# Postmodern Fuzzy System Theory: A Deconstruction Approach Based on Kabbalah

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Figure 1.**Our methodology for building a postmodern fuzzy system theory, KFST, via a Kabbalah based postmodern “deconstruction” of (modern) general system theory (GST) and, of fuzzy sets, fuzzy logic and of fuzzy system theory (FST). The deconstruction incorporates cognitive, emotional and behavioral components in GST and fuzzy sets theory using a Kabbalah Tree of Life possible worlds model. Our previously built Kabbalah system theory (KST) and Kabbalah fuzzy set theory are used in this paper according to this deconstruction chain to construct a Kabbalah fuzzy system theory (KFST).

## 2. The Tree of Life of Kabbalah as a General System Model

**Figure 2.**The fractal sefirotic structure of the Tree of Life of Kabbalah as a general system model: nine interconnected sefirot units organized in three interconnected hierarchical levels or sub-systems corresponding to the cognitive triad of sefirot CBD, emotional triad of sefirot C’GT and behavioral, expressions and actions triad of sefirot NHY. Each sefirot unit is made up similarly of nine sub-sefirot connected according to a Tree of Life structure. (Source: [9]).

**Figure 3.**The hierarchical fractal triadic structure of the Tree of Life, in terms of the sub-triads of each triad of sefirot, corresponds to the fractal sefirotic structure of the Tree of Life in terms of sefirot from Figure 2. The CBD sub-levels of sefirot C, B, D form the CBD sub-triad of CBD denoted by CBD(CBD). Similarly we have C’GT(CBD) and NHY(CBD) which together with CBD(CBD) form the sub-Tree of Life of CBD. The same applies for the sub-Trees of Life of C’GT and NHY. The connections between the sub-Trees of Life of CBD, C’GT, NHY are based on the hierarchical system organization of the Tree of Life of Kabbalah: CBD controls C’GT and NHY by means of its NHY(CBD), the behavior, actions and expressions oriented sub-triad of the cognitive triad CBD. C’GT controls NHY through NHY(C’GT), the behavior, expressions and actions oriented sub-triad of the emotions triad C’GT.

## 3. Kabbalah Fuzzy Set Theory: The Postmodern Deconstruction Based on Kabbalah and on the Possible Worlds Semantic Model of Modal Logic

**Proposition 1. (Postmodern fuzzy set deconstruction canonic form)**Let m(A,⋅) be the membership function of a fuzzy set A (fuzzy subset of universe of discourse X), given by formula (5) relative to a possible worlds model M = < W, R, V, w >, with frame < W, R > of sefirot possible worlds W(i) = S(i), i = 1,…,27 connected like in the graph of the binary accessibility relation R from Figure 3. Then the membership function m (A, x) for all x ϵ X can be canonically decomposed as:

**Kabbalistic fuzzy set (KFS)**.

**CBD**} are the epistemic, cognitive modal truth valuations

**C’GT**} are the doxastic, emotional modal truth valuations

**NHY**} are the behavioral, action modal truth valuations

**CBD**) is the cognitive, epistemic modal estimate of m(A, ⋅), m(A, ⋅ ∣

**C’GT**) is the emotional, doxastic modal estimate while m(A, ⋅ ∣

**NHY**) is the behavioral, action modal estimate of the fuzzy membership function m(A, ⋅). Remark that these estimates or fuzzy subsets are interconnected since weights w(S) at one level are normalized relative to the weights of all 27 sefirot and not just relative to the weights of the nine sefirot forming that level. According to the process described in Figure 1, the postmodern fuzzy set theory, which deconstructs fuzzy sets along the cognitive, emotional and behavioral dimensions, will allow us next to introduce fuzzy systems where the basic concepts of general system theory, input-state-output, are Kabbalistic fuzzy sets.

## 4. Postmodern Fuzzy System Theory

**Proposition 2. (postmodern deconstruction canonic form for fuzzy general systems)**Let < U, X, Y, F, G, m, n, q, p > be a 9-uple of maps and sets as above without relations (19) assumed to hold. Define 9-uples ∑

_{CBD}, ∑

_{C’GT}, ∑

_{NHY}as follows (see Figure 4):

_{CBD}, ∑

_{C’GT}, ∑

_{NHY}are general fuzzy systems as defined above and their diagrams in the right side of Figure 4 commute) then ∑ = <U, X, Y, F, G, m, n, q, p > is a general fuzzy system satisfying the commutativity of the diagram on the left side of Figure 4 given by relations (19).

**Figure 4.**A postmodern deconstruction of a fuzzy general system

**∑**(represented by the diagram on the left side of figure) into its cognitive

**∑**, emotional

_{CBD}**∑**and behavioral/expressions/actions

_{C’GT}**∑**fuzzy subsystems or estimates (represented by the corresponding diagrams on the right side of figure) based on the Kabbalah possible worlds architecture of Figure 3 and Proposition 1 (the postmodern fuzzy set deconstruction canonic form).

_{NHY}**Proof.**If we sum the commutativity inequality conditions of type (19) for ∑

_{CBD}, ∑

_{C’GT}, ∑

_{NHY}which are assumed to hold and we use Equations (23)–(26) then we get exactly the commutativity condition for ∑ = <U, X, Y, F, G, m, n, q, p > proving it is a fuzzy general system.

**∑**

_{CBD},

**∑**

_{C’GT},

**∑**

_{NHY}as a factorization, decomposition or “realization” for

**∑**(not in the classical input-output map realization sense [3]). Note however that

**∑**can have many other possible “realizations”.

**∑**

_{CBD},

**∑**

_{C’GT},

**∑**

_{NHY}are respectively the cognitive/epistemic, emotional/doxastic and behavioral/action fuzzy estimates or fuzzy subsystems of

**∑**.

**Kabbalistic fuzzy system**.

_{CBD}, ∑

_{C’GT}, ∑

_{NHY}and of the Kabbalah based possible world frame in Figure 3. Fuzzy system theory FST was all stopping at ∑ level and its structural properties. Although fuzzy sets and fuzzy systems arise from human subjectivity and knowledge imprecision, this humanistic nature behind fuzzy systems was not visible in the FST but will now become incorporated in KFST according to the deconstruction process in Figure 1 which was carried out in this paper.

_{CBD }, ∑

_{C’GT}, ∑

_{NHY}by using respectively Acc (CBD), Acc (C’GT) and Acc (NHY) instead of just CBD, C’GT and NHY’ respectively. This gives us an idea how the structure of the possible world accessibility relation introduced in Figure 3 allows us to explore a fuzzy system from different partial and progressive information angles, restricting it to just CBD, C’GT, NHY or progressively extending it to Acc (CBD), Acc (C’GT), Acc (NHY). This shows how the Kabbalistic possible world frame <W,R> constructed in Figure 3 is a tool for developing different deconstructed postmodern fuzzy system models based on the structure of information embedded in <W,R>.

## 5. Example: Application of Postmodern Fuzzy System Theory to Create Kabbalistic Fuzzy System Models for Fuzzy Agents

_{CBD}, ∑

_{C’GT}, ∑

_{NHY}displayed on the right side of Figure 4.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- von Bertalanffy, L. General System Theory; George Braziller: New York, NY, USA, 1969. [Google Scholar]
- von Bertalanffy, L. Perspectives on General System Theory; George Braziller: New York, NY, USA, 1975. [Google Scholar]
- Negoita, C.V.; Ralescu, D.A. Applications of Fuzzy Sets to Systems Analysis; John Wiley & Sons: New York, NY, USA, 1975. [Google Scholar]
- Negoita, C.V. Fuzzy Systems; Abacus Press: Tunbridge Wells, UK, 1981. [Google Scholar]
- Negoita, C.V. Expert Systems and Fuzzy Systems; Benjamin/Cummings Publishing Co.: Menlo Park, CA, USA, 1985. [Google Scholar]
- Negoita, C.V. Postmodernism, cybernetics and fuzzy set theory. Kybernetes
**2002**, 31, 1043–1049. [Google Scholar] [CrossRef] - Drob, S. Kabbalah and Postmodernism: A Dialogue; Peter Lang Academic International Publisher: New York, NY, USA, 2009. [Google Scholar]
- Burstein, G.; Negoita, C.V. A Kabbalah System Theory Modelling Framework for Knowledge Based Behavioral Economics and Finance. In Computational Models of Complex Systems; Dabbaghian, V., Mago, V., Eds.; Springer: Zurich, Switzerland, 2013; Volume 53, pp. 5–23. [Google Scholar]
- Burstein, G.; Negoita, C.V.; Kranz, M. Kabbalah logic and semantic foundations for a postmodern fuzzy logic and fuzzy set theory. Appl. Math.
**2014**, 5, 1375–1385. [Google Scholar] [CrossRef] - Menzi, D.W.; Padeh, Z. The Tree of Life: Chayyim Vital’s Introduction to the Kabbalah of Isaac Luria; Arizal Publications Inc.: New York, NY, USA, 2008. [Google Scholar]
- Afilalo, R. The Kabbalah of the Arizal According to the Ramhal; Kabbalah Editions: Montreal, Canada, 2004. [Google Scholar]
- Luzzatto, M.C. 138 Openings of Wisdom; Azamra Institute: Jerusalem, Israel, 2005. [Google Scholar]
- Spielman, Y.M. Tal Orot; (in Hebrew). Yeshiva HaChaim ve HaShalom: Jerusalem, Israel, 1976. [Google Scholar]
- Cresswell, M.J.; Hughes, G.E. A New Introduction to Modal Logic; Routledge: London, UK, 1996. [Google Scholar]
- Resconi, G.; Klir, G.J.; St. Clair, U. Hierarchical uncertainty metatheory based upon modal logic. Int. J. General Systems
**1992**, 21, 23–50. [Google Scholar] [CrossRef] - Turksen, I.B. An Ontological and Epistemological Perspective of Fuzzy Set Theory; Elsevier: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Zadeh, L.A. Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. Syst. Man Cybern.
**1973**, SMC-3, 28–44. [Google Scholar] - Fougeres, A.J. Modelling and simulation of complex systems: An approach based on multi-level agents. IJCSI Int. J. Computer Sci. Issues
**2011**, 8, 8–17. [Google Scholar]

© 2014 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Burstein, G.; Negoita, C.V.; Kranz, M.
Postmodern Fuzzy System Theory: A Deconstruction Approach Based on Kabbalah. *Systems* **2014**, *2*, 590-605.
https://doi.org/10.3390/systems2040590

**AMA Style**

Burstein G, Negoita CV, Kranz M.
Postmodern Fuzzy System Theory: A Deconstruction Approach Based on Kabbalah. *Systems*. 2014; 2(4):590-605.
https://doi.org/10.3390/systems2040590

**Chicago/Turabian Style**

Burstein, Gabriel, Constantin Virgil Negoita, and Menachem Kranz.
2014. "Postmodern Fuzzy System Theory: A Deconstruction Approach Based on Kabbalah" *Systems* 2, no. 4: 590-605.
https://doi.org/10.3390/systems2040590