# Structure and Superstructures in Complex Social Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Main Ideal condition:**The group of actors should be homogenous for the relationships in Figure 2.

## 2. Deontical Impure Systems Theory

- (1)
- A superstructure consisting of concrete, specific and varying beliefs that include: ideologies, values, philosophies, and knowledge systems. We define this as the Doxical Superstructure (DS).
- (2)
- A utopian or ideal superstructure consisting of ideal values and myths, that provides both accounts of them origins and goals of the Structural Base. This superstructure is the Mythical Superstructure (MS).

## 3. A Mathematical Approach

#### 3.1. The Structural Base (SB)

**א**as Reality, with

**ב**as part of this Reality, so that

**ב**$\subset $

**א**. If S is an Observing or Observer Subject during a time interval $\left[{t}_{0},{t}_{f}\right]$, Σ the system’s concept, Σ(

**ב**) the conception that

**ב**is a system (relative beings).

**Note**

**1.**

**ב**) is a social system (DIS).

_{D}(

**ב**) is considered as a DIS composed of a set of relations $\left\{{r}_{i}^{k}\right\}$ that is a subset of the doxical relations that we will represent by ${\Sigma}_{D}=\left(\left\{{r}_{i}^{k}\right\},\wedge \right)$, with $\wedge $ being the function “and”. We define the systemic significance s and there will also be a denotative significance d-s which is what we know of the significance s of an object in Reality that we call an absolute being. We define the set of signifier (signs) of Reality ζ and the set of systemic signifiers ζ

_{Σ}forming a particular CSR, i.e., the signs selected by the Subject to define the boundaries the specific DIS, with the implication that ζ

_{Σ}$\subset $ ζ. If we define ${I}_{\Sigma}=\left\{{i}_{\Sigma}\right\}$ as the set of possible systemic individuals, i.e., every impure object and possible relation included in the boundary of the SB of system $\Sigma $. We are going to define some of the deontic properties of the relations of a system.

**Definition**

**1.**

**Definition**

**2.**

- (1)
- ${r}_{i}^{k}$ is a deontic relation iff for some ${\xi}_{\Sigma}$, ${r}_{i}^{k}$ there is an ${\xi}_{\Sigma}$.
- (2)
- If ${r}_{i}^{k}$ is a deontic relation, there is exactly an ${\xi}_{\Sigma}$.
- (3)
- If ${r}_{i}^{k}$ and ${r}_{j}^{k}$ are deontic relations, ${r}_{i}^{k}$ is composable with ${r}_{j}^{k}$ iff ${\xi}_{\Sigma}^{{\wp}_{i}^{K}}={\xi}_{\Sigma}^{{\wp}_{j}^{K}}.$
- (4)
- The composability is a relation of equivalence in the set of deontic relations of the DIS.

**Definition**

**3.**

_{Σ}is a denotative DIS significance (d-s) iff it is a function defined in $\xi $ so that if ${\xi}_{\Sigma}\subset \xi $ then ${s}_{\Sigma}\left({\xi}_{\Sigma}\right)\subseteq {\xi}_{\Sigma}$.

**Definition**

**4.**

_{Σ}and so that it is the function that maps ${\xi}_{\Sigma}$ to I

_{Σ}.

**Property**

**1.**

**Note**

**2.**

**Theorem**

**1.**

_{Σ}in ${\xi}_{\Sigma}$ if ${s}_{\Sigma}\in {r}_{i}^{k}.$

**Proof.**

_{Σ}in the set of systemic signifiers, as ${s}_{\Sigma}^{C}$ and so that ${i}_{\Sigma}^{C}=\left\{{s}_{\Sigma};{i}_{\Sigma}={s}_{\Sigma}\left({\xi}_{\Sigma}\right)\right\}$. The correlation of i

_{Σ}in ${\xi}_{\Sigma}$ is indeed the set of d-significances that has i

_{Σ}in ${\xi}_{\Sigma}$. This indicates to us that it is a deontic relation.

**Theorem**

**2.**

**Proof.**

_{Σ}. We can form the union of both deontic relations (incomplete and complete), therefore forming a new deontic relation $\mathsf{\Lambda}={r}_{i}^{k}\cup \left\{\theta \right\}$. This deontic relation has the important characteristic of being simultaneously complete, with d-significance s

_{Σ}and also incomplete because it contains $\left\{\theta \right\}$, with d-significance ${s}_{\Sigma}*$, which is absurd.

#### 3.2. The Doxical Superstructure (IDS)

_{Σ}as the set of items in a possible Doxical Superstructure, i.e., all abstract objects and relations belonging to the IDS.

**Definition**

**5.**

_{Σ}.

**Definition**

**6.**

**(a)****Alethic****modalities:**- (1)
- Existence: Existence is the function that maps ${\xi}_{\Sigma}$ to $\left\{{r}_{i}^{k}\right\}$ for some ${i}_{\Sigma}\in {I}_{\Sigma},{r}_{i}^{k}={i}_{\Sigma}^{C}.$
- (2)
- Completeness: Completeness is the function that maps ${\xi}_{\Sigma}$ to $\left\{{r}_{i}^{k}\right\}$ for each ${s}_{\Sigma ,},\left({s}_{\Sigma}\in {r}_{i}^{k}\right)\vee \left(\neg {s}_{\Sigma}\in {r}_{i}^{k}\right)$.
- (3)
- Possibility: Possibility is the function that maps ${\xi}_{\Sigma}$ to $\left\{{r}_{i}^{k}\right\}$ for some other ${\xi}_{\Sigma *}$ and some ${i}_{\Sigma *}\in {I}_{\Sigma *},{r}_{i}^{k}={i}_{\Sigma *}^{C}.$
- (4)
- Necessity: Necessity is the function that maps ${\xi}_{\Sigma}$ to $\left\{{r}_{i}^{k}\right\}$ so that if ${r}_{i}^{k}$ does not exist in ${\xi}_{\Sigma}$ for some ${i}_{\Sigma}\in {I}_{\Sigma},{r}_{i}^{k}\ne {i}_{\Sigma}^{C},$ then ${\xi}_{\Sigma}$ would not exist.

**(b)****Doxical****modalities:**- (1)
- Permission: Permission is the function that maps ${\xi}_{\Sigma}$ to $\left\{{r}_{i}^{k}\right\}$ so that if for some ${i}_{\Sigma}\in {I}_{\Sigma}$ if $\left({r}_{i}^{k}\subseteq {I}_{\Sigma}\right)\wedge \left({r}_{i}^{k}={i}_{\Sigma}^{C}\right)$.
- (2)
- Choice: Choice is the function that maps ${\xi}_{\Sigma}$ to $\left\{{r}_{i}^{k}\right\}$ so that if for some ${i}_{\Sigma}\in {I}_{\Sigma}$ if ${r}_{i}^{k}={i}_{\Sigma}^{C}\wedge {r}_{i}^{k}\ne {i}_{\Sigma}^{C}$.
- (3)
- Obligation: Obligation is the function that maps ${\xi}_{\Sigma}$ to $\left\{{r}_{i}^{k}\right\}$ so that if for some ${i}_{\Sigma}\in {I}_{\Sigma}$, $\left[CAN\left({r}_{i}^{k}={i}_{\Sigma}^{C}\right)\wedge CAN\left({r}_{i}^{k}\ne {i}_{\Sigma}^{C}\right)\right]\wedge \left({r}_{i}^{k}={i}_{\Sigma}^{C}\right)$.

**Definition**

**7.**

**Note**

**3.**

**Note**

**4.**

**L**containing denotative-SB-predicates (d-predicates) and doxical structural predicates (IDS-predicates) according to the definitions in [9].

**Property**

**2.**

**L**, then if

_{M}**υ***names one abstract relation ${R}_{i}^{k}$, ${P}^{DS}{v}^{*}$ is true iff ${P}^{M}{v}^{*}$ is true.

**Property**

**3.**

**L**then if π names one deontic relation ${r}_{i}^{k}$, ${P}^{d}\pi $ is true iff ${P}^{D}\pi $ is true.

**Property**

**4.**

**Theorem**

**3.**

**Proof.**

**L**be a language with ${P}^{d}$ and ${P}^{D}$ its d-predicates and IDS-predicates respectively. By Property 3, ${P}^{d}$ expresses (it names) $\overrightarrow{{s}_{\Sigma}^{D}}$, and by Property 2, ${P}^{d}$, expresses (it names) ${s}_{\Sigma}$. Then $\overrightarrow{{s}_{\Sigma}^{D}}$ expresses (it names) ${s}_{\Sigma}$.

**Theorem**

**4.**

**Proof.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

## 4. Conclusions

_{i.}A belief distance could also be between what is hoped for and what actually happens. The distances vary depending on the contexts: abstract ideals and concrete manifestations of them; ideal values and how values occur in experience; and goals we had in the past compared with our goals today. If we construct a three-dimensional diagram of superstructure with the Doxical Superstructure as a plane, the idea of belief systems can be applied to the Ist-Mythical Superstructure so it is represented as a warped plane (Figure 3). Changes of form on the plane depend on the (belief) distance between experience and the ideals concerned.

- (a)
- Subject A claims that P.
- (b)
- Subject A is untrustworthy.
- (c)
- Therefore, P is false.

- (a)
- Subject A claims that P.
- (b)
- Subject A is particularly trustworthy.
- (c)
- Therefore, P is true.

## Author Contributions

## Conflicts of Interest

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Nescolarde-Selva, J.A.; Usó-Doménech, J.L.; Gash, H.
Structure and Superstructures in Complex Social Systems. *Systems* **2017**, *5*, 28.
https://doi.org/10.3390/systems5020028

**AMA Style**

Nescolarde-Selva JA, Usó-Doménech JL, Gash H.
Structure and Superstructures in Complex Social Systems. *Systems*. 2017; 5(2):28.
https://doi.org/10.3390/systems5020028

**Chicago/Turabian Style**

Nescolarde-Selva, Josué Antonio, José Luis Usó-Doménech, and Hugh Gash.
2017. "Structure and Superstructures in Complex Social Systems" *Systems* 5, no. 2: 28.
https://doi.org/10.3390/systems5020028