model G dichotomy systems can be synthesized in a more encompassing automorphic mathematical system.

____________

Deriving the Tencer-Minaev Dichotomies for Model G

We originally had three separate dichotomy systems working together for Model A: the information/elemental dichotomies, the functional/positional dichotomies and the Reinin dichotomies. Now we have two separate dichotomy systems working together for Model G: the Gulenko/Newman elemental dichotomies, the Gulenko/Newman functional/positional dichotomies, and you guessed it, there is a 3rd!

The reason I call these dichotomies "Tencer-Minaev" is because there are some Socionists who previously discovered them, as I will explain later. Our derivation of these dichotomies is nonetheless somewhat original work since I was the first to discover the GN dichotomies that I will use to derive them. All we need to do to generate the Tencer-Minaev dichotomies is fill the GN positional/functional dichotomies with the GN elemental dichotomies, in analogy to what we did to derive the Reinin dichotomies:

Tencer-Minaev Type Dichotomies (Model G type dichotomies)

Extroverted/Introverted

- Extroverted: Externality positions are filled with Extroverted elements, Internality positions are filled with Introverted elements

- Introverted: Externality positions are filled with Introverted elements, Internality positions are filled with Extroverted elements

Irrational/Rational

- Irrational: Stable positions are filled with Irrational elements, Unstable positions are filled with Rational elements

- Rational: Stable positions are filled with Rational elements, Unstable positions are filled with Irrational elements

Static/Dynamic

- Static: Opening positions are filled with Static elements, Closing positions are filled with Dynamic elements

- Dynamic: Opening positions are filled with Dynamic elements, Closing positions are filled with Static elements

Democratic/Aristocratic

- Democratic: the combined traits of Extroverted+Positivist, or the combined traits of Introverted+Negativist (there is more than one way to form this but it's not straightforward in Model G)

- Aristocratic: the combined traits of Extroverted+Negativist, or the combined traits of Introverted+Positivist (there is more than one way to form this but it's not straightforward in Model G)

Positivist/Negativist

- Positivist: Order Blocks are filled with Positivist Order Blockings, Cross-Order Blocks are filled with Negativist Order Blockings

- Negativist: Order Blocks are filled with Negativist Order Blockings, Cross-Order Blocks are filled with Positivist Order Blockings

Process/Result

- Process: the combined traits of Extroverted+Asking, or the combined traits of Introverted+Declaring (there is more than one way to form this but it's not straightforward in Model G)

- Result: the combined traits of Extroverted+Declaring, or the combined traits of Introverted+Asking (there is more than one way to form this but it's not straightforward in Model G)

Asking/Declaring

- Asking: Revision Blocks are filled with Asking Revision Blockings, Cross-Revision Blocks are filled with Declaring Revision Blockings

- Declaring: Revision Blocks are filled with Declaring Revision Blockings, Cross-Revision Blocks are filled with Asking Revision Blockings

n.b. the first 7 dichotomies here should be familiar, since they are the Orbital dichotomies I mentioned in the section on Reinin dichotomies. These are what the two systems share in structure, just as the elemental and functional/positional dichotomies share the structure of the temperament dichotomies. The other 8 dichotomies are newly derived from the new dichotomy systems we discovered, though they had previously been discovered by other means. Also note that Process/Result is only dichotomy that isn't straightforward in both Model A and Model G; this is no doubt due to its special relationship with the asymmetric relationships in Socionics (Revision and Order), something we will get into in future articles.

1st+6th Alpha Functions + 3rd+8th Gamma Functions / 1st+6th Gamma Functions + 3rd+8th Alpha Functions

- 1st+6th Alpha Functions + 3rd+8th Gamma Functions: the Energetic+Impressionable (Stable) positions are filled with half of the Alpha elements, the Informational+Unimpressionable (Stable) positions are filled with half of the Gamma elements

- 1st+6th Gamma Functions + 3rd+8th Alpha Functions: the Energetic+Impressionable (Stable) positions are filled with half of the Gamma elements, the Informational+Unimpressionable (Stable) positions are filled with half of the Alpha elements

5th+4th Alpha Functions + 2nd+7th Gamma Functions / 5th+4th Gamma Functions + 2nd+7th Alpha Functions

- 5th+4th Alpha Functions + 2nd+7th Gamma Functions: the Excitable+Relaxed (Unstable) positions are filled with half of the Alpha elements, the Inhibitable+Tensioned (Unstable) positions are filled with half of the Gamma elements

- 5th+4th Gamma Functions + 2nd+7th Alpha Functions: the Excitable+Relaxed (Unstable) positions are filled with half of the Gamma elements, the Inhibitable+Tensioned (Unstable) positions are filled with half of the Alpha elements

1st+6th Delta Functions + 3rd+8th Beta Functions / 1st+6th Beta Functions + 3rd+8th Delta Functions

- 1st+6th Delta Functions + 3rd+8th Beta Functions: the Energetic+Impressionable (Stable) positions are filled with half of the Delta elements, the Informational+Unimpressionable (Stable) positions are filled with half of the Beta elements

- 1st+6th Beta Functions + 3rd+8th Delta Functions: the Energetic+Impressionable (Stable) positions are filled with half of the Beta elements, the Informational+Unimpressionable (Stable) positions are filled with half of the Delta elements

5th+4th Beta Functions + 2nd+7th Delta Functions / 5th+4th Delta Functions + 2nd+7th Beta Functions

- 5th+4th Beta Functions + 2nd+7th Delta Functions: the Excitable+Relaxed (Unstable) positions are filled with half of the Beta elements, the Inhibitable+Tensioned (Unstable) positions are filled with half of the Delta elements

- 5th+4th Delta Functions + 2nd+7th Beta Functions: the Excitable+Relaxed (Unstable) positions are filled with half of the Delta elements, the Inhibitable+Tensioned (Unstable) positions are filled with half of the Beta elements

1st+8th Detached Functions + 3rd+6th Involved Functions / 1st+8th Involved Functions + 3rd+6th Detached Functions

- 1st+8th Detached Functions + 3rd+6th Involved Functions: the Excitable+Tensioned (Stable) positions are filled with half of the Detached elements, the Inhibitable+Relaxed (Stable) positions are filled with half of the Involved elements

- 1st+8th Involved Functions + 3rd+6th Detached Functions: the Excitable+Tensioned (Stable) positions are filled with half of the Involved elements, the Inhibitable+Relaxed (Stable) positions are filled with half of the Detached elements

5th+2nd Detached Functions + 4th+7th Involved Functions / 5th+2nd Involved Functions + 4th+7th Detached Functions

- 5th+2nd Detached Functions + 4th+7th Involved Functions: the Energetic+Unimpressionable (Unstable) positions are filled with half of the Detached elements, the Informational+Impressionable (Unstable) positions are filled with half of the Involved elements

- 5th+2nd Involved Functions + 4th+7th Detached Functions: the Energetic+Unimpressionable (Unstable) positions are filled with half of the Involved elements, the Informational+Impressionable (Unstable) positions are filled with half of the Detached elements

1st+8th Implicit Functions + 3rd+6th Explicit Functions / 1st+8th Explicit Functions + 3rd+6th Implicit Functions

- 1st+8th Implicit Functions + 3rd+6th Explicit Functions: the Excitable+Tensioned (Stable) positions are filled with half of the Implicit elements, the Inhibitable+Relaxed (Stable) positions are filled with half of the Explicit elements

- 1st+8th Explicit Functions + 3rd+6th Implicit Functions: the Excitable+Tensioned (Stable) positions are filled with half of the Explicit elements, the Inhibitable+Relaxed (Stable) positions are filled with half of the Implicit elements

5th+2nd Explicit Functions + 4th+7th Implicit Functions / 5th+2nd Implicit Functions + 4th+7th Explicit Functions

- 5th+2nd Explicit Functions + 4th+7th Implicit Functions: the Energetic+Unimpressionable (Unstable) positions are filled with half of the Explicit elements, the Informational+Impressionable (Unstable) positions are filled with half of the Implicit elements

- 5th+2nd Implicit Functions + 4th+7th Explicit Functions: the Energetic+Unimpressionable (Unstable) positions are filled with half of the Implicit elements, the Informational+Impressionable (Unstable) positions are filled with half of the Explicit elements

n.b. yet again we have very long and unwieldly names, so on each of the dichotomies we can shorten it to the 1st function listed since the rest of the name all follows logically from that, e.g. {1stAlpha/1stGamma, 5thAlpha/5thGamma, 1stDelta/1stBeta, 5thBeta/5thDelta, 1stDetached/1stInvolved, 5thDetached/5thInvolved, 1stImplicit/1stExplicit, 5thExplicit/5thImplicit}.

... ... ... ...

And that's it, we have an independent system of 15 type dichotomies which has many important symmetries in Model G which don’t exist in A! It's basically a mirror to the Reinin dichotomies. We can write them in a more elegant way, with shortened names, as follows, just to have them all neatly in one place:

Tencer-Minaev (Model G) Dichotomies

Extroverted/Introverted

Irrational/Rational

Static/Dynamic

Democratic/Aristocratic

Positivist/Negativist

Process/Result

Asking/Declaring

1stAlpha/1stGamma

5thAlpha/5thGamma

1stDelta/1stBeta

5thBeta/5thDelta

1stDetached/1stInvolved

5thDetached/5thInvolved

1stImplicit/1stExplicit

5thExplicit/5thImplicit

... ... ... ...

As I mentioned, the first people to discover these dichotomies are the Socionists Ibrahim Tencer and Yuri Minaev, though apparently they did so through different means since they did not realize the relationship of these dichotomies to Model G. Here is Ibrahim Tencer's paper on the subject in which he also discovered the Tencer-Minaev dichotomies, though his approach was rooted directly in Group Theory and Lattice Theory, which we may explore at a later time (since they relate a lot to the math we have done here and could be used to prove some things more tersely, but in a way less accessible to people who aren't familiar with higher mathematics): https://www.sedecology.com/math. Though, we did make use of some of these disciplines already (e.g. Boolean algebras and their properties like distributivity, structure preserving maps, etc.).

We can easily show that:

- the Tencer-Minaev dichotomies are the same sort of Boolean dichotomy system as the Reinin dichotomies (if it's not already obvious given the proof of Ibrahim Tencer, the same way of deriving it as the Reinin dichotomies, and structural similarity everywhere else in the system)

- with a 4-element basis (which is somewhat arbitrary but choosing one that works is sufficient)

- the ability to add any two other dichotomies together so that they always produce another dichotomy in the system (i.e. it has the property of "closure"). Let's take the following as our basis to use something similar to what we used for the Reinin dichotomies (with the same two temperament dichotomies, and two purely Tencer-Minaev dichotomies):

    E: Extroverted/Introverted

    A: 1stAlpha/1stGamma

    D: 5thDetached/5thInvolved

    P: Irrational/Rational

With this basis and the same type of Boolean system we used for the Reinin dichotomies, you can easily verify for yourself that the following describes all the Boolean structural relationships between the dichotomies (which we’ll prove rigorously by showing all the small groups that have all the relationships between the dichotomies later):

- E: Extroverted/Introverted

- A: 1stAlpha/1stGamma

- D: 5thDetached/1stInvolved

- P: Irrational/Rational

- EA: 1stImplicit/1stExplicit

- ED: 5thBeta/5thDelta

- EP: Static/Dynamic

- AD: Asking/Declaring

- AP: 1stDelta/1stBeta

- DP: 5thExplicit/5thImplicit

- EAD: Process/Result

- EAP: 1stDetached/1stInvolved

- EDP: 5thAlpha/5thGamma

- ADP: Positivist/Negativist

- EADP: Democratic/Aristocratic

... ... ... ...

And yet again we have a completely interconnected system (this time for Model G), with derivations from multiple directions, in a harmonious, elegant structure. In fact, it is even more connected than it seems. So far, the impression I probably gave is something like there being 2 meta-systems (one for Model A and one for Model G) where each has 3 interconnected dichotomy systems which may share some crucial structure but are largely siloed (i.e. segregated) to their particular model. This is not the case! First of all, it is possible to use Model A functional/positional dichotomies to define the Tencer-Minaev dichotomies, and it is possible to use the Model G GN functional/positional dichotomies to define the Reinin dichotomies. We can just give an example of how it would work for one Tencer-Minaev dichotomy and one Reinin dichotomy, since the general structure will be the same for all:

Deriving 1stAlpha/1stGamma (a pure Tencer-Minaev dichotomy) from Model A functional/positional dichotomies

- 1stAlpha: Evaluative+Valued (Accepting) functions are one of two Alpha dual pairs (Ne+Si or Fe+Ti), Situational+Unvalued (Accepting) functions are one of two Gamma dual pairs (Se+Ni or Te+Fi)

- 1stGamma: Evaluative+Valued (Accepting) functions are one of two Gamma dual pairs (Se+Ni or Te+Fi), Situational+Unvalued (Accepting) functions are one of two Alpha dual pairs (Ne+Si or Fe+Ti)

Deriving Intuitive/Sensing (a pure Reinin dichotomy) from Model G GN functional/positional dichotomies

- Intuitive: the Intuitive elements are either Excitable+Tensioned (Stable) or Energetic+Unimpressionable (Unstable), the Sensing elements are either Inhibitable+Relaxed (Stable) or Informational+Impressionable (Unstable)

- Sensing: the Sensing elements are either Excitable+Tensioned (Stable) or Energetic+Unimpressionable (Unstable), the Intuitive elements are either Inhibitable+Relaxed (Stable) or Informational+Impressionable (Unstable)

As you can see, both will have a 50/50 possible split along the lines of the Irrationality/Rationality of the type. However, such 50/50 splits were already how the non-orbital dichotomies looked in all the derivations we did so far, so it's not a problem for the derivation (i.e. with enough dichotomies, the type will still be clear, since one dichotomy isn't supposed to be enough to remove all ambiguity about type anyways, though it still restricts the possible types and thus contributes information). This is seemingly starting to undermine the assertion that the Reinin dichotomies are more associated with Model A, while the Tencer-Minaev dichotomies are more associated with Model G. In fact, it gets worse than that!

Victor Gulenko and his School of Humanitarian Socionics are mostly the only one's to use Model G in any official capacity, yet they have used it (at least in the past) alongside Model A functional/positional dichotomies! You can see exactly those dichotomies defined in this older article describing how Model G works: http://zanoza.socioland.ru/wiki/%D0%9C%D0%BE%D0%B4%D0%B5%D0%BB%D1%8C_G (in particular, the section entitled "Позиционный анализ функций"). This seems to imply that merely because the Model A functional/positional dichotomies are asymmetric in Model G does not prevent their usefulness or them having a sound semantic interpretation (even though some of these definitions need to be refined to current usage, some remain accurate). Likewise, although no one who uses Model A uses the Gulenko-Newman functional/positional dichotomies since they don't even yet know they exist, the possibility remains that they could have a useful (albeit asymmetric) interpretation in Model A!

This trend even extends to the elemental dichotomies! Victor Gulenko, who as you know uses Model G, gave an exhaustive definition of Model A elemental dichotomies many years ago in his article "Disappear to Reappear: Functional States of Personality": https://web.archive.org/web/20140410022344/http://socionics.kiev.ua/articles/methodology/ischez/ (in particular, the section entitled "2. Дихотомическое описание функций"). Again, these definitions aren't all accurate to current usage (though some are), but the point is that a practice based in Model G can still usefully interpret these Model A elemental dichotomies. Likewise, although almost no one who uses Model A is yet aware that there is a set of Model G elemental dichotomies, perhaps they could one day find a useful interpretation for them in their models and practices.

Even further, Victor Gulenko has managed to use several Reinin dichotomies (which are supposed to be more associated with Model A) in his practice, including Intuitive/Sensing (https://socioniks.net/en/article/?id=5), Logical/Ethical (https://socioniks.net/en/article/?id=1), Central/Peripheral (https://socioniks.net/en/article/?id=156) and Tactical/Strategic (https://qfour.blogspot.com/2011/01/blog-post_25.html), and even more besides those. There's little reason why Tencer-Minaev dichotomies could never be used by Model A users, though most aren't even familiar with these dichotomies yet (even though Ibrahim Tencer discovered them more than a decade ago in 2011).

The fact that the Reinin dichotomies and Tencer-Minaev dichotomies can be derived from both Model A and Model G dichotomies and can technically be used with both models, in addition to the fact that the functional/positional and elemental dichotomies of Model A can be used effectively in Model G (albeit asymmetrically) and vice versa, might seem to undermine my assertions that any of these dichotomy systems can be more closely associated with either Model A or Model G in the first place. However, it turns out that my associations are well justified after all, just for a partly different set of reasons.

In the case of the functional/positional dichotomies, the reason we already associated them more with one model than another is because each was only symmetric in one model (the one it is associated with), and asymmetric in the other. So even if both sets of dichotomies can be used in any valid Socionics model, it still seems clear that they have a more elegant, symmetric interpretation in one model, and thus play a greater role in describing the structure that model (which is based on symmetries). Furthermore, the elemental dichotomies are each only mathematically compatible with one set of functional/positional dichotomies, so they are naturally more associated with whichever model their compatible functional/positional dichotomies happen to be. Thus, we still have sufficient reasons for associating the classical elemental dichotomies and functional/positional dichotomies more with Model A, and the Gulenko-Newman dichotomies more with Model G, as long as we are not too strict in this divide.

This phenomenon of being "more associated with one model" may not seem to be the case for the Reinin dichotomies and Tencer-Minaev dichotomies, but we will see that it's very much the case for them when we segue into last topic of this paper, which is "small groups". Each of these dichotomies produces an independent system of small groups, and the Reinin small groups make far more sense in Model A whereas the Tencer-Minaev small groups make far more sense in Model G. Thus, the fact that all of these dichotomy systems can be used in all models does not undermine my assertions. Rather, it shows that both Model A and Model G have the potential to be synthesized into a more powerful and harmonious whole that is beyond any Socionics model that came before. That is truly exciting research!

____________

Small Groups for the Reinin (Model A) dichotomies and Tencer-Minaev (Model G) dichotomies

For both Reinin and Tencer-Minaev dichotomies, we showed that each closed system consists of a Boolean logic with an addition operation whereby we could add together two dichotomies and it would always produce a deterministic third dichotomy based on the structural relationship that the first two dichotomies have within the Boolean system. This operation creates a "small group" of three dichotomies; if any two dichotomies in the group are added (technically we are "interleaving" them), they will always produce the member of the group that was not added (regardless of which two in the group that we pick). One might then wonder what are all possible combinations of three dichotomies, for both the Reinin and Tencer-Minaev dichotomy systems. In turns out that each dichotomy system has 35 possible "small groups" of dichotomies as we have described them. There is some overlap between the small groups in each dichotomy system, which makes sense since they share 7 dichotomies and a lot of structure, however each have 16 completely independent sets of small groups.

First, we should explain a little more about what each small group is. When you figure out two dichotomies in the context of diagnosing the Sociotype of a person, you narrow down the number of possible types from 16 to 4. That's because the first dichotomy divides number of types in half (since each half of a dichotomy applies to only half the types, which is why figuring out which side of a dichotomy someone is on gives us some information about what Sociotype they might be), thus only 8 types are possible. The second dichotomy divides the number of types in half again, so when we figure out two dichotomies, only 4 types are thereby possible. These two dichotomies imply that a third dichotomy must be a certain way (based on strict Boolean logic), but this dichotomy adds no information since it is linearly dependent on the first two dichotomies (if we think about the system of dichotomies as a vector space, which is possible, as Ibrahim Tencer and Andrew Joynton have already showed), so it does not divide the types any further. Because of this, based on which side of the three dichotomies in a small group a person is on (and we only need to manually figure out two of them since the third is logically implied from the other two no matter which two we figure out first), it allows us to assume that a person only has 4 possible types (which is very helpful for diagnostics of Sociotype since it eliminates many possible options). Further, the sets of small groups divide the types into 4 groups of 4 types that share some empirically visible semantic property; some particularly famous small groups include quadra, club and cognitive style. This in turn allows us to identify three dichotomies for a person just by figuring out one property such a small group is expected to have. They are very useful in Socionics.

Also, you can notice that the small groups of the type dichotomies are analogous to the blocking sets in the functional/positional and elemental dichotomies. Both contain three dichotomies, only two of which give information since the third is implied by the other two. However, the blocking sets work on systems of 8 elements or functions (8 divided by 2 twice in a row is 2, which corresponds to their being 4 sets of 2 elements in every blocking set), whereas the small groups work on systems of 16 types (so when 16 is divided by 2 twice, it results in 4 sets of 4 types in every small group set).

Up until now, only the Reinin dichotomies have been exhaustively laid out with some well-known interpretations. My collaborator Andrew Joynton and I were two of the major contributors to this page on WikiSocion which lists and briefly describes all of the small groups in the Reinin dichotomy system. As you can see, the Reinin dichotomies have a set of 35 small group systems, and since the Tencer-Minaev dichotomies number of the same as Reinins and have the same Boolean algebra structure, they will also have 35 small group systems. Let's start with the small group systems they have in common:

Orbital Small Group Systems (Reinin + Tencer-Minaev)

• Temperament
  • Extroverted, Irrational, Static (Flexible-Maneuvering)
    • ILE SLE SEE IEE
  • Extroverted, Rational, Dynamic (Linear-Assertive)
    • ESE EIE LIE LSE
  • Introverted, Irrational, Dynamic (Receptive-Adapative)
    • SEI IEI ILI SLI
  • Introverted, Rational, Static (Balanced-Stable)
    • LII LSI ESI EII
• Positivity Groups
  • Extroverted, Democratic, Positivist (Approaching)
    • ILE ESE SEE LIE
  • Extroverted, Aristocratic, Negativist (Shaping)
    • EIE SLE LSE IEE
  • Introverted, Democratic, Negativist (Abstaining)
    • SEI LII ILI ESI
  • Introverted, Aristocratic, Positivist (Integrating)
    • LSI IEI EII SLI
• Order Rings
  • Extroverted, Process, Asking (Revolutionary)
    • ILE EIE SEE LSE
  • Extroverted, Result, Declaring (Transformative)
    • ESE SLE LIE IEE
  • Introverted, Process, Declaring (Evolutionary)
    • SEI LSI ILI EII
  • Introverted, Result, Asking (Involutionary)
    • LII IEI ESI SLI
• Stress Resistance
  • Irrational, Democratic, Process (Stress-Trained)
    • ILE SEI SEE ILI
  • Irrational, Aristocratic, Result (Stress-Resistant)
    • SLE IEI IEE SLI
  • Rational, Democratic, Result (Stress-Braked)
    • ESE LII LIE ESI
  • Rational, Aristocratic, Process (Stress-Absorbant)
    • EIE LSI LSE EII
• Activity Orientation Shift
  • Irrational, Positivist, Asking (Beneficiaries)
    • ILE IEI SEE SLI
  • Irrational, Negativist, Declaring (Alternatives)
    • SEI SLE ILI IEE
  • Rational, Positivist, Declaring (Streamliners)
    • ESE LSI LIE EII
  • Rational, Negativist, Asking (Iconoclasts)
    • LII EIE ESI LSE
• Challenge-Response Groups
  • Static, Democratic, Asking (Reorienters)
    • ILE LII SEE ESI
  • Static, Aristocratic, Declaring (Foundationalists)
    • LSI SLE EII IEE
  • Dynamic, Democratic, Declaring (Slackeners)
    • SEI ESE ILI LIE
  • Dynamic, Aristocratic, Asking (Sharpeners)
    • EIE IEI LSE SLI
• Revision Rings (Cognitive Styles)
  • Static, Positivist, Process (Causal-Determinist)
    • ILE LSI SEE EII
  • Static, Negativist, Result (Holographic-Panoramic)
    • LII SLE ESI IEE
  • Dynamic, Positivist, Result (Vortical-Synergetic)
    • ESE IEI LIE SLI
  • Dynamic, Negativist, Process (Dialectical-Algorithmic)
    • SEI EIE ILI LSE

The reason that the two dichotomy systems share these 7 small group systems should be obvious: it's because they both share the 7 dichotomies that make up all 7 of these small group systems (they are the Orbital dichotomies). They are very interesting to look at, but since they don't distinguish between the two dichotomy systems, they aren't especially relevant to our topic so we'll just move on for now.

... ... ... ...

The next 2 sets of 12 small group systems are called the Mixed Small Group Systems (abbreviated MSGS), which we will explain briefly. They have similarities and differences in both dichotomy systems, which is why we need 2 sets of 12. Next to the three dichotomies that make up each group of 4 types, I will put a brief description of that small group in parentheses (so that it will serve as similar to a small encyclopedia of small groups).

Mixed Small Group Systems (Reinin)

• Activity Orientation
  • Democratic, Intuitive, Logical (Scientist-Researcher)
    • ILE LII ILI LIE
  • Democratic, Sensing, Ethical (Social-Communicative)
    • SEI ESE SEE ESI
  • Aristocratic, Intuitive, Ethical (Humanitarian-Artisanal)
    • EIE IEI EII IEE
  • Aristocratic, Sensing, Logical (Technical-Managerial)
    • LSI SLE LSE SLI
• Dual Axis (Dimensionality)
  • Democatic, Carefree, Yielding (Evaluative Serious+Judicious)
    • ILE SEI LIE ESI
  • Democratic, Farsighted, Obstinate (Evaluative Merry+Decisive)
    • ESE LII SEE ILI
  • Aristocratic, Carefree, Obstinate (Evaluative Merry+Judicious)
    • EIE LSI IEE SLI
  • Aristocratic, Farsighted, Yielding (Evaluative Serious+Decisive)
    • SLE IEI LSE EII
• Inert Activity Orientation Groups
  • Democratic, Tactical, Constructivist (Inert Humanitarian)
    • ILE ESE ILI ESI
  • Democratic, Strategic, Emotivist (Inert Managerial)
    • SEI LII SEE LIE
  • Aristocratic, Tactical, Emotivist (Inert Scientist)
    • LSI IEI LSE IEE
  • Aristocratic, Strategic, Constructivist (Inert Social)
    • EIE SLE EII SLI
• Quadras
  • Democratic, Judicious, Merry (Alpha)
    • ILE SEI ESE LII
  • Democratic, Decisive, Serious (Gamma)
    • SEE ILI LIE ESI
  • Aristocratic, Judicious, Serious (Delta)
    • LSE EII IEE SLI
  • Aristocratic, Decisive, Merry (Beta)
    • EIE LSI SLE IEI
• Perceptual Groups
  • Irrational, Intuitive, Tactical (Associative)
    • ILE IEI ILI IEE
  • Irrational, Sensing, Strategic (Commutative)
    • SEI SLE SEE SLI
  • Rational, Intuitive, Strategic (Dissociative)
    • LII EIE LIE EII
  • Rational, Sensing, Tactical (Distributive)
    • ESE LSI ESI LSE
• Reasoning Groups
  • Irrational, Logical, Constructivist (Restructurers)
    • ILE SLE ILI SLI
  • Irrational, Ethical, Emotivist (Diplomats)
    • SEI IEI SEE IEE
  • Rational, Ethical, Constructivist (Guardians)
    • ESE EIE ESI EII
  • Rational, Logical, Emotivist (Constructors)
    • LII LSI LIE LSE
• Dual Axis (Irrational)
  • Irrational, Carefree, Judicious (Valued+Evaluative Judicious)
    • ILE SEI IEE SLI
  • Irrational, Farsighted, Decisive (Valued+Evaluatuve Decisive)
    • SLE IEI SEE ILI
  • Rational, Carefree, Decisive (Valued Decisive/Evaluative Judicious)
    • EIE LSI LIE ESI
  • Rational, Farsighted, Judicious (Valued Judicious/Evaluative Decisive)
    • ESE LII LSE EII
• Dual Axis (Rational)
  • Irrational, Yielding, Merry (Valued Merry/Evaluative Serious)
    • ILE SEI SLE IEI
  • Irrational, Obstinate, Serious (Valued Serious/Evaluative Merry)
    • SEE ILI IEE SLI
  • Rational, Yielding, Serious (Valued+Evaluative Serious)
    • LIE ESI LSE EII
  • Rational, Obstinate, Merry (Valued+Evaluative Merry)
    • ESE LII EIE LSI
• Project Groups
  • Process, Intuitive, Constructivist (Inert Ethics/Strong Intuition)
    • ILE EIE ILI EII
  • Process, Sensing, Emotivist (Inert Logic/Strong Sensing)
    • SEI LSI SEE LSE
  • Result, Intuitive, Emotivist (Inert Logic/Strong Intuition)
    • LII IEI LIE IEE
  • Result, Sensing, Constructivist (Inert Ethics/Strong Sensing)
    • ESE SLE ESI SLI
• Implementation Groups
  • Process, Logical, Tactical (Inert Intuition/Strong Logic)
    • ILE LSI ILI LSE
  • Process, Ethical, Strategic (Inert Sensing/Strong Ethics)
    • SEI EIE SEE EII
  • Result, Logical, Strategic (Inert Sensing/Strong Logic)
    • LII SLE LIE SLI
  • Result, Ethical, Tactical (Inert Intuition/Strong Ethics)
    • ESE IEI ESI IEE
• Dual Axis (Passionarity Asymmetric)
  • Process, Carefree, Merry (Valued Merry/Evaluative Judicious)
    • ILE SEI EIE LSI
  • Process, Farsighted, Serious (Valued Serious/Evaluative Decisive)
    • SEE ILI LSE EII
  • Result, Carefree, Serious (Valued Serious/Evaluative Judicious)
    • LIE ESI IEE SLI
  • Result, Farsighted, Merry (Valued Merry/Evaluative Decisive)
    • ESE LII SLE IEI
• Dual Axis (Centrality Asymmetric)
  • Process, Yielding, Judicious (Valued Judicious/Evaluative Serious)
    • ILE SEI LSE EII
  • Process, Obstinate, Decisive (Valued Decisive/Evaluative Merry)
    • EIE LSI SEE ILI
  • Result, Yielding, Decisive (Valued Decisive/Evaluative Serious)
    • SLE IEI LIE ESI
  • Result, Obstinate, Judicious (Valued Judicious/Evaluative Merry)
    • ESE LII IEE SLI

... ... ... ...

Mixed Small Group Systems (Tencer-Minaev)

• Dual Axis (Democratic Elements)
  • Democratic, 1stAlpha, 5thAlpha (Alpha Dualities NOT Informational + Inhibitable, Gamma Dualities NOT Energetic + Excitable)
    • ILE SEI ESE LII
  • Democratic, 1stGamma, 5thGamma (Gamma Dualities NOT Informational + Inhibitable, Alpha Dualities NOT Energetic + Excitable)
    • SEE ILI LIE ESI
  • Aristocratic, 1stAlpha, 5thGamma (Alpha Dualities NOT Informational + Excitable, Gamma Dualities NOT Energetic + Inhibitable)
    • EIE LSI IEE SLI
  • Aristocratic, 1stGamma, 5thAlpha (Gamma Dualities NOT Informational + Excitable, Alpha Dualities NOT Energetic + Inhibitable)
    • SLE IEI LSE EII
• Dual Axis (Aristocratic Elements)
  • Democratic, 1stDelta, 5thBeta (Delta Dualities NOT Informational + Excitable, Beta Dualities NOT Energetic + Inhibitable)
    • ILE SEI LIE ESI
  • Democratic, 1stBeta, 5thDelta (Beta Dualities NOT Informational + Excitable, Delta Dualities NOT Energetic + Inhibitable)
    • ESE LII SEE ILI
  • Aristocratic, 1stDelta, 5thDelta (Delta Dualities NOT Informational + Inhibitable, Beta Dualities NOT Energetic + Excitable)
    • LSE EII IEE SLI
  • Aristocratic, 1stBeta, 5thBeta (Beta Dualities NOT Informational + Inhibitable, Delta Dualities NOT Energetic + Excitable)
    • EIE LSI SLE IEI
• Activity Orientation Axis (Democratic Elements)
  • Democratic, 1stDetached, 5thDetached (Detached NOT Informational + Inhibitable, Involved NOT Energetic + Excitable)
    • ILE LII ILI LIE
  • Democratic, 1stInvolved, 5thInvolved (Involved NOT Informational + Inhibitable, Detached NOT Energetic + Excitable)
    • SEI ESE SEE ESI
  • Aristocratic, 1stDetached, 5thInvolved (Detached NOT Energetic + Inhibitable, Involved NOT Informational + Excitable)
    • LSI IEI LSE IEE
  • Aristocratic, 1stInvolved, 5thDetached (Involved NOT Energetic + Inhibitable, Detached NOT Informational + Excitable)
    • EIE SLE EII SLI
• Activity Orientation Axis (Aristocratic Elements)
  • Democratic, 1stImplicit, 5thExplicit (Implicit NOT Energetic + Inhibitable, Explicit NOT Informational + Excitable)
    • ILE ESE ILI ESI
  • Democratic, 1stExplicit, 5thImplicit (Explicit NOT Energetic + Inhibitable, Implicit NOT Informational + Excitable)
    • SEI LII SEE LIE
  • Aristocratic, 1stImplicit, 5thImplicit (Implicit NOT Informational + Inhibitable, Explicit NOT Energetic + Excitable)
    • EIE IEI EII IEE
  • Aristocratic, 1stExplicit, 5thExplicit (Explicit NOT Informational + Inhibitable, Implicit NOT Energetic + Excitable)
    • LSI SLE LSE SLI
• High/Low Energy Dual Axis
  • Irrational, 1stAlpha, 1stDelta (Peripheral Energetic/Central Informational)
    • ILE SEI IEE SLI
  • Irrational, 1stGamma, 1stBeta (Central Energetic/Peripheral Informational)
    • SLE IEI SEE ILI
  • Rational, 1stAlpha, 1stBeta (Ascending Energetic/Descending Informational)
    • ESE LII EIE LSI
  • Rational, 1stGamma, 1stDelta (Descending Energetic/Ascending Informational)
    • LIE ESI LSE EII
• High/Low Brakes Dual Axis
  • Irrational, 5thAlpha, 5thBeta (Ascending Excitable/Descending Inhibitable)
    • ILE SEI SLE IEI
  • Irrational, 5thGamma, 5thDelta (Descending Excitable/Ascending Inhibitable)
    • SEE ILI IEE SLI
  • Rational, 5thAlpha, 5thDelta (Peripheral Excitable/Central Inhibitable)
    • ESE LII LSE EII
  • Rational, 5thGamma, 5thBeta (Central Excitable/Peripheral Inhibitable)
    • EIE LSI LIE ESI
• High/Low Brakes Redemption Axis
  • Irrational, 1stDetached, 1stImplicit (Intuition Excitable/Sensing Inhititable)
    • ILE IEI ILI IEE
  • Irrational, 1stInvolved, 1stExplicit (Sensing Excitable/Intuitive Inhibitable)
    • SEI SLE SEE SLI
  • Rational, 1stDetached, 1stExplicit (Logic Excitable/Ethics Inhibitable)
    • LII LSI LIE LSE
  • Rational, 1stInvolved, 1stImplicit (Ethics Excitable/Logic Inhibitable)
    • ESE EIE ESI EII
• High/Low Energy Redemption Axis
  • Irrational, 5thDetached, 5thExplicit (Logic Energetic/Ethics Informational)
    • ILE SLE ILI SLI
  • Irrational, 5thInvolved, 5thImplicit (Ethics Energetic/Logic Informational)
    • SEI IEI SEE IEE
  • Rational, 5thDetached, 5thImplicit (Intuition Energetic/Sensing Informational)
    • LII EIE LIE EII
  • Rational, 5thInvolved, 5thExplicit (Sensing Energetic/Intuition Informational)
    • ESE LSI ESI LSE
• Dual Axis (Asymmetric Process Passionarity Result Centrality)
  • Process, 1stAlpha, 5thBeta (Ascending NOT Informational + Inhibitable, Centrality NOT Energetic + Inhibitable)
    • ILE SEI EIE LSI
  • Process, 1stGamma, 5thDelta (Descending NOT Informational + Inhibitable, Peripherality NOT Energetic + Inhibitable)
    • SEE ILI LSE EII
  • Result, 1stAlpha, 5thDelta (Peripherality NOT Informational + Inhibitable, Descending NOT Energetic + Inhibitable)
    • ESE LII IEE SLI
  • Result, 1stGamma, 5thBeta (Centrality NOT Informational + Inhibitable, Ascending NOT Energetic + Inhibitable)
    • SLE IEI LIE ESI
• Dual Axis (Asymmetric Process Centrality Result Passionarity)
  • Process, 1stDelta, 5thAlpha (Peripherality NOT Informational + Inhibitable, Ascending NOT Energetic + Inhibitable)
    • ILE SEI LSE EII
  • Process, 1stBeta, 5thGamma (Centrality NOT Informational + Inhibitable, Descending NOT Energetic + Inhibitable)
    • EIE LSI SEE ILI
  • Result, 1stDelta, 5thGamma (Descending NOT Informational + Inhibitable, Centrality NOT Energetic + Inhibitable)
    • LIE ESI IEE SLI
  • Result, 1stBeta, 5thAlpha (Ascending NOT Informational + Inhibitable, Peripherality NOT Energetic + Inhibitable)
    • ESE LII SLE IEI
• Process Implementation Result Project Groups
  • Process, 1stDetached, 5thExplicit (Logic NOT Informational + Inhibitable, Intuition NOT Energetic + Inhibitable)
    • ILE LSI ILI LSE
  • Process, 1stInvolved, 5thImplicit (Ethics NOT Informational + Inhibitable, Sensing NOT Energetic + Inhibitable)
    • SEI EIE SEE EII
  • Result, 1stDetached, 5thImplicit (Intuition NOT Informational + Inhibitable, Logic NOT Energetic + Inhibitable)
    • LII IEI LIE IEE
  • Result, 1stInvolved, 5thExplicit (Sensing NOT Informational + Inhibitable, Ethics NOT Energetic + Inhibitable)
    • ESE SLE ESI SLI
• Process Project Result Implementation Groups
  • Process, 1stImplicit, 5thDetached (Intuition NOT Informational + Inhibitable, Ethics NOT Energetic + Inhibitable)
    • ILE EIE ILI EII
  • Process, 1stExplicit, 5thInvolved (Sensing NOT Informational + Inhibitable, Logic NOT Energetic + Inhibitable)
    • SEI LSI SEE LSE
  • Result, 1stImplicit, 5thInvolved (Ethics NOT Informational + Inhibitable, Intuition NOT Energetic + Inhibitable)
    • ESE IEI ESI IEE
  • Result, 1stExplicit, 5thDetached (Logic NOT Informational + Inhibitable, Sensing NOT Energetic + Inhibitable)
    • LII SLE LIE SLI

What you should notice about these sets of 4 types is that each appear twice, once in the Reinin small groups and once in the Tencer-Minaev small groups. You can verify this with a close look or your search function. This is what is similar between the two systems of 12 MSGS. They don't contain any fundamentally different small groups of types, so they also don't fully distinguish between the dichotomy systems either, and thus also aren't maximally interesting for our purposes. However, you will notice that they are paired differently into sets of small groups, so a slightly different property of the small groups are emphasized depending on whether we use the Reinin or Tencer-Minaev dichotomies. For example, the small group "ILE LII ILI LIE" is paired with other small groups representing the various "clubs" of Socionics in the Reinin dichotomies. However, this same small group of types is paired with some different groups in the Tencer-Minaev dichotomies which represent something about Detached vs Involved functions.

The reason that these small groups work differently from the rest is that they are all formed with one of the three central dichotomies {Democratic/Aristocratic, Irrational/Rational, Process/Result} which play a different structural role than the rest of the orbital dichotomies in Socionics.

... ... ... ...

Now, we have a total 7+12=19 sets of small groups for each dichotomy system, which means 16 truly unique sets of small groups remain for each.

Pure Reinin Small Group Systems

• Stimulus Seeking
  • Extroverted, Intuitive, Carefree (4D Ne/3D Ni/2D Se/1D Si)
    • ILE EIE LIE IEE
  • Extroverted, Sensing, Farsighted (4D Se/3D Si/2D Ne/1D Ni)
    • ESE SLE SEE LSE
  • Introverted, Intuitive, Farsighted (4D Ni/3D Ne/2D Si/1D Se)
    • LII IEI ILI EII
  • Introverted, Sensing, Carefree (4D Si/3D Se/2D Ni/1D Ne)
    • SEI LSI ESI SLI
• Communication Style
  • Extroverted, Logical, Yielding (4D Te/3D Ti/2D Fe/1D Fi)
    • ILE SLE LIE LSE
  • Extroverted, Ethical, Obstinate (4D Fe/3D Fi/2D Te/1D Ti)
    • ESE EIE SEE IEE
  • Introverted, Logical, Obstinate (4D Ti/3D Te/2D Fi/1D Fe)
    • LII LSI ILI SLI
  • Introverted, Ethical, Yielding (4D Fi/3D Fe/2D Ti/1D Te)
• Centrality Vertness Groups
  • Extroverted, Tactical, Judicious (Inert Bold Valued Ne)
    • ILE ESE LSE IEE
  • Extroverted, Strategic, Decisive (Inert Bold Valued Se)
    • EIE SLE SEE LIE
  • Introverted, Tactical, Decisive (Inert Bold Valued Ni)
    • LSI IEI ILI ESI
  • Introverted, Strategic, Judicious (Inert Bold Valued Si)
    • SEI LII EII SLI
• Passionarity Vertness Groups
  • Extroverted, Constructivist, Merry (Inert Bold Valued Fe)
    • ILE ESE EIE SLE
  • Extroverted, Emotivist, Serious (Inert Bold Valued Te)
    • SEE LIE LSE IEE
  • Introverted, Constructivist, Serious (Inert Bold Valued Fi)
    • ILI ESI EII SLI
  • Introverted, Emotivist, Merry (Inert Bold Valued Ti)
    • SEI LII LSI IEI
• Irrational Ego Block
  • Static, Intuitive, Judicious (Ne Ego)
    • ILE LII EII IEE
  • Static, Sensory, Decisive (Se Ego)
    • LSI SLE SEE ESI
  • Dynamic, Intuitive, Decisive (Ni Ego)
    • EIE IEI ILI LIE
  • Dynamic, Sensory, Judicious (Si Ego)
    • SEI ESE LSE SLI
• Rational Ego Block
  • Static, Logical, Merry (Ti Ego)
    • ILE LII LSI SLE
  • Static, Ethical, Serious (Fi Ego)
    • SEE ESI EII IEE
  • Dynamic, Logical, Serious (Te Ego)
    • ILI LIE LSE SLI
  • Dynamic, Ethical, Merry (Fe Ego)
    • SEI ESE EIE IEI
• Irrational Vertical Block
  • Static, Carefree, Tactical (Inert Mental Evaluative Ne)
    • ILE LSI ESI IEE
  • Static, Farsighted, Strategic (Inert Mental Evaluative Se)
    • LII SLE SEE EII
  • Dynamic, Carefree, Strategic (Inert Mental Evaluative Si)
    • SEI EIE LIE SLI
  • Dynamic, Farsighted, Tactical (Inert Mental Evaluative Ni)
    • ESE IEI ILI LSE
• Rational Vertical Block
  • Static, Yielding, Constructivist (Inert Mental Evaluative Fi)
    • ILE SLE ESI EII
  • Static, Obstinate, Emotivist (Inert Mental Evaluative Ti)
    • LII LSI SEE IEE
  • Dynamic, Yielding, Emotivist (Inert Mental Evaluative Te)
    • SEI IEI LIE LSE
  • Dynamic, Obstinate, Constructivist (Inert Mental Evaluative Fe)
    • ESE EIE ILI SLI
• Same Irrationality Revision
  • Positivist, Intuitive, Yielding (Strong Intuition/Evaluative Serious)
    • ILE IEI LIE EII
  • Positivist, Sensing, Obstinate (Strong Sensing/Evaluative Merry)
    • ESE LSI SEE SLI
  • Negativist, Intuitive, Obstinate (Strong Intuition/Evaluative Merry)
    • LII EIE ILI IEE
  • Negativist, Sensing, Yielding (Strong Sensing/Evaluative Serious)
    • SEI SLE ESI LSE
• Same Rationality Revision
  • Positivist, Logical, Carefree (Strong Logic/Evaluative Judicious)
    • ILE LSI LIE SLI
  • Positivist, Ethical, Farsighted (Strong Ethics/Evaluative Decisive)
    • ESE IEI SEE EII
  • Negativist, Logical, Farsighted (Strong Logic/Evaluative Decisive)
    • LII SLE ILI LSE
  • Negativist, Ethical, Carefree (Strong Ethics/Evaluative Judicious)
    • SEI EIE ESI IEE
• Same Passionarity Revision
  • Positivist, Tactical, Merry (Valued Merry/Inert Intuition)
    • ILE ESE LSI IEI
  • Positivist, Strategic, Serious (Valued Serious/Inert Sensing)
    • SEE LIE EII SLI
  • Negativist, Tactical, Serious (Valued Serious/Inert Intuition)
    • ILI ESI LSE IEE
  • Negativist, Strategic, Merry (Valued Merry/Inert Sensing)
    • SEI LII EIE SLE
• Same Centrality Revision
  • Positivist, Constructivist, Judicious (Valued Judicious/Inert Ethics)
    • ILE ESE EII SLI
  • Positivist, Emotivist, Decisive (Valued Decisive/Inert Logic)
    • LSI IEI SEE LIE
  • Negativist, Constructivist, Decisive (Valued Decisive/Inert Ethics)
    • EIE SLE ILI ESI
  • Negativist, Emotivist, Judicious (Valued Judicious/Inert Logic)
    • SEI LII LSE IEE
• Passionarity Challenge-Response
  • Asking, Intuitive, Merry (Valued Merry/Strong Intuition)
    • ILE LII EIE IEI
  • Asking, Sensing, Serious (Valued Serious/Strong Sensing)
    • SEE ESI LSE SLI
  • Declaring, Intuitive, Serious (Valued Serious/Strong Intuition)
    • ILI LIE EII IEE
  • Declaring, Sensing, Merry (Valued Merry/Strong Sensing)
    • SEI ESE LSI SLE
• Centrality Challenge-Response
  • Asking, Logical, Judicious (Valued Judicious/Strong Logic)
    • ILE LII LSE SLI
  • Asking, Ethical, Decisive (Valued Decisive/Strong Ethics)
    • EIE IEI SEE ESI
  • Declaring, Logical, Decisive (Valued Decisive/Strong Logic)
    • LSI SLE ILI LIE
  • Declaring, Ethical, Judicious (Valued Judicious/Strong Ethics)
    • SEI ESE EII IEE
• Same Irrational Dimension Conflict Challenge-Response
  • Asking, Carefree, Constructivist (Inert Ethics/Evaluative Judicious)
    • ILE EIE ESI SLI
  • Asking, Farsighted, Emotivist (Inert Logic/Evaluative Decisive)
    • LII IEI SEE LSE
  • Declaring, Carefree, Emotivist (Inert Logic/Evaluative Judicious)
    • SEI LSI LIE IEE
  • Declaring, Farsighted, Constructivist (Inert Ethics/Evaluative Decisive)
    • ESE SLE ILI EII
• Same Rational Dimension Conflict Challenge-Response
  • Asking, Yielding, Tactical (Inert Intuition/Evaluative Serious)
    • ILE IEI ESI LSE
  • Asking, Obstinate, Strategic (Inert Sensing/Evaluative Merry)
    • LII EIE SEE SLI
  • Declaring, Yielding, Strategic (Inert Sensing/Evaluative Serious)
    • SEI SLE LIE EII
  • Declaring, Obstinate, Tactical (Inert Intuition/Evaluative Merry)
    • ESE LSI ILI IEE

... ... ... ...

Pure Tencer-Minaev Small Group Systems

• Program Function Aristocratic Order Pairs
  • Extroverted, 1stAlpha, 1stImplicit (Ne or Fe Program)
    • ILE ESE EIE IEE
  • Extroverted, 1stGamma, 1stExplicit (Se or Te Program)
    • SLE SEE LIE LSE
  • Introverted, 1stAlpha, 1stExplicit (Si or Ti Program)
    • SEI LII LSI SLI
  • Introverted, 1stGamma, 1stImplicit (Ni or Fi Program)
    • IEI ILI ESI EII
• Program Function Democratic Order Pairs
  • Extroverted, 1stDelta, 1stDetached (Ne or Te Program)
    • ILE LIE LSE IEE
  • Extroverted, 1stBeta, 1stInvolved (Se or Fe Program)
    • ESE EIE SLE SEE
  • Introverted, 1stDelta, 1stInvolved (Si or Fi Program)
    • SEI ESI EII SLI
  • Introverted, 1stBeta, 1stDetached (Ni or Ti Program)
    • LII LSI IEI ILI
• Demonstrative Function Aristocratic Order Pairs
  • Extroverted, 5thAlpha, 5thExplicit (Si or Ti Demonstrative)
    • ILE ESE SLE LSE
  • Extroverted, 5thGamma, 5thImplicit (Ni or Fi Demonstrative)
    • EIE SEE LIE IEE
  • Introverted, 5thAlpha, 5thImplicit (Ne or Fe Demonstrative)
    • SEI LII IEI EII
  • Introverted, 5thGamma, 5thExplicit (Se or Te Demonstrative)
    • LSI ILI ESI SLI
• Demonstrative Function Democratic Order Pairs
  • Extroverted, 5thBeta, 5thDetached (Ni or Ti Demonstrative)
    • ILE EIE SLE LIE
  • Extroverted, 5thDelta, 5thInvolved (Si or Fi Demonstrative)
    • ESE SEE LSE IEE
  • Introverted, 5thBeta, 5thInvolved (Se or Fe Demonstrative)
    • SEI LSI IEI ESI
  • Introverted, 5thDelta, 5thDetached (Ne or Te Demonstrative)
    • LII ILI EII SLI
• Program Function Democratic Revision Pairs
  • Static, 1stAlpha, 1stDetached (Ne or Ti Program)
    • ILE LII LSI IEE
  • Static, 1stGamma, 1stInvolved (Se or Fi Program)
    • SLE SEE ESI EII
  • Dynamic, 1stAlpha, 1stInvolved (Si or Fe Program)
    • SEI ESE EIE SLI
  • Dynamic, 1stGamma, 1stDetached (Ni or Te Program)
    • IEI ILI LIE LSE
• Program Function Aristocratic Revision Pairs
  • Static, 1stDelta, 1stImplicit (Ne or Fi Program)
    • ILE ESI EII IEE
  • Static, 1stBeta, 1stExplicit (Se or Ti Program)
    • LII LSI SLE SEE
  • Dynamic, 1stDelta, 1stExplicit (Si or Te Program)
    • SEI LIE LSE SLI
  • Dynamic, 1stBeta, 1stImplicit (Ni or Fe Program)
    • ESE EIE IEI ILI
• Demonstrative Function Democratic Revision Pairs
  • Static, 5thAlpha, 5thDetached (Ne or Ti Demonstrative)
    • ILE LII SLE EII
  • Static, 5thGamma, 5thInvolved (Se or Fi Demonstrative)
    • LSI SEE ESI IEE
  • Dynamic, 5thAlpha, 5thInvolved (Si or Fe Demonstrative)
    • SEI ESE IEI LSE
  • Dynamic, 5thGamma, 5thDetached (Ni or Te Demonstrative)
    • EIE ILI LIE SLI
• Demonstrative Function Aristocratic Revision Pairs
  • Static, 5thBeta, 5thExplicit (Se or Ti Demonstrative)
    • ILE LSI SLE ESI
  • Static, 5thDelta, 5thImplicit (Ne or Fi Demonstrative)
    • LII SEE EII IEE
  • Dynamic, 5thBeta, 5thImplicit (Ni or Fe Demonstrative)
    • SEI EIE IEI LIE
  • Dynamic, 5thDelta, 5thExplicit (Si or Te Demonstrative)
    • ESE ILI LSE SLI
• High/Low Energy Aristocratic Order Pairs
  • Positivist, 1stAlpha, 5thExplicit (Energetic SiTi/Informational NiFi)
    • ILE ESE LSI SLI
  • Positivist, 1stGamma, 5thImplicit (Energetic NiFi/Informational SiTi)
    • IEI SEE LIE EII
  • Negativist, 1stAlpha, 5thImplicit (Energetic NeFe/Informational SeTe)
    • SEI LII EIE IEE
  • Negativist, 1stGamma, 5thExplicit (Energetic SeTe/Informational NeFe)
    • SLE ILI ESI LSE
• High/Low Energy Democratic Order Pairs
  • Positivist, 1stDelta, 5thDetached (Energetic NeTe/Informational SeFe)
    • ILE LIE EII SLI
  • Positivist, 1stBeta, 5thInvolved (Energetic SeFe/Informational NeTe)
    • ESE LSI IEI SEE
  • Negativist, 1stDelta, 5thInvolved (Energetic SiFi/Informational NiTi)
    • SEI ESI LSE IEE
  • Negativist, 1stBeta, 5thDetached (Energetic NiTi/Informational SiFi)
    • LII EIE SLE ILI
• High/Low Brakes Aristocratic Order Pairs
  • Positivist, 1stImplicit, 5thAlpha (Excitable NeFe/Inhibitable SeTe)
    • ILE ESE IEI EII
  • Positivist, 1stExplicit, 5thGamma (Excitable SeTe/Inhibitable NeFe)
    • LSI SEE LIE SLI
  • Negativist, 1stImplicit, 5thGamma (Excitable NiFi/Inhibitable SiTi)
    • EIE ILI ESI IEE
  • Negativist, 1stExplicit, 5thAlpha (Excitable SiTi/Inhibitable NiFi)
    • SEI LII SLE LSE
• High/Low Brakes Democratic Order Pairs
  • Positivist, 1stDetached, 5thBeta (Excitable NiTi/Inhibitable SiFi)
    • ILE LSI IEI LIE
  • Positivist, 1stInvolved, 5thDelta (Excitable SiFi/Inhibitable NiTi)
    • ESE SEE EII SLI
  • Negativist, 1stDetached, 5thDelta (Excitable NeTe/Inhibitable SeFe)
    • LII ILI LSE IEE
  • Negativist, 1stInvolved, 5thBeta (Excitable SeFe/Inhibitable NeTe)
    • SEI EIE SLE ESI
• High/Low Energy Democratic Revision Pairs
  • Asking, 1stAlpha, 5thDetached (Energetic NeTi/Informational SeFi)
    • ILE LII EIE SLI
  • Asking, 1stGamma, 5thInvolved (Energetic SeFi/Informational NeTi)
    • IEI SEE ESI LSE
  • Declaring, 1stAlpha, 5thInvolved (Energetic SiFe/Informational NiTe)
    • SEI ESE LSI IEE
  • Declaring, 1stGamma, 5thDetached (Energetic NiTe/Informational SiFe)
    • SLE ILI LIE EII
• High/Low Energy Aristocratic Revision Pairs
  • Asking, 1stDelta, 5thExplicit (Energetic SiTe/Informational NiFe)
    • ILE ESI LSE SLI
  • Asking, 1stBeta, 5thImplicit (Energetic NiFe/Informational SiTe)
    • LII EIE IEI SEE
  • Declaring, 1stDelta, 5thImplicit (Energetic NeFi/Informational SeTi)
    • SEI LIE EII IEE
  • Declaring, 1stBeta, 5thExplicit (Energetic SeTi/Informational NeFi)
    • ESE LSI SLE ILI
• High/Low Brakes Democratic Revision Pairs
  • Asking, 1stDetached, 5thAlpha (Excitable NeTi/Inhibitable SeFi)
    • ILE LII IEI LSE
  • Asking, 1stInvolved, 5thGamma (Excitable SeFi/Inhibitable NeTi)
    • EIE SEE ESI SLI
  • Declaring, 1stDetached, 5thGamma (Excitable NiTe/Inhibitable SiFe)
    • LSI ILI LIE IEE
  • Declaring, 1stInvolved, 5thAlpha (Excitable SiFe/Inhibitable NiTe)
    • SEI ESE SLE EII
• High/Low Brakes Aristocratic Revision Pairs
  • Asking, 1stImplicit, 5thBeta (Excitable NiFe/Inhibitable SiTe)
    • ILE EIE IEI ESI
  • Asking, 1stExplicit, 5thDelta (Excitable SiTe/Inhibitable NiFe)
    • LII SEE LSE SLI
  • Declaring, 1stImplicit, 5thDelta (Excitable NeFi/Inhibitable SeTi)
    • ESE ILI EII IEE
  • Declaring, 1stExplicit, 5thBeta (Excitable SeTi/Inhibitable NeFi)
    • SEI LSI SLE LIE

Here is where we truly see the differences between the dichotomy systems, and why the Reinin dichotomies are primarily associated with Model A whereas the Tencer-Minaev dichotomies are primarily associated with Model G.

The 16 Pure Reinin small group systems directly describe key facets of how the Sociotypes relate to Model A, such as dimensionality, inertness and contactness of functions, Freudian blocks like the ego elements, quadra values and club strengths, etc.

The 16 Pure Tencer-Minaev small group systems directly describe key facets of how the Sociotypes relate to Model G, such as which functions are more energetic and less energetic (i.e. more informational) in Model G, which functions are more excitable (i.e. have low "brakes") and which are more easily braked and inhibited in Model G, and other groups which have no clear interpretation in Model A and mirror the structure of the energy and excitability small groups in Model G (e.g. a group like "ILE LII LSI IEE", which doesn't seem as directly useful in Model A, but has a fractal meaning in Model G that connects it to the energy and excitability small groups since it uses the same pairs of functions on a higher fractal level).

____________

Conclusion and Limitations on the Mathematics

Thus, we have discovered 2 new mutually compatible dichotomy systems for Model G (the Gulenko-Newman functional/positional dichotomies and the Gulenko-Newman elemental dichotomies) and provided some new derivations of the Tencer-Minaev dichotomies (and of the Reinin dichotomies). We also showed that the GN dichotomies and Tencer-Minaev dichotomies are primarily associated with Model G in Socionics, whereas the classical functional/positional dichotomies, elemental dichotomies and Reinin dichotomies are primarily associated with Model A in Socionics. We also saw that, in spite of their primary associations, all of these dichotomies can nonetheless be used with both systems, which shows that they could potentially be united in a more capacious and harmonious synthesis than ever existed previously in Socionics.

Although we showed all of these dichotomy systems to a great deal of logical depth and justification, we will explore even more of the deeper mathematics of Socionics in the future to learn even more.

Before I end this post, I want to address some of the limits of how this mathematics can be used. There are two possible approaches (or at least two extremes) of how to relate this abstract mathematics to practical and empirically-based Socionics:

- One approach is the "mathematically conservative approach", which involves only using the mathematical structures which we have reason to think work empirically (of course, people may differ in opinion on whether there are empirically sufficient reasons to include certain mathematical structures, so there need to be separate criteria to resolve such disagreements and that is beyond the scope of this encyclopedia). The reason this is possible is because doing all this structural abstract mathematics (as we've done in this encyclopedia) doesn't by itself prove the structures with which Socionics operates in empirical reality. That doesn't mean the work we're doing is unimportant, since we are exploring certain necessary logical relationships that Socionics must respect given its use of certain structures (and Socionics has to use some structures so this will always be a significant issue). Further, we already have a large amount of empirical reasons to pursue and research the structures explored in this encyclopedia, based on existing Socionics research and practice. However, the mathematically conservative approach is a quite respectable approach to Socionics math, since it is possible to create additional criteria by which to limit the structures to only a logically consistent subset of the subject of this encyclopedia (on account of the prospective belief that only some are empirically valid). A good example of this approach is taken by the school of Imperative Socionics (BestSocionics), who have their own interpretation of most of the Reinin dichotomies and limit the structures of Socionics to a logically valid subset of Model A and the Reinin dichotomies, but when I spoke to them, they clearly indicated an open-mindedness to further logically consistent research. A more negative example of this approach is taken by Ibrahim Tencer, who disparages Model G with a variety of unfounded claims, even though he hasn’t taken the time to understand its mathematical structure or how it operates empirically (as I explore in this article: https://varlawend.blogspot.com/2020/09/you-cant-refute-model-g-with-model.html).

- Another approach is the "mathematically speculative approach", which involves taking mathematical structures consistent with current Socionics theory seriously as representing possible features of Socionics theory which could later be empirically supported or refuted (and generally investigated). The fact that this is possible could not be more obvious, and in this very encyclopedia we are showing an example of this approach in discovering new dichotomy systems that can serve as a foundation of Model G, in addition to using previously unused speculative dichotomy systems like the Tencer-Minaev dichotomies. Victor Gulenko and Ausra Augustinaviciute also used this approach in founding their own models of Socionics (Model G and Model A respectively). Ibrahim Tencer himself used to take this approach to Socionics back when he used more Reinin dichotomies and discovered his eponymous dichotomy system over a decade ago. He even has an older article which I wholeheartedly agree with in which he explains the limits of empiricism that make the mathematically speculative approach necessary: https://sedecology.blogspot.com/2017/12/empiricism.html. Mathematics may not tell us what structures that empirical reality conforms to, but it is very useful for giving a foundation to empirical theories and for suggesting future research directions in a structured way (e.g. we may not know the exact empirical structure of reality, but we may know enough it that we can limit the possibilities to certain mathematical structures or approaches).

- For a balanced approach to the research of Socionics, combining open-mindedness with rigor, both of these approaches need to be used. We can't just accept any theoretical mathematics that relates to Socionics, since it need to be tested empirically, but we also need to remain open to ways of broadening and extending our understanding of Socionics in ways not predicted by current structures (yet remain logically consistent with the structures that work).

There is another question as to whether some of these dichotomy systems are more fundamental than others in the structure. For example, there are Model A theorists who argue that the Reinin dichotomies and other type dichotomies are less fundamental than the combination of functional/positional dichotomies and elemental dichotomies, since the latter represent "simpler" hypotheses (clear distinctions in dichotomies like Detached/Involved) as compared to more complex Reinin dichotomies that can be derived from these simpler distinctions. To be sure, I do think it is valid and sometimes useful to employ this kind of reductionistic thinking (i.e. nomothetic thinking) where the whole (a type, its Reinin dichotomies and small group properties) can be broken down entirely into its parts and thus is to be derived from these smaller parts.

However, the reverse approach of holistic thinking (i.e. ideographic thinking) is also very important in science and in the study of complex systems generally (especially living systems, like those studied in Socionics). I highly recommend reading this wonderful article by the Complex Systems Researcher Joe Normal to understand the importance of holistic thinking in the study of living systems (including Socionics): https://thesideview.co/journal/generating-wholes/. The way this may apply to Socionics is that there may be groupings which empirically are very clear, distinct and observable, yet have many composite parts, such as the temperaments, clubs or cognitive styles (some may dispute some of these groups in particular but this does not refute the general point). There is nothing contradictory about a whole being clearer and easier to observe than its parts; after all, humans and different sexes were quite easy to observe before we understood all of the reductionistic body parts and internal organs involved in their functioning. Likewise, as an example that Victor Gulenko gave when justifying his use of Gestalt Psychology (an important branch of scientific psychology) in his practice of Humanitarian Socionics, he pointed out that the study of individual ants (i.e. dichotomies) will never give us an idea of what an anthill (i.e. the type) is by itself.

On the other hand, just because a whole is clearer and easier to see than its parts doesn't mean that the parts are unimportant; they still play clear and important structural roles in the greater whole, as we can see with all the internal systems and organs of the body that comprise humans. We may first clearly observe a whole like the temperaments (linear assertive, balanced stable, flexible maneuvering, and receptive adaptive) or even the humorism of historical figures like Hippocrates and Galen that inspired them, and only later figure out the dichotomies that comprise them (extroverted/introverted, irrational/rational, static/dynamic). Or we may clearly see parts like High Energy/Low Energy functions (Energetic/Informational in the GN dichotomies) and High Brakes/Low Brakes functions (Excitable/Inhibitable in the GN dichotomies), and then later use them in the construction of a more holistic Socionics model (Model G), as Victor Gulenko once did. The point is that the whole and the parts are both irreplaceably important, and in this static system, it is potentially useful for us to make derivations in both directions, so neither should necessarily be treated as simpler or more fundamental. A model which only focuses on consistent reductionistic thinking is poor in semantic consistency and descriptiveness, and a model which only focuses on providing holistic semantic descriptions is poor in logical and mathematical consistency.

Lastly, since this is an encyclopedia, if it turns out to be incomplete in any way, I'm happy to add more to it in a postscript. I may also furnish it with further mathematical details and empirical descriptions in the future, but only in a way consistent with the current content of the post.