a mathematical theory of sustainability and

A MATHEMATICAL THEORY OF SUSTAINABILITY AND SUSTAINABLE DEVELOPMENT


Ricardo Alvira Baeza

Cover. Background photograph Malabaristas da Cidade, by Tiago Celestino [License CC BY 2.0] 1st Edition April 2014. Rev: 2.00/ October 2017 ISBN-13: 978-1497439375 ISBN-10: 149743937X ©Ricardo Alvira Baeza

This book is of course, dedicated to Monica.

Also to all those who in one way or another have helped make this work possible: family, friends, teachers, colleagues ..., not forgetting those whose previous work facilitates the task of those who arrived later. .

SUMMARY AND TABLE OF CONTENTS


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SUMMARY In recent times the problems associated with the development of human society have made clear both that the current model is seriously inefficient in the use of scarce and necessary resources, and that it may be impossible to sustain it even in the medium term. As a consequence different approaches have emerged, grouped under the terms Sustainability and Sustainable Development, proposing solutions or perspectives that seek maximizing the efficient use of resources, increasing both the positive effects of development and the capacity to endure. And though the abundant scientific production shows the importance given to this issue, its review also shows a tremendously fragmented knowledge, something troubling for several reasons: 



From a scientific perspective, the lack of connection between different knowledge reduces its heuristic capacity; non-relatable statements are hardly re-combined, so their ability to generate new knowledge is reduced. From the sustainability perspective, the disconnection between different sectorial proposals prevents their consideration altogether, undermining the main quality of sustainability: a holistic or systemic approach in which the whole is always different from the sum of the parts.

This disconnection and fragmentation of the conceptual models is transmitted to the mathematical models that measure different issues and in different ways, providing results that do not match even though they refer to the same reality. And this is a major issue, because human societies are continuously deciding their actions, and Sustainable Development will only happen if Sustainability is a central parameter in decision making, but ... how can it be if different models provide different results leading towards different decisions? The abundance of unordered proposals is creating a situation of uncertainty, in which models’ ability of leading society towards sustainability is considerably reduced by the fact that different models imply different statements [hence courses of action], leading us to a situation in which a lot of knowledge has very low efficiency. Disorder among sustainability statements is generating noise rather than a range of options which brings us to the main hypothesis of this text; it is necessary to undertake a scientific formalization of sustainability knowledge.

In order to do this, we use the Socio Ecological System [SES] model which allows us modeling a wide range of systems [a neighborhood association, a nation, a company, a municipality, etc...]. However, SES are real systems and any theory referred to them incorporates the subjectivity and inaccuracy inherent to factual sciences [science referred to reality], which could make the conclusions of this theory not fully accepted. To prevent this, we divide the theory in two parts:

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In the first part, we formulate a theory referring to a type of conceptual systems sufficiently similar to SES; Adaptive Systems [AS]. This allows us to fully develop a formal or mathematical theory, including a clear methodology and mathematical principles to measure the sustainability of these systems. To do so we follow these three steps:   

Firstly, we revise various scientific theories, through a process that allows us to progressively characterize both the concepts of Sustainability and Adaptive Systems. Secondly, we propose a System of Definitions and a System of Axioms, which serve as Premises for the formalization of the theory. And thirdly, we develop the Theory, deducing all its propositions from the Definitions and Axioms Systems. The Theory is considered to be proven in relation to the Premises,

In the second part, we review the issues that need to be considered when applying the mathematical theory to Socio Ecological Systems [SES]:  

We review a model that allows us to see the ease of designing assessment models applied to SES following the principles of the mathematical theory. We review in detail the relevant issues for collective decision making processes.

Some of the main contributions of this text are: •

•

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Detailed methodology for the design of sustainability assessment models including: o Logical decomposition of sustainability. o Detecting relevant variables. o Sustainability indicator design. o Indicator aggregation formulations. Detailed listing of issues that SES assessment models shall meet: o Logical decomposition of Sustainability in dimensions. o Methodology to use operational models in collective decision making processes.


INDEX SUMMARY AND TABLE OF CONTENTS ___________________________________________________________ 7 SUMMARY _________________________________________________________________________________ 9 INDEX ____________________________________________________________________________________ 11 FIGURES, DIAGRAMS AND ILLUSTRATIONS ______________________________________________________ 14

FIGURES ______________________________________________________________________ 14 DIAGRAMS ____________________________________________________________________ 16 ILLUSTRATIONS ________________________________________________________________ 16 1

INTRODUCTION AND GENERAL APPROACH __________________________________________________ 17

1.1

INTRODUCTION ___________________________________________________________ 19

1.2

GENERAL APPROACH ______________________________________________________ 21

PART I MATHEMATICAL THEORY OF SUSTAINABILITY AND SUSTAINABLE DEVELOPMENT [OF ADAPTATIVE SYSTEMS] _________________________________________________________________________________ 23 2

CHARACTERIZING ADAPTIVE SYSTEM’S SUSTAINABILITY: THEORETICAL FRAMEWORK _______________ 25

2.1

A ‘LOGICAL’ APPROACH TO SUSTAINABILITY CONCEPT ____________________________ 29

2.1.1 2.1.1.1 2.1.1.2 2.1.1.3 2.1.1.4

2.1.2

SETS PROPERTIES AND OPERATIONS: RULES OF LOGICAL INFERENCE _____________________ 29 PROPERTIES OF FUZZY SETS ____________________________________________________________ GRAPHICAL REPRESENTATION OF MEMBERSHIP FUNCTIONS _________________________________ FUZZY SETS AGGREGATION ____________________________________________________________ FUZZY SETS DECOMPOSITION___________________________________________________________

30 31 32 33

A LOGICAL APPROACH TO THE CONCEPT OF SUSTAINABILITY ___________________________ 35

2.1.2.1 CLASSIC SET THEORY: SUSTAINABILITY AND UNSUSTAINABILITY AS MUTUALLY EXCLUSIVE SETS _____ 35 2.1.2.2 FUZZY SETS THEORY: SUSTAINABILITY DEGREE AS A SYSTEM’S GRADE OF MEMBERSHIP TO SUSTAINABILITY CLASS _________________________________________________________________________ 35

2.1.3 2.1.3.1

2.1.4

2.2

CONCEPTS OF RELEVANT VARIABLE AND INDICATOR ________________________________________ 38

CONCLUSION __________________________________________________________________ 39

A SYSTEMIC APPROACH TO ADAPTIVE SYSTEMS’ SUSTAINABILITY ___________________ 40

2.2.1 2.2.1.1 2.2.1.2 2.2.1.3 2.2.1.4

2.2.2 2.2.2.1 2.2.2.2

2.3

LOGICAL DECOMPOSITION OF SUSTAINABILITY ______________________________________ 37

ADAPTIVE SYSTEMS CHARACTERIZATION ___________________________________________ 40 CONCEPT AND MODELLING OF SYSTEMS _________________________________________________ THE ‘COMPLEXITY’ OF SYSTEMS _________________________________________________________ ADAPTIVE SYSTEMS: SYSTEMS THAT EVOLVE ______________________________________________ THE ENVIRONMENT OF SYSTEMS ________________________________________________________

40 48 55 59

THE UNPREDICTABILITY OF SYSTEMS _______________________________________________ 62 CHAOS THEORY: SENSITIVE DEPENDENCE AND NONLINEARITY ________________________________ 62 AS’ UNPREDICTABILITY: SYSTEMS THAT LEARN, MAKE DECISIONS AND HAVE TELEOLOGY __________ 66

A PROBABILISTIC APPROACH TO SUSTAINABILITY ________________________________ 69

2.3.1 2.3.1.1 2.3.1.2

2.3.2

SUSTAINABILITY AS THE PROBABILITY TO ENDURE ____________________________________ 69 PROPERTIES OF EVENTS AND RELATIONS BETWEEN EVENTS __________________________________ 70 SUSTAINABILITY DEGREE OF A SYSTEM AS ITS PROBABILITY TO ENDURE ________________________ 74

SUSTAINABILITY AS SUBJECTIVE PROBABILITY OR DEGREE OF BELIEF _____________________ 75

2.3.2.1 SUSTAINABILITY DEGREE AS DEGREE OF RATIONAL BELIEF ABOUT THE TRUTH OF THE STATEMENT ‘THE SYSTEM IS SUSTAINABLE’ _______________________________________________________________________ 76

2.3.3

LOGICAL DECOMPOSITION AS PROBABILITY ASSIGNMENT _____________________________ 76

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A MATHEMATICAL THEORY OF SUSTAINABILITY AND SUSTAINABLE DEVELOPMENT 2.3.3.1 2.3.3.2

3

OBJECTIVE PROBABILITY OF EACH SUBCLASS OR INDICATOR __________________________________ 77 SUBJECTIVE PROBABILITY OF EACH INDICATOR_____________________________________________ 78

DEFINITIONS AND AXIOMS SYSTEMS _______________________________________________________ 79

3.1

DEFINITIONS _____________________________________________________________ 81 DEFINITION 01_COMPLETE UNSUSTAINABILITY _____________________________________________________ DEFINITION 02_COMPLETE SUSTAINABILITY ________________________________________________________ DEFINITION 03_SUSTAINABILITY DEGREE __________________________________________________________ DEFINITION 04_UNSUSTAINABILITY DEGREE ________________________________________________________ DEFINITION 05_HIERARCHICAL DECOMPOSITION OF THE SUSTAINABILITY OF A SYSTEM ____________________ DEFINITION 06_RELEVANT VARIABLE FOR THE SUSTAINABILITY OF A SYSTEM _____________________________ DEFINITION 07_SUSTAINABILITY LIMITS OF A RELEVANT VARIABLE _____________________________________ DEFINITION 08_SUSTAINABILITY INDICATOR ________________________________________________________ DEFINITION 09_GLOBAL ACCESSIBLE ENVIRONMENT OF A SYSTEM _____________________________________

3.2

AXIOMATIC SYSTEM _______________________________________________________ 85 AXIOM 01_UNIVERSALITY AND INVARIANCE ________________________________________________________ AXIOM 02_LIMITS _____________________________________________________________________________ AXIOM 03_COMPLEMENTARITY OF SUSTAINABILITY AND UNSUSTAINABILITY _____________________________ AXIOM 04_COMPLETE UNSUSTAINABILITY _________________________________________________________ AXIOM 05_COMPLETE SUSTAINABILITY ____________________________________________________________ AXIOM 06_EQUIVALENCE AND INTERCHANGEABILITY OF SUSTAINABILITY INDICATORS _____________________ AXIOM 07_MONOTONICITY _____________________________________________________________________ AXIOM 08_ GLOBAL ACCESSIBLE ENVIRONMENT ____________________________________________________

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81 82 82 83 83 84 84 84 84

85 85 85 85 85 86 86 86

MATHEMATICAL THEORY OF ADAPTIVE SYSTEMS’ SUSTAINABILITY ______________________________ 87

4.1

LOGICAL DECOMPOSITION OF SYSTEM-ENVIRONMENT SUSTAINABILITY ______________ 89

4.1.1 TOP-BOTTOM APPROACH: LOGICAL DECOMPOSITION _________________________________ 90 4.1.2 BOTTOM-UP APPROACH: COMPLETENESS OF THE DESCRIPTION ________________________ 91 4.1.3 IDENTIFICATION OF INDICATORS CAPABLE OF IMPLYING VALUE ZERO [NO-TRUTH] IN THE AGGREGATE LEVEL _____________________________________________________________________ 92 4.1.4 VERIFICATION OF THE GLOBAL CONSISTENCY OF THE HIERARCHICAL REPRESENTATION______ 93 4.1.4.1 4.1.4.2 4.1.4.3

4.2

GLOBAL VERIFICATION OF THE HIERARCHY STRUCTURE______________________________________ 93 CHECKING THE GLOBAL SIGNIFICANCE [IMPORTANCE] OF EACH INDICATOR _____________________ 94 VERIFICATION OF EQUAL SIGNIFICANCE OF INDICATORS IN EACH SUBSYSTEM ___________________ 95

FORMULATION OF SUSTAINABILITY INDICATORS ________________________________ 96

4.2.1 4.2.2

ELEMENTARY INDICATORS FORMULATION __________________________________________ 96 FORMULATION OF AGGREGATED INDICATORS _______________________________________ 97

4.3 SUSTAINABLE DEVELOPMENT: SUSTAINABILITY INCREASE AS A VARIATION IN THE UTILITY OF A SYSTEM STATE _____________________________________________________________ 99 5

PROOFING THE THEORY ________________________________________________________________ 101

5.1

PREMISES SYSTEM: AXIOMS AND DEFINITIONS ________________________________ 103

5.1.1

5.2

CONSISTENCY OF THE MATHEMATICAL THEORY ________________________________ 104

5.2.1

5.3

CONSISTENCY WITH THE UNIFIED COMPLEXITY THEORY ______________________________ 103 THE RELIABILITY OF THE OBTAINED VALUE _________________________________________ 104

OTHER ISSUES ___________________________________________________________ 106

5.3.1 5.3.2

MODELING STABLE SYSTEMS ____________________________________________________ 106 MODELING TRUTH DEGREE OF NEARLY DECOMPOSABLE COMPLEX CONCEPTS ____________ 106

PART II SUSTAINABILITY AND SUSTAINABLE DEVELOPMENT OF REAL SYSTEMS ________________________ 109

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SUMMARY AND TABLE OF CONTENTS 6

THEORY APPLICATION TO ECOLOGICAL AND SOCIO ECOLOGICAL SYSTEMS _______________________ 111

6.1

SUSTAINABILITY ASSESSMENT MODELS _______________________________________ 113

6.2

OPERATIONAL MODELS ORIENTED TO SES COLLECTIVE DECISION-MAKING ___________ 116

6.3

CONCLUSIONS ___________________________________________________________ 119

6.3.1 6.3.2 6.3.3

TESTING THE MODELS __________________________________________________________ 119 CAN THE SUSTAINABILITY DEGREE OF A SES BE ACCURATELY MEASURED? _______________ 120 TWO ISSUES SUSTAINABILITY ASSESSMENT MODELS SHALL VERIFY _____________________ 121

6.3.3.1 6.3.3.2

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ASSESSMENT MODELS COMPLETENESS __________________________________________________ 121 HIERARCHICAL REPRESENTATION OF SES: MODEL SYSTEM ENVIRONMENT _____________________ 123

REFERENCES __________________________________________________________________________ 125

7.1

REFERENCES ____________________________________________________________ 127

7.2

SOURCES OF FIGURES, DIAGRAMS AND IMAGES ________________________________ 132

ANNEXES ____________________________________________________________________________ 133

ANNEX I A-I.1 A-I.2

FORMULATION AND FORMALIZATION OF SCIENTIFIC THEORIES ________________ 135 THE PURPOSE OF THE THEORY _____________________________________________________ 135 LAWS AND PROPOSITIONS ________________________________________________________ 135

A.I.2.1

A-I.3

FORMULATION OF SYSTEMS OF AXIOMS _________________________________________________ 136

TESTING A THEORY ______________________________________________________________ 136

ANNEX II

TYPES OF SYSTEMS __________________________________________________ 138

ANNEX III

STATISTICS AND FRACTALS; PATTERNS IN SYSTEMS’ INFORMATION ___________ 140

A-III.1 A-III.1.1 A-III.1.2

A-III.2 A.III.2.1

STATISTICS: FINDING PATTERNS IN CHAOTIC/EVOLUTIONARY SYSTEMS __________________ 140 INFERRING ORGANIZATIONAL PATTERNS IN A CLASS: VARIABLE ANALYSIS _____________________ 140 INFERRING INTER-TEMPORAL PATTERNS: TIME SERIES ANALYSIS _____________________________ 141

FRACTAL GEOMETRY: ORDER INTO DISORDER ______________________________________ 142 SYSTEMS’ HIERARCHICAL ORGANIZATION AS FRACTAL PATTERN _____________________________ 143

ANNEX IV

SYSTEMS EFFICIENCY: EFFICIENCY VS DEGREE OF EFFICIENCY ________________ 145

ANNEX V

FORMULATION OF SUSTAINABILITY INDICATORS OF A SYSTEM _______________ 151

A-V.1 A-V.1.1 A-V.1.2 A-V.1.3 A-V.1.4

A-V.2 A-V.3.1 A-V.3.2 A-V.3.3 A-V.3.4

A-V.3

ANNEX VI A-VI.1 A-VI.1.1 A-VI.1.2 A-VI.1.3

A-VI.2 A-VI.2.1

MATHEMATICAL FUNCTIONS FOR SUSTAINABILITY INDICATORS ________________________ 151 LINEAR FUNCTIONS __________________________________________________________________ NON-LINEAR FUNCTIONS _____________________________________________________________ INDICATORS WITH CUT VALUES ________________________________________________________ INACCURACY OF SUSTAINABILITY INDICATORS ____________________________________________

151 153 153 154

THE DIFFERENT MEANING OF INDICATORS _________________________________________ 155 EXAMPLE 1: TWO INDICATORS REFERRED TO NOISE _______________________________________ EXAMPLE 2: TWO INDICATORS RELATING TO EMPLOYMENT RATE ____________________________ EXAMPLE 3: AN INDICATOR REFERRED TO WATER CONSUMPTION OF A POPULATION ____________ CONCLUSIONS ______________________________________________________________________

155 157 159 160

INFORMATION THAT ANY INDICATOR SHOULD PROVIDE ______________________________ 161

INDICATOR AGGREGATION ___________________________________________ 162 MATHEMATICAL FORMULATIONS FOR INDICATOR AGGREGATION ______________________ 162 FORMULATION AS A MEASURE OF POSITION O DISTANCE: RESILIENCE DEGREE Rs[I]: __________ 162 FORMULATION AS DEGREE OF CERTAINTY / NEGENTROPY ________________________________ 167 FORMULATION AS A CENTRAL TENDENCY MEASURE _____________________________________ 171

COMPARISON OF RESULTS USING THE DIFFERENT FORMULATIONS _____________________ 172 AGGREGATED VALUE OF A SUBSYSTEM WITH FIVE INDICATORS ____________________________ 173

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A MATHEMATICAL THEORY OF SUSTAINABILITY AND SUSTAINABLE DEVELOPMENT A-VI.2.2

A-VI.3

AGGREGATED VALUE OF A HIERARCHY WITH SEVERAL PARTIAL AGGREGATIONS ______________ 175

REVIEW OF INFLUENCE OF EACH INDICATOR ON THE AGGREGATED VALUE _______________ 176

A-VI.3.1 A-VI-3.2

INFLUENCE OF EACH ELEMENTARY INDICATOR ON THE AGGREGATED VALUE_________________ 176 DECREASING MARGINAL INFLUENCE OF SUSTAINABILITY INDICATORS ______________________ 178

ANNEX VII A-VII.1 A-VII.2

SUSTAINABILITY OF THE DEVELOPMENT OF A SYSTEM______________________ 180 LOGICAL APPROACH: MEMBERSHIPS FUNCTIONS AGGREGATION _______________________ 180 UTILITY APPROACH: TOTAL UTILITY COMPARISON BETWEEN TWO STATES OF A SYSTEM ____ 180

ANNEX VIII

DECISION MAKING __________________________________________________ 181

A-VIII.1

RATIONAL DECISION MAKING: EXPECTED UTILITY MAXIMIZATION ____________________ 181

A-VIII.1.1 A-VIII.1.2

A-VIII.2 A-VIII.2.1 A-VIII.2.2 A-VIII.2.3 A-VIII.2.4

THE SUSTAINABILITY DEGREE AS A UTILITY FUNCTION ___________________________________ 182 UTILITY AS ‘EXPECTED UTILITY’ ______________________________________________________ 183

DECISIONS IN SOCIO ECOLOGICAL SYSTEMS ______________________________________ 183 COLLECTIVE DECISIONS: COLLECTIVE VS INDIVIDUAL UTILITY ______________________________ EXPECTED UTILITY MAXIMIZATION: MULTICRITERIA ANALYSIS ____________________________ OPPORTUNITY COST AND COMPLETENESS OF MODELS ___________________________________ RESTRICTIVE CONDITIONS __________________________________________________________

183 185 186 187

FIGURES, DIAGRAMS AND ILLUSTRATIONS FIGURES Figure 01: Binary membership function, ________________________________________________________ 29 Figure 02: Fuzzy membership function __________________________________________________________ 31 Figure 03: Membership function of x to class A fa[x] and to its complementary class ¬A f¬A [x] _____________ 32 Figure 04: Fuzzy signature or Fuzzy Hierarchy ____________________________________________________ 34 Figure 05: Sustainability and Unsustainability as mutually exclusive concepts __________________________ 35 Figure 06: The Sustainability Degree of a system as its Grade of Membership to Sustainability set or class ___ 36 Figure 07: Membership of I system to sets Sustainability and Unsustainable for variable 'i'________________ 36 Figure 08: Decomposition of the Sustainability concept S ___________________________________________ 37 Figure 09: Membership function of I to Si, _______________________________________________________ 38 Figure 10: A system _________________________________________________________________________ 40 Figure 11: The concept of hierarchy ____________________________________________________________ 41 Figure 12: The Sustainability Degree as relative distance of a system to its unsustainability threshold _______ 42 Figure 13: Composition of a nested hierarchy ____________________________________________________ 45 Figure 14: Hierarchical decomposition of a system ________________________________________________ 46 Figure 15: Interactions in a nested hierarchical description of a system _______________________________ 46 Figure 16: Sustainability Degree as a measure of the degree to which a structure matches its optimal ______ 50 Figure 17: Implication of two events ___________________________________________________________ 71 Figure 18: Unsustainability is the opposite event of Sustainability ____________________________________ 72 Figure 19: Intersection of two events: __________________________________________________________ 73 Figure 20: Sustainability Degree of a system as the probability of its intersection with Sustainability ________ 74

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SUMMARY AND TABLE OF CONTENTS Figure 21: Two-level hierarchy ________________________________________________________________ 77 Figure 22: Two-level hierarchy with equally likely indicators ________________________________________ 77 Figure 23: Decomposition of SES sustainability into three equally important dimensions _________________ 90 Figure 24: Hierarchical decomposition of Sustainability ____________________________________________ 91 Figure 25: Review of the Hierarchical Organization ________________________________________________ 92 Figure 26: Modeling non-truth conditions in the hierarchical organization _____________________________ 92 Figure 27: Range of influence in equilibrium _____________________________________________________ 94 Figure 28: Checking indicators significance level [01] ______________________________________________ 95 Figure 29: Checking indicators significance level [02] ______________________________________________ 95 Figure 30: Sustainability limits of a variable ______________________________________________________ 96 Figure 31: Graphic representation of a four limits variable membership function: _______________________ 97 Figure 32: Example of logical decomposition with non-truth conditions _______________________________ 98 Figure 34: Decomposition as a fractal__________________________________________________________ 143 Figure 35: Nested Hierarchy as an iteration of invariant functions ___________________________________ 144 Figure 36: Entropy flows between a system and its environment, ___________________________________ 145 Figure 37: The review of the system-environment in terms Complexity-flows _________________________ 146 Figure 38: Non-relatedness of Efficiency and Sustainability Degree __________________________________ 147 Figure 39: Relation between Efficiency and Sustainability Degrees of a system ________________________ 148 Figure 40: Sustainability Degree as the aggregated indicator _______________________________________ 162 Figure 41: Hierarchical representation of a five-indicators subsystem ________________________________ 173 Figure 42: Example of hierarchy with several partial aggregations ___________________________________ 175 Figure 43: Diminishing marginal influence of sustainability indicators. _______________________________ 178

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DIAGRAMS Diagram 01: Process of theory formulation ______________________________________________________ 21 Diagram 02: Process for AS’ Sustainability characterization _________________________________________ 27 Diagram 03: Composition of a nested hierarchy __________________________________________________ 44 Diagram 04: Hierarchical Decomposition of the sustainability of a class of systems ______________________ 90 Diagram 05: Design process for a sustainability assessment model __________________________________ 113 Diagram 06: Process for formulating a formal Theory _____________________________________________ 135

ILLUSTRATIONS Image 01: St. Peter's Piazza ___________________________________________________________________ 41 Image 02: An elective parliamentary system _____________________________________________________ 43 Image 03: A Company _______________________________________________________________________ 49 Image 04: Sports facility _____________________________________________________________________ 51 Image 05: Paris_____________________________________________________________________________ 52 Image 06: Lorenz Attractor ___________________________________________________________________ 63 Image 07: Precipitation and temperature charts __________________________________________________ 64 Image 08: Feigenbaum Fractal ________________________________________________________________ 67 Image 09: Oasis of Huacachica _______________________________________________________________ 124 Image 10: Solar System _____________________________________________________________________ 138 Image 11: Moon-Earth System _______________________________________________________________ 139 Image 12: Menger sponge ___________________________________________________________________ 143 Image 13: Madrid Rio ______________________________________________________________________ 186 Image 14: Central Park _____________________________________________________________________ 188

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INTRODUCTION AND GENERAL APPROACH

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1.1 INTRODUCTION In recent times the problems associated with the development of human society have made clear both that the current model is seriously inefficient in the use of scarce and necessary resources and that it may be impossible to sustain it even in the medium term. As a consequence, different approaches have emerged grouped under the terms Sustainability and Sustainable Development, proposing solutions or perspectives that seek maximizing the efficient use of resources, increasing the positive effects of development and the capacity to endure. And though the abundant scientific production and sustainability assessment models proposals can be considered a progress towards achieving instruments that help us reduce the unsustainability of our development, their review also shows some disturbing aspects: The first is the excessive fragmentation of knowledge; the diversity of approaches reveals different perceptions of what the key issues are, and leads to statements that cannot be related to each other, precluding their joint use1. Although in certain questions the existence of different points of view is convenient [as it provides a range of available options], we can say that the current excessive differentiation without structure in relation to Sustainability is not generating a range of options but noise:  

It reduces its heuristic power2. It is directly translated into the mathematical models that provide different results even when applied to the same reality.

In addition, it is extremely difficult to analyze this heterogeneity of results following a rigorous criterion since few models sufficiently explicit their departing assumptions. The second is the often lack of sufficient testing degree of the statements, largely as a direct result of the previous fragmentation, it is not possible to compare the results of models that consider the key issues to be different. But also due to something internalized by many scientists, the conviction that statements relating social sciences do not support [or may be do not require] to be tested3. When reviewing sustainability statements, many of them lack any testing degree. And the third is that most proposals are not operational, which contrasts with the fact that development of Socio Ecological Systems [SES]4 is essentially directed; it involves continuous decision making and Sustainability requires being included in most decision processes.

1

Among other issues, because it is not clear what the rules to combine these statements in order to jointly use them might be.

2

The lack of structure reduces the positive effect of two valuable features of shared knowledge: feedback and exponential effect of its accumulation; its capacity to become a common base for building more knowledge. 3

According to Popper [1935, p.28] this would be the essential feature of a scientific statement “their susceptibility to revision; i.e.: they can be criticized and superseded by better ones” 4

We define Socio Ecological Systems as any form of stable organization involving 'people' that interact with each other and with their environment, and currently we can consider the whole Earth to be a big SES.

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But also the few existing operational proposals incorporate the former issue; they are built on different assumptions and consequently they lead towards different elections, making impossible [or greatly hampering] their use in rational decision making processes5.

Insufficient testing, lack of relation among different proposals and the fact that few models are operational, have taken us to a situation where the ability of sustainability to be a rational decision-making criterion is greatly diminished. To solve it, we need to act on three lines corresponding to the issues discussed above, which can be briefly summarized as:   

At a global level, it is necessary to develop a comprehensive unifying framework that allows relating the different proposals regarding Sustainability. At an individual proposal level, it is necessary that assessment models and indicators test their results and incorporate feedback. Transversely, it is necessary to enhance its operational nature integrating sustainability with Decision Theory, designing methodologies to make it a parameter readily usable in most decision making processes.

We can summarize it by saying that a scientific formalization of sustainability is necessary, which should be undertaken seeking consistency and integrating it with Decision Theory, so models can be designed to be used in SES’ usual decision making processes. This theory intends to be a first step in this direction by establishing principles that allow understating and modeling Sustainability, including also some guidelines for modeling SES sustainability and for designing decision-making models. Let us start by briefly reviewing the characteristics and formalization process we undertake for developing the Theory.

5

The problem is obvious: if two models proposed by equally prestigious organizations/authors lead to different preference orders, the decision is necessarily contrary to the ‘advise’ of one of these models. According to one of such models, it is not the right decision.

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1.2 GENERAL APPROACH The approach we follow in the present text differentiates two parts: In the first part, we develop a formal/mathematical theory which reviews the sustainability of Adaptive Systems [AS], conceptual model sufficiently similar to SES, which consequently will allow us to apply the conclusions of this theory to such systems. It is the main part of the text and we undertake it in the following order6:

Problem Aim/Purpose Hypothesis

Characterization of AS Sustainability

Axioms and Definitions System

Theory

Proof / Test

Diagram 01: Process of theory formulation

Let us review each part of the process in detail: This theory is formulated with the aim of contributing to sustainable development of SES, and because there is a knowledge problem; there is no formal theory that provides mathematical models to measure their sustainability. For the greatest certainty on the conclusions of the theory, we focus on a conceptual object sufficiently similar to SES; Adaptive Systems [AS], whose conceptual nature allows us to make more categorical statements. We can therefore re-state the purpose of the theory, which we divide into two partial purposes:  

Providing mathematical models that allow to measure AS sustainability. Providing guidelines for developing models referred to SES sustainability.

The hypotheses are deducted from the purposes of the theory, and are the following three:   

The Sustainability of any AS can be fuzzily quantified as its Sustainability Degree. The Sustainability of its development can be assessed reviewing the variation in time of its Sustainability Degree. Both issues can be referred to SES.

Based on the knowledge problem, purpose and hypotheses, we propose a body of knowledge to be reviewed for characterizing sustainability of AS, which include a large number of theories that we group into three main perspectives: conceptual, systemic and probabilistic. From this characterization, we propose a Premises System on which we set the grounds for the theory which includes two types of propositions:

6

For a brief explanation of the process of formulation and formalization of theories, see ANNEX I FORMULATION AND FORMALIZATION OF SCIENTIFIC THEORIES

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



A System of Definitions that seeks to ensure two issues: o The consistency of different definitions of sustainability based on different theories [multi and transdisciplinary approach]. o The consistency of the proposals of the theory with other revised theories A System of Axioms that allows us to support all formulations and statements.

From this Premises System all the statements of the Theory are deducted, ensuring their consistency at two levels:  

All statements must be deducted from the Axiomatic System through logical or mathematical transformations [internal consistency] All statements must be consistent with the Definitions System [external consistency with the theoretical framework]

Finally we test the theory, comprising the following steps: consistency and empirical verification. The internal and external consistency of the theory is deemed incorporated in the formulation process allowing us to consider the theory to be ‘true’ or tautological in relation to its premises7. The empirical verification of the theory is not strictly required as the mathematical theory is mainly of a formal nature; AS are conceptual objects with no real existence.

In the second part, we review some key issues for the development of models applied to SES and their use in collective decision making processes. This review includes the conclusions of the review of a large number of SES' sustainability assessment models, which can be easily adapted according to the principles proposed by the mathematical theory in a majority of cases, pointing to its practical applicability in two broad areas:  

It is possible to model many different SES types according the criteria provided in the Theory. It is possible to design operational models applicable to different SES decision making processes.

The wide range of issues reviewed in the present text has made preferable to separate some parts as annexes in order to maintain clarity. Nevertheless, it is important to say that it does not mean annexes are less important than the main text. This Theory can only be fully understood if all parts are read.

7

The proof of a theory does not imply truth in an absolute sense but in relation to the premises assumed to be true and from our current knowledge.

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CHARACTERIZING ADAPTIVE SYSTEM’S SUSTAINABILITY

PART I MATHEMATICAL THEORY OF SUSTAINABILITY AND SUSTAINABLE DEVELOPMENT [OF ADAPTATIVE SYSTEMS]

23


24


2 CHARACTERIZING ADAPTIVE SYSTEM’S SUSTAINABILITY: THEORETICAL FRAMEWORK

25


26


In this chapter we review different existing theories related both to Sustainability and Adaptive Systems [AS], and at the same time we start setting the grounds for the present Theory. Therefore, it may be considered both a theoretical framework review and the initial development of the theory. Different conceptualizations of Systems’ Sustainability and Sustainability Degree are proposed in relation to each of the revised theories, and definitions and formulations are presented in advance.

Characterizing AS’ Sustainability is going to require reviewing a large number of quite different theories that for clarity we group into three general perspectives or approaches:

Logical Approach

Systemic Approach

Probabilistic Approach

Diagram 02: Process for AS’ Sustainability characterization

First we undertake a ‘logical’ approach to the concept of Sustainability based on Logic and Set Theory, with two different purposes: 



Establishing the inference rules to be used for deducing statements from the Premises, which relates to two approaches of Set Theory/Logic: o Classic Set Theory/Boolean logic. o Fuzzy Set Theory/Fuzzy Logic. Proposing a conceptual approach to sustainability, that guides us through the development of the theory.

Second we undertake a ‘systemic’ approach to AS’ Sustainability, based on different theories:  

 

General Systems and Hierarchy Theories review allow us to understand and describe hierarchical Systems Complexity Theory review provides tools to understand systems’ complexity from two perspectives: o as Organization, which relates to Communication Theory. o as Emergent property, which relates to non-equilibrium thermodynamics and Entropy Ecology review approaches us to the System-Environment model and provides us the concept of system’s efficiency. Adaptive Systems Theory review introduces us AS’ specificities as systems that: o Evolve, which relates to Complexity Theory as organization o Decide, which relates to Decision Theory

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

o Interact with each other, which relates to the Theory of Games8. Chaos theory review allow us to understand aperiodic systems with nonlinear feedback from two different perspectives: o Disorganized complexity, which relates to Statistical Theory o Self-similarity and strange attractors, which relates to Fractals Theory

Finally, the review of systems’ unpredictability leads us to a ‘probabilistic’ approach to Sustainability, which we undertake based on the two conceptual approaches of Probability theory:  

Probability a Stable Frequency Probability as Degree of Belief

The intentionality of the following review requires a preliminary definition of Sustainability / Sustainable, for which we consider the following: Sustainability: “Quality of Sustainable” Sustainable9:

“Able to continue over a period of time” “Causing little or no damage to the environment and therefore able to continue for a long time”10

These preliminary definitions are completed with different definitions proposed from each of the theories that we review, providing a criterion to validate formulations. Different approaches may take to different formulations but consistency requires their results to be coincident. Thus, we do not try to reach a unified definition of Sustainability or Sustainability Degree of a system, but instead to propose the joint consideration of the different perspectives altogether as the way to thoroughly assess systems’ sustainability11. Let us begin with a logical approach to the concept of Sustainability.

8

To avoid excessive length of the present text, a review of Theory of Games is not included [only some isolated references are pointed out]. However, many issues related to Decision Theory can be easily translated into the language of the Theory of Games. 9

Both definitions are from Cambridge Dictionaries Online [from now on CDO]

10

The vagueness of the statement “long time” coincide in part with the sense with which herein we will use the expression ‘indefinitely endure’, in which the term indefinitely must be understood in a relative way, referred to the timescales we usually use. 11

The Definition System allows us to compare them, noting that some of them are different ways of saying the same thing; but every one adds something new in relation to the other definitions.

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2.1 A ‘LOGICAL’ APPROACH TO SUSTAINABILITY CONCEPT We approach the concept and measurement of systems sustainability from a logic perspective based on Set Theory, and our review undertakes three steps:  



Firstly, we review the basic properties and operations between sets, which serve as inference rules to deduce propositions from the premises. Secondly, we review sustainability and unsustainability as complementary concepts, classes or sets, and define the sustainability degree of a system as its Grade of membership to the set of sustainable systems or Sustainability class. And thirdly, we propose a logical decomposition of sustainability concept as the procedure for calculating system’s global Grade of membership to Sustainability class, based on two additional conceptualizations: o Sustainability indicators as membership functions of a system to classes [or concepts] contained [or implicit] in Sustainability class [or concept]. o Relevant variables for the sustainability of a system as any system’s information which variation can modify system’s Grade of membership to the set of sustainable systems or Sustainability class.

Let us begin by reviewing some properties and operations between Sets. 2.1.1 SETS PROPERTIES AND OPERATIONS: RULES OF LOGICAL INFERENCE There are two approaches to Set Theory that relate differently to the present Theory:  

Classic Set Theory, which corresponds with binary or Boolean Logic. Fuzzy Sets Theory, which corresponds with Fuzzy Logic.

Classic Set Theory addresses classification of mathematical objects, assigning to each object x a binary membership function fA[x] to a class or set A; i.e.: x membership to A can only have two values; 0 if x does not belong to A or 1 if x belongs to A:

𝑓 [𝑥] =

0 𝑖𝑓 𝑥 ∉ 𝐴 1 𝑖𝑓 𝑥 ∈ 𝐴

Figure 01: Binary membership function, where ‘i’ is a variable that relates the state of x to its membership to A

Fuzzy Set Theory arises to characterize classes or sets that support Grades of membership and Exclusion, and proposes the concept of fuzzy set as a class of objects characterized by a continuous membership function fA[x] that assigns each object ‘x’ a Grade of membership ranging between zero and one [adapted from Zadeh 1965]12: 𝐴=

12

𝑥, 𝑓 [𝑥] |𝑥 ∈ 𝑋 → 𝑓 [𝑥] → [0,1]

(1)

Another possible definition of fuzzy set is "a set without a well-defined boundary" [Goguen 1967, p.146].

29


While binary classes only allow for complete membership or exclusion which is reached at specific values of i, fuzzy sets allow partial membership or exclusion which progressively happens in relation to one or more ranges of ‘i’ values. Objects present a Grade of membership to a particular set or class and we can relate it to Fuzzy Logic. The degree of membership of an object x to a class A coincides with the degree to which the concept A is true referred to x:   

If the Grade of membership is zero, x does not belong at all to class A [or concept A is totally false referred to x]. If the Grade of membership is one, x totally belongs to class A [or concept A is totally true referred to x]. If the Grade of membership is between zero and one, it is equivalent to the degree that x belongs to class A [or the degree to which A referred to x is true].

We have hypothesized that AS’ sustainability can be fuzzily quantified in terms of Sustainability degree, and therefore our review focus on fuzzy sets properties and operations. 2.1.1.1 PROPERTIES OF FUZZY SETS13 EQUALITY Two fuzzy sets A and B are equal if and only if their membership functions fA[x] y fB[x] are equal for any x possible:

Equality

∀𝑥 ∈ 𝑋: 𝐴 = 𝐵 ⟷ 𝑓 [𝑥] = 𝑓 [𝑥]

(2)

COMPLEMENT The complement of a set A is written as ¬A and is defined as:

Complement

𝑓 [𝑥] = 1 − 𝑓¬ [𝑥]

(3)

CONTAINMENT A is contained in B if and only if for every possible x its membership function fA[x] is less than B’s fB[x]

Containment

∀𝑥 ∈ 𝑋: 𝐴 ⊆ 𝐵 ⟷ 𝑓 [𝑥] ≤ 𝑓 [𝑥]

(4)

This condition is very important at two levels: 

13

At the physical level, it allows us to set an upper bound to the maximum Sustainability Degree of a system; it is always equal or lower than the Sustainability Degree of its accessible environment.

Compilation from Zadeh [1965, p.340/343]

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

At the conceptual level, it allows us to establish a condition that any sustainability indicator shall meet; it must refer to a concept contained in Sustainability concept.

CUT VALUES It is equivalent to consider that above or below a certain value of fA[x], x membership to A becomes full or nil, which for a function fA with two cut-values of α and β can be expressed as:   

a value of fA[x] less than α means nil membership of x to A a value of fA[x] greater than β means full membership of x to A any value of fA[x] between α and β means a 'grade of membership' of x to A

Cut Values

𝑓 [𝑥] < 𝛼 ⟶ 𝑓 [𝑥] = 0 ∧ 𝑓¬ [𝑥] = 1 𝑓 [𝑥] > 𝛽 ⟶ 𝑓 [𝑥] = 1 ∧ 𝑓¬ [𝑥] = 0 𝛼 < 𝑓 [𝑥] < 𝛽 ⟶ 𝑓 [𝑥] ∈ [0,1] ∧ 𝑓¬ [𝑥] ∈ [0,1]

(5)

Cut-values have two main applications:  

They allow decomposing fuzzy sets in several binary sets [however this issue is not relevant for the present theory]. They allow incorporating restraining conditions into indicators in operational models, an issue that is interesting and we later develop.

2.1.1.2 GRAPHICAL REPRESENTATION OF MEMBERSHIP FUNCTIONS Graphical representation of fuzzy membership functions allows us to observe the existence of at least two singular points i1 and i2 that limit the range of x fuzzy membership to A14.

𝑓 [𝑥] = 𝑚𝑎𝑥 𝑚𝑖𝑛

𝑖−𝑖 ,1 ,0 𝑖 −𝑖

Figure 02: Fuzzy membership function, where ‘i’ is a variable that relates the state of x with its membership to A.

We see that an object x can only belong to a class A if there is at least one i1-i2 range of values for which a variation of i modifies x Grade of membership to A:   

14

ii2 indicates complete membership of x to A [and therefore null membership of x to ¬ A] values of i between i1 and i2 indicate a grade of membership of x to A between 0 and 1 [and therefore a grade of membership of x to ¬A between 1 and 0]

In some rare occasions, i1 or i2 could be equal to infinite, and therefore may be represented or not, depending on the scale.

31


Another interesting issue is that we can summarize in a single graph the grade of membership of x to both a class A and its complementary class ¬A:

𝑓 [𝑥] = 𝑚𝑎𝑥 𝑚𝑖𝑛

𝑖−𝑖 ,1 ,0 𝑖 −𝑖

𝑓¬ [𝑥] = 1 − 𝑓 [𝑥] 𝑓¬ [𝑥] = 𝑚𝑎𝑥 𝑚𝑖𝑛 1 −

𝑖−𝑖 ,1 ,0 𝑖 −𝑖

Figure 03: Membership function of x to class A fa[x] and to its complementary class ¬A f¬A [x]

The graphic presents a horizontal symmetry at fA[x]=0.5, an important threshold because if fA[x]>=0.5 then x belongs more to A than to ¬ A, while if it is lower the opposite holds. 2.1.1.3 FUZZY SETS AGGREGATION An aggregation of fuzzy sets is any operation that combines several fuzzy sets into one fuzzy set, and must meet the following two conditions:

Boundary conditions Monotonicity

𝐴[0, … ,0] = 0 𝑦 𝐴[1, … ,1] = 1

(6)

∀𝑖 ∈ [1, . . . , 𝑛]; 𝑥 ≥ 𝑦 ⟷ 𝐴[𝑥 , . . . , 𝑥 ] ≥ 𝐴[𝑦 , . . . , 𝑦 ]

(7)

UNION The union of two fuzzy sets A and B with respective membership functions fA[x] y fB[x] is a fuzzy set C, whose membership function fC[x] is:

Union

𝐶 = 𝐴 ∪ 𝐵 → ∀𝑥 ∈ 𝑋: 𝑓 [𝑥] = 𝑚𝑎𝑥 𝑓 [𝑥], 𝑓 [𝑥]

(8)

INTERSECTION The intersection of two fuzzy sets A and B with respective membership functions fA[x] y fB[x] is a fuzzy set C, whose membership function fC[x] is:

Intersection

𝐶 = 𝐴 ∩ 𝐵 → ∀𝑥 ∈ 𝑋: 𝑓 [𝑥] = 𝑚𝑖𝑛 𝑓 [𝑥], 𝑓 [𝑥]

(9)

ALGEBRAIC PRODUCT The algebraic product of A and B is denoted AB and is calculated as:

Algebraic Product

𝑓 [𝑥] = 𝑓 [𝑥] ∗ 𝑓 [𝑥]

(10)

𝐴𝐵 ⊂ 𝐴 ∩ 𝐵

(11)

And it satisfies the condition:

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CONVEX COMBINATION A convex combination of two fuzzy sets is a linear combination with the form: ∀𝑥 ∈ 𝑋: 𝑓[

, ; ] [𝑥]

= 𝛬 ∗ 𝑓 [𝑥] + [1 − 𝛬] ∗ 𝑓 [𝑥]

(12)

A convex combination is a centroid of the membership functions fA[x] y fB[x], being the first located at a distance Λ and the second at a distance 1-Λ, and its value is always between the intersection and the union of both sets: 𝑚𝑖𝑛 𝑓 [𝑥], 𝑓 [𝑥] ≤ 𝑓[

, ; ] [𝑥]

≤ 𝑚𝑎𝑥 𝑓 [𝑥], 𝑓 [𝑥]

𝐴 ∩ 𝐵 ≤ [𝐴, 𝐵; 𝛬] ≤ 𝐴 ∪ 𝐵

(13) (14)

AVERAGES Other possible fuzzy aggregation functions’ formulations are averages that may be Arithmetic, Geometric and Harmonic means, weighted or unweighted.

Aggregation operations reviewed above, allow us to combine fuzzy sets or concepts in order to obtain an aggregated or composed fuzzy set. Yet for some sets or concepts we may be interested in undertaking the complementary operation; break them down [or decompose them] in multiple sets or fuzzy concepts, what we review next15.

2.1.1.4 FUZZY SETS DECOMPOSITION Decomposition is the reverse process of combining/aggregation operations and to undertake it we build on three concepts or ideas: First, the concept of L-Fuzzy Sets [Goguen 1967] which suggests that majority of fuzzy sets or concepts can be understood as fuzzy sets of fuzzy sets or concepts: 𝐴 = {𝐴 , 𝐴 , … , 𝐴 }

(15)

Where AL is a fuzzy set and Ai – An the concepts or sets that fuzzily belong to A Second, the concept of Vector Valued Fuzzy Sets [VVFS]16 which is proposed as a special case of the above, composed by grades of membership that can be expressed as a vector relating x grade of membership to concepts Ai-An, to its grade of membership to A: 𝐴: 𝑥 → [0,1] : ∀𝑥 ∈ 𝑋

(16)

A is a VVFS and n_ the number of fuzzy sets or concepts Ai – An implied [contained] in A

15

This is especially interesting relating concepts for which we cannot directly propose membership functions.

16

Kóczy et Al 1980, cited in Mendis 2008

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And third, the concept of Fuzzy Signature which arises when each concept or set implied in A is recursively modeled as a VVFS, turning into another nested vector [branch or signature] or into an end node [leaf], obtaining an overall hierarchical decomposition that can be expressed as:

𝐴 : 𝑥 → [𝑆 ] [0,1] 𝑤ℎ𝑒𝑟𝑒 𝑆 =

𝑆

≡ 𝑓[𝑆 ] ; 𝑖𝑓 𝑙𝑒𝑎𝑓

; 𝑖𝑓 𝑏𝑟𝑎𝑛𝑐ℎ 𝑜𝑟 𝑠𝑖𝑔𝑛𝑎𝑡𝑢𝑟𝑒

Figure 04: Fuzzy signature or Fuzzy Hierarchy. Each branch or signature may have a different aggregation function.

Fuzzy Signatures are a conceptual tool highly useful for decomposing concepts which membership is determined by many interdependent features that show an underlying hierarchical organization17, and we can understand them from two perspectives: 



From an information perspective, a Fuzzy Signature is a representation of the patterns existing in a dataset structured in a complex way that can be detected through the separability of the data18. From a logical perspective it may be understood as a representation of a problem in a similar manner to the way people approach [analyze or decompose] real issues.

A especially interesting type of Fuzzy Signature are Fuzzy Signature Sets [Tamás and Koczy 2007] whose end nodes or leaves are not variable values but membership functions, and therefore match the ideas of organization of a class of systems and logical decomposition of a concept that we review later. A Fuzzy Signature Set is essentially a hierarchical organization of the information we associate with a global concept, class or set A, characterized as a structure of membership functions to different concepts or classes Ai implied/contained in A:  

Leaf concepts or classes are membership functions in the range 0-1. Branch concepts or classes are fuzzy aggregation of membership functions [leaves and branches] they contain at lower levels19.

Therefore Fuzzy Signature Sets allow us to calculate the Grade of membership of an object to any decomposable class as a recursive aggregation process of its Grade of membership to each branch and leaf it contains. 17

Wong et Al 2004; Mendis and Gedeon 2008. Corresponds to our intuition of Organized Complexity

18

Wong et Al 2004. The concept of separability relates to those of decomposability and modularity that we review later.

19

Mendis [2008] proposes as typical aggregation functions: minimum, harmonic mean, geometric mean, arithmetic mean and maximum. Also different types of weighted aggregations can be used [e.g.: GOWA; WRAO, …]

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2.1.2 A LOGICAL APPROACH TO THE CONCEPT OF SUSTAINABILITY The semantic/logical analysis of the terms Sustainability and Unsustainability leads us to Unsustainability as the non-sustainability, i.e. as the opposite concept or complementary class of Sustainability, and this allows us its conceptualization from the two approaches of Set Theory: 2.1.2.1 CLASSIC SET THEORY: SUSTAINABILITY AND UNSUSTAINABILITY AS MUTUALLY EXCLUSIVE SETS Classic Set Theory /Binary or Boolean logic allows us a first approach to sustainability and unsustainability as mutually exclusive concepts or classes, i.e., concepts or classes whose intersection is empty and whose union provides a universe of discourse. Let us take the class of all Adaptive Systems [AS] and divide it into two sub-classes:  

We call S or Sustainability to the class composed by all sustainable AS. Likewise we call ¬S or Unsustainability to the complementary class of S, composed by all nonsustainable [unsustainable] AS. 𝑆 ∪ ¬𝑆 = 𝐴𝑆 = 𝛺 [𝑅]

(17)

𝑆 ∩ ¬𝑆 = ∅20 ↔ 𝑆 = 1 − ¬𝑆

(18)

Therefore, if we divide all AS between S and ¬S the union of both sets must include all AS, while its intersection must be empty.

Figure 05: Sustainability and Unsustainability as mutually exclusive concepts or classes in the space AS.

The drawback of this interpretation is that even being theoretically correct, it is too restrictive because it does not support Grades of Membership which characterize most real systems To solve it, we review it from Fuzzy Logic or Fuzzy Sets Theory. 2.1.2.2 FUZZY SETS THEORY: SUSTAINABILITY DEGREE AS A SYSTEM’S GRADE OF MEMBERSHIP TO SUSTAINABILITY CLASS The concept of Grade of Membership of Fuzzy Set Theory allows us a first characterization [or definition] of Sustainability and Unsustainability Degrees of a system:

20

This formula is derived from Duality Law proposed by Boole [1854, p.35] as an 'interpretability condition for logical functions' [which in turn refers to Aristotelian Contradiction Principle] as X [1-X] = 0

35


 

The Sustainability Degree of a system I is its Grade of membership to Sustainability set or class fs[I]. The Unsustainability Degree of a system I is its Grade of membership to Unsustainability set or class f¬s[I].

Figure 06: The Sustainability Degree of a system as its Grade of Membership to Sustainability set or class, and Unsustainability Degree as its Grade of Membership to Unsustainability set or class.

𝑆[𝐼] = 𝑓 [𝐼] = 𝐼 ∩ 𝑆

(19)

¬𝑆[𝐼] = 𝑓¬ [𝐼] = 𝐼 ∩ ¬𝑆

(20)

¬𝑆[𝐼] = 1 − 𝑆[𝐼]

(21)

The value of S[I] ranges between 0 and 1, so we can distinguish three values:   

S[I]=1 membership to S set is complete [and therefore membership to ¬S set is null]. 0