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Reconsideration of Neutrosophic Social Science and Neutrosophic
Phenomenology with Non-classical logic
Takaaki Fujita 1∗
1∗Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan.
Abstract
Body-Mind-Soul-Spirit Fluidity is a concept rooted in psychology and phenomenology, oering signicant
insights into human decision-making and well-being. Similarly, in social analysis and social sciences, frame-
works such as PDCA, DMAIC, SWOT, and OODA have been established to enable structured evaluation and
eective problem-solving. Furthermore, in phenomenology and social sciences, various logical systems have
been developed to address specic objectives and practical applications.
This paper extends these concepts using the Neutrosophic theory, revisiting their mathematical denitions and
exploring their properties. The Neutrosophic Set, an extension of the Fuzzy Set, is a highly exible framework
that has been widely studied in elds such as social sciences. By incorporating Neutrosophic Sets, we aim to
improve their suitability for programming and mathematical analysis, providing advanced methods to tackle
complex, multi-dimensional problems.
We hope that this research will inspire further studies and foster the development of practical applications across
various related disciplines.
Keywords: Neutrosophic Set, plithogenic set, fuzzy set, Phenomenology
1 Short Introduction
1.1 Phenomenology: Body-Mind-Soul-Spirit Fluidity
Phenomenology is a philosophical approach that investigates conscious experiences as they are perceived, fo-
cusing on intentionality, subjective interpretation, and the suspension of preconceived notions to reveal the
essence of phenomena and lived experiences [110, 139, 244, 345, 398, 411]. Its relevance spans disciplines
such as psychology, sociology [87,107], education [98], and healthcare [297], highlighting the importance of
continued research in phenomenological studies.
Body-Mind-Soul-Spirit Fluidity is a concept originating from psychology and phenomenology (cf. [105,112,
183, 195, 360]). It reects the interconnected dimensions of human existence: the physical body, mental pro-
cesses, emotional soul, and spiritual awareness. Recently, this concept has been extended through the frame-
work of Neutrosophic Sets, giving rise to Neutrosophic Body-Mind-Soul-Spirit Fluidity, a more exible and
robust model for understanding the dynamics of these interconnected dimensions [337].
1.2 Social Analysis: PDCA, DMAIC, SWOT, OODA, and Five Forces Analysis
Social Science studies human behavior, societies, and cultures using systematic research and interdisciplinary
approaches [142,393]. Social Analysis examines societal structures, relationships, and processes to understand
social dynamics and address challenges [34,138]. In the eld of Social Analysis and Social Sciences, various
frameworks have been established to facilitate structured evaluation and problem-solving [138]. Notable ex-
amples include the following frameworks, which are widely recognized for their practical applications. In this
paper, these concepts will be extended using the Neutrosophic Set framework discussed later.
•PDCA (Plan-Do-Check-Act): A cyclical framework designed for continuous improvement. It involves
planning strategies, executing actions, evaluating results, and rening processes to achieve better out-
comes [133,173, 256,291].
•DMAIC (Dene-Measure-Analyze-Improve-Control): A methodology derived from Six Sigma that em-
phasizes dening problems, collecting and measuring data, analyzing root causes, implementing im-
provements, and controlling processes to maintain quality [224, 242, 285,286,300, 356].
1
•SWOT (Strengths-Weaknesses-Opportunities-Threats): A strategic planning tool used to assess internal
strengths and weaknesses, as well as external opportunities and threats, for eective organizational anal-
ysis [93,140, 237, 305,311,391].
•OODA (Observe-Orient-Decide-Act): A decision-making process that focuses on observing situations,
orienting oneself to the context, making informed decisions, and acting promptly, particularly in dynamic
or competitive environments [131,198, 236,282, 298,415].
•Porter’s Five Forces Analysis: A framework for analyzing industry competition. It examines ve key
forces: industry rivalry, buyer power, supplier power, the threat of substitutes, and the threat of new
entrants [94,150, 278].
1.3 Neutrosophic Set and Related Set Theories
Psychology, Phenomenology, and Social Analysis are inherently intertwined with uncertainty. The Neutro-
sophic Set provides a comprehensive framework for eectively addressing and managing these uncertainties.
This subsection explains the Neutrosophic Set and its related concepts.
Set theory is a foundational branch of mathematics that focuses on the study of ”sets,” which are collections of
objects [90, 180, 382,385]. Over time, extensions of classical set theory have been developed to better handle
the complexities and uncertainties encountered in real-world scenarios. These include Fuzzy Sets [88, 358,403,
405–408,417], Vague Sets [9,58,63, 165, 412], Soft Sets [14,15, 120, 222,241,400], Hypersoft Sets [332, 333],
Rough Sets [266–272], Hyperfuzzy Sets [119,136,182,349], and Neutrosophic Sets [11,54,100, 115,251,319,
320,323, 324,340, 390].
Each of these frameworks addresses specic forms of ambiguity or uncertainty. For example, Fuzzy Sets as-
sign to each element a membership degree within the interval [0,1], representing partial rather than binary
membership [403]. Neutrosophic Sets extend this concept by assigning three independent degrees—truth, in-
determinacy, and falsity—to each element, making them particularly suitable for managing complex uncertain-
ties [319,320].
1.4 Our Contribution in This Paper
In this paper, we extend the concepts of Body-Mind-Soul-Spirit Fluidity, PDCA, DMAIC, SWOT, OODA, and
Five Forces Analysis within the framework of Neutrosophic theory and provide a brief exploration of their
properties. Furthermore, we investigate various types of logic in the contexts of Neutrosophic Phenomenology
and Neutrosophic Social Science. It is important to note that the term ”logic” here refers specically to non-
classical logic. While some of these concepts are already established, we revisit their mathematical denitions
to facilitate programming and mathematical analysis using Neutrosophic Sets.
We hope that this research will inspire further studies and encourage the development of practical applications
in this emerging eld.
1.5 The Structure of the Paper
The structure of this paper is as follows.
1 Short Introduction 1
1.1 Phenomenology: Body-Mind-Soul-Spirit Fluidity . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Social Analysis: PDCA, DMAIC, SWOT, OODA, and Five Forces Analysis . . . . . . . . . . 1
1.3 Neutrosophic Set and Related Set Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 OurContributioninThisPaper.................................. 2
1.5 TheStructureofthePaper .................................... 2
2 Preliminaries and Denitions 3
2.1 CoreConceptsinSetTheory................................... 3
2.2 Fuzzy Sets and Neutrosophic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Plithogenic Set: A Generalization of Uncertain Sets . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 UncertainLogic.......................................... 7
2
3 Result and Discussion in this Paper 8
3.1 Neutrosophic Phenomenology: Neutrosophic Body-Mind-Soul-Spirit Fluidity . . . . . . . . . 8
3.2 LogicofPhenomenology..................................... 14
3.2.1 Neutrosophic Intentional Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.2 Neutrosophic Ontological Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 NeutrosophicSocialAnalysis................................... 19
3.3.1 PDCA Cycle with Neutrosophic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3.2 DMAIC Cycle with Neutrosophic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.3 SWOT Analysis with Neutrosophic Sets . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.4 OODA Cycle with Neutrosophic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.5 Neutrosophic Porter’s Five Forces Analysis . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Some Neutrosophic (Social or Business) Logic . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4.1 Neutrosophic Institutional Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4.2 Dominant Neutrosophic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4.3 Service-Dominant Neutrosophic Logic . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4.4 Neutrosophic Critical Thinking (Neutrosophic Critical Logic) . . . . . . . . . . . . . 35
3.4.5 Neutrosophic Climate Change Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.6 Neutrosophic Social Media Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.7 Neutrosophic Critical Service Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Future Tasks: Various Extensions 49
4.1 Real-World Applications within a New Social Framework . . . . . . . . . . . . . . . . . . . . 49
4.1.1 Plithogenic Social Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.2 Hyperanalysis and Hypercycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.3 OtherFrameworks .................................... 53
4.2 NewStrategicLeadership..................................... 54
4.2.1 Neutrosophic Strategic Leadership . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.2 HyperLeadership..................................... 61
4.3 NewNegotiationTheory ..................................... 62
4.3.1 Neutrosophic Negotiation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4 NewFraming ........................................... 65
4.4.1 NeutrosophicFraming .................................. 65
4.4.2 Hyperframing....................................... 68
4.5 NewMentoringMethod...................................... 70
4.5.1 NeutrosophicMentoring ................................. 70
4.5.2 HyperMentoring ..................................... 72
4.6 NewStorytellingDenition.................................... 74
4.6.1 Neutrosophic Storytelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.6.2 HyperStorytelling .................................... 76
4.6.3 Neutrosophic Work-Life Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2 Preliminaries and Denitions
This section introduces essential concepts from set theory that are used throughout this work. For a deeper
exploration of these concepts and their applications, readers are encouraged to consult the cited references as
necessary [113,159,167,180,207]. Detailed discussions on related operations and extensions are also available
in the listed references.
2.1 Core Concepts in Set Theory
The following are foundational principles in set theory. For additional insights and examples, readers may refer
to the recommended references [180].
Denition 2.1 (Set).[180] A set is dened as a well-determined collection of distinct elements. These elements
are either included in or excluded from the set. If 𝐴is a set and 𝑥is one of its elements, this is expressed as
𝑥∈𝐴. Sets are typically denoted using curly braces, e.g., 𝐴={𝑎, 𝑏, 𝑐 }.
Denition 2.2 (Subset).[180] A set 𝐴is said to be a subset of another set 𝐵, written 𝐴⊆𝐵, if all elements of
𝐴are also elements of 𝐵. Formally, this is expressed as:
𝐴⊆𝐵⇐⇒ ∀𝑥(𝑥∈𝐴=⇒𝑥∈𝐵).
3
We also use the following concepts.
Denition 2.3. (cf. [168]) The set of real numbers Rincludes all rational and irrational numbers. Formally, it
is dened as a complete, ordered eld that satises the completeness property:
Every non-empty subset of Rthat is bounded above has a least upper bound in R.
Denition 2.4. (cf. [194]) The set of integers Zconsists of all whole numbers, including positive, negative,
and zero:
Z={. . . , −2,−1,0,1,2, . . . }.
2.2 Fuzzy Sets and Neutrosophic Sets
Fuzzy Sets and Neutrosophic Sets are often introduced in relation to their foundational counterpart, the Crisp
Set. Below are formal denitions to establish this context.
Denition 2.5 (Universe Set).(cf. [252]) A universe set, denoted as 𝑈, is the complete set of all elements
relevant to a particular discussion or problem. It serves as the universal context, encompassing every element
that could be considered within a given framework. For any subset 𝐴, the relationship 𝐴⊆𝑈holds, meaning
all elements of 𝐴must belong to 𝑈.
The universe set 𝑈is foundational in set theory, acting as the domain of discourse within which all subsets are
dened. It is synonymous with concepts such as the underlying set or total set.
Denition 2.6 (Crisp Set).[259] Let 𝑋be a universe set, and let 𝑃(𝑋)represent the power set of 𝑋, which
includes all subsets of 𝑋. A crisp set 𝐴⊆𝑋is dened by its characteristic function 𝜒𝐴:𝑋→ {0,1}, where:
𝜒𝐴(𝑥)=(1if 𝑥∈𝐴,
0if 𝑥∉𝐴.
The characteristic function 𝜒𝐴assigns a value of 1to elements belonging to 𝐴and 0to those outside it, creating
a clear and denitive boundary. Crisp sets adhere strictly to binary logic, distinguishing whether an element is
inside or outside the set.
A Fuzzy Set assigns each element a degree of membership between 0 and 1, representing partial truth and
handling uncertainty.
Denition 2.7 (Fuzzy Set).[403–408] A fuzzy set 𝜏in a non-empty universe 𝑌is a function 𝜏:𝑌→ [0,1],
where each element 𝑦∈𝑌is assigned a degree of membership in the interval [0,1].
Afuzzy relation 𝛿is a fuzzy subset of 𝑌×𝑌. If 𝜏is a fuzzy set in 𝑌and 𝛿is a fuzzy relation on 𝑌,𝛿is called a
fuzzy relation on 𝜏if:
𝛿(𝑦, 𝑧) ≤ min{𝜏(𝑦), 𝜏(𝑧)} for all 𝑦, 𝑧 ∈𝑌 .
Example 2.8 (Temperature Perception).(cf. [64]) Consider the fuzzy set 𝜏of ‘‘warm temperatures” in a uni-
verse 𝑌=R(all temperatures in Celsius). The membership function 𝜏could be dened as:
𝜏(𝑦)=
0,if 𝑦≤15 (cold);
𝑦−15
10 ,if 15 < 𝑦 < 25;
1,if 𝑦≥25 (warm).
For example, at 𝑦=20◦𝐶, the membership degree of ‘‘warm” is 0.5.
Example 2.9 (Tall People).(cf. [196]) In a population where height is measured, the fuzzy set 𝜏of ‘‘tall people”
can assign membership values based on height ℎ:
𝜏(ℎ)=
0,if ℎ≤150 cm (not tall);
ℎ−150
30 ,if 150 < ℎ < 180;
1,if ℎ≥180 cm (tall).
Here, a person of height 165 cm has a membership degree of 0.5.
4
Example 2.10 (Risk Level in Investments).(cf. [158]) The fuzzy set 𝜏of ‘‘high-risk investments” in a universe
𝑌of possible investments may assign degrees of risk based on volatility or expected return. For example:
𝜏(𝑟)=
0,if volatility 𝑟≤5%;
𝑟−5
10 ,if 5% < 𝑟 < 15%;
1,if 𝑟≥15%.
An investment with volatility 𝑟=10% would have a membership degree of 0.5in the ‘‘high-risk” category.
Neutrosophic Set extends Fuzzy Set by introducing truth, indeterminacy, and falsity, each independently in
[0,1], handling uncertainty and contradictions more comprehensively [319]. Unlike Fuzzy Sets, Neutrosophic
Sets model indeterminacy explicitly, enabling greater exibility for uncertain, inconsistent, or ambiguous data
representation.
Denition 2.11 (Neutrosophic Set).[319, 321, 322, 339, 340] Let 𝑋be a non-empty set. A (single-valued)
Neutrosophic Set 𝐴on 𝑋is characterized by three membership functions:
𝑇𝐴:𝑋→ [0,1], 𝐼 𝐴:𝑋→ [0,1], 𝐹𝐴:𝑋→ [0,1],
where for every 𝑥∈𝑋,𝑇𝐴(𝑥),𝐼𝐴(𝑥), and 𝐹𝐴(𝑥)denote the degrees of truth, indeterminacy, and falsity, respec-
tively. These functions satisfy the following condition:
0≤𝑇𝐴(𝑥) + 𝐼𝐴(𝑥) + 𝐹𝐴(𝑥) ≤ 3.
Example 2.12 (Analysis of Tasks).”Analysis of Tasks” systematically examines tasks by breaking them into
components, evaluating resources, priorities, dependencies, and performance for optimization (cf. [70, 232]).
Let 𝑈={𝑎, 𝑏, 𝑐}be a set of tasks. A Neutrosophic Set 𝑆can assign the following degrees of truth, indetermi-
nacy, and falsity to each task:
• Task 𝑎:𝑇(𝑎)=0.8,𝐼(𝑎)=0.1,𝐹(𝑎)=0.1
• Task 𝑏:𝑇(𝑏)=0.5,𝐼(𝑏)=0.3,𝐹(𝑏)=0.2
• Task 𝑐:𝑇(𝑐)=0.6,𝐼(𝑐)=0.2,𝐹(𝑐)=0.2
This setup illustrates a scenario where task 𝑎has a high likelihood of success, task 𝑏is relatively uncertain,
and task 𝑐has a moderate chance of being true.
Example 2.13 (Analysis of Consumer Sentiment).”Consumer Sentiment” measures individuals’ attitudes,
condence, and feelings about economic conditions, inuencing spending behavior and market trends [59,134].
Consider a product review 𝑥. The sentiment of the review can be quantied using Neutrosophic Sets as follows:
•𝑇𝐴(𝑥)=0.6: 60% of users convey positive feedback.
•𝐼𝐴(𝑥)=0.3: 30% of users exhibit neutral or uncertain opinions.
•𝐹𝐴(𝑥)=0.1: 10% of users express negative feedback.
Neutrosophic Sets have been widely applied in sentiment analysis to handle uncertainty and partial truths in
user opinions [31,164, 190,210, 283].
Theorem 2.14. A Neutrosophic Set can generalize both Fuzzy Sets and Crisp Sets.
Proof. This follows directly from the denition, as a Neutrosophic Set encompasses the structures of Fuzzy
Sets and Crisp Sets as special cases.
5
As related concepts of Fuzzy Sets, the following are well-known: Hesitant Fuzzy Sets [72, 366,367], Picture
Fuzzy Sets [6, 6, 80, 81, 253], Bipolar Fuzzy Sets [12, 13, 18, 61, 152, 248], Hyperfuzzy set [119, 136, 182,
349], Spherical fuzzy sets [25, 199, 200, 221, 338], and Tripolar Fuzzy Sets [288–290]. Additionally, related
concepts of the Neutrosophic Set include the Bipolar Neutrosophic Set [1,239,372], Neutrosophic Soft Set [7,
18,53,188,191], Hyperneutrosophic set [119], Neutrosophic oset [115,323,324,328–330, 342] and Complex
Neutrosophic Set [16,17], among others.
Furthermore, Fuzzy and Neutrosophic concepts have been studied not only in the context of sets but also in
various elds such as Graph Theory and Algebra [8,10, 114, 116,123, 126,254]. Therefore, research on Fuzzy
and Neutrosophic frameworks is of great signicance.
2.3 Plithogenic Set: A Generalization of Uncertain Sets
The Plithogenic Set is recognized as a type of set capable of generalizing Neutrosophic Sets, Fuzzy Sets, and
other similar uncertain sets [326, 327]. The denition of the Plithogenic Set is provided below.
Denition 2.15. [326, 327] Let 𝑆be a universal set, and 𝑃⊆𝑆. A Plithogenic Set 𝑃𝑆 is dened as:
𝑃𝑆 =(𝑃, 𝑣, 𝑃𝑣, 𝑝 𝑑𝑓 , 𝑝𝐶 𝐹)
where:
•𝑣is an attribute.
•𝑃𝑣 is the range of possible values for the attribute 𝑣.
•𝑝𝑑 𝑓 :𝑃×𝑃𝑣 → [0,1]𝑠is the Degree of Appurtenance Function (DAF).
•𝑝𝐶 𝐹 :𝑃𝑣 ×𝑃𝑣 → [0,1]𝑡is the Degree of Contradiction Function (DCF).
These functions satisfy the following axioms for all 𝑎, 𝑏 ∈𝑃𝑣:
1. Reexivity of Contradiction Function:
𝑝𝐶 𝐹 (𝑎, 𝑎)=0
2. Symmetry of Contradiction Function:
𝑝𝐶 𝐹 (𝑎, 𝑏)=𝑝 𝐶𝐹 (𝑏, 𝑎)
Example 2.16. (cf. [125]) The following examples of Plithogenic sets are provided.
• When 𝑠=𝑡=1,𝑃𝑆 is called a Plithogenic Fuzzy Set.
• When 𝑠=2, 𝑡 =1,𝑃𝑆 is called a Plithogenic Intuitionistic Fuzzy Set.
• When 𝑠=3, 𝑡 =1,𝑃𝑆 is called a Plithogenic Neutrosophic Set.
• When 𝑠=4, 𝑡 =1,𝑃𝑆 is called a Plithogenic quadripartitioned Neutrosophic Set (cf. [171,287, 310]).
• When 𝑠=5, 𝑡 =1,𝑃𝑆 is called a Plithogenic pentapartitioned Neutrosophic Set (cf. [46,83, 223]).
• When 𝑠=6, 𝑡 =1,𝑃𝑆 is called a Plithogenic hexapartitioned Neutrosophic Set (cf. [265]).
• When 𝑠=7, 𝑡 =1,𝑃𝑆 is called a Plithogenic heptapartitioned Neutrosophic Set (cf. [52,249]).
• When 𝑠=8, 𝑡 =1,𝑃𝑆 is called a Plithogenic octapartitioned Neutrosophic Set.
• When 𝑠=9, 𝑡 =1,𝑃𝑆 is called a Plithogenic nonapartitioned Neutrosophic Set.
The Plithogenic Set can generalize various sets that handle uncertainty, including Neutrosophic Sets and Fuzzy
Sets [119, 326]. Several derived concepts of the Plithogenic Set have been studied [91, 124, 229, 230, 315,
352], along with its applications in graph theory and related elds [125, 128, 129,316]. Therefore, research on
Plithogenic Sets is as signicant as that on Fuzzy Sets and Neutrosophic Sets.
6
2.4 Uncertain Logic
This subsection explains Uncertain Logic. Various types of logic, such as Fuzzy Logic [247,404,409], Intuition-
istic Fuzzy Logic [27,69, 359], Neutrosophic Logic [130,319,321], Plithogenic Logic [327], and Upside-Down
Logic [127,336], have been studied under the umbrella of Uncertain Logic. Below, we introduce some of these
logics.
Denition 2.17 (Classical Logic).(cf. [75,79,101,312]) Classical Logic is a formal system of reasoning based
on binary truth values: true (1) and false (0). It operates under the principles of the law of identity, the law of
non-contradiction, and the law of excluded middle, ensuring that every proposition is either true or false, with
no intermediate states.
Denition 2.18 (Fuzzy Logic).[403] Fuzzy Logic is an extension of classical logic designed to handle rea-
soning under uncertainty and vagueness. It assigns a degree of truth to each proposition, rather than a binary
value (true or false). Formally, Fuzzy Logic is dened as a system:
F=(X, 𝜇, R ),
where:
•X: A universal set of discourse, representing all possible elements under consideration.
•𝜇:X → [0,1]: A membership function that maps each element 𝑥∈ X to a degree of truth in the interval
[0,1], where:
𝜇(𝑥)=1if 𝑥is fully true,
𝜇(𝑥)=0if 𝑥is fully false.
Intermediate values (0< 𝜇(𝑥)<1) represent partial truth.
•R: A set of fuzzy rules or relations, typically of the form:
If 𝐴is 𝑋then 𝐵is 𝑌,
where 𝐴, 𝐵 ∈ X and 𝑋 , 𝑌 are fuzzy sets dened on X.
Denition 2.19 (Neutrosophic Logic).[319] Neutrosophic Logic extends classical logic by assigning to each
proposition a truth value comprising three components:
𝑣(𝐴)=(𝑇, 𝐼, 𝐹),
where 𝑇, 𝐼, 𝐹 ∈ [0,1]represent the degrees of truth, indeterminacy, and falsity, respectively.
Remark 2.20. Fuzzy logic is a special case of Neutrosophic Logic where both indeterminacy and falsity are
set to zero. Moreover, Plithogenic Logic is known for its ability to generalize both Neutrosophic Logic and
Fuzzy Logic.
Denition 2.21 (Plithogenic Logic).[326,327] Plithogenic Logic extends classical and fuzzy logic by incor-
porating the concepts of contradiction and attribute values to model uncertainty and decision-making under
complex conditions. Formally, let 𝑆be a universal set, and 𝑃⊆𝑆. A Plithogenic Set 𝑃𝑆 is dened as:
𝑃𝑆 =(𝑃, 𝑣, 𝑃𝑣, 𝑝 𝑑𝑓 , 𝑝𝐶 𝐹),
where:
•𝑣: An attribute describing elements of 𝑃.
•𝑃𝑣: The range of possible values for the attribute 𝑣.
•𝑝𝑑 𝑓 :𝑃×𝑃𝑣 → [0,1]𝑠: The Degree of Appurtenance Function (DAF), which assigns a degree of
belonging for an element of 𝑃based on the attribute 𝑣.
7
•𝑝𝐶 𝐹 :𝑃𝑣 ×𝑃𝑣 → [0,1]𝑡: The Degree of Contradiction Function (DCF), which measures the degree of
contradiction between pairs of attribute values.
The following axioms must hold for all 𝑎, 𝑏 ∈𝑃𝑣:
1. Reexivity of Contradiction Function:
𝑝𝐶 𝐹 (𝑎, 𝑎)=0
2. Symmetry of Contradiction Function:
𝑝𝐶 𝐹 (𝑎, 𝑏)=𝑝 𝐶𝐹 (𝑏, 𝑎)
Example 2.22. (cf. [125]) The following examples of Plithogenic Logic are provided.
• When 𝑠=𝑡=1,𝑃 𝐿 is called a Plithogenic Fuzzy Logic.
• When 𝑠=2, 𝑡 =1,𝑃𝐿 is called a Plithogenic Intuitionistic Fuzzy Logic.
• When 𝑠=3, 𝑡 =1,𝑃𝐿 is called a Plithogenic Neutrosophic Logic.
• When 𝑠=4, 𝑡 =1,𝑃𝐿 is called a Plithogenic Quadripartitioned Neutrosophic Logic.
• When 𝑠=5, 𝑡 =1,𝑃𝐿 is called a Plithogenic Pentapartitioned Neutrosophic Logic.
• When 𝑠=6, 𝑡 =1,𝑃𝐿 is called a Plithogenic Hexapartitioned Neutrosophic Logic.
• When 𝑠=7, 𝑡 =1,𝑃𝐿 is called a Plithogenic Heptapartitioned Neutrosophic Logic.
• When 𝑠=8, 𝑡 =1,𝑃𝐿 is called a Plithogenic Octapartitioned Neutrosophic Logic.
• When 𝑠=9, 𝑡 =1,𝑃𝐿 is called a Plithogenic Nonapartitioned Neutrosophic Logic.
3 Result and Discussion in this Paper
This section provides a concise explanation of the mathematical denitions and properties of Neutrosophic
Phenomenology and Neutrosophic Social Science discussed in this paper.
3.1 Neutrosophic Phenomenology: Neutrosophic Body-Mind-Soul-Spirit Fluidity
Neutrosophic Body-Mind-Soul-Spirit Fluidity is a novel concept introduced in [337]. This concept extends the
traditional idea of Body-Mind-Soul-Spirit Fluidity by incorporating the principles of the Neutrosophic Set. If
we attempt to dene it mathematically, it can be expressed as follows.
Denition 3.1 (Neutrosophic Phenomenology).Neutrosophic Phenomenology is the study of phenomena and
consciousness under uncertainty, incorporating neutrosophic components of truth (𝑇), indeterminacy (𝐼), and
falsity (𝐹). It provides a framework to model subjective experiences where information is incomplete, ambigu-
ous, or contradictory.
Denition 3.2 (Components of Neutrosophic Body-Mind-Soul-Spirit Fluidity).The Neutrosophic Body-Mind-
Soul-Spirit Fluidity (NBMSSF) integrates the four fundamental aspects of human existence—Body,Mind,Soul,
and Spirit—within the neutrosophic framework. Each component is dened as follows:
1. Body: Represents the physical aspect of a person, characterized by biological processes. In neutrosophy,
the body exists not merely in health or illness but also in neutral states, reecting the dynamic balance and
transition between wellness, growth, and decay.
2. Mind: Encompasses cognitive functions like reasoning and memory. The mind, in neutrosophic terms, tran-
scends a binary rational/irrational framework, allowing for indeterminate states where beliefs and perceptions
coexist in varying degrees of clarity, ambiguity, and inuence.
8
3. Soul: Represents the essence or immaterial core of a person. In neutrosophy, the soul is not limited to good
or evil but uctuates between true identity (𝑇), uncertain beliefs (𝐼), and societal misconceptions (𝐹), reecting
the full spectrum of human emotional and spiritual experiences.
4. Spirit: Associated with transcendence and connection to the divine. Neutrosophy views the spirit as existing
in transitional states, balancing truths of divine experience (𝑇), uncertainties in belief (𝐼), and misconceptions
about spiritual practices (𝐹).
Example 3.3 (Real-Life Intuitive and Mathematically Correct Illustration of NBMSSF).Consider the case of
an individual recovering from a serious illness (cf. [85,86]), reecting the interplay of Body, Mind, Soul, and
Spirit within the Neutrosophic Body-Mind-Soul-Spirit Fluidity (NBMSSF) framework:
1. Body: The individual’s physical state uctuates between health and illness. For instance, while the immune
system is actively recovering, the body exists in a dynamic state, not fully healthy (𝑇), not completely ill (𝐹),
and in a transitional phase (𝐼) as new treatments are being adapted.
2. Mind: Cognitively, the individual experiences varying degrees of clarity and confusion. For example,
optimism about recovery (𝑇) may coexist with doubts about treatment ecacy (𝐼) or fear of relapse (𝐹), creating
a nuanced mental state.
3. Soul: Emotionally, the person may feel both gratitude for life (𝑇) and unresolved pain from the illness (𝐹),
alongside uncertainty about their spiritual purpose (𝐼). These uctuations represent the complexity of the soul
in navigating existential questions.
4. Spirit: Spiritually, the person seeks connection with the divine or higher purpose. They may experience mo-
ments of profound clarity and faith (𝑇), intermixed with uncertainties about their beliefs (𝐼), or misconceptions
about spiritual practices (𝐹), especially during challenging times.
This example illustrates the NBMSSF concept by highlighting how each component operates within neutro-
sophic parameters, oering a more comprehensive understanding of human experiences in real-life situations.
Taking the above components into consideration, Neutrosophic Body-Mind-Soul-Spirit Fluidity is dened as
follows.
Denition 3.4 (Neutrosophic Body-Mind-Soul-Spirit Fluidity).Neutrosophic Body-Mind-Soul-Spirit Fluidity
(NBMSSF) is dened as a mathematical structure consisting of four interacting components Body,Mind,Soul,
and Spirit. Each component 𝑋∈ {Body, Mind, Soul, Spirit}is characterized by the Neutrosophic Triad 𝑇(𝑋),
𝐼(𝑋),𝐹(𝑋), which satises the following conditions:
1. Neutrosophic Triad:
𝑇(𝑋) ∈ [0,1](Degree of Truth)
𝐼(𝑋) ∈ [0,1](Degree of Indeterminacy)
𝐹(𝑋) ∈ [0,1](Degree of Falsehood)
𝑇(𝑋) + 𝐼(𝑋) + 𝐹(𝑋)=1.
2. Dynamics: Each component 𝑋’s state is inuenced by the other three components 𝑌, 𝑍, 𝑊 , expressed as a
uidity function F ( 𝑋):
F (𝑋)=𝑓𝑋(𝑇(𝑌), 𝐼 (𝑌), 𝐹 (𝑌), 𝑇 (𝑍), 𝐼 (𝑍), 𝐹 (𝑍), 𝑇 (𝑊), 𝐼 (𝑊), 𝐹 (𝑊)) ,
where 𝑓𝑋is the inuence function, determined by the specic application.
3. Interdependency Model: The state of each component evolves as a system of dierential equations:
𝑑𝑇 (𝑋)
𝑑𝑡 =𝑔𝑇 ,𝑋 (𝑇 , 𝐼, 𝐹),
𝑑𝐼 (𝑋)
𝑑𝑡 =𝑔𝐼 ,𝑋 (𝑇 , 𝐼, 𝐹),
𝑑𝐹 (𝑋)
𝑑𝑡 =𝑔𝐹 ,𝑋 (𝑇 , 𝐼, 𝐹),
9
where 𝑔𝑇, 𝑋 ,𝑔𝐼 , 𝑋, and 𝑔𝐹, 𝑋 describe the rate of change for each state. The explicit forms of these functions
can include interactions between the components, such as 𝑔𝑇, 𝑋 =𝛼𝑋𝑇(𝑌) − 𝛽𝑋𝐹(𝑊), where 𝛼𝑋and 𝛽𝑋are
sensitivity coecients.
4. Global Fluidity Matrix: The overall state of the system is represented as a matrix:
S(𝑡)=
𝑇(Body)𝐼(Body)𝐹(Body)
𝑇(Mind)𝐼(Mind)𝐹(Mind)
𝑇(Soul)𝐼(Soul)𝐹(Soul)
𝑇(Spirit)𝐼(Spirit)𝐹(Spirit)
,
which evolves over time 𝑡.
5. Characteristic Function: Each component’s state transition is described by:
Φ𝑋(𝑇, 𝐼, 𝐹)=𝛼𝑋𝑇(𝑋) + 𝛽𝑋𝐼(𝑋) + 𝛾𝑋𝐹(𝑋),
where 𝛼𝑋, 𝛽𝑋, 𝛾𝑋are context-dependent weights.
6. Constraints: To ensure global balance, the following constraint holds:
Õ
𝑋∈ {Body, Mind, Soul, Spirit}
𝑇(𝑋) + 𝐼(𝑋) + 𝐹(𝑋)=4.
Remark 3.5. Fuzzy Body-Mind-Soul-Spirit Fluidity is a special case of Neutrosophic Body-Mind-Soul-Spirit
Fluidity where both indeterminacy and falsity are set to zero. Furthermore, Plithogenic Body-Mind-Soul-Spirit
Fluidity is notable for its ability to generalize both Neutrosophic and Fuzzy Body-Mind-Soul-Spirit Fluidity.
Example 3.6. Consider an individual who is generally healthy, mentally active, and emotionally balanced but
experiencing some uncertainty in spiritual matters. Their states are as follows:
•Body:𝑇(Body)=0.7,𝐼(Body)=0.2,𝐹(Body)=0.1(indicating good physical health).
•Mind:𝑇(Mind)=0.5,𝐼(Mind)=0.3,𝐹(Mind)=0.2(a mixture of clarity and indecision).
•Soul:𝑇(Soul)=0.6,𝐼(Soul)=0.2,𝐹(Soul)=0.2(emotional stability but with some conicting
emotions).
•Spirit:𝑇(Spirit)=0.4,𝐼(Spirit)=0.4,𝐹(Spirit)=0.2(reecting spiritual uncertainty).
The global uidity matrix at this moment is:
S(𝑡)=
0.7 0.2 0.1
0.5 0.3 0.2
0.6 0.2 0.2
0.4 0.4 0.2
.
Example 3.7. Consider an individual recovering from stress (cf. [350, 371]), where fatigue and indecision
dominate their state. Their characteristics are:
•Body:𝑇(Body)=0.6,𝐼(Body)=0.3,𝐹(Body)=0.1(recovering from physical exhaustion).
•Mind:𝑇(Mind)=0.4,𝐼(Mind)=0.5,𝐹(Mind)=0.1(struggling with mental clarity).
•Soul:𝑇(Soul)=0.5,𝐼(Soul)=0.4,𝐹(Soul)=0.1(seeking emotional balance).
•Spirit:𝑇(Spirit)=0.3,𝐼(Spirit)=0.5,𝐹(Spirit)=0.2(spiritually uncertain and seeking direction).
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The uidity matrix for this scenario is:
S(𝑡)=
0.6 0.3 0.1
0.4 0.5 0.1
0.5 0.4 0.1
0.3 0.5 0.2
.
This example highlights how improving one aspect, such as practicing mindfulness to reduce 𝐼(Mind), can
create cascading positive eects, improving both emotional balance (𝑇(Soul)) and spiritual clarity (𝑇(Spirit)).
Example 3.8. Suppose an individual achieves signicant spiritual clarity and emotional stability after a trans-
formative event, such as a retreat or life-changing realization. Their states are:
•Body:𝑇(Body)=0.8,𝐼(Body)=0.1,𝐹(Body)=0.1(excellent physical health).
•Mind:𝑇(Mind)=0.7,𝐼(Mind)=0.2,𝐹(Mind)=0.1(sharp mental focus).
•Soul:𝑇(Soul)=0.9,𝐼(Soul)=0.05,𝐹(Soul)=0.05 (peaceful emotional state).
•Spirit:𝑇(Spirit)=0.85,𝐼(Spirit)=0.1,𝐹(Spirit)=0.05 (strong spiritual connection).
The uidity matrix is:
S(𝑡)=
0.8 0.1 0.1
0.7 0.2 0.1
0.9 0.05 0.05
0.85 0.1 0.05
.
This scenario models a person who has realigned their physical, mental, and spiritual dimensions, leading to a
harmonious state.
The theorems that hold in Neutrosophic Body-Mind-Soul-Spirit Fluidity are presented below.
Theorem 3.9. Neutrosophic Body-Mind-Soul-Spirit Fluidity has the structure of a Neutrosophic Set.
Proof. This follows directly from the denition of Neutrosophic Body-Mind-Soul-Spirit Fluidity.
Theorem 3.10 (Invariant Triad Property).In the context of Neutrosophic Body-Mind-Soul-Spirit Fluidity, un-
der the specied dynamics, suppose the initial condition satises:
𝑇(𝑋0) + 𝐼(𝑋0) + 𝐹(𝑋0)=1at 𝑡=0.
Then for all 𝑡≥0, the following invariant holds:
𝑇(𝑋𝑡) + 𝐼(𝑋𝑡) + 𝐹(𝑋𝑡)=1.
Proof. The dynamics of the system are governed by the dierential equations for 𝑇(𝑋),𝐼(𝑋), and 𝐹(𝑋).
Summing these equations, we have:
𝑑
𝑑𝑡 𝑇(𝑋) + 𝐼(𝑋) + 𝐹(𝑋)=𝑑 𝑇 (𝑋)
𝑑𝑡 +𝑑 𝐼 (𝑋)
𝑑𝑡 +𝑑𝐹 (𝑋)
𝑑𝑡 .
By the interdependency model, the rates of change satisfy:
𝑑𝑇 (𝑋)
𝑑𝑡 +𝑑 𝐼 (𝑋)
𝑑𝑡 +𝑑𝐹 (𝑋)
𝑑𝑡 =𝑔𝑇 , 𝑋 +𝑔𝐼 ,𝑋 +𝑔𝐹 , 𝑋 .
From the system denition, 𝑔𝑇, 𝑋 +𝑔𝐼 ,𝑋 +𝑔𝐹 , 𝑋 =0. Thus:
𝑑
𝑑𝑡 𝑇(𝑋) + 𝐼(𝑋) + 𝐹(𝑋)=0.
Integrating over time, the sum 𝑇(𝑋) + 𝐼(𝑋) + 𝐹(𝑋)remains constant, and given the initial condition 𝑇(𝑋0) +
𝐼(𝑋0) + 𝐹(𝑋0)=1, the result follows:
𝑇(𝑋𝑡) + 𝐼(𝑋𝑡) + 𝐹(𝑋𝑡)=1for all 𝑡≥0.
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Theorem 3.11 (Non-Negativity and Boundedness).In the context of Neutrosophic Body-Mind-Soul-Spirit Flu-
idity, assume the initial condition:
𝑇(𝑋0), 𝐼 (𝑋0), 𝐹 (𝑋0) ∈ [0,1].
Then for any 𝑡≥0, the following holds:
𝑇(𝑋𝑡), 𝐼 (𝑋𝑡), 𝐹 (𝑋𝑡) ∈ [0,1].
Proof. The invariant property (Theorem 3.10) ensures that the sum 𝑇(𝑋𝑡) + 𝐼(𝑋𝑡) + 𝐹(𝑋𝑡)=1holds for all
𝑡≥0. Assume by contradiction that one of the components, say 𝑇(𝑋𝑡), leaves the interval [0,1].
If 𝑇(𝑋𝑡)>1, then 𝐼(𝑋𝑡) + 𝐹(𝑋𝑡)<0, which violates non-negativity. Similarly, if 𝑇(𝑋𝑡)<0, then 𝐼(𝑋𝑡) +
𝐹(𝑋𝑡)>1, which is also impossible.
Using standard comparison theorems for dierential equations and ensuring non-negativity through Grönwall’s
inequality, the components 𝑇(𝑋𝑡),𝐼(𝑋𝑡), and 𝐹(𝑋𝑡)are bounded within [0,1]. Hence:
𝑇(𝑋𝑡), 𝐼 (𝑋𝑡), 𝐹 (𝑋𝑡) ∈ [0,1]for all 𝑡≥0.
Theorem 3.12 (Global Balance Constraint).In the context of Neutrosophic Body-Mind-Soul-Spirit Fluidity,
let the Global Fluidity Matrix at time 𝑡be:
S(𝑡)=
𝑇(Body𝑡)𝐼(Body𝑡)𝐹(Body𝑡)
𝑇(Mind𝑡)𝐼(Mind𝑡)𝐹(Mind𝑡)
𝑇(Soul𝑡)𝐼(Soul𝑡)𝐹(Soul𝑡)
𝑇(Spirit𝑡)𝐼(Spirit𝑡)𝐹(Spirit𝑡)
.
Then the total balance constraint holds:
Õ
𝑋∈ {Body,Mind,Soul ,Spirit}𝑇(𝑋𝑡) + 𝐼(𝑋𝑡) + 𝐹(𝑋𝑡)=4for all 𝑡≥0.
Proof. From Theorem 3.10, each component 𝑋satises 𝑇(𝑋𝑡) + 𝐼(𝑋𝑡) + 𝐹(𝑋𝑡)=1for all 𝑡≥0. Summing
over all components: Õ
𝑋∈ {Body,Mind,Soul,Spirit}𝑇(𝑋𝑡) + 𝐼(𝑋𝑡) + 𝐹(𝑋𝑡)=4·1=4.
This holds for all 𝑡≥0, completing the proof.
Based on the discussion above, we redene Dynamic Neutrosophic Body-Mind-Soul-Spirit Fluidity. This
model allows the observation and analysis of changes in the Body, Mind, Soul, and Spirit over time. The
formal denitions and associated properties are presented below.
Denition 3.13 (Dynamic Neutrosophic Body-Mind-Soul-Spirit Fluidity).Dynamic Neutrosophic Body-Mind-
Soul-Spirit Fluidity (Dynamic NBMSSF) extends the static NBMSSF framework by incorporating time-dependent
changes and interactions among its four components: Body,Mind,Soul, and Spirit. Each component 𝑋∈
{Body,Mind,Soul,Spirit}evolves over time according to the following properties:
•Neutrosophic Triad Dynamics: For each component 𝑋, the Truth 𝑇(𝑋𝑡), Indeterminacy 𝐼(𝑋𝑡), and
Falsity 𝐹(𝑋𝑡)values vary with time 𝑡and satisfy:
𝑇(𝑋𝑡), 𝐼 (𝑋𝑡), 𝐹 (𝑋𝑡) ∈ [0,1]and 𝑇(𝑋𝑡) + 𝐼(𝑋𝑡) + 𝐹(𝑋𝑡)=1∀𝑡≥0.
•Inuence Function: Each component 𝑋is inuenced by the other three components 𝑌, 𝑍 , 𝑊 through a
uidity function F ( 𝑋𝑡):
F (𝑋𝑡)=𝑓𝑋𝑇(𝑌𝑡), 𝐼 (𝑌𝑡), 𝐹 (𝑌𝑡), 𝑇 (𝑍𝑡), 𝐼 (𝑍𝑡), 𝐹 (𝑍𝑡), 𝑇 (𝑊𝑡), 𝐼 (𝑊𝑡), 𝐹 (𝑊𝑡),
where 𝑓𝑋is an application-specic function describing how other components aect 𝑋.
12
•Time Evolution Equations: The temporal behavior of each component is modeled by a system of dier-
ential equations:
𝑑𝑇 (𝑋)
𝑑𝑡 =𝑔𝑇 ,𝑋 (𝑇 , 𝐼, 𝐹 , 𝑡),
𝑑𝐼 (𝑋)
𝑑𝑡 =𝑔𝐼 ,𝑋 (𝑇 , 𝐼, 𝐹 , 𝑡),
𝑑𝐹 (𝑋)
𝑑𝑡 =𝑔𝐹 ,𝑋 (𝑇 , 𝐼, 𝐹 , 𝑡),
where 𝑔𝑇, 𝑋 , 𝑔𝐼, 𝑋 , 𝑔𝐹 , 𝑋 capture the rates of change for Truth, Indeterminacy, and Falsity, potentially de-
pending on all components and external factors.
•Global Dynamics Matrix: The overall system state at time 𝑡is represented by the matrix:
S(𝑡)=
𝑇(Body𝑡)𝐼(Body𝑡)𝐹(Body𝑡)
𝑇(Mind𝑡)𝐼(Mind𝑡)𝐹(Mind𝑡)
𝑇(Soul𝑡)𝐼(Soul𝑡)𝐹(Soul𝑡)
𝑇(Spirit𝑡)𝐼(Spirit𝑡)𝐹(Spirit𝑡)
.
This matrix evolves over time according to the system’s dynamics.
•Invariant Properties and Constraints:
–Invariant Triad Property: For each component 𝑋,𝑇(𝑋𝑡) + 𝐼(𝑋𝑡) + 𝐹(𝑋𝑡)=1remains true for all
𝑡≥0.
–Global Balance Constraint: Summing over all four components at any time 𝑡yields
Õ
𝑋∈ {Body,Mind,Soul,Spirit}𝑇(𝑋𝑡) + 𝐼(𝑋𝑡) + 𝐹(𝑋𝑡)=4.
•Characteristic Dynamics Function: Each component’s combined state can be expressed by a character-
istic function:
Φ𝑋𝑇, 𝐼 , 𝐹, 𝑡 =𝛼𝑋𝑇(𝑋𝑡) + 𝛽𝑋𝐼(𝑋𝑡) + 𝛾𝑋𝐹(𝑋𝑡),
where 𝛼𝑋, 𝛽𝑋, 𝛾𝑋are context-dependent parameters indicating the relative signicance of each dimen-
sion.
Example 3.14 (Rehabilitation Scenario).Consider an individual undergoing rehabilitation for a sports injury
(cf. [201]):
•Body:𝑇(Body𝑡)represents the probability of full physical recovery, 𝐼(Body𝑡)indicates uncertainty dur-
ing the healing process, and 𝐹(Body𝑡)accounts for residual impairment or setbacks.
•Mind:𝑇(Mind𝑡)measures mental clarity and optimism, 𝐼(Mind𝑡)captures confusion or doubts, and
𝐹(Mind𝑡)reects negative beliefs about the rehabilitation process.
•Soul:𝑇(Soul𝑡)represents personal resilience or spiritual harmony, 𝐼(Soul𝑡)signies existential uncer-
tainty, and 𝐹(Soul𝑡)might correspond to cultural misconceptions or conicts.
•Spirit:𝑇(Spirit𝑡)denotes moments of profound insight or faith, 𝐼(Spirit𝑡)covers spiritual ambiguity,
and 𝐹(Spirit𝑡)indicates doubts or misunderstandings about spiritual practices.
As rehabilitation progresses over time 𝑡, each triad 𝑇(𝑋𝑡), 𝐼 (𝑋𝑡), 𝐹 (𝑋𝑡)evolves dynamically based on the
individual’s physical therapy, mental training, emotional support, and spiritual practices. The Invariant Triad
Property ensures 𝑇(𝑋𝑡) + 𝐼(𝑋𝑡) + 𝐹(𝑋𝑡)=1for each component, while the Global Balance Constraint enforces
the total sum to remain 4 at any time 𝑡.
Theorem 3.15. Dynamic Neutrosophic Body-Mind-Soul-Spirit Fluidity possesses the structure of a Neutro-
sophic Set.
13
Proof. This result follows directly from the denition of Neutrosophic Body-Mind-Soul-Spirit Fluidity, as each
component (Body,Mind,Soul, and Spirit) is represented using the Neutrosophic Triad (𝑇 , 𝐼, 𝐹), which satises
the axioms of a Neutrosophic Set.
Theorem 3.16. Dynamic Neutrosophic Body-Mind-Soul-Spirit Fluidity can be transformed into Neutrosophic
Body-Mind-Soul-Spirit Fluidity by omitting temporal dependencies.
Proof. This follows from the denition of Dynamic Neutrosophic Body-Mind-Soul-Spirit Fluidity. By setting
the time-dependent functions 𝑇(𝑋𝑡), 𝐼 (𝑋𝑡), 𝐹 (𝑋𝑡)to their initial values at 𝑡=0, the model reduces to the static
form of Neutrosophic Body-Mind-Soul-Spirit Fluidity.
Question 3.17. Related concepts such as Holistic Well-Being [410], Embodied Cognition [308, 397], Mind-
fulness and Meditation Practices [82,418], and Psychoneuroimmunology [234,245] are well-known.
Is it possible to extend these concepts using Fuzzy Sets and Neutrosophic Sets? Furthermore, what would their
applications and mathematical structures entail?
3.2 Logic of Phenomenology
There is a deep connection between phenomenology and logic, and several logical systems have been studied in
this context. This subsection explores the logic within phenomenology, including considerations of its potential
extension to Neutrosophic Logic.
3.2.1 Neutrosophic Intentional Logic
Intentional concepts in phenomenology describe how consciousness always aims at or is directed toward ob-
jects, revealing the relationship between subject and object in experience (cf. [65,89, 374,375,413]). Intentional
Logic studies the structure of intentionality, analyzing how mental states are directed toward objects, contents,
or propositions systematically (cf. [343,383]).
Denition 3.18 (Intentional Logic).Intentional Logic formalizes the structure of intentionality, dened as the
directedness of mental states toward objects or contents. Let:
•𝑊: the set of all possible worlds.
•𝑆: the set of subjects (agents).
•𝑂: the set of objects (including abstract entities).
•B={0,1}: the Boolean domain indicating intentional states.
The intentionality of a subject 𝑠∈𝑆toward an object 𝑜∈𝑂in a world 𝑤∈𝑊is modeled as a relation:
𝐼:𝑆×𝑂×𝑊→B,
where 𝐼(𝑠, 𝑜, 𝑤)=1indicates that 𝑠intentionally directs their mental state toward 𝑜in 𝑤.
Intentional Content. The intentional content of a subject 𝑠is dened as:
I
𝑠={( 𝑜, 𝑤) | 𝐼(𝑠, 𝑜, 𝑤)=1}.
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Axioms. Intentional Logic satises the following properties:
1. Existence: For all 𝑠∈𝑆, there exists at least one 𝑜∈𝑂and 𝑤∈𝑊such that 𝐼(𝑠, 𝑜, 𝑤)=1.
2. Consistency: For any 𝑠∈𝑆, if 𝐼(𝑠, 𝑜1, 𝑤 )=1and 𝐼(𝑠, 𝑜2, 𝑤)=1, then 𝑜1=𝑜2(if exclusivity is
assumed).
3. Higher-Order Intentionality: If 𝑜is an intentional state itself, then 𝑜∈ P (𝑆×𝑂), allowing for recursive
representation of intentions.
Denition 3.19 (Neutrosophic Intentional Logic).Neutrosophic Intentional Logic extends classical Intentional
Logic by incorporating the neutrosophic components of truth (𝑇), indeterminacy (𝐼), and falsity (𝐹). Let:
•𝑊: the set of all possible worlds.
•𝑆: the set of subjects (agents).
•𝑂: the set of objects (including abstract entities).
•N=[0,1]3: the neutrosophic domain, where each component (𝑇 , 𝐼, 𝐹 )satises 0≤𝑇+𝐼+𝐹≤1.
The intentionality of a subject 𝑠∈𝑆toward an object 𝑜∈𝑂in a world 𝑤∈𝑊is modeled as:
𝐼𝑁:𝑆×𝑂×𝑊→N,
where 𝐼𝑁(𝑠, 𝑜, 𝑤)=(𝑇 , 𝐼 , 𝐹)indicates the degrees of truth (𝑇), indeterminacy (𝐼), and falsity (𝐹) of 𝑠’s
intentional state toward 𝑜in 𝑤.
Neutrosophic Intentional Content. The neutrosophic intentional content of a subject 𝑠is dened as:
I𝑁
𝑠={( 𝑜, 𝑤, (𝑇, 𝐼, 𝐹)) | 𝐼𝑁(𝑠, 𝑜, 𝑤)=(𝑇, 𝐼, 𝐹)}.
Axioms. Neutrosophic Intentional Logic satises the following properties:
1. Existence: For all 𝑠∈𝑆, there exists at least one 𝑜∈𝑂and 𝑤∈𝑊such that 𝐼𝑁(𝑠, 𝑜, 𝑤)=(𝑇 , 𝐼, 𝐹)with
𝑇 > 0.
2. Consistency: For any 𝑠∈𝑆, if 𝐼𝑁(𝑠, 𝑜1, 𝑤)=(𝑇1, 𝐼1, 𝐹1)and 𝐼𝑁(𝑠, 𝑜2, 𝑤)=(𝑇2, 𝐼2, 𝐹2), then 𝑜1=𝑜2if
𝑇1+𝑇2=1and 𝐼1=𝐼2=0.
3. Higher-Order Neutrosophic Intentionality: If 𝑜is an intentional state, then 𝑜∈ P (𝑆×𝑂×N), allowing
recursive representation of neutrosophic intentionality.
Remark 3.20 (Neutrosophic Intentional Logic).Fuzzy Intentional Logic is a special case of Neutrosophic
Intentional Logic where both indeterminacy and falsity are set to zero. Furthermore, Plithogenic Intentional
Logic is notable for its ability to generalize both Neutrosophic and Fuzzy Intentional Logic.
Example 3.21 (Neutrosophic Intentional Logic).Consider an agent 𝑠thinking about the proposition 𝑜: ”The
market will grow by 10% next year” in the world 𝑤1. The intentionality is modeled as:
𝐼𝑁(𝑠, 𝑜, 𝑤1)=(𝑇 , 𝐼 , 𝐹),
where 𝑇=0.6,𝐼=0.3, and 𝐹=0.1. This means:
• The agent believes the proposition is 60% true (𝑇=0.6).
• There is a 30% level of uncertainty or indeterminacy due to insucient data (𝐼=0.3).
• The agent believes the proposition is 10% false (𝐹=0.1).
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Higher-Order Intentionality. If the agent 𝑠also contemplates their own belief about 𝑜, this is represented
as:
𝐼𝑁(𝑠, 𝐼 𝑁(𝑠, 𝑜, 𝑤1), 𝑤2)=(𝑇0, 𝐼 0, 𝐹 0),
where 𝑤2is a meta-level world reecting the agent’s introspection.
Visualization of Content. The neutrosophic intentional content of 𝑠is:
I𝑁
𝑠={( 𝑜, 𝑤1,(0.6,0.3,0.1))}.
This captures the agent’s nuanced and uncertain attitude toward the proposition 𝑜in 𝑤1.
3.2.2 Neutrosophic Ontological Logic
Ontology is the study of existence and reality, exploring entities, their properties, relationships, and categories
[99, 149, 151, 160, 344]. Ontology is often studied in connection with phenomenology [261]. Concepts like
Ontological Logic [262,302] are also recognized within ontology.
To dene this within the framework of Neutrosophic Logic, we rst mathematically dene Ontological Logic
and then extend it. The denition is provided below.
Denition 3.22 (Ontological Logic).Ontological Logic formalizes the relationships, properties, and existence
of entities. Let:
•𝑈: the universe of discourse, partitioned into:
𝑈=𝐸∪𝑃∪𝑅∪𝑇,
where 𝐸: entities, 𝑃: properties, 𝑅: relations, and 𝑇: time.
•𝜎:𝑃×𝐸×𝑇→B: a function assigning truth values to properties of entities at specic times.
•𝑅:𝐸×𝐸→B: a function dening binary relations between entities.
The ontological structure is dened as a tuple:
O=(𝐸 , 𝑃, 𝑅, 𝑇 , 𝜎).
Core Axioms. Ontological Logic satises the following axioms:
1. Identity: For every entity 𝑒∈𝐸, there exists at least one property 𝑝∈𝑃and time 𝑡∈𝑇such that
𝜎(𝑝, 𝑒, 𝑡)=1.
2. Non-Contradiction: For any 𝑒∈𝐸, 𝑝 ∈𝑃, 𝑡 ∈𝑇,𝜎(𝑝, 𝑒, 𝑡)=1implies 𝜎(¬𝑝, 𝑒, 𝑡)=0.
3. Temporal Consistency: For persistent properties 𝑝∈𝑃, if 𝜎(𝑝 , 𝑒, 𝑡1)=1, then 𝜎(𝑝, 𝑒 , 𝑡2)=1for all
𝑡2≥𝑡1.
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Mereological Relations. Part-whole relationships are formalized as:
𝑃𝑊⊆𝐸×𝐸,
where (𝑒1, 𝑒2) ∈ 𝑃𝑊indicates that 𝑒1is a part of 𝑒2. The following properties hold:
•Transitivity: (𝑒1, 𝑒2),(𝑒2, 𝑒3) ∈ 𝑃𝑊=⇒ (𝑒1, 𝑒3) ∈ 𝑃𝑊.
•Antisymmetry: (𝑒1, 𝑒2) ∈ 𝑃𝑊∧ (𝑒2, 𝑒1) ∈ 𝑃𝑊=⇒𝑒1=𝑒2.
Example 3.23 (Ontological Logic in Healthcare System).Consider a healthcare system (cf. [44, 416]) where
entities, properties, relations, and time are formalized as follows:
•𝐸={Patient,Doctor,Medication,Treatment Plan}: A set of entities.
•𝑃={isHealthy,isPrescribed,isAdministered,isEective}: A set of properties.
•𝑅={treats,prescribes,monitors}: A set of relations between entities.
•𝑇={Day 1,Day 2, . . . , Day 30}: A set of time points.
Property Assignment. The property function 𝜎assigns truth values to properties of entities over time:
𝜎(isPrescribed,Medication, 𝑡)=(1if the medication is prescribed at time 𝑡,
0otherwise.
Relations. The relation function 𝑅formalizes interactions between entities. For example:
𝑅(Doctor,Patient)=treats, 𝑅(Doctor,Medication)=prescribes.
Core Axioms in Context. The core axioms of Ontological Logic can be applied to this healthcare example:
•Identity: Every patient 𝑒∈𝐸must have at least one property 𝑝∈𝑃at a specic time 𝑡:
∃𝑝∈𝑃, 𝑡 ∈𝑇s.t. 𝜎(𝑝, Patient, 𝑡)=1.
Example: 𝜎(isHealthy,Patient,Day 10)=1.
•Non-Contradiction: A medication cannot simultaneously be prescribed and not prescribed at the same
time:
𝜎(isPrescribed,Medication, 𝑡)=1=⇒𝜎(¬isPrescribed,Medication, 𝑡)=0.
•Temporal Consistency: If a treatment plan is eective on Day 5, it must remain eective for subsequent
days unless modied:
𝜎(isEective,Treatment Plan,Day 5)=1=⇒𝜎(isEective,Treatment Plan, 𝑡)=1∀𝑡≥Day 5.
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Mereological Relations. Part-whole relationships in the healthcare system are dened as follows:
𝑃𝑊={(Medication,Treatment Plan)}.
Here, medication 𝑒1is a part of the treatment plan 𝑒2. The transitivity and antisymmetry properties hold:
•Transitivity: If Medication A is part of Treatment Plan X, and Treatment Plan X is part of Healthcare
Protocol Y, then Medication A is part of Healthcare Protocol Y.
(Medication A,Treatment Plan X) ∈ 𝑃𝑊∧ (Treatment Plan X,Healthcare Protocol Y) ∈ 𝑃𝑊
=⇒ (Medication A,Healthcare Protocol Y) ∈ 𝑃𝑊.
•Antisymmetry: If Medication A is part of Treatment Plan X and vice versa, then Medication A and
Treatment Plan X are identical:
(Medication A,Treatment Plan X) ∈ 𝑃𝑊∧ (Treatment Plan X,Medication A) ∈ 𝑃𝑊
=⇒Medication A =Treatment Plan X.
The denition of Neutrosophic Ontological Logic, which incorporates the principles of Neutrosophic Logic
into Ontological Logic, is provided below.
Denition 3.24 (Neutrosophic Ontological Logic).Neutrosophic Ontological Logic extends classical Onto-
logical Logic by incorporating the neutrosophic components of truth (𝑇), indeterminacy (𝐼), and falsity (𝐹).
Let:
•𝑈: the universe of discourse, partitioned as:
𝑈=𝐸∪𝑃∪𝑅∪𝑇,
where 𝐸: entities, 𝑃: properties, 𝑅: relations, and 𝑇: time.
•N=[0,1]3: the neutrosophic domain, where (𝑇, 𝐼, 𝐹)satises 0≤𝑇+𝐼+𝐹≤1.
•𝜎𝑁:𝑃×𝐸×𝑇→N: a function assigning neutrosophic truth values to properties of entities at specic
times.
•𝑅𝑁:𝐸×𝐸×𝑇→N: a function assigning neutrosophic truth values to binary relations between entities.
The neutrosophic ontological structure is dened as a tuple:
O𝑁=(𝐸 , 𝑃, 𝑅, 𝑇 , 𝜎 𝑁, 𝑅𝑁).
Core Axioms. Neutrosophic Ontological Logic satises the following axioms:
1. Neutrosophic Identity: For every entity 𝑒∈𝐸, there exists at least one property 𝑝∈𝑃and time 𝑡∈𝑇
such that:
𝜎𝑁(𝑝, 𝑒, 𝑡)=(𝑇 , 𝐼, 𝐹),where 𝑇 > 0.
2. Neutrosophic Non-Contradiction: For any 𝑒∈𝐸, 𝑝 ∈𝑃, 𝑡 ∈𝑇,𝜎𝑁(𝑝 , 𝑒, 𝑡)=(𝑇 , 𝐼, 𝐹)implies that
𝜎𝑁(¬𝑝, 𝑒, 𝑡)=(𝐹, 𝐼 , 𝑇).
3. Neutrosophic Temporal Consistency: For persistent properties 𝑝∈𝑃, if 𝜎𝑁(𝑝, 𝑒, 𝑡1)=(𝑇1, 𝐼1, 𝐹1)and
𝑡2≥𝑡1, then:
𝜎𝑁(𝑝, 𝑒, 𝑡2)=(𝑇2, 𝐼2, 𝐹2),where 𝑇2≤𝑇1and 𝐹2≥𝐹1.
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4. Neutrosophic Mereological Relations: Part-whole relationships 𝑃𝑊⊆𝐸×𝐸are assigned neutrosophic
truth values:
𝑅𝑁(𝑒1, 𝑒2, 𝑡)=(𝑇, 𝐼 , 𝐹),
where (𝑇, 𝐼, 𝐹)represents the degree to which 𝑒1is a part of 𝑒2at time 𝑡.
Remark 3.25 (Neutrosophic Ontological Logic).Fuzzy Ontological Logic is a special case of Neutrosophic
Ontological Logic where both indeterminacy and falsity are set to zero. Furthermore, Plithogenic Ontological
Logic is notable for its ability to generalize both Neutrosophic and Fuzzy Ontological Logic.
Example 3.26 (Neutrosophic Ontological Logic for healthcare system).Consider a healthcare system modeled
as O𝑁with:
•𝑒1: a hospital.
•𝑒2: a healthcare network to which the hospital belongs.
•𝑝: the property ”provides emergency services.”
•𝑡: the current time.
Property Evaluation. The property 𝑝for the entity 𝑒1at 𝑡is evaluated as:
𝜎𝑁(𝑝, 𝑒1, 𝑡 )=(0.8,0.1,0.1),
indicating that:
• There is an 80% certainty (𝑇=0.8) that the hospital provides emergency services.
• There is a 10% uncertainty (𝐼=0.1) due to incomplete data.
• There is a 10% falsity (𝐹=0.1) based on occasional service disruptions.
Mereological Relation. The hospital’s membership in the healthcare network is represented as:
𝑅𝑁(𝑒1, 𝑒2, 𝑡)=(0.9,0.05,0.05),
indicating a 90% certainty (𝑇=0.9) that the hospital is part of the network, with 5% uncertainty (𝐼=0.05)
and 5% falsity (𝐹=0.05) due to occasional administrative errors.
Temporal Consistency. If 𝑝represents ”provides emergency services,” and at a later time 𝑡0> 𝑡, the hospi-
tal’s performance declines, the evaluation might adjust to:
𝜎𝑁(𝑝, 𝑒1, 𝑡 0)=(0.6,0.2,0.2).
This reects reduced certainty (𝑇=0.6) and increased falsity (𝐹=0.2) due to degraded service.
3.3 Neutrosophic Social Analysis
This subsection provides a mathematical denition of the PDCA (Plan-Do-Check-Act), DMAIC (Dene-Measure-
Analyze-Improve-Control), SWOTAnalysis (Strengths, Weaknesses, Opportunities, Threats), and OODA Loop
(Observe, Orient, Decide, Act) cycles using the concept of Neutrosophic Sets, incorporating truth, indetermi-
nacy, and falsehood degrees for decision-making under uncertainty.
First, as a broad perspective, Neutrosophic Social Analysis is roughly dened as follows. It extends Social
Analysis by incorporating the principles of Neutrosophic Logic.
Denition 3.27 (Neutrosophic Social Analysis).Neutrosophic Social Analysis is the evaluation of social sys-
tems, behaviors, and relationships under uncertainty. It incorporates neutrosophic components of truth (𝑇),
indeterminacy (𝐼), and falsity (𝐹) to model complex, ambiguous, or conicting social dynamics.
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3.3.1 PDCA Cycle with Neutrosophic Sets
The PDCA (Plan-Do-Check-Act) cycle is a continuous improvement framework consisting of four stages: plan-
ning, implementing, evaluating results, and rening processes [133,173, 256,291]. These four stages are ex-
tended within the framework of Neutrosophic Sets as follows. It is worth noting that several studies have
explored the application of the PDCA cycle in the Fuzzy domain and the Neutrosophic domain [24,137, 392].
Denition 3.28 (Neutrosophic PDCA cycle).The Neutrosophic PDCA cycle is an extension of the traditional
Plan-Do-Check-Act (PDCA) cycle, incorporating Neutrosophic Sets to model uncertainty, indeterminacy, and
truth. The cycle consists of four stages:
1. Plan (P): Represented by a Neutrosophic Set 𝑃:
𝑃={(𝑥 , 𝑇𝑃(𝑥), 𝐼𝑃(𝑥), 𝐹𝑃(𝑥)) | 𝑥∈Planning Elements},
where:
•𝑇𝑃(𝑥): Degree to which the plan is expected to succeed.
•𝐼𝑃(𝑥): Degree of uncertainty associated with the plan.
•𝐹𝑃(𝑥): Degree to which the plan is expected to fail.
2. Do (D): Represented by a Neutrosophic Set 𝐷:
𝐷={( 𝑦 , 𝑇𝐷(𝑦), 𝐼𝐷(𝑦), 𝐹𝐷(𝑦)) | 𝑦∈Execution Elements},
where:
•𝑇𝐷(𝑦): Degree to which the execution is successful.
•𝐼𝐷(𝑦): Degree of uncertainty during execution.
•𝐹𝐷(𝑦): Degree to which the execution is unsuccessful.
3. Check (C): Represented by a Neutrosophic Set 𝐶:
𝐶={( 𝑧 , 𝑇𝐶(𝑧), 𝐼𝐶(𝑧), 𝐹𝐶(𝑧)) | 𝑧∈Evaluation Criteria},
where:
•𝑇𝐶(𝑧): Degree to which evaluation criteria are met.
•𝐼𝐶(𝑧): Degree of uncertainty in the evaluation process.
•𝐹𝐶(𝑧): Degree to which evaluation criteria are not met.
4. Act (A): Represented by a Neutrosophic Set 𝐴:
𝐴={( 𝑤 , 𝑇𝐴(𝑤), 𝐼𝐴(𝑤), 𝐹𝐴(𝑤)) | 𝑤∈Improvement Elements},
where:
•𝑇𝐴(𝑤): Degree to which the improvement is eective.
•𝐼𝐴(𝑤): Degree of uncertainty in the improvement’s impact.
•𝐹𝐴(𝑤): Degree to which the improvement is ineective.
Example 3.29. Consider applying the Neutrosophic PDCA cycle to a marketing campaign (cf. [250, 279]):
•Plan (P): Tasks such as ”Develop Ad Content” and ”Set Budget” might have the following values:
–Develop Ad Content: 𝑇𝑃=0.7,𝐼𝑃=0.2,𝐹𝑃=0.1
–Set Budget: 𝑇𝑃=0.6,𝐼𝑃=0.3,𝐹𝑃=0.1
•Do (D): Execution tasks such as ”Run Ad Campaign” and ”Monitor Metrics”:
20
–Run Ad Campaign: 𝑇𝐷=0.8,𝐼𝐷=0.1,𝐹𝐷=0.1
–Monitor Metrics: 𝑇𝐷=0.6,𝐼𝐷=0.3,𝐹𝐷=0.1
•Check (C): Evaluation criteria such as ”ROI Improvement [206]” and ”Engagement Increase [157]”:
–ROI Improvement: 𝑇𝐶=0.7,𝐼𝐶=0.2,𝐹𝐶=0.1
–Engagement Increase: 𝑇𝐶=0.5,𝐼𝐶=0.4,𝐹𝐶=0.1
•Act (A): Improvement actions such as ”Adjust Budget” and ”Redesign Ad Content”:
–Adjust Budget: 𝑇𝐴=0.6,𝐼𝐴=0.3,𝐹𝐴=0.1
–Redesign Ad Content: 𝑇𝐴=0.8,𝐼𝐴=0.1,𝐹𝐴=0.1
This demonstrates how the Neutrosophic PDCA cycle integrates uncertainty and truth degrees into planning,
execution, evaluation, and improvement stages.
Theorem 3.30. Neutrosophic PDCA cycle has the structure of a Neutrosophic Set.
Proof. This follows directly from the denition of Neutrosophic PDCA cycle.
Question 3.31. Numerous derived concepts of PDCA, such as the PDSA Cycle (Plan-Do-Study-Act) [66,102,
205,292], OPDCA Cycle (Observe-Plan-Do-Check-Act) [179,351], and SDCA Cycle (Standardize-Do-Check-
Act) [22,104, 211], are widely known.
What characteristics emerge when concepts like Neutrosophic Sets are applied to these derived cycles? Fur-
thermore, what potential applications could result from such adaptations?
3.3.2 DMAIC Cycle with Neutrosophic Sets
The DMAIC Cycle is a Six Sigma methodology [231,263] designed for process improvement [242]. It consists
of ve phases: Dene, Measure, Analyze, Improve, and Control, aiming to optimize processes systemati-
cally [224, 242, 285,286, 300, 356]. This framework is widely utilized in business management and has also
been explored in Fuzzy and Neutrosophic contexts [141, 143, 402]. The following outlines an extension of the
DMAIC Cycle using Neutrosophic Sets.
Denition 3.32 (Neutrosophic DMAIC cycle).The Neutrosophic DMAIC cycle is an extension of the tradi-
tional Dene-Measure-Analyze-Improve-Control (DMAIC) cycle, incorporating Neutrosophic Sets to model
uncertainty, indeterminacy, and truth. The cycle consists of ve stages:
1. D ene (D): Represented by a Neutrosophic Set 𝐷𝑓:
𝐷𝑓={(𝑥 , 𝑇𝐷𝑓(𝑥), 𝐼𝐷𝑓(𝑥), 𝐹𝐷𝑓(𝑥)) | 𝑥∈Denition Elements},
where:
•𝑇𝐷𝑓(𝑥): Degree to which the denition is accurate.
•𝐼𝐷𝑓(𝑥): Degree of uncertainty in the denition.
•𝐹𝐷𝑓(𝑥): Degree to which the denition is inaccurate.
2. Measure (M): Represented by a Neutrosophic Set 𝑀:
𝑀={( 𝑦 , 𝑇𝑀(𝑦), 𝐼𝑀(𝑦), 𝐹𝑀(𝑦)) | 𝑦∈Measurement Elements},
where:
•𝑇𝑀(𝑦): Degree of reliability of the measurement.
•𝐼𝑀(𝑦): Degree of uncertainty in the measurement process.
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•𝐹𝑀(𝑦): Degree to which the measurement is unreliable.
3. Analyze (A): Represented by a Neutrosophic Set 𝐴𝑛:
𝐴𝑛={( 𝑧 , 𝑇𝐴𝑛(𝑧), 𝐼𝐴𝑛(𝑧), 𝐹𝐴𝑛(𝑧) ) | 𝑧∈Analysis Elements},
where:
•𝑇𝐴𝑛(𝑧): Degree to which the analysis results are correct.
•𝐼𝐴𝑛(𝑧): Degree of uncertainty in the analysis.
•𝐹𝐴𝑛(𝑧): Degree to which the analysis results are incorrect.
4. Improve (I): Represented by a Neutrosophic Set 𝐼𝑚:
𝐼𝑚={( 𝑤 , 𝑇𝐼𝑚(𝑤), 𝐼𝐼𝑚(𝑤), 𝐹𝐼𝑚(𝑤)) | 𝑤∈Improvement Actions},
where:
•𝑇𝐼𝑚(𝑤): Degree to which the improvement is successful.
•𝐼𝐼𝑚(𝑤): Degree of uncertainty about the improvement’s eectiveness.
•𝐹𝐼𝑚(𝑤): Degree to which the improvement fails.
5. Control (C): Represented by a Neutrosophic Set 𝐶𝑡:
𝐶𝑡={( 𝑣 , 𝑇𝐶𝑡(𝑣), 𝐼𝐶𝑡(𝑣), 𝐹𝐶𝑡(𝑣)) | 𝑣∈Control Elements},
where:
•𝑇𝐶𝑡(𝑣): Degree to which control is eective.
•𝐼𝐶𝑡(𝑣): Degree of uncertainty in the control process.
•𝐹𝐶𝑡(𝑣): Degree to which control is ineective.
Example 3.33. A production process is a sequence of operations transforming raw materials into nished
products eciently [197, 313]. Consider applying the Neutrosophic DMAIC cycle to improve a production
process:
•Dene (D): Tasks such as ”Identify Core Needs” and ”Set Goals”:
–Identify Core Needs: 𝑇𝐷𝑓=0.8,𝐼𝐷𝑓=0.1,𝐹𝐷𝑓=0.1
–Set Goals: 𝑇𝐷𝑓=0.7,𝐼𝐷𝑓=0.2,𝐹𝐷𝑓=0.1
•Measure (M): Measuring performance metrics like ”Production Eciency” and ”Customer Satisfaction”:
–Production Eciency: 𝑇𝑀=0.9,𝐼𝑀=0.05,𝐹𝑀=0.05
–Customer Satisfaction: 𝑇𝑀=0.7,𝐼𝑀=0.2,𝐹𝑀=0.1
•Analyze (A): Analyzing issues such as ”Supply Chain Delays” and ”Equipment Downtime”:
–Supply Chain Delays: 𝑇𝐴𝑛=0.6,𝐼𝐴𝑛=0.3,𝐹𝐴𝑛=0.1
–Equipment Downtime: 𝑇𝐴𝑛=0.7,𝐼𝐴𝑛=0.2,𝐹𝐴𝑛=0.1
•Improve (I): Improvement actions such as ”Add New Suppliers” and ”Upgrade Machinery”:
–Add New Suppliers: 𝑇𝐼𝑚=0.7,𝐼𝐼𝑚=0.2,𝐹𝐼𝑚=0.1
–Upgrade Machinery: 𝑇𝐼𝑚=0.8,𝐼𝐼𝑚=0.1,𝐹𝐼𝑚=0.1
•Control (C): Control measures like ”Real-Time Monitoring” and ”Automated Alerts”:
–Real-Time Monitoring: 𝑇𝐶𝑡=0.9,𝐼𝐶𝑡=0.05,𝐹𝐶𝑡=0.05
–Automated Alerts: 𝑇𝐶𝑡=0.8,𝐼𝐶𝑡=0.1,𝐹𝐶𝑡=0.1
22
This demonstrates how the Neutrosophic DMAIC cycle integrates uncertainty and truth degrees into dening,
measuring, analyzing, improving, and controlling stages.
Theorem 3.34. Neutrosophic DMAIC cycle has the structure of a Neutrosophic Set.
Proof. This follows directly from the denition of Neutrosophic DMAIC cycle.
Question 3.35. Several derived concepts of the DMAIC cycle are widely recognized, including the DMADV
Cycle (Dene-Measure-Analyze-Design-Verify) [38, 155, 348] and the DCOV Cycle (Dene-Characterize-
Optimize-Verify) [30,203].
What unique characteristics arise when concepts such as Neutrosophic Sets are incorporated into these derived
cycles? Additionally, what potential applications might be enabled by such adaptations?
3.3.3 SWOT Analysis with Neutrosophic Sets
SWOT Analysis is a strategic planning tool used to assess a project or organization’s internal and external
factors. It identies four key dimensions: Strengths, Weaknesses, Opportunities, and Threats, aiming to develop
eective strategies [2,93,140, 237,305, 311,391].
This framework is widely applied across various industries, including business, education [2], and healthcare
[281], and has also been studied within Fuzzy and Neutrosophic contexts [37,163,309]. The following outlines
an extension of SWOT Analysis using Neutrosophic Sets.
Denition 3.36. The Neutrosophic SWOT Analysis extends the traditional Strengths-Weaknesses-Opportunities-
Threats framework by incorporating Neutrosophic Sets to model uncertainty, indeterminacy, and truth. The
analysis consists of four components:
1. Strengths (S): Represented by a Neutrosophic Set 𝑆:
𝑆={(𝑥 , 𝑇𝑆(𝑥), 𝐼𝑆(𝑥), 𝐹𝑆(𝑥)) | 𝑥∈Strength Elements},
where:
•𝑇𝑆(𝑥): Degree to which 𝑥is a strength.
•𝐼𝑆(𝑥): Degree of uncertainty in determining 𝑥as a strength.
•𝐹𝑆(𝑥): Degree to which 𝑥is not a strength.
2. Weaknesses (W): Represented by a Neutrosophic Set 𝑊:
𝑊={( 𝑦 , 𝑇𝑊(𝑦), 𝐼𝑊(𝑦), 𝐹𝑊(𝑦)) | 𝑦∈Weakness Elements},
where:
•𝑇𝑊(𝑦): Degree to which 𝑦is a weakness.
•𝐼𝑊(𝑦): Degree of uncertainty in determining 𝑦as a weakness.
•𝐹𝑊(𝑦): Degree to which 𝑦is not a weakness.
3. Opportunities (O): Represented by a Neutrosophic Set 𝑂:
𝑂={( 𝑧 , 𝑇𝑂(𝑧), 𝐼𝑂(𝑧), 𝐹𝑂(𝑧)) | 𝑧∈Opportunity Elements},
where:
•𝑇𝑂(𝑧): Degree to which 𝑧is an opportunity.
•𝐼𝑂(𝑧): Degree of uncertainty in determining 𝑧as an opportunity.
•𝐹𝑂(𝑧): Degree to which 𝑧is not an opportunity.
23
4. Threats (T): Represented by a Neutrosophic Set 𝑇:
𝑇={( 𝑤 , 𝑇𝑇(𝑤), 𝐼𝑇(𝑤), 𝐹𝑇(𝑤)) | 𝑤∈Threat Elements},
where:
•𝑇𝑇(𝑤): Degree to which 𝑤is a threat.
•𝐼𝑇(𝑤): Degree of uncertainty in determining 𝑤as a threat.
•𝐹𝑇(𝑤): Degree to which 𝑤is not a threat.
Example 3.37. Consider applying Neutrosophic SWOT Analysis to evaluate a company:
•Strengths (S): ”Brand Recognition [153]” and ”Skilled Workforce [376]”:
–Brand Recognition: 𝑇𝑆=0.9,𝐼𝑆=0.05,𝐹𝑆=0.05
–Skilled Workforce: 𝑇𝑆=0.8,𝐼𝑆=0.1,𝐹𝑆=0.1
•Weaknesses (W): ”High Operational Costs” and ”Limited Market Presence”:
–High Operational Costs: 𝑇𝑊=0.7,𝐼𝑊=0.2,𝐹𝑊=0.1
–Limited Market Presence: 𝑇𝑊=0.6,𝐼𝑊=0.3,𝐹𝑊=0.1
•Opportunities (O): ”Emerging Markets” and ”Technological Advancements”:
–Emerging Markets: 𝑇𝑂=0.8,𝐼𝑂=0.15,𝐹𝑂=0.05
–Technological Advancements: 𝑇𝑂=0.9,𝐼𝑂=0.05,𝐹𝑂=0.05
•Threats (T): ”Economic Recession [346]” and ”New Competitors”:
–Economic Recession: 𝑇𝑇=0.7,𝐼𝑇=0.2,𝐹𝑇=0.1
–New Competitors: 𝑇𝑇=0.6,𝐼𝑇=0.3,𝐹𝑇=0.1
This analysis demonstrates how Neutrosophic Sets model strengths, weaknesses, opportunities, and threats with
varying degrees of truth, uncertainty, and falsehood.
Theorem 3.38. Neutrosophic SWOT Analysis has the structure of a Neutrosophic Set.
Proof. This follows directly from the denition of Neutrosophic SWOT Analysis.
Question 3.39. Several extended concepts of SWOT Analysis are widely recognized, including SWOC Anal-
ysis (Strengths, Weaknesses, Opportunities, Challenges) [33, 55, 177, 264, 284], SOAR Analysis (Strengths,
Opportunities, Aspirations, Results) [174,354, 355,357], and Dynamic SWOT Analysis [45,178, 396].
What mathematical characteristics and potential applications could emerge if these frameworks were extended
using Neutrosophic Sets?
3.3.4 OODA Cycle with Neutrosophic Sets
The OODA Cycle (Observe-Orient-Decide-Act) is a decision-making framework designed to enable eective
responses in dynamic and competitive environments [131,198,236,282,298,415]. It emphasizes observing the
situation, orienting oneself based on the context, making informed decisions, and taking timely actions. The
following outlines an extension of the OODA Cycle using Neutrosophic Sets.
Denition 3.40. The Neutrosophic OODA Loop extends the traditional Observe-Orient-Decide-Act framework
by incorporating Neutrosophic Sets to model uncertainty, indeterminacy, and truth. The loop consists of four
stages:
24
1. Observe (O): Represented by a Neutrosophic Set 𝑂𝑏:
𝑂𝑏={(𝑥 , 𝑇𝑂𝑏(𝑥), 𝐼𝑂𝑏(𝑥), 𝐹𝑂𝑏(𝑥)) | 𝑥∈Observation Elements},
where:
•𝑇𝑂𝑏(𝑥): Degree to which 𝑥is accurately observed.
•𝐼𝑂𝑏(𝑥): Degree of uncertainty in observing 𝑥.
•𝐹𝑂𝑏(𝑥): Degree to which 𝑥is inaccurately observed.
2. Orient (O): Represented by a Neutrosophic Set 𝑂𝑟:
𝑂𝑟={( 𝑦 , 𝑇𝑂𝑟(𝑦), 𝐼𝑂𝑟(𝑦), 𝐹𝑂𝑟(𝑦)) | 𝑦∈Orientation Elements},
where:
•𝑇𝑂𝑟(𝑦): Degree to which orientation is correct.
•𝐼𝑂𝑟(𝑦): Degree of uncertainty in orientation.
•𝐹𝑂𝑟(𝑦): Degree to which orientation is incorrect.
3. Decide (D): Represented by a Neutrosophic Set 𝐷𝑐:
𝐷𝑐={( 𝑧 , 𝑇𝐷𝑐(𝑧), 𝐼𝐷𝑐(𝑧), 𝐹𝐷𝑐(𝑧)) | 𝑧∈Decision Elements},
where:
•𝑇𝐷𝑐(𝑧): Degree to which the decision is correct.
•𝐼𝐷𝑐(𝑧): Degree of uncertainty in the decision.
•𝐹𝐷𝑐(𝑧): Degree to which the decision is incorrect.
4. Act (A): Represented by a Neutrosophic Set 𝐴𝑐:
𝐴𝑐={( 𝑤 , 𝑇𝐴𝑐(𝑤), 𝐼𝐴𝑐(𝑤), 𝐹𝐴𝑐(𝑤)) | 𝑤∈Action Elements},
where:
•𝑇𝐴𝑐(𝑤): Degree to which the action is eective.
•𝐼𝐴𝑐(𝑤): Degree of uncertainty in the action.
•𝐹𝐴𝑐(𝑤): Degree to which the action is ineective.
Example 3.41. Consider applying the Neutrosophic OODA Loop to a business decision:
•Observe (O): Observing market trends such as ”Customer Preferences [347]” and ”Competitor Actions
[26]”:
–Customer Preferences: 𝑇𝑂𝑏=0.8,𝐼𝑂𝑏=0.1,𝐹𝑂𝑏=0.1
–Competitor Actions: 𝑇𝑂𝑏=0.7,𝐼𝑂𝑏=0.2,𝐹𝑂𝑏=0.1
•Orient (O): Orienting strategies based on ”Market Positioning [60]” and ”Customer Segmentation [74]”:
–Market Positioning: 𝑇𝑂𝑟=0.7,𝐼𝑂𝑟=0.2,𝐹𝑂𝑟=0.1
–Customer Segmentation: 𝑇𝑂𝑟=0.8,𝐼𝑂𝑟=0.1,𝐹𝑂𝑟=0.1
•Decide (D): Making decisions on ”Budget Allocation [414]” and ”Market Entry [186]”:
–Budget Allocation: 𝑇𝐷𝑐=0.7,𝐼𝐷𝑐=0.2,𝐹𝐷𝑐=0.1
–Market Entry: 𝑇𝐷𝑐=0.6,𝐼𝐷𝑐=0.3,𝐹𝐷𝑐=0.1
•Act (A): Implementing actions like ”Launch New Product” and ”Improve Distribution Channels”:
–Launch New Product: 𝑇𝐴𝑐=0.8,𝐼𝐴𝑐=0.1,𝐹𝐴𝑐=0.1
–Improve Distribution Channels: 𝑇𝐴𝑐=0.7,𝐼𝐴𝑐=0.2,𝐹𝐴𝑐=0.1
This example illustrates how the Neutrosophic OODA Loop integrates truth, uncertainty, and falsehood degrees
into observing, orienting, deciding, and acting stages.
Theorem 3.42. Neutrosophic OODA Loop has the structure of a Neutrosophic Set.
Proof. This follows directly from the denition of Neutrosophic OODA Loop.
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3.3.5 Neutrosophic Porter’s Five Forces Analysis
Neutrosophic Porter’s Five Forces Analysis is an extended framework based on the classic Porter’s Five Forces
Analysis. This approach evaluates industry competition through ve key factors: rivalry among existing com-
petitors, bargaining power of buyers, bargaining power of suppliers, threat of substitutes, and threat of new
entrants [94,150, 277,278].
Several related studies have been conducted within the contexts of Fuzzy Sets and Neutrosophic Sets [240].
The formal denition is provided below.
Denition 3.43. The Neutrosophic Porter’s Five Forces Analysis extends the traditional framework by incor-
porating Neutrosophic Sets to model uncertainty, indeterminacy, and truth across the ve competitive forces:
1. Threat of New Entrants (N): Represented by a Neutrosophic Set 𝑁:
𝑁={(𝑥 , 𝑇𝑁(𝑥), 𝐼𝑁(𝑥), 𝐹𝑁(𝑥)) | 𝑥∈New Entrant Factors},
where:
•𝑇𝑁(𝑥): Degree to which 𝑥increases the threat of new entrants.
•𝐼𝑁(𝑥): Degree of uncertainty regarding the inuence of 𝑥.
•𝐹𝑁(𝑥): Degree to which 𝑥does not inuence the threat of new entrants.
2. Bargaining Power of Suppliers (S): Represented by a Neutrosophic Set 𝑆:
𝑆={( 𝑦 , 𝑇𝑆(𝑦), 𝐼𝑆(𝑦), 𝐹𝑆(𝑦)) | 𝑦∈Supplier Factors},
where:
•𝑇𝑆(𝑦): Degree to which 𝑦increases supplier bargaining power.
•𝐼𝑆(𝑦): Degree of uncertainty regarding the inuence of 𝑦.
•𝐹𝑆(𝑦): Degree to which 𝑦does not inuence supplier bargaining power.
3. Bargaining Power of Buyers (B): Represented by a Neutrosophic Set 𝐵:
𝐵={( 𝑧 , 𝑇𝐵(𝑧), 𝐼𝐵(𝑧), 𝐹𝐵(𝑧)) | 𝑧∈Buyer Factors},
where:
•𝑇𝐵(𝑧): Degree to which 𝑧increases buyer bargaining power.
•𝐼𝐵(𝑧): Degree of uncertainty regarding the inuence of 𝑧.
•𝐹𝐵(𝑧): Degree to which 𝑧does not inuence buyer bargaining power.
4. Threat of Substitutes (U): Represented by a Neutrosophic Set 𝑈:
𝑈={( 𝑤 , 𝑇𝑈(𝑤), 𝐼𝑈(𝑤), 𝐹𝑈(𝑤)) | 𝑤∈Substitute Factors},
where:
•𝑇𝑈(𝑤): Degree to which 𝑤increases the threat of substitutes.
•𝐼𝑈(𝑤): Degree of uncertainty regarding the inuence of 𝑤.
•𝐹𝑈(𝑤): Degree to which 𝑤does not inuence the threat of substitutes.
5. Industry Rivalry (R): Represented by a Neutrosophic Set 𝑅:
𝑅={( 𝑣 , 𝑇𝑅(𝑣), 𝐼𝑅(𝑣), 𝐹𝑅(𝑣)) | 𝑣∈Rivalry Factors},
where:
•𝑇𝑅(𝑣): Degree to which 𝑣intensies industry rivalry.
26
•𝐼𝑅(𝑣): Degree of uncertainty regarding the inuence of 𝑣.
•𝐹𝑅(𝑣): Degree to which 𝑣does not inuence industry rivalry.
Example 3.44. Consider applying Neutrosophic Porter’s Five Forces Analysis to a retail business (cf. [208]):
•Threat of New Entrants (N): Factors such as ”Low Capital Requirements” and ”Lack of Brand Loyalty”:
–Low Capital Requirements: 𝑇𝑁=0.8,𝐼𝑁=0.15,𝐹𝑁=0.05
–Lack of Brand Loyalty: 𝑇𝑁=0.7,𝐼𝑁=0.2,𝐹𝑁=0.1
•Bargaining Power of Suppliers (S): Factors such as ”Few Suppliers” and ”High Switching Costs”:
–Few Suppliers: 𝑇𝑆=0.9,𝐼𝑆=0.05,𝐹𝑆=0.05
–High Switching Costs: 𝑇𝑆=0.8,𝐼𝑆=0.1,𝐹𝑆=0.1
•Bargaining Power of Buyers (B): Factors such as ”Availability of Alternatives” and ”Price Sensitivity”:
–Availability of Alternatives: 𝑇𝐵=0.7,𝐼𝐵=0.2,𝐹𝐵=0.1
–Price Sensitivity: 𝑇𝐵=0.8,𝐼𝐵=0.1,𝐹𝐵=0.1
•Threat of Substitutes (U): Factors such as ”Ease of Switching” and ”Low Cost of Substitutes”:
–Ease of Switching: 𝑇𝑈=0.8,𝐼𝑈=0.1,𝐹𝑈=0.1
–Low Cost of Substitutes: 𝑇𝑈=0.7,𝐼𝑈=0.2,𝐹𝑈=0.1
•Industry Rivalry (R): Factors such as ”High Number of Competitors” and ”Slow Market Growth”:
–High Number of Competitors: 𝑇𝑅=0.9,𝐼𝑅=0.05,𝐹𝑅=0.05
–Slow Market Growth: 𝑇𝑅=0.8,𝐼𝑅=0.1,𝐹𝑅=0.1
This example illustrates how Neutrosophic Sets can quantify and model the dynamics of Porter’s Five Forces
in the context of a competitive market.
Theorem 3.45. Neutrosophic Porter’s Five Forces has the structure of a Neutrosophic Set.
Proof. This follows directly from the denition of Neutrosophic Porter’s Five Forces.
Question 3.46. As a related concept, frameworks such as six-forces analysis have been studied [19,48,49,189].
Can the principles of Neutrosophic Logic be applied to these frameworks, and what potential applications might
emerge?
3.4 Some Neutrosophic (Social or Business) Logic
In the eld of Social Science, various logics have been studied (e.g., [21,176,257]). This paper aims to explore
potential extensions of these logics, including their expansion into Neutrosophic Logic.
3.4.1 Neutrosophic Institutional Logics
Institutional Logics are frameworks guiding behavior within societal institutions, integrating material practices
and symbolic systems to shape actions and norms [43,362–364].
Denition 3.47 (Institutional Logics).[363] Institutional logics are formalized as a structure L=(I,S,R,C ),
where:
1. I={𝐼1, 𝐼2, . . . , 𝐼𝑘}is a nite set of institutional orders. Each 𝐼𝑖∈ I corresponds to a domain such as
markets, states, families, or religions.
27
2. Sis the set of structural-symbolic systems, dened as:
S={𝑆𝑖=(𝑀𝑖, 𝐶𝑖) | 𝑖∈ {1,2, . . . , 𝑘}},
where:
•𝑀𝑖is a set of material practices, formalized as a function 𝑀𝑖:𝑋→𝑌, where 𝑋represents resource
inputs and 𝑌represents outputs.
•𝐶𝑖is a symbolic system, dened as a tuple 𝐶𝑖=(Σ,G), where Σis a set of cultural symbols and
G:Σ→ [0,1]is a probability distribution encoding the salience of each symbol.
3. R ⊆ I × S is a relation mapping institutional orders 𝐼𝑖∈ I to their corresponding structural-symbolic
systems 𝑆𝑖∈ S.
4. Cis a set of constraints, where C:𝐴× I → Bmaps actions 𝐴and institutional orders Ito a boolean
domain B={0,1}, enforcing domain-specic norms and rules.
Denition 3.48 (Behavior under Institutional Logics).The behavior of an actor 𝑎∈𝐴within an institutional
logic Lis dened as a function:
𝐵L(𝑎)=arg max
𝑏∈𝐵
𝑈(𝑏| L),
where 𝐵is the set of all possible behaviors, and 𝑈:𝐵× L → Ris a utility function dened as:
𝑈(𝑏| L) =
𝑘
Õ
𝑖=1𝜔𝑖·𝑓𝑀(𝑏, 𝑀𝑖) + 𝑓𝐶(𝑏 , 𝐶𝑖),
with:
•𝜔𝑖∈ [0,1]representing the weight of the 𝑖-th institutional order.
•𝑓𝑀(𝑏, 𝑀𝑖)quantifying the compatibility of behavior 𝑏with material practices 𝑀𝑖.
•𝑓𝐶(𝑏, 𝐶𝑖)quantifying the alignment of behavior 𝑏with symbolic systems 𝐶𝑖.
Denition 3.49 (Institutional Change).Institutional change occurs when the relation Ror constraints Care
updated due to exogenous events or endogenous contradictions. Formally, institutional change is a process:
Φ:L𝑡→ L𝑡+1,
where L𝑡and L𝑡+1represent institutional logics at time 𝑡and 𝑡+1, respectively, and Φsatises:
Φ(L𝑡)=I,S0,R0,C0,
with S0,R0,C0reecting updated material practices, symbolic systems, or constraints.
Remark 3.50 (Neutrosophic Institutional Logic).Fuzzy Institutional Logic is a special case of Neutrosophic
Institutional Logic where both indeterminacy and falsity are set to zero. Furthermore, Plithogenic Institutional
Logic is notable for its ability to generalize both Neutrosophic and Fuzzy Institutional Logic.
Example 3.51 (Market Logic).Consider a market logic Lmarket =(𝐼market, 𝑆market,Rmarket,Cmarket), where:
•𝐼market represents the institutional order of markets.
•𝑆market =(𝑀exchange, 𝐶prot ), with:
–𝑀exchange formalized as a function 𝑀exchange(𝑝, 𝑞)=𝑝·𝑞, where 𝑝is price and 𝑞is quantity.
–𝐶prot representing the cultural schema of prot maximization, encoded as G(prot)=1.
•Cmarket (𝑎, 𝐼market)=1if 𝑎adheres to legal and competitive norms, otherwise 0.
The following describes Institutional Neutrosophic Logics, which extend this concept using Neutrosophic
Logic.
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Denition 3.52 (Institutional Neutrosophic Logics).Institutional Neutrosophic Logics extend classical Insti-
tutional Logics by incorporating uncertainty, represented by the neutrosophic components of truth (T), indeter-
minacy (I), and falsity (F). Formally, an Institutional Neutrosophic Logic is dened as:
L𝑁=(I,S,R,C,N ),
where Nmaps each proposition 𝑃about an institutional action or state to a neutrosophic value:
N (𝑃)=(𝑇 , 𝐼, 𝐹),
with 𝑇, 𝐼, 𝐹 ∈ [0,1]satisfying 0≤𝑇+𝐼+𝐹≤1.
•𝑇: Degree to which 𝑃is true within the institutional logic.
•𝐼: Degree to which 𝑃is indeterminate due to conicting or insucient evidence.
•𝐹: Degree to which 𝑃is false.
The behavior under Institutional Neutrosophic Logics is dened by a neutrosophic utility function:
𝑈𝑁(𝑏| L𝑁)=
𝑘
Õ
𝑖=1𝜔𝑖·𝑓𝑁
𝑀(𝑏, 𝑀𝑖) + 𝑓𝑁
𝐶(𝑏, 𝐶𝑖),
where 𝑓𝑁
𝑀and 𝑓𝑁
𝐶incorporate neutrosophic evaluations of material practices and symbolic systems.
Remark 3.53. Institutional Fuzzy Logic is a special case of Institutional Neutrosophic Logic where both in-
determinacy and falsity are set to zero. Furthermore, Institutional Plithogenic Logic can also be dened using
Plithogenic Logic.
Example 3.54 (Neutrosophic Market Logic).Consider a neutrosophic market logic
L𝑁
market =(𝐼market, 𝑆market,Rmarket,Cmarket,N)
, where:
•N (𝑃)=(𝑇 , 𝐼, 𝐹)evaluates propositions such as ”The market will grow by 10% next year” with𝑇=0.6,
𝐼=0.3, and 𝐹=0.1. This reects a moderately condent prediction with some uncertainty and minimal
falsity.
•𝑈𝑁incorporates these neutrosophic values into decision-making. For example, an investor uses 𝑇, 𝐼 , 𝐹
to decide whether to allocate resources, balancing the condence (𝑇) against the uncertainty (𝐼) and risk
(𝐹).
• Material practices 𝑀market include pricing strategies modeled as 𝑀market (𝑝, 𝑞)=𝑝·𝑞, where 𝑝is the
price per unit and 𝑞is the quantity sold.
• Symbolic systems 𝐶prot prioritize prot maximization, encoded as G (prot)=1.
Theorem 3.55. Institutional Neutrosophic Logics naturally incorporate the structure of Neutrosophic Logic.
Proof. This follows directly from the denition of Institutional Neutrosophic Logics, as they extend the prin-
ciples and framework of Neutrosophic Logic to institutional contexts.
Theorem 3.56. Institutional Neutrosophic Logics naturally incorporate the structure of Institutional Logics.
Proof. This follows directly from the denition of Institutional Neutrosophic Logics, as they integrate the
fundamental aspects of traditional Institutional Logics into a neutrosophic framework.
Theorem 3.57. Every Institutional Neutrosophic Logic L𝑁is a superset of Institutional Fuzzy Logic L𝐹.
29
Proof. By denition, an Institutional Neutrosophic Logic L𝑁=(I,S,R,C,N ) includes a neutrosophic map-
ping:
N (𝑃)=(𝑇 , 𝐼, 𝐹),
where 𝑇, 𝐼, 𝐹 ∈ [0,1]and 0≤𝑇+𝐼+𝐹≤1. In Institutional Fuzzy Logic L𝐹=(I,S,R,C,F ), the mapping:
F (𝑃)=𝑇 ,
can be viewed as a special case of N (𝑃)where 𝐼=0and 𝐹=0. Since L𝐹is dened within the constraints of
L𝑁, every Institutional Fuzzy Logic is inherently embedded within an Institutional Neutrosophic Logic. Thus,
L𝑁is a superset of L𝐹.
Theorem 3.58. Institutional Neutrosophic Logic can model multiple institutional orders simultaneously, pre-
serving independence and interdependence of I
𝑖.
Proof. Let L𝑁=(I,S,R,C,N ), where I={𝐼1, 𝐼2, . . . , 𝐼𝑘}is the set of institutional orders. The neutro-
sophic mapping N (𝑃)=(𝑇 , 𝐼, 𝐹)applies independently to propositions 𝑃𝑖within each institutional order
𝐼𝑖. Additionally, interdependencies between institutional orders are encoded in the relation R ⊆ I × S. The
independence of I
𝑖is preserved by maintaining separate evaluations for each 𝐼𝑖, while interdependencies are
modeled via shared structural-symbolic systems Sand constraints C. Thus, L𝑁accommodates both indepen-
dence and interdependence among multiple institutional orders.
3.4.2 Dominant Neutrosophic Logic
Dominant Logic refers to the mindset or cognitive framework organizations use to make decisions, allocate
resources, and interpret information, shaping strategy and performance [162,193, 204,280, 295,378].
Denition 3.59 (Dominant Logic).(cf. [378]) Let 𝐹be a rm operating a portfolio of businesses B={𝐵1, 𝐵2, . . . , 𝐵𝑛}.
The Dominant Logic Lof the rm is a cognitive and operational framework dened as:
L=(S,D,K,P) ,
where:
•S={𝑆1, 𝑆2, . . . , 𝑆𝑚}: A set of schemas, where each schema 𝑆𝑖is a mapping:
𝑆𝑖:E → A,
that transforms environmental inputs E(e.g., market trends) into actionable decisions A.
•D: A decision-making function dened as:
D:V × C → R+,
where Vrepresents strategic variables (e.g., product pricing, market share), Crepresents organizational
capabilities, and D (v,c)is the resource allocation decision.
•K: A knowledge structure represented as a directed graph (N ,E), where Nis the set of knowledge nodes
and Eare directed edges encoding the relationships among knowledge components.
•P={𝑃1, 𝑃2, . . . , 𝑃𝑘}: A set of performance metrics, where each 𝑃𝑗:O → Rmaps observable out-
comes O(e.g., revenue, market share) to a real-valued evaluation.
Remark 3.60 (Neutrosophic Dominant Logic).Fuzzy Dominant Logic is a special case of Neutrosophic Dom-
inant Logic where both indeterminacy and falsity are set to zero. Furthermore, Plithogenic Dominant Logic is
notable for its ability to generalize both Neutrosophic and Fuzzy Dominant Logic.
Example 3.61 (Application of Dominant Logic).Consider a rm 𝐹with two businesses:
B={𝐵1:Consumer Electronics, 𝐵2:Healthcare Products}.
The rm’s Dominant Logic Lis described as follows:
30
•Schemas (S): 𝑆1is a schema that responds to market trends by adjusting product pricing. For instance:
𝑆1(Increase in demand)=Increase price by 10%.
•Decision-making (D): The rm allocates R&D resources to maximize revenue. For example:
D(Budget Share: 0.6,Capabilities: Advanced R&D)=0.8,
indicating 80% of the R&D budget is allocated to Consumer Electronics.
•Knowledge Structure (K): Nodes represent domain expertise such as ‘‘Electronics Design’’ and ‘‘Health-
care Regulation,’’ with directed edges denoting knowledge dependencies.
•Performance Metrics (P): Metrics include 𝑃1=Revenue Growth and 𝑃2=Customer Retention,mea-
sured as:
𝑃1=Revenuecurrent −Revenueprevious
Revenueprevious
.
Through this Dominant Logic, the rm evaluates whether R&D investments optimize the metrics 𝑃1and 𝑃2,
adapting to feedback from market performance.
Denition 3.62 (Strategic Fit).A Dominant Logic Lachieves strategic t if, for each business 𝐵𝑖∈ B, there
exists a schema 𝑆𝑖∈ S and a decision D(v,c)such that:
P𝑗(𝐵𝑖)is maximized for all 𝑃𝑗∈ P.
Example 3.63 (Strategic Fit).In the earlier example, the rm aligns Dwith Pby prioritizing R&D spending
in Consumer Electronics, where revenue growth (𝑃1) shows the highest marginal return per unit investment. If
the healthcare business (𝐵2) exhibits diminishing returns, resources are reallocated to 𝐵1to maximize overall
rm performance.
Next, the following describes Dominant Neutrosophic Logic, which extends Dominant Logic using Neutro-
sophic Logic.
Denition 3.64 (Dominant Neutrosophic Logic).Dominant Neutrosophic Logic is an extension of Dominant
Logic that incorporates neutrosophic components of truth (𝑇), indeterminacy (𝐼), and falsity (𝐹) to handle
uncertainty and incomplete information in decision-making processes. It is formally dened as a tuple:
L𝑁=(S 𝑁,D𝑁,K𝑁,P𝑁),
where:
1. S𝑁={𝑆𝑁
1, 𝑆𝑁
2, . . . , 𝑆𝑁
𝑚}is a set of neutrosophic schemas. Each schema 𝑆𝑁
𝑖is a mapping:
𝑆𝑁
𝑖:E → A 𝑁,
where Eis the space of environmental inputs, and A𝑁is the space of neutrosophic-valued actions dened
as:
A𝑁={(𝑇, 𝐼, 𝐹) | 𝑇, 𝐼, 𝐹 ∈ [0,1], 𝑇 +𝐼+𝐹≤1}.
Here, 𝑇represents the degree of truth, 𝐼represents the degree of indeterminacy, and 𝐹represents the
degree of falsity.
2. D𝑁is a neutrosophic decision-making function:
D𝑁:V × C → A 𝑁,
where Vis the space of strategic variables, Cis the space of organizational capabilities, and D𝑁(v,c)
assigns a neutrosophic value to each decision.
31
3. K𝑁is a neutrosophic knowledge structure, represented as a directed graph (N ,E), where:
N={𝐾𝑁
1, 𝐾 𝑁
2, . . . , 𝐾 𝑁
𝑝},
is a set of knowledge nodes, and each 𝐾𝑁
𝑖is associated with a neutrosophic value (𝑇𝑖, 𝐼𝑖, 𝐹𝑖). The edges
in Erepresent knowledge dependencies, each assigned a neutrosophic weight.
4. P𝑁={𝑃𝑁
1, 𝑃𝑁
2, . . . , 𝑃𝑁
𝑘}is a set of neutrosophic performance metrics. Each metric 𝑃𝑁
𝑗is a function:
𝑃𝑁
𝑗:O → A 𝑁,
where Ois the space of observable outcomes, and 𝑃𝑁
𝑗(𝑜)=(𝑇𝑗, 𝐼 𝑗, 𝐹𝑗)evaluates the outcome 𝑜in terms
of truth, indeterminacy, and falsity.
Remark 3.65. Dominant Neutrosophic Logic generalizes Dominant Logic by explicitly modeling uncertainty
and conict through the neutrosophic components (𝑇 , 𝐼, 𝐹). Fuzzy Dominant Logic is a special case where
indeterminacy (𝐼) and falsity (𝐹) are zero, i.e., (𝑇, 𝐼, 𝐹)=(𝑇, 0,0).
Example 3.66 (Application of Dominant Neutrosophic Logic).Consider a rm 𝐹managing two business
domains:
B={𝐵1:Articial Intelligence, 𝐵2:Healthcare Devices}.
The Dominant Neutrosophic Logic L𝑁for 𝐹can be described as follows:
1. Neutrosophic Schema (S𝑁): A schema 𝑆𝑁
1evaluates the proposition ”Invest in AI R&D” based on market
trends:
𝑆𝑁
1(Positive market trend)=(𝑇=0.8, 𝐼 =0.15, 𝐹 =0.05).
2. Neutrosophic Decision-making (D𝑁): Allocates resources with uncertainty in mind. For example:
D𝑁(R&D Budget: 60%,Capabilities: AI Research)=(𝑇=0.7, 𝐼 =0.2, 𝐹 =0.1).
3. Neutrosophic Knowledge Structure (K𝑁): Nodes include ”Market Trends” and ”Technology Readiness,”
with neutrosophic weights:
(Market Trends) → (AI Research)=(𝑇=0.9, 𝐼 =0.05, 𝐹 =0.05).
4. Neutrosophic Performance Metrics (P𝑁): Metrics include revenue growth, evaluated as:
𝑃𝑁
1(Revenue Growth)=(𝑇=0.75, 𝐼 =0.2, 𝐹 =0.05).
The neutrosophic framework helps the rm balance condence (𝑇), uncertainty (𝐼), and risk (𝐹).
Theorem 3.67. Dominant Neutrosophic Logic naturally incorporates the structure of Neutrosophic Logic.
Proof. This follows directly from the denition of Dominant Neutrosophic Logic, as it extends the principles
and framework of Neutrosophic Logic to dominant logical structures and reasoning processes.
Theorem 3.68. Dominant Neutrosophic Logic naturally incorporates the structure of Dominant Logic.
Proof. This follows directly from the denition of Dominant Neutrosophic Logic, as it integrates the funda-
mental aspects of traditional Dominant Logic into a neutrosophic framework.
Theorem 3.69. Dominant Neutrosophic Logic is a superset of Fuzzy Dominant Logic.
Proof. Fuzzy Dominant Logic is a special case of Dominant Neutrosophic Logic where 𝐼=0and 𝐹=0,
reducing the neutrosophic value (𝑇 , 𝐼 , 𝐹)to (𝑇 , 0,0). Since Dominant Neutrosophic Logic allows 𝑇 , 𝐼 , 𝐹 to
independently range within [0,1]under the constraint 𝑇+𝐼+𝐹≤1, Fuzzy Dominant Logic is fully embedded
within this broader framework. Thus, Dominant Neutrosophic Logic generalizes Fuzzy Dominant Logic.
32
Theorem 3.70. Dominant Neutrosophic Logic accommodates multiple schemas, preserving independence and
interdependence among decision components.
Proof. Let L𝑁=(S𝑁,D𝑁,K𝑁,P𝑁), where S𝑁={𝑆𝑁
1, 𝑆𝑁
2, . . . , 𝑆𝑁
𝑚}represents neutrosophic schemas.
Each schema 𝑆𝑁
𝑖operates independently as a mapping 𝑆𝑁
𝑖:E → A 𝑁, where A𝑁={ (𝑇 , 𝐼, 𝐹 )}. Interde-
pendence is introduced through shared resources or dependencies represented in K𝑁, a directed graph linking
knowledge nodes. This structure preserves independence at the schema level while modeling interdependen-
cies through relationships in K𝑁. Thus, Dominant Neutrosophic Logic eectively manages independent and
interdependent components.
Theorem 3.71. Dominant Neutrosophic Logic enhances decision-making by explicitly modeling uncertainty
and conict through (𝑇, 𝐼 , 𝐹).
Proof. In traditional Dominant Logic, decisions rely on deterministic or probabilistic values, lacking explicit
representation of indeterminacy or falsity. Dominant Neutrosophic Logic extends this framework by incor-
porating neutrosophic values (𝑇 , 𝐼 , 𝐹 ), allowing decisions to account for truth, uncertainty, and conict si-
multaneously. This enriched representation improves decision-making in complex scenarios with incomplete
or conicting information, as each decision component D𝑁evaluates strategic variables and organizational
capabilities under neutrosophic uncertainty.
3.4.3 Service-Dominant Neutrosophic Logic
Service-Dominant Logic emphasizes value co-creation through service exchange, viewing goods as service
delivery mechanisms, focusing on relationships, collaboration, and customer-centricity in value creation [147,
215–219,260, 317,379–381].
Denition 3.72 (Service-Dominant Logic).(cf. [379, 380]) Service-Dominant Logic (S-D Logic) is a theo-
retical framework that conceptualizes value creation as a collaborative process among multiple actors within a
service ecosystem. Formally, it is dened as a tuple:
L𝑆𝐷 =(A,R,I,V) ,
where:
1. A={𝐴1, 𝐴2, . . . , 𝐴𝑛}: A set of actors in the service ecosystem, where each actor 𝐴𝑖is a resource
integrator.
2. R={𝜌𝑖 𝑗 }: A set of resource exchanges between actors 𝐴𝑖and 𝐴𝑗, where:
𝜌𝑖 𝑗 =(𝑅𝑖 𝑗 , 𝐸𝑖 𝑗 ),
with 𝑅𝑖 𝑗 being the resource provided by 𝐴𝑖to 𝐴𝑗, and 𝐸𝑖 𝑗 being the corresponding value exchange.
3. I={𝐼1, 𝐼2, . . . , 𝐼𝑚}: A set of institutional arrangements, where each 𝐼𝑘denes the rules, norms, and
practices governing resource exchanges within the ecosystem.
4. V={𝑉1, 𝑉2, . . . , 𝑉𝑝}: A set of value cocreation processes, where each 𝑉𝑙is a mapping:
𝑉𝑙:A × R → R,
assigning a value 𝑣∈Rto each interaction based on the integration of resources by the actors.
Example 3.73 (Service Ecosystem).Consider a healthcare service ecosystem:
A={Patients,Doctors,Pharmacies,Insurers}.
Here:
33
• Resource exchanges (R) include the transfer of medical knowledge (𝑅𝑖 𝑗 ) from doctors to patients and
nancial resources (𝑅𝑗 𝑖 ) from insurers to healthcare providers.
• Institutional arrangements (I) include healthcare regulations and insurance policies.
• Value cocreation processes (V) evaluate outcomes such as patient health improvement or cost-eectiveness.
Through this framework, the ecosystem collectively cocreates value.
Denition 3.74 (Service-Dominant Neutrosophic Logic).Service-Dominant Neutrosophic Logic (SDN Logic)
extends Service-Dominant Logic by incorporating neutrosophic components of truth (𝑇), indeterminacy (𝐼),
and falsity (𝐹) to address uncertainty and incomplete information within a service ecosystem. Formally, SDN
Logic is dened as:
L𝑆𝐷 𝑁 =(A ,R𝑁,I,V𝑁),
where:
1. A={𝐴1, 𝐴2, . . . , 𝐴𝑛}: A set of actors in the service ecosystem. Each actor 𝐴𝑖is a resource integrator
and decision-maker.
2. R𝑁={𝜌𝑁
𝑖 𝑗 }: A set of neutrosophic resource exchanges between actors 𝐴𝑖and 𝐴𝑗, where:
𝜌𝑁
𝑖 𝑗 =(𝑅𝑖 𝑗 ,(𝑇𝑖 𝑗 , 𝐼𝑖 𝑗 , 𝐹𝑖 𝑗 )),
with 𝑅𝑖 𝑗 being the resource provided by 𝐴𝑖to 𝐴𝑗, and (𝑇𝑖 𝑗 , 𝐼𝑖 𝑗 , 𝐹𝑖 𝑗 )representing the neutrosophic truth,
indeterminacy, and falsity values of the resource exchange.
3. I={𝐼1, 𝐼2, . . . , 𝐼𝑚}: A set of institutional arrangements dening the rules, norms, and practices gov-
erning interactions and exchanges within the ecosystem.
4. V𝑁={𝑉𝑁
1, 𝑉 𝑁
2, . . . , 𝑉 𝑁
𝑝}: A set of neutrosophic value cocreation processes, where each 𝑉𝑁
𝑙is a map-
ping:
𝑉𝑁
𝑙:A × R 𝑁→ A𝑁,
assigning a neutrosophic value (𝑇 , 𝐼 , 𝐹 )to each interaction based on the integration of resources by the
actors.
The neutrosophic constraints require that:
𝑇, 𝐼, 𝐹 ∈ [0,1], 𝑇 +𝐼+𝐹≤1.
Remark 3.75 (Neutrosophic Service-Dominant Logic).Fuzzy Service-Dominant Logic is a special case of
Neutrosophic Service-Dominant Logic where both indeterminacy and falsity are set to zero. Furthermore,
Plithogenic Service-Dominant Logic is notable for its ability to generalize both Neutrosophic and Fuzzy Service-
Dominant Logic.
Example 3.76 (Healthcare Service Ecosystem).A Healthcare Service Ecosystem is a dynamic network of
interconnected stakeholders collaboratively co-creating value through services, resources, and relationships to
improve health outcomes (cf. [4,67, 399]). Consider a healthcare service ecosystem:
A={Patients,Doctors,Pharmacies,Insurers}.
Here:
• Neutrosophic resource exchanges (R𝑁) include the transfer of medical advice (𝑅𝑖 𝑗 ) from doctors to pa-
tients with:
𝜌𝑁
Doctor, Patient =(Medical Advice,(𝑇=0.9, 𝐼 =0.05, 𝐹 =0.05)).
• Institutional arrangements (I) include healthcare regulations and insurance policies.
34
• Neutrosophic value cocreation processes (V𝑁) evaluate outcomes such as patient health improvement.
For instance:
𝑉𝑁
Health Improvement(Doctor,Patient)=(𝑇=0.85, 𝐼 =0.1, 𝐹 =0.05).
Through this framework, the ecosystem balances condence (𝑇), uncertainty (𝐼), and risk (𝐹) in resource
exchanges and value creation.
Theorem 3.77. Service-Dominant Neutrosophic Logic generalizes Service-Dominant Logic by incorporating
neutrosophic components (𝑇 , 𝐼 , 𝐹).
Proof. Service-Dominant Logic L𝑆𝐷 is dened as L𝑆 𝐷 =(A ,R,I,V), where Rrepresents deterministic
resource exchanges and Vdeterministic value cocreation processes. Service-Dominant Neutrosophic Logic
L𝑆𝐷 𝑁 =(A ,R𝑁,I,V𝑁)extends L𝑆𝐷 by introducing R𝑁and V𝑁, where resource exchanges and value
cocreation processes are represented with neutrosophic components (𝑇 , 𝐼, 𝐹). These components allow L𝑆 𝐷 𝑁
to explicitly model uncertainty (𝐼) and conict (𝐹), which are absent in L𝑆 𝐷 . Thus, Service-Dominant Neu-
trosophic Logic generalizes Service-Dominant Logic.
Theorem 3.78. Service-Dominant Neutrosophic Logic inherently possesses the structure of Neutrosophic
Logic.
Proof. Service-Dominant Neutrosophic Logic L𝑆𝐷 𝑁 incorporates neutrosophic resource exchanges R𝑁and
neutrosophic value cocreation processes V𝑁, which map interactions and outcomes to neutrosophic values
(𝑇, 𝐼, 𝐹). These mappings align directly with the principles of Neutrosophic Logic, where 𝑇, 𝐼, 𝐹 represent
truth, indeterminacy, and falsity, respectively. As 𝑇+𝐼+𝐹≤1is a fundamental constraint in both frameworks,
L𝑆𝐷 𝑁 naturally inherits the structure of Neutrosophic Logic.
Theorem 3.79. Service-Dominant Neutrosophic Logic balances resource exchanges and value cocreation un-
der uncertainty, enabling robust decision-making.
Proof. In Service-Dominant Neutrosophic Logic L𝑆𝐷 𝑁 , resource exchanges 𝜌𝑁
𝑖 𝑗 =(𝑅𝑖 𝑗 ,(𝑇𝑖 𝑗 , 𝐼𝑖 𝑗 , 𝐹𝑖 𝑗 )) ex-
plicitly account for uncertainty (𝐼) and falsity (𝐹) in interactions. Neutrosophic value cocreation processes
𝑉𝑁
𝑙:A × R 𝑁→ A𝑁integrate these components to evaluate outcomes with condence (𝑇) while accom-
modating uncertainty and conict. This balanced approach ensures that decision-making within the service
ecosystem is robust, adapting to incomplete or conicting information.
Theorem 3.80. Service-Dominant Neutrosophic Logic enables dynamic optimization of resource exchanges
and value cocreation processes in complex ecosystems.
Proof. The neutrosophic components (𝑇 , 𝐼 , 𝐹)in Service-Dominant Neutrosophic Logic allow dynamic as-
sessment of resource exchanges 𝜌𝑁
𝑖 𝑗 and value processes 𝑉𝑁
𝑙. By continuously updating (𝑇 , 𝐼 , 𝐹 )based on
new information, the framework adapts to changes in the service ecosystem, optimizing interactions and out-
comes. This exibility supports decision-making in complex and evolving environments, where uncertainty
and conicting information are prevalent.
3.4.4 Neutrosophic Critical Thinking (Neutrosophic Critical Logic)
Critical Thinking is the objective analysis and evaluation of information to form reasoned judgments, empha-
sizing logic, and evidence [35,103, 170, 202,243,275].
Denition 3.81 (Critical Thinking).Critical thinking is the systematic, recursive, and logical process of ana-
lyzing, evaluating, and synthesizing information to derive coherent conclusions and self-reectively improve
reasoning. Mathematically, it can be represented as:
C(𝑋)=R ◦ F ◦ E ◦ A ◦ I (𝑋),
where:
35
1. I:X → R (Interpretation Function): A function that maps raw data 𝑋∈ X into a structured represen-
tation 𝑅∈ R, capturing its semantic meaning. Formally:
I( 𝑋)=𝑅, where 𝑅is a structured framework.
2. A:R → P ( E) (Analysis Operator): A function that decomposes 𝑅into its atomic elements or sub-
components {𝑒1, 𝑒2, . . . , 𝑒𝑛} ⊆ E, where P(E ) denotes the power set of E. Formally:
A(𝑅)={𝑒𝑖|𝑒𝑖represents an atomic element of 𝑅}.
3. E:E → [0,1](Evaluation Metric): A function that assigns a weight 𝑤(𝑒𝑖)to each element 𝑒𝑖, quanti-
fying its credibility or logical strength. Formally:
E(𝑒𝑖)=𝑤(𝑒𝑖), 𝑤(𝑒𝑖)indicates the reliability of 𝑒𝑖.
4. F:P(E) → C (Inference Function): A function that aggregates weighted elements {(𝑒𝑖, 𝑤(𝑒𝑖)) } into
a conclusion 𝐶∈ C based on logical or probabilistic rules. Formally:
F ({ (𝑒𝑖, 𝑤(𝑒𝑖)) }) =𝐶 , where 𝐶is logically consistent.
5. R:C → C0(Self-Regulation Operator): A recursive function that reassesses and renes all preceding
steps, resulting in an improved critical thinking process C0. Formally:
R(C) =C0,C0is an updated process.
Remark 3.82. The critical thinking process Cis inherently recursive, as the self-regulation operator Rallows
iterative improvement. This ensures both logical rigor and adaptability to new information.
Example 3.83. Consider 𝑋as a dataset of experimental observations supporting a scientic hypothesis. The
process proceeds as follows:
1. Interpretation (I): Organize 𝑋into a structured hypothesis 𝑅.
2. Analysis (A): Decompose 𝑅into key premises {𝑒1, 𝑒2, . . . , 𝑒𝑛}.
3. Evaluation (E): Assign weights 𝑤(𝑒𝑖)to each premise based on empirical evidence.
4. Inference (F): Derive a conclusion 𝐶by combining weighted premises.
5. Regulation (R): Reassess I,A,E,Fand rene the conclusion 𝐶.
Denition 3.84 (Neutrosophic Critical Thinking).Neutrosophic Critical Thinking (NCT) is an extension of
classical critical thinking that operates under the framework of neutrosophic logic, incorporating degrees of
truth (𝑇), indeterminacy (𝐼), and falsity (𝐹). This enables reasoning and decision-making in the presence of
uncertainty and contradictions. Formally, NCT is a structured process dened as:
N CT ( 𝑋)=R𝑁◦ F 𝑁◦ E𝑁◦ A 𝑁◦ I 𝑁(𝑋),
where 𝑋∈ X is the input data or information, and the components are dened as follows:
1. I𝑁:X → R 𝑁(Neutrosophic Interpretation): A mapping that converts raw data 𝑋into a neutrosophic
representation 𝑅𝑁∈ R𝑁. For each proposition 𝑃in 𝑅𝑁, a neutrosophic truth value is assigned:
N (𝑃)=(𝑇 , 𝐼, 𝐹), 𝑇 , 𝐼, 𝐹 ∈ [0,1],0≤𝑇+𝐼+𝐹≤1,
where:
•𝑇: Degree to which 𝑃is true.
•𝐼: Degree to which 𝑃is indeterminate (uncertain or conicting).
•𝐹: Degree to which 𝑃is false.
36
2. A𝑁:R𝑁→ P(E 𝑁)(Neutrosophic Analysis): Decomposes 𝑅𝑁into atomic components {𝑒1, 𝑒2, . . . , 𝑒𝑛} ⊆
E𝑁, where each component 𝑒𝑖represents a fundamental unit of 𝑅𝑁. Each 𝑒𝑖is associated with a neu-
trosophic evaluation:
A𝑁(𝑅𝑁)={( 𝑒𝑖,N ( 𝑒𝑖)) | 𝑒𝑖is an atomic component of 𝑅𝑁}.
3. E𝑁:E𝑁→ [0,1]3(Neutrosophic Evaluation): Assigns a neutrosophic truth value N (𝑒𝑖)=(𝑇𝑒𝑖, 𝐼𝑒𝑖, 𝐹𝑒𝑖)
to each atomic component 𝑒𝑖, quantifying its truth, indeterminacy, and falsity. Formally:
E𝑁(𝑒𝑖)=(𝑇𝑒𝑖, 𝐼𝑒𝑖, 𝐹𝑒𝑖), 𝑇𝑒𝑖, 𝐼𝑒𝑖, 𝐹𝑒𝑖∈ [0,1], 𝑇𝑒𝑖+𝐼𝑒𝑖+𝐹𝑒𝑖≤1.
4. F𝑁:P(E 𝑁) → C 𝑁(Neutrosophic Inference): Synthesizes the neutrosophic evaluations { (𝑒𝑖,N (𝑒𝑖) ) }
into a conclusion 𝐶𝑁, represented as:
N (𝐶𝑁)=(𝑇𝐶, 𝐼𝐶, 𝐹𝐶),
where:
𝑇𝐶=
𝑛
Õ
𝑖=1
𝑤𝑖𝑇𝑒𝑖, 𝐼𝐶=
𝑛
Õ
𝑖=1
𝑤𝑖𝐼𝑒𝑖, 𝐹𝐶=
𝑛
Õ
𝑖=1
𝑤𝑖𝐹𝑒𝑖,
and 𝑤𝑖are weights such that Í𝑛
𝑖=1𝑤𝑖=1.
5. R𝑁:C𝑁→ C𝑁(Neutrosophic Self-Regulation): A recursive operator that re-evaluates and renes C𝑁
by iteratively applying the process to updated information or revised assumptions. Formally:
R𝑁(C 𝑁)=C𝑁
updated,
where C𝑁
updated incorporates new evaluations or corrections.
Remark 3.85 (Neutrosophic Critical Thinking).Fuzzy Critical Thinking is a special case of Neutrosophic Crit-
ical Thinking where both indeterminacy and falsity are set to zero. Furthermore, Plithogenic Critical Thinking
is notable for its ability to generalize both Neutrosophic and Fuzzy Critical Thinking.
Example 3.86 (Neutrosophic Decision-Making in Scientic Hypotheses).Let 𝑋represent experimental data
supporting a scientic hypothesis. The process unfolds as follows:
1. Interpretation (I𝑁): Convert 𝑋into 𝑅𝑁, where each proposition 𝑃(e.g., ”The hypothesis holds under
condition A”) is assigned N (𝑃)=(𝑇 , 𝐼, 𝐹).
2. Analysis (A𝑁): Decompose 𝑅𝑁into atomic premises {𝑒1, 𝑒2, . . . , 𝑒𝑛}, with N ( 𝑒𝑖)=(𝑇𝑒𝑖, 𝐼𝑒𝑖, 𝐹𝑒𝑖).
3. Evaluation (E𝑁): Assign 𝑇𝑒𝑖, 𝐼𝑒𝑖, 𝐹𝑒𝑖values to each 𝑒𝑖based on empirical evidence and logical consis-
tency.
4. Inference (F𝑁): Compute N (𝐶𝑁)=(𝑇𝐶, 𝐼𝐶, 𝐹𝐶)as the weighted aggregate of N (𝑒𝑖).
5. Self-Regulation (R𝑁): Reassess N (𝐶𝑁)and update components based on additional data or new hy-
potheses.
For instance, a hypothesis with N (𝐶𝑁)=(0.7,0.2,0.1)indicates 70% condence, 20% uncertainty, and 10%
falsity.
Theorem 3.87. Neutrosophic Critical Thinking inherently extends classical critical thinking by modeling un-
certainty and contradiction through (𝑇, 𝐼, 𝐹).
Proof. In classical critical thinking, each proposition 𝑃is evaluated as either true or false, lacking an explicit
representation of uncertainty or contradiction. Neutrosophic Critical Thinking extends this framework by as-
signing to each proposition N (𝑃)=(𝑇 , 𝐼, 𝐹), where:
𝑇, 𝐼, 𝐹 ∈ [0,1], 𝑇 +𝐼+𝐹≤1.
This representation allows propositions to simultaneously have degrees of truth (𝑇), indeterminacy (𝐼), and
falsity (𝐹). By incorporating 𝐼and 𝐹, Neutrosophic Critical Thinking explicitly accounts for uncertainty and
contradictions, providing a more comprehensive framework for reasoning in ambiguous or complex scenarios.
37
Theorem 3.88. Neutrosophic Critical Thinking improves decision-making by balancing condence, uncer-
tainty, and falsity in evaluations.
Proof. In Neutrosophic Critical Thinking, the inference process F𝑁aggregates neutrosophic evaluations:
N (𝐶𝑁)=(𝑇𝐶, 𝐼𝐶, 𝐹𝐶),
where:
𝑇𝐶=
𝑛
Õ
𝑖=1
𝑤𝑖𝑇𝑒𝑖, 𝐼𝐶=
𝑛
Õ
𝑖=1
𝑤𝑖𝐼𝑒𝑖, 𝐹𝐶=
𝑛
Õ
𝑖=1
𝑤𝑖𝐹𝑒𝑖.
The weights 𝑤𝑖are adjusted based on the importance or reliability of atomic components 𝑒𝑖. This balanced
approach enables decision-making that considers condence (𝑇𝐶), uncertainty (𝐼𝐶), and falsity (𝐹𝐶), allowing
for nuanced conclusions that classical frameworks cannot achieve.
Theorem 3.89. Neutrosophic Critical Thinking provides a self-regulating mechanism for iterative reasoning
and decision-making.
Proof. The self-regulation operator R𝑁in Neutrosophic Critical Thinking re-evaluates and renes conclusions
C𝑁by incorporating new data or updated assumptions:
R𝑁(C 𝑁)=C𝑁
updated.
This recursive process ensures that decisions and conclusions remain adaptive to evolving information, im-
proving robustness and accuracy over time. Such iterative renement is absent in classical critical thinking,
highlighting the advanced capabilities of the neutrosophic approach.
Theorem 3.90. Neutrosophic Critical Thinking is applicable to systems with incomplete or conicting data,
where classical critical thinking fails.
Proof. In systems with incomplete or conicting data, propositions 𝑃cannot be fully classied as true or false.
Neutrosophic Critical Thinking assigns N (𝑃)=(𝑇 , 𝐼, 𝐹 ), where indeterminacy (𝐼) captures the ambiguity or
conict. By explicitly modeling 𝐼alongside 𝑇and 𝐹, the framework accommodates incomplete or contradictory
information, enabling reasoning and decision-making where classical approaches are inadequate.
3.4.5 Neutrosophic Climate Change Logic
In social science, Climate Change Logic models the interplay between human behavior, policies, and environ-
mental impacts, analyzing strategies to mitigate climate change while accounting for societal, economic, and
regulatory factors [23,42, 77,166, 373].
Denition 3.91 (Climate Change Logic).Climate Change Logic is a formal mathematical system for modeling,
evaluating, and optimizing the dynamic interactions between environmental states, human activities, and their
impacts on climate systems under uncertainty. It is formally expressed as:
L𝐶𝐶 =(𝑆 , 𝐴, 𝑇 , 𝐹 , 𝑃 , 𝑉, 𝐶 , 𝑅 , U),
where:
•𝑆={𝑠1, 𝑠2, . . . , 𝑠𝑛}: A nite or innite set of environmental states representing measurable climate-
related variables such as CO2concentration, global temperature rise, or sea-level rise.
•𝐴={𝑎1, 𝑎2, . . . , 𝑎𝑚}: A nite set of human activities or interventions inuencing the state transitions,
such as emissions, deforestation, industrial output, or renewable energy adoption.
•𝑇={𝑡0, 𝑡1, . . . , 𝑡𝑝}: A discrete or continuous time horizon over which environmental dynamics and
human interventions are observed.
38
•𝐹:𝑆×𝐴×𝑇→Δ(𝑆): The state transition function, where 𝐹(𝑠, 𝑎, 𝑡 )gives the probability distribution
over 𝑆at time 𝑡+1, conditioned on the current state 𝑠∈𝑆and activity 𝑎∈𝐴. Here, Δ(𝑆)is the set of
probability distributions on 𝑆.
•𝑃:𝑆→ [0,1]: The risk function, assigning the probability of adverse events (e.g., natural disasters,
economic damage) occurring at state 𝑠.
•𝑉:𝑆×𝑇→R+: The valuation function, quantifying the severity of impacts or costs (e.g., economic
losses, biodiversity loss, or health damage) associated with state 𝑠at time 𝑡.
•𝐶:𝐴×𝑇→R+: The activity cost function, representing the cost associated with implementing activity
𝑎at time 𝑡.
•𝑅:𝑆×𝐴→R+: The regulation function, dening the regulatory or mitigation costs required to control
the state transition induced by activity 𝑎from state 𝑠.
•U:P(𝑆×𝐴) → R: The utility function, capturing the decision-maker’s preferences over states and
actions, accounting for both immediate and future impacts.
Climate Impact Evaluation. The cumulative climate impact 𝐼over a time horizon 𝑇is expressed as:
𝐼=∫𝑡𝑝
𝑡0Õ
𝑠∈𝑆Õ
𝑎∈𝐴
𝑃(𝑠) · 𝐹(𝑠, 𝑎, 𝑡) · 𝑉(𝑠, 𝑡)𝑑𝑡 .
Optimal Climate Policy. The optimal climate policy 𝜋∗is a strategy that minimizes the cumulative impact
𝐼and total costs 𝐶, while accounting for regulatory constraints:
𝜋∗=arg min
𝜋∈Π"∫𝑡𝑝
𝑡0 𝐼+Õ
𝑎∈𝐴
𝐶(𝑎, 𝑡) + Õ
𝑠∈𝑆
𝑅(𝑠, 𝑎)!𝑑𝑡#,
subject to:
𝐹(𝑠, 𝑎, 𝑡) ∈ Δ(𝑆),∀𝑡∈𝑇, 𝑠 ∈𝑆, 𝑎 ∈𝐴.
Here, Πis the set of all feasible policies mapping states 𝑆to actions 𝐴.
Uncertainty in Climate Change Logic. If uncertainty in state transitions or valuations is represented by a
neutrosophic framework, the state transition function 𝐹𝑁and valuation function 𝑉𝑁are extended as follows:
𝐹𝑁(𝑠, 𝑎, 𝑡)=(𝑇𝐹, 𝐼𝐹, 𝐹𝐹), 𝑉 𝑁(𝑠, 𝑡)=(𝑇𝑉, 𝐼𝑉, 𝐹𝑉),
where 𝑇,𝐼, and 𝐹denote the truth, indeterminacy, and falsity components, respectively.
The cumulative neutrosophic impact 𝐼𝑁becomes:
𝐼𝑁=∫𝑡𝑝
𝑡0Õ
𝑠∈𝑆Õ
𝑎∈𝐴
(𝑇𝐹−𝐹𝐹)·𝑃(𝑠) · 𝑉𝑁(𝑠, 𝑡 )𝑑𝑡.
Example 3.92 (Climate Change Logic: Renewable Energy vs. Forest Regeneration).Consider a climate policy
scenario where policymakers aim to reduce greenhouse gas (GHG) emissions [258] over a time horizon 𝑇=
{𝑡0, 𝑡1, 𝑡2}. The components of the Climate Change Logic are as follows:
•𝑆={𝑠1, 𝑠2, 𝑠3}: Environmental states.
–𝑠1: Low GHG emissions.
–𝑠2: Moderate GHG emissions.
–𝑠3: High GHG emissions.
39
•𝐴={𝑎1, 𝑎2, 𝑎3}: Climate mitigation activities.
–𝑎1: Adoption of renewable energy (solar, wind).
–𝑎2: Forest regeneration programs.
–𝑎3: No intervention.
•𝐹:𝑆×𝐴×𝑇→Δ(𝑆): State transition probabilities under mitigation activities.
𝐹(𝑠3, 𝑎1, 𝑡1)=0.7, 𝐹 (𝑠3, 𝑎 2, 𝑡1)=0.6.
Interpretation: At 𝑡1, adopting 𝑎1reduces 𝑠3to lower states with 70% probability, while 𝑎2achieves a
60% reduction probability.
•𝑉:𝑆×𝑇→R+: Climate impact valuation.
𝑉(𝑠3, 𝑡2)=100, 𝑉 (𝑠2, 𝑡2)=50, 𝑉 (𝑠1, 𝑡2)=10.
Interpretation: The cost of high emissions (𝑠3) at 𝑡2is 100, while moderate (𝑠2) and low emissions (𝑠1)
cost 50 and 10, respectively.
•𝐶:𝐴×𝑇→R+: Activity cost function.
𝐶(𝑎1, 𝑡1)=30, 𝐶 (𝑎2, 𝑡1)=20, 𝐶 (𝑎3, 𝑡1)=0.
Interpretation: The implementation costs of 𝑎1(renewable energy) and 𝑎2(forest regeneration) at 𝑡1are
30 and 20, respectively. 𝑎3(no intervention) incurs no cost.
•𝑅:𝑆×𝐴→R+: Regulatory compliance cost.
𝑅(𝑠3, 𝑎3)=50.
Interpretation: Maintaining high emissions (𝑠3) under no intervention (𝑎3) incurs regulatory penalties
of 50.
Cumulative Climate Impact. The total climate impact 𝐼over 𝑇is calculated as:
𝐼=∫𝑡2
𝑡0Õ
𝑠∈𝑆Õ
𝑎∈𝐴
𝐹(𝑠, 𝑎, 𝑡) · 𝑉(𝑠, 𝑡)𝑑 𝑡 .
Cost-Benet Comparison. The total costs 𝐶total for each policy (activity) include implementation costs and
regulatory penalties:
𝐶total(𝑎1)=30, 𝐶total (𝑎2)=20, 𝐶total (𝑎3)=50.
Optimal Policy. The optimal activity 𝑎∗minimizes the sum of cumulative climate impact and total costs:
𝑎∗=arg min
𝑎∈𝐴[𝐼+𝐶total].
Results.
•𝑎1(renewable energy) achieves the largest emission reduction probability (70%), reducing 𝐼signicantly,
but incurs higher upfront costs.
•𝑎2(forest regeneration) provides a lower reduction probability (60%) but is more cost-eective.
•𝑎3(no intervention) results in the highest regulatory penalties and climate impact, making it the least
optimal choice.
40
Thus, policymakers must evaluate trade-os between emission reductions and associated costs to determine
the optimal climate mitigation policy.
Denition 3.93 (Neutrosophic Climate Change Logic).Neutrosophic Climate Change Logic is a formalized
framework that models climate systems, human activities, and their interactions under uncertainty, indetermi-
nacy, and falsity. It is dened as:
L𝑁
𝐶𝐶 =(𝑆 , 𝐴, 𝑇 , 𝐹 𝑁, 𝑃 𝑁, 𝑉 𝑁, 𝐶 𝑁, 𝑅𝑁,U𝑁),
where:
•𝑆={𝑠1, 𝑠2, . . . , 𝑠𝑛}: A nite or innite set of environmental states (e.g., temperature rise, CO2concen-
tration, sea-level change).
•𝐴={𝑎1, 𝑎2, . . . , 𝑎𝑚}: A nite set of human activities or mitigation strategies that inuence state transi-
tions, such as energy consumption, reforestation, or carbon capture.
•𝑇={𝑡0, 𝑡1, . . . , 𝑡𝑝}: A time domain (discrete or continuous) representing the temporal evolution of
climate states.
•𝐹𝑁:𝑆×𝐴×𝑇→ [0,1]3: The neutrosophic state transition function, where:
𝐹𝑁(𝑠, 𝑎, 𝑡)=(𝑇𝐹, 𝐼𝐹, 𝐹𝐹),
assigns the degrees of truth (𝑇𝐹), indeterminacy (𝐼𝐹), and falsity (𝐹𝐹) for the probability of transitioning
to a new state 𝑠∈𝑆under activity 𝑎at time 𝑡.
•𝑃𝑁:𝑆→ [0,1]3: The neutrosophic risk function, where:
𝑃𝑁(𝑠)=(𝑇𝑃, 𝐼𝑃, 𝐹𝑃),
represents the neutrosophic probabilities of risks (e.g., disasters or adverse eects) occurring at state 𝑠.
•𝑉𝑁:𝑆×𝑇→R3: The neutrosophic valuation function, where:
𝑉𝑁(𝑠, 𝑡)=(𝑇𝑉, 𝐼𝑉, 𝐹𝑉),
gives the truth (𝑇𝑉), indeterminacy (𝐼𝑉), and falsity (𝐹𝑉) components of the impacts or costs associated
with state 𝑠at time 𝑡(e.g., economic loss, biodiversity decline).
•𝐶𝑁:𝐴×𝑇→R3: The neutrosophic cost function, where:
𝐶𝑁(𝑎, 𝑡)=(𝑇𝐶, 𝐼𝐶, 𝐹𝐶),
quanties the truth, indeterminacy, and falsity of the costs incurred by implementing activity 𝑎at time 𝑡.
•𝑅𝑁:𝑆×𝐴→R3: The neutrosophic regulation function, where:
𝑅𝑁(𝑠, 𝑎)=(𝑇𝑅, 𝐼𝑅, 𝐹𝑅),
represents the costs or regulatory constraints (with uncertainty) for controlling state 𝑠under activity 𝑎.
•U𝑁:P(𝑆×𝐴) → R3: The neutrosophic utility function, evaluating the decision-maker’s preferences
over states and actions, incorporating truth, indeterminacy, and falsity.
Neutrosophic Climate Impact. The cumulative neutrosophic climate impact 𝐼𝑁over a time horizon 𝑇is
dened as:
𝐼𝑁=∫𝑡𝑝
𝑡0Õ
𝑠∈𝑆Õ
𝑎∈𝐴
(𝑇𝐹−𝐹𝐹) · 𝑃𝑁(𝑠) · 𝑉𝑁(𝑠, 𝑡)𝑑 𝑡,
where 𝑇𝐹and 𝐹𝐹are the truth and falsity degrees from 𝐹𝑁.
41
Optimal Neutrosophic Climate Policy. The optimal policy 𝜋∗
𝑁minimizes the cumulative neutrosophic im-
pact and costs while respecting regulatory constraints:
𝜋∗
𝑁=arg min
𝜋∈Π"∫𝑡𝑝
𝑡0 𝐼𝑁+Õ
𝑎∈𝐴
𝐶𝑁(𝑎, 𝑡) + Õ
𝑠∈𝑆
𝑅𝑁(𝑠, 𝑎)!𝑑𝑡#,
subject to:
𝐹𝑁(𝑠, 𝑎, 𝑡) ∈ [0,1]3,∀𝑡∈𝑇, 𝑠 ∈𝑆, 𝑎 ∈𝐴.
Neutrosophic Uncertainty Representation. In this framework, uncertainty is explicitly represented through
neutrosophic triplets:
(𝑇, 𝐼, 𝐹),
where:
•𝑇: Degree of truth, reecting known and veried information.
•𝐼: Degree of indeterminacy, accounting for ambiguity or incomplete information.
•𝐹: Degree of falsity, representing contradictory or false information.
The neutrosophic extension allows for a comprehensive evaluation of climate change processes, enabling decision-
makers to handle uncertain, incomplete, and conicting data eectively.
Remark 3.94 (Neutrosophic Climate Change Logic).Fuzzy Climate Change Logic is a special case of Neutro-
sophic Climate Change Logic where both indeterminacy and falsity are set to zero. Furthermore, Plithogenic
Climate Change Logic is notable for its ability to generalize both Neutrosophic and Fuzzy Climate Change
Logic.
Example 3.95 (Neutrosophic Climate Change Logic: GHG Emission Reduction).Consider a scenario where
policymakers aim to reduce greenhouse gas (GHG) emissions to mitigate climate change. Let the components
of Neutrosophic Climate Change Logic be dened as follows:
•𝑆={𝑠1, 𝑠2, 𝑠3}: Set of environmental states.
–𝑠1: Low GHG emission level (below target threshold).
–𝑠2: Moderate GHG emission level.
–𝑠3: High GHG emission level.
•𝐴={𝑎1, 𝑎2, 𝑎3}: Set of mitigation actions.
–𝑎1: Implementation of renewable energy (solar, wind).
–𝑎2: Industrial carbon capture and storage (CCS).
–𝑎3: No intervention (business as usual).
•𝑇={𝑡0, 𝑡1, 𝑡2}: Discrete time steps 𝑡0(initial), 𝑡1(mid-term), 𝑡2(long-term).
•𝐹𝑁:𝑆×𝐴×𝑇→ [0,1]3: Neutrosophic state transition function.
𝐹𝑁(𝑠2, 𝑎1, 𝑡1)=(0.8,0.1,0.1), 𝐹 𝑁(𝑠3, 𝑎 2, 𝑡2)=(0.6,0.3,0.1).
Interpretation: Implementing 𝑎1at 𝑡1reduces emissions to 𝑠2with 80% certainty (𝑇=0.8), 10% inde-
terminacy (𝐼=0.1), and 10% falsity (𝐹=0.1).
•𝑃𝑁:𝑆→ [0,1]3: Neutrosophic risk function.
𝑃𝑁(𝑠3)=(0.9,0.05,0.05),
indicating a 90% chance of severe climate risks under high emissions (𝑠3), with 5% indeterminacy and
5% falsity.
42
•𝑉𝑁:𝑆×𝑇→R3: Neutrosophic valuation function for environmental impact.
𝑉𝑁(𝑠3, 𝑡2)=(−100,0.2,−5),
meaning the environmental cost of state 𝑠3at 𝑡2is highly negative (𝑇=−100), with 20% uncertainty and
−5representing an overestimated loss.
•𝐶𝑁:𝐴×𝑇→R3: Neutrosophic cost function for mitigation actions.
𝐶𝑁(𝑎1, 𝑡1)=(30,0.1,2), 𝐶 𝑁(𝑎2, 𝑡2)=(50,0.2,5),
where implementing 𝑎1at 𝑡1incurs a cost of 30 with 10% indeterminacy, while 𝑎2at 𝑡2incurs a higher
cost of 50 with 20% uncertainty.
•𝑅𝑁:𝑆×𝐴→R3: Neutrosophic regulation function for compliance costs.
𝑅𝑁(𝑠3, 𝑎3)=(0,0.1,0),
indicating no additional regulation cost under 𝑠3with no intervention (𝑎3).
Neutrosophic Climate Impact. The cumulative neutrosophic climate impact 𝐼𝑁over 𝑇is calculated as:
𝐼𝑁=Õ
𝑡∈𝑇Õ
𝑠∈𝑆Õ
𝑎∈𝐴
(𝑇𝐹−𝐹𝐹) · 𝑃𝑁(𝑠) · 𝑉𝑁(𝑠, 𝑡),
where 𝑇𝐹and 𝐹𝐹are the truth and falsity degrees, respectively.
Optimal Policy. The optimal mitigation policy 𝜋∗
𝑁minimizes the total neutrosophic impact and associated
costs:
𝜋∗
𝑁=arg min
𝜋∈Π"𝐼𝑁+Õ
𝑡∈𝑇Õ
𝑎∈𝐴
𝐶𝑁(𝑎, 𝑡) + 𝑅𝑁(𝑠, 𝑎)#.
Interpretation. Based on the neutrosophic values:
• Implementing 𝑎1(renewable energy) reduces emissions to 𝑠2with high certainty and low indeterminacy,
making it a cost-eective option.
•𝑎2(carbon capture) achieves results with moderate certainty but incurs higher costs.
•𝑎3(no intervention) results in severe climate risks (𝑠3) with high probability.
The decision-maker uses the neutrosophic framework to weigh uncertainties, evaluate trade-os, and determine
the most eective policy.
3.4.6 Neutrosophic Social Media Logic
Social Media Logic refers to the principles driving social media platforms, focusing on programmability, pop-
ularity, connectivity, and datacation to shape user interactions and content dynamics [76,187, 296,368, 377].
This is extended using Neutrosophic Logic. The denition is provided below.
Denition 3.96 (Social Media Logic).(cf. [76,187,296, 368,377]) Social Media Logic (SML) is a mathematical
framework that models the underlying principles governing social media platforms. It is dened as:
SML =(P ,L,C,D),
where:
43
•Programmability (P): A bidirectional function P:(𝑈×𝐴) → ( 𝑅×𝐴0), where:
–𝑈: Set of users,
–𝐴: Set of algorithms,
–𝑅: Set of platform responses,
–𝐴0: Updated state of algorithms based on user interactions.
•Popularity (L): A scalar function L:𝐶→R+, where 𝐶is the set of content items, and L ( 𝑐)quanties
the popularity of content 𝑐using a weighted sum of metrics.
•Connectivity (C): A dynamic graph 𝐺=(𝑉 , 𝐸), where:
–𝑉=𝑈∪𝐶: Set of users and content,
–𝐸⊆𝑉×𝑉: Set of directed edges representing relationships or interactions.
•Datacation (D): A function D:𝐸→R𝑛, mapping each edge 𝑒∈𝐸to a vector of numerical features
describing interaction attributes.
Example 3.97 (Components of Social Media Logic).Consider a simplied social media scenario:
•Programmability (P): User 𝑢1interacts with algorithm 𝑎1, resulting in a response 𝑟1(e.g., recommended
content), and updates the algorithm to state 𝑎0
1:
P(𝑢1, 𝑎1)=(𝑟1, 𝑎0
1).
•Popularity (L): The popularity of a post 𝑐1is calculated as:
L(𝑐1)=𝑤1·Likes +𝑤2·Shares +𝑤3·Comments,
where 𝑤1, 𝑤2, 𝑤3are weights assigned to each metric.
•Connectivity (C): The platform is represented as a graph 𝐺=(𝑉, 𝐸), where:
𝑉={𝑢1, 𝑢2, 𝑐1, 𝑐2}, 𝐸 ={(𝑢1, 𝑐1),(𝑐1, 𝑢2) }.
•Datacation (D): An edge 𝑒=(𝑢1, 𝑐1)is mapped to a vector representing interaction attributes:
D(𝑒)=[time_spent,clicks,likes].
Denition 3.98 (Social Media Neutrosophic Logic).Social Media Neutrosophic Logic (SMNL) is a frame-
work for analyzing the uncertainty, indeterminacy, and truthfulness of propositions on social media. It extends
classical Social Media Logic by incorporating neutrosophic components. Formally, SMNL is dened as:
SMNL =(P ,L,C,D,N ) ,
where:
•Programmability (P): A bidirectional function P:(𝑈×𝐴) → ( 𝑅×𝐴0), where:
–𝑈: Set of users,
–𝐴: Set of algorithms,
–𝑅: Set of platform responses,
–𝐴0: Updated state of algorithms inuenced by user interactions.
•Popularity (L): A neutrosophic scalar function L:𝐶→R3, where 𝐶is the set of content items, and:
L(𝑐)=(𝑇𝑐, 𝐼𝑐, 𝐹𝑐),
where 𝑇𝑐, 𝐼𝑐, 𝐹𝑐∈ [0,1]represent the truth, indeterminacy, and falsity of content 𝑐, satisfying:
0≤𝑇𝑐+𝐼𝑐+𝐹𝑐≤1.
44
•Connectivity (C): A dynamic graph 𝐺=(𝑉 , 𝐸), where:
–𝑉=𝑈∪𝐶: Set of users and content,
–𝐸⊆𝑉×𝑉: Set of directed edges representing relationships or interactions,
–Each edge 𝑒∈𝐸is assigned a neutrosophic value N (𝑒)=(𝑇𝑒, 𝐼𝑒, 𝐹𝑒).
•Datacation (D): A function D:𝐸→R𝑛, mapping edges 𝑒∈𝐸to feature vectors of quantied
interaction data.
•Neutrosophic Evaluation (N): A mapping N:𝑃→R3, where 𝑃represents propositions about user
interactions, platform algorithms, or content. For any 𝑃, we have:
N (𝑃)=(𝑇 , 𝐼, 𝐹),
where 𝑇, 𝐼, 𝐹 ∈ [0,1]denote the degrees of truth, indeterminacy, and falsity, satisfying 0≤𝑇+𝐼+𝐹≤1.
Remark 3.99 (Neutrosophic Social Media Logic).Fuzzy Social Media Logic is a special case of Neutrosophic
Social Media Logic where both indeterminacy and falsity are set to zero. Furthermore, Plithogenic Social
Media Logic is notable for its ability to generalize both Neutrosophic and Fuzzy Social Media Logic.
Example 3.100 (Application of SMNL Components).A social media platform is a digital environment en-
abling users to create, share, and interact with content, fostering communication, networking, and engagement
(cf. [62,68]).
Consider a social media platform evaluating a post 𝑐1:
•Programmability (P): User 𝑢1interacts with the platform’s algorithm 𝑎1, which generates a response 𝑟1
(e.g., content recommendation) and updates itself to state 𝑎0
1:
P(𝑢1, 𝑎1)=(𝑟1, 𝑎0
1).
•Popularity (L): The post 𝑐1is evaluated as:
L(𝑐1)=(𝑇𝑐1, 𝐼𝑐1, 𝐹𝑐1)=(0.7,0.2,0.1),
indicating high truthfulness, moderate indeterminacy, and low falsity.
•Connectivity (C): The platform graph 𝐺=(𝑉, 𝐸)includes nodes 𝑉={𝑢1, 𝑢 2, 𝑐1, 𝑐2}and edges 𝐸=
{(𝑢1, 𝑐1),(𝑐1, 𝑢 2)}. The edge (𝑢1, 𝑐1)has a neutrosophic value:
N ( (𝑢1, 𝑐1)) =(𝑇𝑒, 𝐼𝑒, 𝐹𝑒)=(0.8,0.1,0.1).
•Datacation (D): The edge (𝑢1, 𝑐1)is mapped to a feature vector:
D( (𝑢1, 𝑐1)) =[time_spent,clicks,likes]=[300,5,10].
•Neutrosophic Evaluation (N): A proposition 𝑃: ”Post 𝑐1is reliable” is evaluated as:
N (𝑃)=(𝑇 , 𝐼, 𝐹)=(0.7,0.2,0.1).
Theorem 3.101. Social Media Neutrosophic Logic inherently possesses the structure of a Neutrosophic Logic.
Proof. This result follows directly from the denition of Social Media Neutrosophic Logic, as it extends the
principles and components of Neutrosophic Logic to the domain of social media.
Theorem 3.102. Social Media Neutrosophic Logic inherently possesses the structure of a Social Media Logic.
Proof. This result follows directly from the denition of Social Media Neutrosophic Logic, as it adapts the
principles and mechanisms of Social Media Logic within a neutrosophic framework.
45
Theorem 3.103. The neutrosophic popularity function L (𝑐)in SMNL balances truth, indeterminacy, and
falsity to model content evaluation.
Proof. The popularity function L (𝑐)in SMNL maps each content item 𝑐∈𝐶to a neutrosophic value:
L(𝑐)=(𝑇𝑐, 𝐼𝑐, 𝐹𝑐), 𝑇𝑐, 𝐼𝑐, 𝐹𝑐∈ [0,1], 𝑇𝑐+𝐼𝑐+𝐹𝑐≤1.
This representation balances:
•𝑇𝑐: The degree to which the content is truthful or reliable.
•𝐼𝑐: The degree of uncertainty or ambiguity in evaluating the content.
•𝐹𝑐: The degree to which the content is false or unreliable.
The constraint 𝑇𝑐+𝐼𝑐+𝐹𝑐≤1ensures that the evaluation is consistent and accounts for all available information.
By incorporating 𝐼𝑐, SMNL captures ambiguity that deterministic or probabilistic models cannot, providing a
nuanced evaluation of content.
Theorem 3.104. SMNL explicitly models uncertainty and conict in social media interactions through neu-
trosophic connectivity C.
Proof. The connectivity component Cin SMNL is a dynamic graph 𝐺=(𝑉, 𝐸), where:
N (𝑒)=(𝑇𝑒, 𝐼𝑒, 𝐹𝑒), 𝑇𝑒, 𝐼𝑒, 𝐹𝑒∈ [0,1], 𝑇𝑒+𝐼𝑒+𝐹𝑒≤1.
For each edge 𝑒∈𝐸, the neutrosophic value N (𝑒)represents:
•𝑇𝑒: The degree to which the interaction is meaningful or reliable.
•𝐼𝑒: The degree of uncertainty or ambiguity in the interaction.
•𝐹𝑒: The degree to which the interaction is misleading or false.
By modeling interactions with N (𝑒), SMNL captures the uncertainty and conict inherent in social media
interactions, enabling a more accurate analysis of network dynamics.
Theorem 3.105. SMNL enhances decision-making by integrating neutrosophic evaluations into the programma-
bility component P.
Proof. In SMNL, the programmability function Pis dened as:
P:(𝑈×𝐴) → (𝑅×𝐴0),
where 𝑈is the set of users, 𝐴is the set of algorithms, 𝑅is the set of platform responses, and 𝐴0is the updated
state of algorithms. By incorporating neutrosophic evaluations N (𝑃)=(𝑇 , 𝐼 , 𝐹)for propositions 𝑃about user
interactions or algorithm behavior, Penables algorithms to:
• Prioritize responses with high 𝑇(truthfulness).
• Mitigate decisions with high 𝐼(uncertainty).
• Avoid actions with high 𝐹(falsity).
This integration ensures that platform decisions are robust and adaptive to uncertainty and conicting informa-
tion.
46
3.4.7 Neutrosophic Critical Service Logic
In the eld of social science, Service Logic is well recognized. As a related concept, Critical Service Logic is
also known. Critical Service Logic focuses on understanding value creation through interactions, emphasizing
customer experiences, resources, and context within service ecosystems [148]. This framework is extended
using Neutrosophic Logic to incorporate uncertainty, indeterminacy, and falsity into the analysis of value co-
creation processes. Denitions and formalizations are provided below.
Denition 3.106 (Neutrosophic Critical Service Logic).Neutrosophic Critical Service Logic (NCSL) is a
mathematical framework for value creation and co-creation under uncertainty, ambiguity, and conict, using
neutrosophic components of truth (𝑇), indeterminacy (𝐼), and falsity (𝐹). Formally, NCSL is dened as:
N CS L =(A,R,V𝑁,E𝑁,D𝑁,N ),
where:
1. A={𝐴1, 𝐴2, . . . , 𝐴𝑛}: A set of actors in the service system, where each actor 𝐴𝑖integrates resources
for value creation.
𝐴𝑖=(role,capabilities,N).
2. R={𝑅1, 𝑅2, . . . , 𝑅𝑝}: A set of resources, where each resource 𝑅𝑘includes:
𝑅𝑘=(nancial,human,technological,N),
and each resource eectiveness is evaluated as:
N ( 𝑅𝑘)=(𝑇𝑅𝑘, 𝐼𝑅𝑘, 𝐹𝑅𝑘), 𝑇𝑅𝑘+𝐼𝑅𝑘+𝐹𝑅𝑘≤1.
3. V𝑁={𝑉𝑁
1, 𝑉 𝑁
2, . . . , 𝑉 𝑁
𝑚}: A set of neutrosophic value functions. Each value function 𝑉𝑁
𝑗maps time
horizons to neutrosophic evaluations:
𝑉𝑁
𝑗:T → R3, 𝑉 𝑁
𝑗(𝑡)=(𝑇𝑗(𝑡), 𝐼 𝑗(𝑡), 𝐹𝑗(𝑡)).
4. E𝑁={𝐸𝑁
1, 𝐸 𝑁
2, . . . , 𝐸 𝑁
𝑞}: A set of neutrosophic environmental states aecting value co-creation,
where:
𝐸𝑁
ℎ:T → R3, 𝐸 𝑁
ℎ(𝑡)=(𝑇𝐸ℎ(𝑡), 𝐼𝐸ℎ(𝑡), 𝐹𝐸ℎ(𝑡)).
5. D𝑁={𝐷𝑁
1, 𝐷 𝑁
2, . . . , 𝐷𝑁
𝑟}: A set of neutrosophic decisions, where each decision 𝐷𝑁
𝑙is dened as:
𝐷𝑁
𝑙:(R,E𝑁) → V 𝑁.
6. N: A neutrosophic evaluation function assigning a truth value to propositions 𝑃about actors, resources,
or environmental states:
N (𝑃)=(𝑇𝑃, 𝐼 𝑃, 𝐹𝑃), 𝑇𝑃, 𝐼𝑃, 𝐹𝑃∈ [0,1], 𝑇𝑃+𝐼𝑃+𝐹𝑃≤1.
Remark 3.107 (Neutrosophic Critical Service Logic).Fuzzy Critical Service Logic is a special case of Neutro-
sophic Critical Service Logic where both indeterminacy and falsity are set to zero. Furthermore, Plithogenic
Critical Service Logic is notable for its ability to generalize both Neutrosophic and Fuzzy Critical Service
Logic.
Example 3.108 (Neutrosophic Critical Service Logic in Renewable Energy).Renewable energy is energy
derived from naturally replenishing sources like sunlight, wind, water, and biomass, providing sustainable,
eco-friendly power [56, 92, 214, 369]. Consider a renewable energy service ecosystem where stakeholders
collaborate to create sustainable energy solutions under uncertainty.
•Actors (A):
A={Energy Providers,Governments,Investors,Consumers}.
Each actor 𝐴𝑖integrates resources for value creation. For instance:
𝐴Investors =(Financial support,Capital allocation,N=(𝑇=0.8, 𝐼 =0.15, 𝐹 =0.05)) .
47
•Resources (R): Resources include nancial investments, technological infrastructure, and human exper-
tise:
𝑅1=($10 Million,20 Engineers,Solar Panels,(𝑇=0.9, 𝐼 =0.05, 𝐹 =0.05)).
•Neutrosophic Value Functions (V𝑁): The value ”energy production eciency” is measured over a year
as:
𝑉𝑁
eciency (𝑡)=(𝑇eciency, 𝐼eciency, 𝐹eciency),
where:
𝑉𝑁
eciency(12 months)=(0.75,0.2,0.05).
•Neutrosophic Environmental States (E𝑁): External factors such as government subsidies and climate
conditions inuence outcomes:
𝐸𝑁
subsidy (𝑡)=(0.7,0.2,0.1).
•Neutrosophic Decisions (D𝑁): A decision to invest in solar energy is evaluated based on resources and
environmental states:
𝐷𝑁
solar(𝑅1, 𝐸 𝑁
subsidy)=𝑉𝑁
eciency.
In this example, NCSL quanties the uncertainties (𝐼) and risks (𝐹) involved in renewable energy investments,
allowing stakeholders to make informed and balanced decisions.
Theorem 3.109. The Neutrosophic Critical Service Logic exhibits the structure of a Neutrosophic Set.
Proof. The result follows directly from the denition.
Theorem 3.110. The Neutrosophic Critical Service Logic exhibits the structure of a Classic Critical Service
Logic.
Proof. The result follows directly from the denition.
Theorem 3.111 (Non-negativity of Neutrosophic Components).For any neutrosophic evaluation N ( 𝑃)=
(𝑇𝑃, 𝐼𝑃, 𝐹𝑃)in NCSL, the components 𝑇𝑃,𝐼𝑃, and 𝐹𝑃are non-negative:
𝑇𝑃≥0, 𝐼𝑃≥0, 𝐹𝑃≥0.
Proof. By the denition of the neutrosophic evaluation function:
N (𝑃)=(𝑇𝑃, 𝐼 𝑃, 𝐹𝑃),where 𝑇𝑃, 𝐼𝑃, 𝐹𝑃∈ [0,1].
The interval [0,1]imposes the lower bound 0for 𝑇𝑃,𝐼𝑃, and 𝐹𝑃. Hence, the components are non-negative:
𝑇𝑃≥0, 𝐼𝑃≥0, 𝐹𝑃≥0.
Theorem 3.112 (Bounded Sum of Neutrosophic Components).For any neutrosophic evaluation N (𝑃)=
(𝑇𝑃, 𝐼𝑃, 𝐹𝑃)in NCSL, the sum of components is bounded:
𝑇𝑃+𝐼𝑃+𝐹𝑃≤1.
Proof. By the denition of the neutrosophic evaluation function, we have:
N (𝑃)=(𝑇𝑃, 𝐼 𝑃, 𝐹𝑃), 𝑇𝑃, 𝐼𝑃, 𝐹𝑃∈ [0,1].
The condition 𝑇𝑃+𝐼𝑃+𝐹𝑃≤1ensures that the total evaluation remains within the valid range. If any of the
components 𝑇𝑃,𝐼𝑃, or 𝐹𝑃increase, the other components must decrease to satisfy this bound. Thus:
𝑇𝑃+𝐼𝑃+𝐹𝑃≤1.
48
Theorem 3.113 (Optimal Neutrosophic Decision-Making).Given a set of resources Rand environmental
states E𝑁, a neutrosophic decision 𝐷𝑁is optimal if it maximizes the truth component 𝑇while minimizing
indeterminacy 𝐼and falsity 𝐹:
𝐷𝑁
optimal =arg max
𝐷𝑁
𝑙∈ D 𝑁𝑇𝐷𝑙−𝐼𝐷𝑙−𝐹𝐷𝑙.
Proof. Let 𝐷𝑁
𝑙be a neutrosophic decision such that:
𝐷𝑁
𝑙:(R,E𝑁) → V 𝑁, 𝐷 𝑁
𝑙=(𝑇𝐷𝑙, 𝐼𝐷𝑙, 𝐹𝐷𝑙).
The optimal decision 𝐷𝑁
optimal seeks to balance the neutrosophic components by maximizing the truth 𝑇𝐷𝑙and
simultaneously minimizing the indeterminacy 𝐼𝐷𝑙and falsity 𝐹𝐷𝑙. Formally:
𝐷𝑁
optimal =arg max
𝐷𝑁
𝑙∈ D 𝑁𝑇𝐷𝑙−𝐼𝐷𝑙−𝐹𝐷𝑙,
subject to the constraint:
𝑇𝐷𝑙+𝐼𝐷𝑙+𝐹𝐷𝑙≤1.
This ensures that the decision 𝐷𝑁
optimal satises the neutrosophic bounds while optimizing the value for the
decision-maker.
4 Future Tasks: Various Extensions
This section provides a brief overview of the future prospects of this research.
It is important to note that the concepts dened in this Future Tasks section are merely examples and hold
signicant potential for improvement depending on the objectives and perspectives involved. However, by
engaging in such mathematical modeling, we believe that these concepts can be analyzed using various existing
mathematical frameworks and logics.
Further exploration of these denitions, their applications, and related research developments are expected to
progress in the future.
4.1 Real-World Applications within a New Social Framework
In this subsection, we discuss potential real-world applications within an uncertain social framework.
4.1.1 Plithogenic Social Framework
As previously mentioned, the plithogenic set is widely recognized for its exibility and its ability to general-
ize Fuzzy Sets and Neutrosophic Sets [20, 304, 314,325–327, 341, 352]. Owing to its versatile structure, the
plithogenic set holds signicant potential for real-world applications. In this study, we propose extending the
plithogenic set framework to established methodologies such as Neutrosophic Psychology, PDCA, DMAIC,
SWOT, and OODA. By integrating plithogenic sets into these frameworks, our aim is to explore their inter-
connections and enhance their capability to address complex, multi-dimensional, and contradictory scenarios
eectively.
Below, we outline conceptual denitions for applying plithogenic sets to these systems:
•Plithogenic Body-Mind-Soul-Spirit Fluidity: A framework capturing multi-attribute dynamics in psycho-
logical and spiritual contexts, enabling nuanced assessments of human decision-making and well-being.
•Plithogenic PDCA: An extension of the Plan-Do-Check-Act cycle that incorporates multi-criteria and
contradictory attributes for more eective quality improvement and problem-solving.
49
•Plithogenic DMAIC: A generalized approach to Dene-Measure-Analyze-Improve-Control, leveraging
plithogenic attributes to address complex operational challenges in Six Sigma processes.
•Plithogenic SWOT: An enriched version of Strengths-Weaknesses-Opportunities-Threats analysis, inte-
grating multi-dimensional perspectives and contradictions for strategic decision-making.
•Plithogenic OODA: A plithogenic adaptation of the Observe-Orient-Decide-Act loop, enabling exible
and adaptive responses in dynamic and uncertain environments.
•Plithogenic Five Forces: An extension of Porter’s Five Forces framework, incorporating multi-attribute
and contradictory factors to analyze industry competition with greater exibility and precision.
Theorem 4.1 (Generalization of Fuzzy and Neutrosophic Concepts in Plithogenic Frameworks).The frame-
works of Plithogenic Body-Mind-Soul-Spirit Fluidity, Plithogenic PDCA, Plithogenic DMAIC, Plithogenic
SWOT, Plithogenic OODA, and Plithogenic Five Forces extend the Fuzzy and Neutrosophic concepts by in-
tegrating multi-attribute, multi-criteria, and contradictory characteristics. These generalizations facilitate the
modeling and analysis of complex, multi-dimensional, and dynamic systems.
Proof. The claim is evident from the denitions of the Plithogenic frameworks. Similar proofs have been
provided in the literature [119,125, 129]. Readers may refer to these works for detailed justications if needed.
By applying plithogenic sets to these widely used frameworks, we hope to provide more robust tools for
decision-making, strategic planning, and continuous improvement in diverse real-world contexts.
4.1.2 Hyperanalysis and Hypercycle
We also hope that concepts such as Hyperanalysis/Hypercycle and Superhyperanalysis/SuperHypercycle, which
hierarchically represent the ideas presented in this paper, will be explored as needed. These approaches are
envisioned as applications of hyperstructure [41,51] and superhyperstructure [117,118,121,122,154, 331, 334]
principles to the concepts introduced in this study.
First, we provide the denitions related to hyperstructure and superhyperstructure below. In set theory, hyper-
structure and superhyperstructure can be viewed as the power set and nth-superhyperset, respectively. Intu-
itively, they represent iterative structures. For detailed denitions of Hyperstructure and Superhyperstructure,
readers are encouraged to refer to relevant works such as [122,318, 335] as needed.
Denition 4.2 (Powerset).[118] The powerset of a set 𝑆, denoted by P(𝑆), is the set of all subsets of 𝑆,
including both the empty set and 𝑆itself. Formally:
P(𝑆)={𝐴|𝐴⊆𝑆}.
Denition 4.3 (Hyperoperation).(cf. [294,386–388]) A hyperoperation is an extension of a traditional binary
operation where the result of applying the operation to two elements is a subset of the base set rather than a
single element. Formally, given a set 𝑆, a hyperoperation ◦is dened as:
◦:𝑆×𝑆→ P(𝑆),
where P(𝑆)is the powerset of 𝑆.
Denition 4.4 (Hyperstructure).(cf. [118, 318, 335]) A Hyperstructure is a mathematical construct that gen-
eralizes operations on a set using its powerset. Formally, it is dened as:
H=(P (𝑆),◦),
where:
•𝑆is the underlying base set.
50
•P(𝑆)denotes the powerset of 𝑆, which includes all subsets of 𝑆.
•◦is an operation acting on the elements of P ( 𝑆).
Denition 4.5 (𝑛-th Powerset).(cf. [118, 318, 335]) The 𝑛-th powerset of a set 𝐻, denoted as P𝑛(𝐻), is con-
structed recursively through successive powerset operations. Specically:
P1(𝐻)=P(𝐻),P𝑛+1(𝐻)=P (P𝑛(𝐻)) for 𝑛≥1.
Similarly, the 𝑛-th non-empty powerset, denoted as P∗
𝑛(𝐻), excludes the empty set at each level and is dened
as:
P∗
1(𝐻)=P∗(𝐻),P∗
𝑛+1(𝐻)=P∗(P ∗
𝑛(𝐻)),
where P∗(𝐻)represents the standard powerset P (𝐻)with the empty set removed.
Denition 4.6 (SuperHyperOperations).(cf. [335]) Let 𝐻be a non-empty set, and let P (𝐻)represent the
powerset of 𝐻. The 𝑛-th powerset, denoted as P𝑛(𝐻), is recursively dened as:
P0(𝐻)=𝐻, P𝑘+1(𝐻)=P (P 𝑘(𝐻)),∀𝑘≥0.
ASuperHyperOperation of order (𝑚, 𝑛)is an 𝑚-ary operation expressed as:
◦(𝑚,𝑛 ):𝐻𝑚→ P𝑛
∗(𝐻),
where P𝑛
∗(𝐻)denotes the 𝑛-th powerset of 𝐻, with two variations depending on inclusion or exclusion of the
empty set:
• If the codomain excludes the empty set, the operation is referred to as a classical-type (𝑚, 𝑛)-SuperHyperOperation.
• If the codomain includes the empty set, it is termed a Neutrosophic (𝑚, 𝑛)-SuperHyperOperation.
These SuperHyperOperations generalize hyperoperations to higher-order structures, accommodating multi-
layered relationships through iterative powerset constructions.
Denition 4.7 (𝑛-Superhyperstructure).(cf. [318,335]) An 𝑛-Superhyperstructure is an advanced extension of
a hyperstructure that incorporates 𝑛-fold iterations of the powerset operation. It is dened as:
SH 𝑛=(P𝑛(𝑆),◦),
where:
•𝑆is the base set.
•P𝑛(𝑆)represents the 𝑛-th powerset of 𝑆, obtained through recursive applications of the powerset opera-
tion.
•◦is an operation dened on elements of P𝑛(𝑆).
The aforementioned concepts of hyperstructure and superhyperstructure can be applied not only to various
mathematical frameworks but also to concepts beyond pure mathematics. Consequently, it is natural to consider
their applicability to the ideas presented in this paper. For instance, the denitions of the PDCA Hypercycle
and PDCA n-SuperhyperCycle are provided above. We anticipate further exploration of these frameworks and
their potential applications to other models.
Denition 4.8 (PDCA Hypercycle).APDCA Hypercycle is dened as:
H𝑃𝐷 𝐶 𝐴 =(P (𝑆),◦),
where 𝑆is a set of system states, and ◦maps:
H𝑃𝐷 𝐶 𝐴(𝑋)=𝐴(𝐶(𝐷(𝑃(𝑋)))), 𝑋 ⊆𝑆 .
51
Example 4.9 (PDCA Hypercycle in Quality Management).Consider a manufacturing process aimed at im-
proving product quality using the PDCA (Plan-Do-Check-Act) Hypercycle framework. The process can be
described as follows:
•𝑆={𝑠1, 𝑠2, . . . , 𝑠5}: A set of system states, where each 𝑠𝑖represents a dierent stage of product quality,
such as:
𝑠1=Initial state, 𝑠2=Design stage, 𝑠3=Production stage, 𝑠4=Quality inspection, 𝑠5=Defect correction.
•P(𝑆): The powerset of 𝑆, capturing all subsets of system states 𝑋⊆𝑆, such as:
𝑋={𝑠2, 𝑠3},P ( 𝑋)={ {𝑠2},{𝑠3},{𝑠2, 𝑠3}}.
• The PDCA Hypercycle operates through the following steps:
1. 𝑃(𝑋):Plan phase – Dene quality objectives and prepare production plans for the subset of states
𝑋={𝑠2, 𝑠3}. For example, improving the defect rate by optimizing production parameters.
2. 𝐷(𝑋):Do phase – Implement the plans, such as testing new production methods or upgrading
machinery in states 𝑠2and 𝑠3.
3. 𝐶(𝑋):Check phase – Evaluate the outcomes of the Do phase by inspecting the quality results and
collecting metrics, such as:
Defect rate reduced from 5% to 3%.
4. 𝐴(𝑋):Act phase – Adjust processes based on the Check phase results. For instance, ne-tune the
machine settings further or update training protocols for workers.
The PDCA Hypercycle iteratively renes 𝑋, evolving system states through higher-order feedback loops. The
process can be expressed mathematically as:
H𝑃𝐷 𝐶 𝐴(𝑋)=𝐴(𝐶(𝐷(𝑃(𝑋)))).
Outcome. After multiple iterations of the PDCA Hypercycle, the system achieves an improved state with a
defect rate of 1%, meeting the quality target.
Denition 4.10 (PDCA 𝑛-SuperhyperCycle).APDCA 𝑛-SuperhyperCycle is dened as:
SH 𝑛
𝑃𝐷 𝐶 𝐴 =(P 𝑛(𝑆),◦(4, 𝑛)),
where P𝑛(𝑆)is the 𝑛-th powerset of 𝑆, and:
SH 𝑛
𝑃𝐷 𝐶 𝐴(𝑋)=𝐴𝑛◦𝐶𝑛◦𝐷𝑛◦𝑃𝑛(𝑋), 𝑋 ∈ P 𝑛(𝑆).
Example 4.11 (PDCA 𝑛-SuperhyperCycle in Project Management).In a complex project management sce-
nario:
•𝑆: Tasks {𝑇1,𝑇2, . . . , 𝑇5}.
•P2(𝑆): Powerset of subsets of tasks, capturing interdependent subtasks and their groupings.
The PDCA 2-SuperhyperCycle proceeds as follows:
1. 𝑃2(𝑋): Generates plans across grouped subtasks. For example:
𝑃2(𝑋)={{𝑇1, 𝑇2},{𝑇3, 𝑇4}}.
52
2. 𝐷2(𝑋): Executes actions on these subsets, producing partial results.
3. 𝐶2(𝑋): Evaluates subset outcomes, such as task completion percentages.
4. 𝐴2(𝑋): Adjusts task groupings and priorities based on evaluation.
The process evolves 𝑋iteratively through multi-level renement, achieving higher-order optimization.
For clarication, as with the PDCA Hypercycle and n-SuperhyperCycle, the following concepts are dened.
We look forward to further research and advancements in these areas.
•DMAIC Hypercycle: A generalized approach to Dene-Measure-Analyze-Improve-Control, leveraging
hyperstructure attributes to address complex operational challenges within Six Sigma processes.
•SWOT Hyperanalysis: An enhanced version of the Strengths-Weaknesses-Opportunities-Threats anal-
ysis, integrating multi-dimensional perspectives and interdependencies to improve strategic decision-
making.
•OODA Hypercycle: A hyperstructure-based adaptation of the Observe-Orient-Decide-Act loop, enabling
exible and adaptive responses in dynamic and uncertain environments.
•Five Forces Hyperanalysis: An extended version of Porter’s Five Forces framework, incorporating multi-
attribute and interdependent factors to analyze industry competition with greater precision and adaptabil-
ity.
•DMAIC 𝑛-Superhypercycle: A higher-order extension of the Dene-Measure-Analyze-Improve-Control
process, addressing 𝑛-fold complexities through multi-level operational analysis.
•SWOT 𝑛-Superhyperanalysis: A multi-level enhancement of SWOT analysis, incorporating 𝑛-fold di-
mensions and contradictions to enable comprehensive strategic planning and decision-making.
•OODA 𝑛-Superhypercycle: A higher-order adaptation of the Observe-Orient-Decide-Act loop, capturing
𝑛-fold interdependencies to support adaptive and resilient decision-making in uncertain environments.
•Five Forces 𝑛-Superhyperanalysis: An advanced extension of Porter’s Five Forces model, integrating
𝑛-fold multi-attribute and hierarchical structures to analyze industry competition with greater depth and
exibility.
4.1.3 Other Frameworks
In addition to the frameworks discussed in this paper, numerous others are developed dailyacross various elds.
For example:
•COBIT (Control Objectives for Information and Related Technologies) [172,273],
•BADIR (Business Question, Analysis Plan, Data Collection, Insights Derivation, Recommendations)
[175],
•ITIL (Information Technology Infrastructure Library) [135,227],
•Five Whys [39,303, 353],
•Kanban [5,365],
•VRIO (Value, Rarity, Imitability, Organization) [192,235],
•OGSM (Objectives, Goals, Strategies, and Measures) [213,246, 274],
•PEST Analysis [78,106, 220].
We hope to explore the potential for extending these frameworks using concepts such as Neutrosophic Struc-
tures, Uncertain Structures, and Superhyperstructures. Future research may focus on examining the mathemat-
ical structures of these extended frameworks and exploring their applications in elds such as social sciences.
53
4.2 New Strategic Leadership
4.2.1 Neutrosophic Strategic Leadership
In addition to the concepts discussed in this paper, the neutrosophic framework can be applied to a variety of
other elds and ideas. As an example, we introduce the concept of Neutrosophic Strategic Leadership.
Leadership refers to the ability to inuence, guide, and inspire individuals or groups to achieve objectives
through eective communication, motivation, and vision [40,57]. Strategic Leadership, in particular, focuses
on balancing short-term goals with long-term vision, emphasizing resource allocation, adaptability, and orga-
nizational alignment to ensure sustained success [108,109, 384].
The related denitions and formalizations are presented below. It is important to note that leadership itself is
a multifaceted concept that can be dened and studied from various perspectives, depending on the context or
scope of analysis. The denitions provided here represent only one example among many.
We hope that future research will further explore concepts like Neutrosophic Strategic Leadership and its appli-
cations. Additionally, many related leadership frameworks have been studied extensively in existing literature,
including examples such as servant leadership [299], meta-leadership [95,225,226], e-leadership [29,84,184],
Agile leadership [28,181], and followership [36, 370], among others.
Denition 4.12 (Classic Leadership).Classic Leadership is a structured decision-making framework that for-
malizes the process of directing, inuencing, and coordinating individuals or groups to achieve organizational
goals. It is mathematically dened as a tuple:
L𝐶 𝐿 =(A,T,R,S,P),
where:
1. A={𝐴1, 𝐴2, . . . , 𝐴𝑛}: A set of agents (leaders and followers), where each agent 𝐴𝑖has attributes:
𝐴𝑖=(role,capabilities,preferences).
Here:
• role ∈ {Leader,Follower}denes the agent’s position.
• capabilities ∈R𝑑represents the skillset or competence vector in 𝑑-dimensional space.
• preferences ∈R𝑘indicates the agent’s goals or utility preferences.
2. T={𝑇1, 𝑇2, . . . , 𝑇𝑚}: A set of tasks to be accomplished, where each task 𝑇𝑗is dened as:
𝑇𝑗=(R 𝑗,O𝑗,C𝑗),
with:
•R𝑗: Resource requirements for 𝑇𝑗.
•O𝑗: The output or measurable outcome of 𝑇𝑗.
•C𝑗: Constraints, such as deadlines or quality thresholds.
3. R={𝑅1, 𝑅2, . . . , 𝑅𝑝}: A set of resources required to execute tasks, where each resource 𝑅𝑘has a nite
capacity:
𝑅𝑘=(type,capacity),capacity ∈R+.
4. S={𝑆1, 𝑆2, . . . , 𝑆𝑞}: A set of strategies for resource allocation and task assignment, where each strategy
𝑆𝑙maps agents and resources to tasks:
𝑆𝑙:A × R → T .
54
5. P={𝑃1, 𝑃2, . . . , 𝑃𝑟}: A set of performance metrics to evaluate leadership eectiveness. Each perfor-
mance metric 𝑃ℎis a mapping:
𝑃ℎ:T → R,
where 𝑃ℎ(𝑇𝑗)measures the success or eciency of completing task 𝑇𝑗.
Remark 4.13 (Components of Classic Leadership).Classic Leadership focuses on task execution and organi-
zational performance by:
• Aligning agents (A) with appropriate tasks (T) using their capabilities.
• Optimizing resource allocation (R) under constraints.
• Selecting strategies (S) to achieve goals eciently.
• Evaluating performance (P) based on measurable outcomes.
Example 4.14 (Classic Leadership in a Project Management Scenario).Consider a project with three agents,
two tasks, and nite resources:
A={𝐴1:Leader, 𝐴2:Follower, 𝐴3:Follower},
T={𝑇1:Design Phase, 𝑇2:Implementation Phase},
R={𝑅1:Budget = $10,000, 𝑅2:Human Resources = 5 engineers}.
The leader 𝐴1assigns resources and strategies:
𝑆1(𝐴2, 𝑅1) → 𝑇1, 𝑆1(𝐴3, 𝑅2) → 𝑇2.
The performance metrics 𝑃1evaluate success:
𝑃1(𝑇1)=90% completion, 𝑃1(𝑇2)=80% completion.
This example demonstrates how Classic Leadership optimizes task execution and resource utilization.
The concept of Neutrosophic Leadership, which integrates the principles of Neutrosophic Logic into the above
denition of leadership, is presented below.
Denition 4.15 (Neutrosophic Leadership).Neutrosophic Leadership is a mathematical framework that models
leadership under uncertainty, ambiguity, and contradiction by extending classical leadership principles with
neutrosophic logic. It incorporates truth (𝑇), indeterminacy (𝐼), and falsity (𝐹) to evaluate decisions, strategies,
and resource allocations. Formally, it is dened as:
L𝑁 𝐿 =(A,T,R𝑁,S𝑁,P𝑁),
where:
1. A={𝐴1, 𝐴2, . . . , 𝐴𝑛}: A set of agents (leaders and followers), where each agent 𝐴𝑖is dened as:
𝐴𝑖=(role,capabilities,N𝐴),
with:
• role ∈ {Leader,Follower}: The position of the agent.
• capabilities ∈R𝑑: The agent’s skillset or competence vector in 𝑑-dimensional space.
•N𝐴=(𝑇𝐴𝑖, 𝐼𝐴𝑖, 𝐹𝐴𝑖): A neutrosophic evaluation of the agent’s eectiveness, where:
𝑇𝐴𝑖(truth), 𝐼𝐴𝑖(indeterminacy), 𝐹𝐴𝑖(falsity) ∈ [0,1], 𝑇𝐴𝑖+𝐼𝐴𝑖+𝐹𝐴𝑖≤1.
2. T={𝑇1, 𝑇2, . . . , 𝑇𝑚}: A set of tasks, where each task 𝑇𝑗is described as:
𝑇𝑗=(R 𝑁
𝑗,O𝑗,C𝑗),
with:
55
•R𝑁
𝑗: Neutrosophic resource requirements evaluated as:
N ( 𝑅𝑗)=(𝑇𝑅𝑗, 𝐼𝑅𝑗, 𝐹𝑅𝑗).
•O𝑗: The outcome of task 𝑇𝑗.
•C𝑗: Task constraints such as deadlines or priorities.
3. R𝑁={𝑅𝑁
1, 𝑅𝑁
2, . . . , 𝑅𝑁
𝑝}: A set of neutrosophic resources, where each resource 𝑅𝑁
𝑘includes:
𝑅𝑁
𝑘=(type,capacity,N𝑅),N𝑅=(𝑇𝑅𝑘, 𝐼𝑅𝑘, 𝐹𝑅𝑘).
4. S𝑁={𝑆𝑁
1, 𝑆𝑁
2, . . . , 𝑆𝑁
𝑞}: A set of neutrosophic strategies that allocate agents and resources to tasks
under uncertainty:
𝑆𝑁
𝑙:A × R 𝑁→ T .
Each strategy 𝑆𝑁
𝑙is evaluated as:
N (𝑆𝑁
𝑙)=(𝑇𝑆𝑙, 𝐼𝑆𝑙, 𝐹𝑆𝑙).
5. P𝑁={𝑃𝑁
1, 𝑃𝑁
2, . . . , 𝑃𝑁
𝑟}: A set of neutrosophic performance metrics to evaluate leadership eective-
ness. Each performance metric 𝑃𝑁
ℎmaps tasks to neutrosophic evaluations:
𝑃𝑁
ℎ:T → R3, 𝑃 𝑁
ℎ(𝑇𝑗)=(𝑇𝑃ℎ, 𝐼𝑃ℎ, 𝐹𝑃ℎ).
Remark 4.16 (Characteristics of Neutrosophic Leadership).Neutrosophic Leadership extends classical lead-
ership by:
• Incorporating truth, indeterminacy, and falsity components into agents, resources, tasks, and strategies.
• Managing ambiguity and uncertainty in decision-making.
• Balancing resource allocation and performance evaluations under incomplete information.
Remark 4.17 (Neutrosophic Leadership).Fuzzy Leadership is a special case of Neutrosophic Leadership
where both indeterminacy and falsity are set to zero. Furthermore, Plithogenic Leadership is notable for its
ability to generalize both Neutrosophic and Fuzzy Leadership.
Example 4.18 (Neutrosophic Leadership in a Construction Project).Consider a construction project with three
agents, two tasks, and limited resources:
A={𝐴1:Leader, 𝐴2:Engineer, 𝐴3:Worker}.
The resources and tasks are as follows:
𝑅𝑁
1=(Budget = $50,000,N=(𝑇=0.8, 𝐼 =0.15, 𝐹 =0.05)), 𝑇1=(R 𝑁
1,Foundation work,C1).
The leader 𝐴1evaluates the strategy 𝑆𝑁
1as:
𝑆𝑁
1(𝐴2, 𝑅𝑁
1) → 𝑇1,N (𝑆𝑁
1)=(𝑇𝑆1=0.85, 𝐼𝑆1=0.1, 𝐹𝑆1=0.05).
The performance metric 𝑃𝑁
1for task 𝑇1is evaluated as:
𝑃𝑁
1(𝑇1)=(𝑇𝑃1=0.8, 𝐼𝑃1=0.15, 𝐹𝑃1=0.05).
This example demonstrates how neutrosophic leadership handles uncertainty and evaluates performance with
truth, indeterminacy, and falsity components.
Theorem 4.19. Neutrosophic Leadership generalizes Classical Leadership by incorporating uncertainty, in-
determinacy, and falsity into all components of leadership.
Proof. By denition, Classical Leadership uses precise values for agents, resources, and tasks. In Neutrosophic
Leadership, these components are extended to include neutrosophic evaluations (𝑇 , 𝐼, 𝐹). Since 𝑇+𝐼+𝐹≤1,
Neutrosophic Leadership preserves the classical framework while accommodating uncertainty and contradic-
tion. Hence, Classical Leadership is a special case of Neutrosophic Leadership when 𝐼=0and 𝐹=0.
56
Theorem 4.20. Neutrosophic Leadership improves decision-making under uncertainty compared to Classical
Leadership.
Proof. In Classical Leadership, decisions are based solely on precise values. In Neutrosophic Leadership,
decisions incorporate uncertainty (𝐼) and falsity (𝐹) to provide a more robust evaluation. For any strategy 𝑆𝑁,
the neutrosophic evaluation:
N (𝑆𝑁)=(𝑇𝑆, 𝐼𝑆, 𝐹𝑆),
allows leaders to account for ambiguity and risk. By assigning weights to 𝐼and 𝐹, decisions reect a realistic
assessment of uncertain environments, which improves outcomes.
Next, the denition of Strategic Leadership is provided below.
Denition 4.21 (Strategic Leadership).Strategic Leadership is a mathematical framework for decision-making
and organizational guidance, balancing short-term and long-term goals through resource allocation, environ-
mental analysis, and stakeholder alignment. Formally, it is dened as a tuple:
L𝑆𝐿 =(V,O,R,D,E),
where:
1. V={𝑉1, 𝑉2, . . . , 𝑉𝑛}: A set of organizational visions or objectives, where each 𝑉𝑖is a function:
𝑉𝑖:T → R,
mapping time horizons Tto a measurable outcome, such as prot, market share, or sustainability.
2. O={𝑂1, 𝑂2, . . . , 𝑂𝑚}: A set of operational strategies, where each 𝑂𝑗is dened as:
𝑂𝑗:R → V ,
mapping resources Rto organizational objectives.
3. R={𝑅1, 𝑅2, . . . , 𝑅𝑝}: A set of resources, where each 𝑅𝑘is a tuple:
𝑅𝑘=(nancial,human,technological),
representing resource allocations across critical categories.
4. D={𝐷1, 𝐷2, . . . , 𝐷 𝑞}: A set of strategic decisions, where each 𝐷𝑙is dened as:
𝐷𝑙:(O ,E) → V ,
mapping operational strategies and environmental states to organizational objectives.
5. E={𝐸1, 𝐸2, . . . , 𝐸𝑟}: A set of environmental states, where each 𝐸ℎrepresents external conditions,
modeled as:
𝐸ℎ:T → S,
with Sbeing a set of state variables, such as market trends, regulatory changes, or competitive dynamics.
Remark 4.22 (Components and Relationships).The framework integrates the following key components:
•Vision Alignment: Leaders optimize:
max
𝑂𝑗∈ O
𝑛
Õ
𝑖=1
𝛼𝑖𝑉𝑖(𝑇),
where 𝛼𝑖represents the weight assigned to each objective 𝑉𝑖at time 𝑇.
57
•Resource Allocation: Resources Rare allocated by solving:
min
𝑅𝑘∈ R ©«
𝑚
Õ
𝑗=1
𝛽𝑗𝑂𝑗(𝑅𝑘) − 𝛾C ( 𝑅𝑘)ª®¬,
where 𝛽𝑗is the importance of strategy 𝑂𝑗,𝛾is a penalty factor, and C( 𝑅𝑘)is the cost function of resource
𝑅𝑘.
•Adaptability: Strategic decisions Dadapt to environmental states by satisfying:
𝐷𝑙(𝑂𝑗, 𝐸ℎ)=arg max
𝑂𝑗
𝑛
Õ
𝑖=1
𝛿𝑖𝑉𝑖(𝐸ℎ(𝑇)),
where 𝛿𝑖represents the sensitivity of 𝑉𝑖to 𝐸ℎ(𝑇).
Remark 4.23 (The dierences between Strategic Leadership and Classical Leadership ).The dierences be-
tween Strategic Leadership and Classical Leadership are summarized as follows:
1. Focus and Goals:
•Classical Leadership: Task-oriented, focusing on short-term objectives and immediate resource
utilization.
•Strategic Leadership: Balances short-term goals and long-term visions by aligning resources and
strategies for sustainability.
2. Decision-Making Framework:
•Classical Leadership: Uses predened roles and strategies for decision-making.
•Strategic Leadership: Incorporates exibility by dynamically adapting to external conditions.
3. Resource Management:
•Classical Leadership: Focuses on resource allocation for immediate task execution.
•Strategic Leadership: Dynamically allocates resources to achieve broader, long-term organizational
goals.
4. Environmental Adaptability:
•Classical Leadership: Assumes a static environment with limited external inuence.
•Strategic Leadership: Explicitly models external conditions (E) and adapts to changing environ-
ments.
5. Evaluation:
•Classical Leadership: Evaluates performance using task-specic metrics (P).
•Strategic Leadership: Measures success using broader, vision-oriented metrics (V).
Example 4.24 (Strategic Leadership in Renewable Energy Development).Consider a renewable energy com-
pany aiming to expand its operations by balancing short-term protability with long-term sustainability. The
components of Strategic Leadership L𝑆𝐿 are instantiated as follows:
1. V={𝑉1, 𝑉2,𝑉3}: The organizational visions are dened as:
•𝑉1(𝑇): Short-term protability, measured in millions of dollars over time 𝑇.
•𝑉2(𝑇): Long-term sustainability, quantied as the percentage of energy sourced from renewable
resources over time 𝑇.
•𝑉3(𝑇): Market share in the renewable energy sector, measured as a percentage over time 𝑇.
2. O={𝑂1, 𝑂2, 𝑂 3}: The operational strategies are:
58
•𝑂1(𝑅): Investing in wind energy infrastructure.
•𝑂2(𝑅): Developing solar energy projects.
•𝑂3(𝑅): Marketing campaigns to promote renewable energy solutions.
Each strategy 𝑂𝑗maps resource allocations 𝑅to organizational objectives V.
3. R={𝑅1, 𝑅2, 𝑅3}: The resource allocations are:
𝑅1=($50 M,200 employees,wind turbines),
𝑅2=($30 M,150 employees,solar panels),
𝑅3=($20 M,50 employees,marketing tools).
4. D={𝐷1, 𝐷2}: The strategic decisions are:
•𝐷1(𝑂, 𝐸 ): Allocating 60% of resources to 𝑂1and 40% to 𝑂2, based on favorable environmental
conditions 𝐸.
•𝐷2(𝑂, 𝐸 ): Shifting resources to 𝑂3during periods of high public demand for renewable energy
awareness.
5. E={𝐸1, 𝐸2}: The environmental states are:
•𝐸1(𝑇): Government incentives for renewable energy projects.
•𝐸2(𝑇): Fluctuations in fossil fuel prices aecting market dynamics.
These states are modeled as functions of time, inuencing operational strategies and resource allocations.
Optimization: The company optimizes its strategies by solving:
max
𝑂𝑗∈ O (𝛼1𝑉1(𝑇) + 𝛼2𝑉2(𝑇) + 𝛼3𝑉3(𝑇)),
where 𝛼1=0.4,𝛼2=0.4, and 𝛼3=0.2reect the relative importance of each objective.
Adaptability: Strategic decisions are adjusted dynamically based on environmental changes. For instance, when
𝐸1(𝑇)increases government subsidies, the company prioritizes 𝑂1and 𝑂2, maximizing long-term sustainabil-
ity.
Resource Allocation: Resources 𝑅𝑘are allocated to minimize costs:
min
𝑅𝑘∈ R ©«
3
Õ
𝑗=1
𝛽𝑗𝑂𝑗(𝑅𝑘) − 𝛾C ( 𝑅𝑘)ª®¬,
where 𝛽𝑗is the importance of each strategy, and C( 𝑅𝑘)represents resource costs.
This framework ensures the company achieves its objectives while remaining responsive to market and envi-
ronmental dynamics.
We extend the above framework using Neutrosophic Sets to introduce Neutrosophic Strategic Leadership. The
followingoutlines this concept. We anticipate that further research and validation of this approach will progress
in the future.
Denition 4.25 (Neutrosophic Strategic Leadership).Neutrosophic Strategic Leadership (NSL) extends clas-
sical Strategic Leadership by incorporating uncertainty, indeterminacy, and falsity into the decision-making
process. It is dened as:
L𝑁 𝑆𝐿 =( V 𝑁,O𝑁,R𝑁,D𝑁,E𝑁),
where:
59
1. V𝑁={𝑉𝑁
1, 𝑉 𝑁
2, . . . , 𝑉 𝑁
𝑛}: A set of neutrosophic organizational visions or objectives, where each 𝑉𝑁
𝑖
is a mapping:
𝑉𝑁
𝑖:T → R3,
such that:
𝑉𝑁
𝑖(𝑇)=(𝑇𝑉𝑖, 𝐼𝑉𝑖, 𝐹𝑉𝑖),
where 𝑇𝑉𝑖, 𝐼𝑉𝑖, 𝐹𝑉𝑖∈ [0,1]represent the truth, indeterminacy, and falsity of achieving 𝑉𝑖over the time
horizon 𝑇, satisfying 𝑇𝑉𝑖+𝐼𝑉𝑖+𝐹𝑉𝑖≤1.
2. O𝑁={𝑂𝑁
1, 𝑂 𝑁
2, . . . , 𝑂 𝑁
𝑚}: A set of neutrosophic operational strategies, where each 𝑂𝑁
𝑗is dened as:
𝑂𝑁
𝑗:R𝑁→ V 𝑁,
mapping neutrosophic resource allocations R𝑁to neutrosophic organizational objectives V𝑁.
3. R𝑁={𝑅𝑁
1, 𝑅𝑁
2, . . . , 𝑅𝑁
𝑝}: A set of neutrosophic resources, where each 𝑅𝑁
𝑘is a tuple:
𝑅𝑁
𝑘=(nancial,human,technological,N),
and Nassigns a neutrosophic value:
N ( 𝑅𝑁
𝑘)=(𝑇𝑅𝑘, 𝐼𝑅𝑘, 𝐹𝑅𝑘),
representing the truth, indeterminacy, and falsity of the eectiveness of resource 𝑅𝑁
𝑘.
4. D𝑁={𝐷𝑁
1, 𝐷 𝑁
2, . . . , 𝐷𝑁
𝑞}: A set of neutrosophic strategic decisions, where each 𝐷𝑁
𝑙is dened as:
𝐷𝑁
𝑙:(O 𝑁,E𝑁) → V 𝑁,
mapping neutrosophic operational strategies and neutrosophic environmental states to neutrosophic or-
ganizational objectives.
5. E𝑁={𝐸𝑁
1, 𝐸 𝑁
2, . . . , 𝐸 𝑁
𝑟}: A set of neutrosophic environmental states, where each 𝐸𝑁
ℎrepresents ex-
ternal conditions, modeled as:
𝐸𝑁
ℎ:T → R3,
such that:
𝐸𝑁
ℎ(𝑇)=(𝑇𝐸ℎ, 𝐼𝐸ℎ, 𝐹𝐸ℎ),
where 𝑇𝐸ℎ, 𝐼𝐸ℎ, 𝐹𝐸ℎ∈ [0,1]denote the truth, indeterminacy, and falsity of the state variables at time 𝑇,
satisfying 𝑇𝐸ℎ+𝐼𝐸ℎ+𝐹𝐸ℎ≤1.
Remark 4.26 (Neutrosophic Strategic Leadership).Fuzzy Strategic Leadership is a special case of Neutro-
sophic Strategic Leadership where both indeterminacy and falsity are set to zero. Furthermore, Plithogenic
Strategic Leadership is notable for its ability to generalize both Neutrosophic and Fuzzy Strategic Leadership.
Example 4.27 (Application in Corporate Sustainability).A company evaluates its sustainability strategy (cf.
[132,161, 389]) under uncertain environmental regulations:
•V𝑁: The objective ”achieve 50% renewable energy usage by 2030” is represented as:
𝑉𝑁
1(𝑇)=(𝑇𝑉1, 𝐼𝑉1, 𝐹𝑉1)=(0.6,0.3,0.1).
•O𝑁: Operational strategies include investments in solar and wind energy, each evaluated with neutro-
sophic values:
𝑂𝑁
solar (𝑅)=(𝑇=0.7, 𝐼 =0.2, 𝐹 =0.1).
•R𝑁: Resources for solar investments have a neutrosophic eectiveness:
𝑅𝑁
solar =($100𝑀 , 500 employees,solar panels,(𝑇=0.8, 𝐼 =0.1, 𝐹 =0.1)).
•E𝑁: Environmental state ”government incentives for renewables” is represented as:
𝐸𝑁
incentives(𝑇)=(𝑇𝐸, 𝐼𝐸, 𝐹𝐸)=(0.7,0.2,0.1).
The framework ensures robust decision-making by balancing 𝑇,𝐼, and 𝐹across all components.
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4.2.2 HyperLeadership
Furthermore, we anticipate future advancements in the research on the applications and validity of HyperLead-
ership and n-SuperhyperLeadership, which extend the principles of hyperstructure and superhyperstructure to
leadership. Although these ideas remain at the conceptual stage, the denitions are outlined below. As previ-
ously mentioned, for detailed denitions of Hyperstructure and Superhyperstructure, readers are encouraged to
consult relevant works such as [119,122, 318,335] as needed.
Denition 4.28 (HyperLeadership).HyperLeadership is an extended leadership framework that operates on
the powerset of agents, tasks, and resources, capturing hierarchical, multi-level, and interdependent leadership
dynamics. It is formally dened as a tuple:
H𝐻 𝐿 =(P (A),P (T ),P (R ),P (S),P ( P)) ,
where:
1. P(A): The powerset of agents, including individual agents and their groupings:
P(A) ={𝐴, 𝐴0⊆ A | 𝐴≠∅}.
2. P (T ) : The powerset of tasks, capturing interdependencies between tasks:
P (T ) ={𝑇, 𝑇 0⊆ T | 𝑇≠∅}.
3. P(R): The powerset of resources, representing combinations and allocations:
P(R) ={𝑅, 𝑅0⊆ R | 𝑅≠∅}.
4. P(S ): The powerset of strategies, where each strategy subset assigns resources and agents to task subsets:
𝑆:P (A ) × P (R ) → P ( T ).
5. P(P): The powerset of performance metrics, evaluating leadership eectiveness at various levels:
𝑃ℎ:P (T ) → R.
Remark 4.29. HyperLeadership extends Classic Leadership by incorporating higher-order interactions among
agents, tasks, and resources. It enables hierarchical grouping and complex interrelations across organizational
levels.
Denition 4.30 (𝑛-SuperhyperLeadership).𝑛-SuperhyperLeadership is a higher-order generalization of Hy-
perLeadership achieved through 𝑛-fold applications of the powerset operation. It is formally dened as:
SH L 𝑛=(P 𝑛(A),P𝑛( T ),P𝑛(R),P𝑛(S),P𝑛( P)),
where:
1. P𝑛(A): The 𝑛-th powerset of agents, recursively dened as:
P0(A) =A,P𝑘+1(A) =P(P𝑘(A)), 𝑘 ≥0.
2. P𝑛(T ) : The 𝑛-th powerset of tasks, capturing multi-layered task hierarchies:
P𝑛(T ) =P ( P𝑛−1( T )).
3. P𝑛(R): The 𝑛-th powerset of resources, describing higher-order combinations and allocations:
P𝑛(R) =P ( P 𝑛−1(R) ).
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4. P𝑛(S): The 𝑛-th powerset of strategies, mapping higher-order subsets of agents and resources to task
hierarchies:
𝑆𝑛:P𝑛(A) × P 𝑛(R) → P𝑛( T ).
5. P𝑛(P ): The 𝑛-th powerset of performance metrics, evaluating leadership eectiveness at multi-level
task structures:
𝑃𝑛:P𝑛(T ) → R.
Remark 4.31. 𝑛-SuperhyperLeadership provides a comprehensive framework for analyzing and managing
leadership dynamics across multiple organizational layers, accounting for interdependencies, feedback loops,
and iterative renements.
Example 4.32 (HyperLeadership in Multi-Team Project Management).Consider a project with three teams of
agents (A), six tasks (T), and three types of resources (R):
A={Team 1,Team 2,Team 3},T={𝑇1, 𝑇2, . . . ,𝑇6},R={𝑅1, 𝑅2, 𝑅3}.
• HyperLeadership generates subsets of agents, tasks, and resources:
P(A) ={{Team 1},{Team 2, Team 3}, . . . }.
• A strategy 𝑆maps teams and resources to tasks:
𝑆({Team 1, Team 2},{𝑅1, 𝑅2}) → {𝑇1, 𝑇3,𝑇4}.
• Performance metrics evaluate task outcomes:
𝑃(𝑇1, 𝑇3, 𝑇4)=85% completion.
This example illustrates the role of HyperLeadership in managing interdependent teams, tasks, and resources.
4.3 New Negotiation Theory
4.3.1 Neutrosophic Negotiation Theory
Negotiation Theory is the study of strategies and processes that parties use to reach agreements, focusing on
balancing interests, alternatives, and outcomes [3, 96, 361, 394,395]. In Negotiation Theory, the frameworks
of BATNA and ZOPA are well known. BATNA refers to the best outcome a party can achieve if negotiations
fail, serving as their most advantageous alternative or fallback option [50,73, 228,293,301]. ZOPA is the range
of possible agreements where both parties’ outcomes overlap, enabling a mutually benecial deal; outside this
range, no rational agreement can be reached [185,212, 238, 401].
Denition 4.33 (Best Alternative to a Negotiated Agreement (BATNA)).Let 𝑁represent a negotiation between
two parties, 𝐴(Agent 1) and 𝐵(Agent 2), where the set of all possible deals is D ⊆ R2.
The Best Alternative to a Negotiated Agreement (BATNA) for each party is the utility associated with their best
achievable outcome if no agreement is reached. Formally:
BATNA𝑖=max
𝛼∈ A𝑖
𝑈𝑖(𝛼), 𝑖 ∈ { 𝐴, 𝐵},
where:
•A𝑖: The set of alternatives available to party 𝑖outside the current negotiation 𝑁(e.g., other partners,
fallback options).
•𝑈𝑖:A𝑖→R: The utility function of party 𝑖, representing their valuation for each alternative outcome.
• BATNA𝑖: The maximum utility value party 𝑖can achieve independently of the current negotiation.
62
Interpretation. The BATNA represents the threshold utility for each party to accept any negotiated deal
𝑑∈ D. Specically, party 𝑖will accept a deal 𝑑only if:
𝑈𝑖(𝑑) ≥ BATNA𝑖.
Denition 4.34 (Zone of Possible Agreement (ZOPA)).The Zone of Possible Agreement (ZOPA) is the set of
feasible deals where both parties’ utilities meet or exceed their respective BATNAs.
Let 𝑈𝐴:D → Rand 𝑈𝐵:D → Rrepresent the utility functions of parties 𝐴and 𝐵, respectively. Then the
ZOPA is dened as:
ZOPA ={𝑑∈ D | 𝑈𝐴(𝑑) ≥ BATNA𝐴and 𝑈𝐵(𝑑) ≥ BATNA𝐵},
where:
•D ⊆ R2: The set of all possible deals 𝑑=(𝑑𝐴, 𝑑𝐵), where 𝑑𝐴and 𝑑𝐵represent the utilities for parties
𝐴and 𝐵, respectively.
•𝑈𝐴(𝑑)and 𝑈𝐵(𝑑): The utilities for parties 𝐴and 𝐵when deal 𝑑is agreed upon.
• BATNA𝐴and BATNA𝐵: The BATNAs for parties 𝐴and 𝐵, as dened earlier.
Conditions for ZOPA Existence. The ZOPA exists if and only if there exists a deal 𝑑∈ D such that:
𝑈𝐴(𝑑) ≥ BATNA𝐴and 𝑈𝐵(𝑑) ≥ BATNA𝐵.
The conditions for the existence of a ZOPA can be expressed as:
max
𝑑∈ D 𝑈𝐴(𝑑) ≥ BATNA𝐴and max
𝑑∈ D 𝑈𝐵(𝑑) ≥ BATNA𝐵.
Negative Bargaining Zone. If no such 𝑑∈ D exists where both conditions hold, then the ZOPA does not
exist, and the negotiation is said to have a Negative Bargaining Zone (NBZ).
Example 4.35 (ZOPA in Practice).Suppose two parties 𝐴and 𝐵negotiate over the price of a car. Let:
BATNA𝐴=5,000 and BATNA𝐵=4,500.
The possible deals 𝑑(prices) are represented by 𝑑∈ D =[4,000,6,000], where:
𝑈𝐴(𝑑)=6,000 −𝑑and 𝑈𝐵(𝑑)=𝑑−4,000.
The ZOPA is the set of prices 𝑑where both utilities exceed their BATNAs:
6,000 −𝑑≥5,000 and 𝑑−4,000 ≥4,500.
Simplifying these conditions gives:
𝑑≤5,000 and 𝑑≥4,500.
Therefore, the ZOPA is:
ZOPA =[4,500,5,000].
The above concepts are extended by incorporating the conditions of the Neutrosophic Set.
Denition 4.36 (Neutrosophic Best Alternative to a Negotiated Agreement (Neutrosophic BATNA)).Let 𝑁
represent a negotiation between two parties 𝐴(Agent 1) and 𝐵(Agent 2), where the set of all possible deals is
D ⊆ R2. The Neutrosophic Best Alternative to a Negotiated Agreement (Neutrosophic BATNA) incorporates
the degrees of truth (𝑇), indeterminacy (𝐼), and falsity (𝐹) into the evaluation of alternatives.
Formally, the Neutrosophic BATNA for each party 𝑖∈ {𝐴, 𝐵}is dened as:
NBATNA𝑖=max
𝛼∈ A𝑖
N𝑖(𝛼),N𝑖(𝛼)=(𝑇𝑖(𝛼), 𝐼𝑖(𝛼), 𝐹𝑖(𝛼)),
where:
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•A𝑖: The set of alternatives available to party 𝑖outside the current negotiation 𝑁(e.g., fallback options,
external agreements).
•N𝑖:A𝑖→ [0,1]3: The neutrosophic utility function of party 𝑖, mapping each alternative 𝛼to a tuple:
N𝑖(𝛼)=(𝑇𝑖(𝛼), 𝐼𝑖(𝛼), 𝐹𝑖(𝛼)),
where:
𝑇𝑖(𝛼) + 𝐼𝑖(𝛼) + 𝐹𝑖(𝛼) ≤ 1, 𝑇𝑖, 𝐼𝑖, 𝐹𝑖∈ [0,1].
• NBATNA𝑖: The maximum neutrosophic utility for party 𝑖, which quanties the best outcome they can
achieve independently.
Acceptance Condition. For any negotiated deal 𝑑∈ D, party 𝑖will only accept 𝑑if:
N𝑖(𝑑) NBATNA𝑖,
where denotes a partial order such that:
(𝑇𝑖(𝑑), 𝐼𝑖(𝑑), 𝐹𝑖(𝑑)) (𝑇𝑖, 𝐼𝑖, 𝐹𝑖) ⇐⇒ 𝑇𝑖(𝑑) ≥ 𝑇𝑖, 𝐼𝑖(𝑑) ≤ 𝐼𝑖,and 𝐹𝑖(𝑑) ≤ 𝐹𝑖.
Remark 4.37 (Neutrosophic BATNA).Fuzzy BATNA is a special case of Neutrosophic BATNA where both
indeterminacy and falsity are set to zero. Furthermore, Plithogenic BATNA is notable for its ability to generalize
both Neutrosophic and Fuzzy BATNA.
Theorem 4.38. The Neutrosophic BATNA exhibits the structure of a Neutrosophic Set.
Proof. The result follows directly from the denition.
Theorem 4.39. The Neutrosophic BATNA exhibits the structure of a Classic BATNA.
Proof. The result follows directly from the denition.
Denition 4.40 (Neutrosophic Zone of Possible Agreement (Neutrosophic ZOPA)).The Neutrosophic Zone
of Possible Agreement (Neutrosophic ZOPA) is the set of feasible deals where the neutrosophic utility of both
parties meets or exceeds their respective Neutrosophic BATNAs.
Let N𝐴:D → [0,1]3and N𝐵:D → [0,1]3represent the neutrosophic utility functions of parties 𝐴and 𝐵,
respectively. Then the Neutrosophic ZOPA is dened as:
NZOPA ={𝑑∈ D | N𝐴(𝑑) NBATNA𝐴and N𝐵(𝑑) NBATNA𝐵}.
Existence Condition. The Neutrosophic ZOPA exists if and only if there exists a deal 𝑑∈ D such that:
N𝐴(𝑑) NBATNA𝐴and N𝐵(𝑑) NBATNA𝐵.
If no such deal 𝑑exists, the negotiation is said to have a Neutrosophic Negative Bargaining Zone (NNBZ).
Remark 4.41 (Neutrosophic ZOPA).Fuzzy ZOPA is a special case of Neutrosophic ZOPA where both inde-
terminacy and falsity are set to zero. Furthermore, Plithogenic ZOPA is notable for its ability to generalize both
Neutrosophic and Fuzzy ZOPA.
Example 4.42 (Neutrosophic ZOPA in Practice).Suppose two parties 𝐴and 𝐵negotiate over a service fee.
Their Neutrosophic BATNAs are:
NBATNA𝐴=(0.8,0.1,0.1),NBATNA𝐵=(0.7,0.2,0.1).
The possible deals 𝑑∈ D are represented by their Neutrosophic utility values:
N𝐴(𝑑)=(𝑇𝐴(𝑑), 𝐼𝐴(𝑑), 𝐹𝐴(𝑑)) ,N𝐵(𝑑)=(𝑇𝐵(𝑑), 𝐼𝐵(𝑑), 𝐹𝐵(𝑑)).
64
For a deal 𝑑to belong to the Neutrosophic ZOPA, the following conditions must hold:
N𝐴(𝑑) (0.8,0.1,0.1)and N𝐵(𝑑) (0.7,0.2,0.1).
Assume a deal 𝑑1has the following utilities:
N𝐴(𝑑1)=(0.85,0.08,0.07),N𝐵(𝑑1)=(0.75,0.15,0.1).
Since both conditions are satised:
0.85 ≥0.8,0.08 ≤0.1,0.07 ≤0.1,and 0.75 ≥0.7,0.15 ≤0.2,0.1≤0.1,
we conclude that 𝑑1∈NZOPA.
Theorem 4.43. The Neutrosophic ZOPA exhibits the structure of a Neutrosophic Set.
Proof. The result follows directly from the denition.
Theorem 4.44. The Neutrosophic ZOPA exhibits the structure of a Classic ZOPA.
Proof. The result follows directly from the denition.
4.4 New Framing
4.4.1 Neutrosophic Framing
Framing is the presentation of identical information in dierent ways, inuencing decision-making behavior
by altering perception of outcomes and choices.
Denition 4.45 (Framing).Framing is a representation of a decision problem where the same problem is
presented in dierent ways, inuencing decision-making behavior and preferences. Mathematically, a frame 𝐹
is dened as:
𝐹=(𝐴, 𝑂 , 𝑃, 𝑉 , 𝑈),
where:
•𝐴={𝑎1, 𝑎2, . . . , 𝑎𝑛}: The set of available actions or choices.
•𝑂={𝑜1, 𝑜2, . . . , 𝑜𝑚}: The set of possible outcomes.
•𝑃:𝐴×𝑂→ [0,1]: The probability function, assigning a probability 𝑃(𝑜𝑗|𝑎𝑖)to each outcome 𝑜𝑗∈𝑂
for a given action 𝑎𝑖∈𝐴, satisfying:
∀𝑎𝑖∈𝐴, Õ
𝑜𝑗∈𝑂
𝑃(𝑜𝑗|𝑎𝑖)=1.
•𝑉:𝑂→R: The valuation function, assigning a numerical value 𝑉(𝑜𝑗)(e.g., gain or loss) to each
outcome 𝑜𝑗∈𝑂.
•𝑈:𝐴→R: The utility function, dened as:
𝑈(𝑎𝑖)=Õ
𝑜𝑗∈𝑂
𝑃(𝑜𝑗|𝑎𝑖) · 𝑉(𝑜𝑗).
The decision-maker selects the action 𝑎∗∈𝐴that maximizes their perceived utility:
𝑎∗=arg max
𝑎𝑖∈𝐴𝑈(𝑎𝑖).
65
Remark 4.46 (Impact of Framing).Framing inuences 𝑉, the valuation of outcomes, depending on how the
outcomes are presented. Specically:
• A positive frame presents outcomes as gains, leading to risk-averse behavior.
• A negative frame presents outcomes as losses, leading to risk-seeking behavior.
Thus, the same 𝐴,𝑂, and 𝑃may yield dierent decisions due to changes in 𝑉.
Example 4.47 (Framing Eect: Risk Preferences).Consider two equivalent frames for a medical treatment
decision:
•Positive Frame: ”200 lives will be saved.”
•Negative Frame: ”400 people will die.”
The outcomes 𝑂={𝑜1, 𝑜2}are identical, with:
𝑉(𝑜1)=200 lives saved, 𝑉 (𝑜2)=400 lives lost.
Given the same probabilities 𝑃, a decision-maker under the positive frame tends to be risk-averse, favoring a
certain outcome (e.g., saving 200 lives). Under the negative frame, the decision-maker becomes risk-seeking,
preferring uncertain options to avoid losses.
Theorem 4.48 (Framing-Induced Preference Reversal).Let 𝐹1and 𝐹2represent two frames of the same deci-
sion problem with identical 𝐴,𝑂, and 𝑃, but dierent valuations 𝑉1and 𝑉2. Then:
𝑈1(𝑎𝑖)≠𝑈2(𝑎𝑖)for some 𝑎𝑖∈𝐴=⇒preference reversal.
Proof. The utility 𝑈depends on 𝑉, the valuation of outcomes:
𝑈𝑘(𝑎𝑖)=Õ
𝑜𝑗∈𝑂
𝑃(𝑜𝑗|𝑎𝑖) · 𝑉𝑘(𝑜𝑗), 𝑘 =1,2.
If 𝑉1(𝑜𝑗)≠𝑉2(𝑜𝑗)for at least one 𝑜𝑗∈𝑂, then:
𝑈1(𝑎𝑖)≠𝑈2(𝑎𝑖).
This dierence in utilities alters the decision-maker’s ranking of actions 𝐴, leading to a preference reversal.
Denition 4.49 (Neutrosophic Framing).Neutrosophic Framing is a mathematical representation of a deci-
sion problem where uncertainty, ambiguity, and contradiction are explicitly incorporated into the evaluation of
outcomes. A Neutrosophic frame 𝐹𝑁is dened as:
𝐹𝑁=(𝐴, 𝑂 , 𝑃, 𝑉 𝑁, 𝑈𝑁),
where:
•𝐴={𝑎1, 𝑎2, . . . , 𝑎𝑛}: The set of available actions or choices.
•𝑂={𝑜1, 𝑜2, . . . , 𝑜𝑚}: The set of possible outcomes.
•𝑃:𝐴×𝑂→ [0,1]: The probability function, which assigns a probability 𝑃(𝑜𝑗|𝑎𝑖)to each outcome
𝑜𝑗∈𝑂given action 𝑎𝑖∈𝐴. The function satises:
∀𝑎𝑖∈𝐴, Õ
𝑜𝑗∈𝑂
𝑃(𝑜𝑗|𝑎𝑖)=1.
66
•𝑉𝑁:𝑂→ [0,1]3: The neutrosophic valuation function, which assigns a triple 𝑉𝑁(𝑜𝑗)=(𝑇𝑜𝑗, 𝐼𝑜𝑗, 𝐹𝑜𝑗)
to each outcome 𝑜𝑗∈𝑂, where:
–𝑇𝑜𝑗: The degree of truth (positive evaluation) of the outcome 𝑜𝑗.
–𝐼𝑜𝑗: The degree of indeterminacy (uncertainty or ambiguity) of the outcome 𝑜𝑗.
–𝐹𝑜𝑗: The degree of falsity (negative evaluation) of the outcome 𝑜𝑗.
–𝑇𝑜𝑗+𝐼𝑜𝑗+𝐹𝑜𝑗≤1: Consistency condition ensuring the total evaluation remains bounded.
•𝑈𝑁:𝐴→ [0,1]3: The neutrosophic utility function, dened for each action 𝑎𝑖∈𝐴as:
𝑈𝑁(𝑎𝑖)=𝑇𝑎𝑖, 𝐼𝑎𝑖, 𝐹𝑎𝑖,
where:
𝑇𝑎𝑖=Õ
𝑜𝑗∈𝑂
𝑃(𝑜𝑗|𝑎𝑖) · 𝑇𝑜𝑗, 𝐼𝑎𝑖=Õ
𝑜𝑗∈𝑂
𝑃(𝑜𝑗|𝑎𝑖) · 𝐼𝑜𝑗, 𝐹𝑎𝑖=Õ
𝑜𝑗∈𝑂
𝑃(𝑜𝑗|𝑎𝑖) · 𝐹𝑜𝑗.
The decision-maker selects the action 𝑎∗∈𝐴that maximizes the truth utility 𝑇𝑎𝑖while considering the inde-
terminacy 𝐼𝑎𝑖and falsity 𝐹𝑎𝑖:
𝑎∗=arg max
𝑎𝑖∈𝐴𝑇𝑎𝑖.
Remark 4.50 (Neutrosophic Valuation and Decision-Making).Neutrosophic framing allows for a richer eval-
uation of decision problems by incorporating:
• Positive outcomes (𝑇) that contribute directly to utility.
• Uncertain or ambiguous outcomes (𝐼), which reect incomplete or unclear information.
• Negative outcomes (𝐹) that reect losses or contradictions.
This framework can model real-world scenarios where outcomes are not purely true or false but lie within a
range of truth, uncertainty, and falsity.
Remark 4.51 (Neutrosophic framing).Fuzzy framing is a special case of Neutrosophic framing where both
indeterminacy and falsity are set to zero. Furthermore, Plithogenic framing is notable for its ability to generalize
both Neutrosophic and Fuzzy framing.
Example 4.52 (Neutrosophic Framing in Decision-Making).Consider a decision-maker choosing between two
investment options 𝐴={𝑎1, 𝑎 2}with uncertain outcomes 𝑂={𝑜1, 𝑜2}.
•Action 𝑎1leads to outcome 𝑜1with:
𝑉𝑁(𝑜1)=(𝑇𝑜1, 𝐼𝑜1, 𝐹𝑜1)=(0.7,0.2,0.1), 𝑃(𝑜1|𝑎1)=0.8.
•Action 𝑎2leads to outcome 𝑜2with:
𝑉𝑁(𝑜2)=(𝑇𝑜2, 𝐼𝑜2, 𝐹𝑜2)=(0.6,0.3,0.1), 𝑃(𝑜2|𝑎2)=0.9.
The neutrosophic utilities for each action are calculated as:
𝑈𝑁(𝑎1)=𝑇𝑎1, 𝐼𝑎1, 𝐹𝑎1=(0.8·0.7,0.8·0.2,0.8·0.1)=(0.56,0.16,0.08),
𝑈𝑁(𝑎2)=𝑇𝑎2, 𝐼𝑎2, 𝐹𝑎2=(0.9·0.6,0.9·0.3,0.9·0.1)=(0.54,0.27,0.09).
The decision-maker compares the truth utilities:
𝑇𝑎1=0.56, 𝑇𝑎2=0.54.
Since 𝑇𝑎1> 𝑇𝑎2, the decision-maker selects 𝑎1as the optimal action.
67
Theorem 4.53. The Neutrosophic Frames exhibits the structure of a Neutrosophic Set.
Proof. The result follows directly from the denition.
Theorem 4.54. The Neutrosophic Frames exhibits the structure of a Classic Frames.
Proof. The result follows directly from the denition.
Theorem 4.55 (Preference Reversal in Neutrosophic Frames).Let 𝐹1
𝑁and 𝐹2
𝑁be two Neutrosophic frames of
the same decision problem with identical 𝐴,𝑂, and 𝑃but dierent neutrosophic valuations 𝑉1
𝑁and 𝑉2
𝑁. Then:
𝑈1
𝑁(𝑎𝑖)≠𝑈2
𝑁(𝑎𝑖)for some 𝑎𝑖∈𝐴=⇒preference reversal.
Proof. The neutrosophic utility 𝑈𝑁depends on the valuation 𝑉𝑁. If 𝑉1
𝑁(𝑜𝑗)≠𝑉2
𝑁(𝑜𝑗)for at least one 𝑜𝑗∈𝑂,
then:
𝑈1
𝑁(𝑎𝑖)≠𝑈2
𝑁(𝑎𝑖).
This change in utility leads to a dierent ranking of actions 𝐴, resulting in a preference reversal.
4.4.2 Hyperframing
Additionally, we introduce the concepts of Hyperframing and Superhyperframing, which incorporate hierar-
chical structures into traditional framing. While these concepts are currently at the conceptual stage, their
denitions are outlined below.
We hope that future research will explore and develop these frameworks further.
Denition 4.56 (Hyperframing).Hyperframing extends the classical framing concept into a hyperstructure
framework, allowing multi-level relationships between actions, outcomes, and utilities. A Hyperframe 𝐹𝐻is
dened as:
𝐹𝐻=(P( 𝐴),P (𝑂), 𝑃𝐻, 𝑉𝐻, 𝑈 𝐻),
where:
•P( 𝐴): The powerset of the set of available actions 𝐴={𝑎1, 𝑎2, . . . , 𝑎𝑛}, representing multi-level or
grouped actions.
•P(𝑂): The powerset of the set of outcomes 𝑂={𝑜1, 𝑜2, . . . , 𝑜𝑚}, representing interconnected or com-
bined outcomes.
•𝑃𝐻:P( 𝐴) × P(𝑂) → [0,1]: The hyperprobability function, which assigns probabilities to outcomes
𝑋⊆𝑂given hyper-actions 𝑌⊆𝐴, satisfying:
∀𝑌∈ P ( 𝐴),Õ
𝑋∈ P (𝑂)
𝑃𝐻(𝑋|𝑌)=1.
•𝑉𝐻:P(𝑂) → R: The hypervaluation function, which assigns numerical values to subsets of outcomes
𝑋∈ P(𝑂).
•𝑈𝐻:P( 𝐴) → R: The hyperutility function, dened as:
𝑈𝐻(𝑌)=Õ
𝑋∈ P (𝑂)
𝑃𝐻(𝑋|𝑌) · 𝑉𝐻(𝑋), 𝑌 ∈ P ( 𝐴).
The decision-maker selects the hyper-action 𝑌∗∈ P ( 𝐴)that maximizes the hyperutility:
𝑌∗=arg max
𝑌∈ P ( 𝐴)𝑈𝐻(𝑌).
68
Remark 4.57 (Hyperstructure in Hyperframing).Hyperframing incorporates multiple layers of choices and
outcomes, where actions and outcomes are represented as subsets rather than individual elements. This allows
for a more exible and interconnected decision-making process.
Example 4.58 (Hyperframing in a Project Management Context).Consider a project with two main tasks
𝐴={𝑎1, 𝑎2}and two outcomes 𝑂={𝑜1, 𝑜2}. The hyperstructure allows grouping of actions and outcomes as
subsets:
P( 𝐴)={ {𝑎1},{𝑎2},{𝑎1, 𝑎2}},P (𝑂)={{𝑜1},{𝑜2},{𝑜1, 𝑜2}}.
Suppose:
𝑃𝐻({𝑜1}|{𝑎1, 𝑎2}) =0.7, 𝑉𝐻( {𝑜1}) =10.
The hyperutility is:
𝑈𝐻({𝑎1, 𝑎2}) =𝑃𝐻({𝑜1}|{𝑎1, 𝑎2}) · 𝑉𝐻({𝑜1}) =0.7·10 =7.
Denition 4.59 (𝑛-Superhyperframing).𝑛-Superhyperframing is a higher-order generalization of hyperframing
using 𝑛-th powersets, enabling multi-level hierarchies of actions, outcomes, and utilities. An 𝑛-Superhyperframe
𝐹𝑛
𝑆𝐻 is dened as:
𝐹𝑛
𝑆𝐻 =P𝑛(𝐴),P𝑛(𝑂), 𝑃𝑛
𝑆𝐻 , 𝑉 𝑛
𝑆𝐻 , 𝑈 𝑛
𝑆𝐻 ,
where:
•P𝑛(𝐴): The 𝑛-th powerset of the set of available actions 𝐴, capturing 𝑛-level groupings of actions.
•P𝑛(𝑂): The 𝑛-th powerset of the set of outcomes 𝑂, capturing 𝑛-level interdependencies of outcomes.
•𝑃𝑛
𝑆𝐻 :P𝑛(𝐴) × P 𝑛(𝑂) → [0,1]: The 𝑛-superhyperprobability function, satisfying:
∀𝑌∈ P𝑛(𝐴),Õ
𝑋∈ P𝑛(𝑂)
𝑃𝑛
𝑆𝐻 (𝑋|𝑌)=1.
•𝑉𝑛
𝑆𝐻 :P𝑛(𝑂) → R: The 𝑛-superhypervaluation function, assigning a value to 𝑋∈ P𝑛(𝑂).
•𝑈𝑛
𝑆𝐻 :P𝑛(𝐴) → R: The 𝑛-superhyperutility function, dened as:
𝑈𝑛
𝑆𝐻 (𝑌)=Õ
𝑋∈ P𝑛(𝑂)
𝑃𝑛
𝑆𝐻 (𝑋|𝑌) · 𝑉𝑛
𝑆𝐻 (𝑋), 𝑌 ∈ P𝑛(𝐴).
The decision-maker selects the 𝑛-superhyperaction 𝑌∗∈ P𝑛(𝐴)that maximizes the 𝑛-superhyperutility:
𝑌∗=arg max
𝑌∈ P𝑛(𝐴)𝑈𝑛
𝑆𝐻 (𝑌).
Example 4.60 (𝑛-Superhyperframing in Complex Decision-Making).Consider three actions 𝐴={𝑎1, 𝑎2, 𝑎3}
and outcomes 𝑂={𝑜1, 𝑜2, 𝑜3}. The 2-Superhyperframe includes:
P2(𝐴)={{{𝑎1}},{{𝑎1, 𝑎2},{𝑎3}}},P2(𝑂)={{{𝑜1}},{{𝑜2, 𝑜3}}}.
Suppose the probabilities and valuations are:
𝑃2
𝑆𝐻 ( {{𝑜1}}|{{𝑎1, 𝑎2}}) =0.8, 𝑉 2
𝑆𝐻 ( {{𝑜1}}) =15.
The 2-superhyperutility is:
𝑈2
𝑆𝐻 ( {{𝑎1, 𝑎2}}) =𝑃2
𝑆𝐻 ( {{𝑜1}}|{{𝑎1, 𝑎2}}) · 𝑉2
𝑆𝐻 ( {{𝑜1}}) =0.8·15 =12.
69
4.5 New Mentoring Method
4.5.1 Neutrosophic Mentoring
Mentoring is a structured process where an experienced mentor guides, supports, and transfers knowledge to a
less experienced protégé for skill and personal development [97,156, 233,255].
Denition 4.61 (Mentoring).Mentoring is a structured knowledge transfer process between two agents, dened
as a tuple:
𝑀=(𝐸 , 𝑃, 𝐾 , 𝑇 , 𝐺),
where:
•𝐸={𝑒1, 𝑒2}: A set of agents where 𝑒1is the mentor (knowledge provider) and 𝑒2is the protege (knowl-
edge receiver), such that 𝑒1≠𝑒2.
•𝐾={𝑘1, 𝑘2, . . . , 𝑘 𝑛}: A nite set of knowledge components shared in the mentoring process.
•𝑃:𝐸×𝐾×𝑇→ [0,1]: The knowledge transfer function, where 𝑃(𝑒1, 𝑘𝑖, 𝑡)represents the degree of
knowledge 𝑘𝑖∈𝐾transferred from 𝑒1to 𝑒2at time 𝑡∈𝑇, satisfying:
Õ
𝑘𝑖∈𝐾
𝑃(𝑒1, 𝑘𝑖, 𝑡 ) ≤ 1,∀𝑡∈𝑇 .
•𝑇={𝑡0, 𝑡1, . . . , 𝑡𝑚}: A nite or innite set of discrete or continuous time steps during which mentoring
occurs.
•𝐺:𝐾→R+: The goal attainment function, mapping knowledge 𝑘𝑖to a measurable value 𝑔𝑖indicating
the protege’s learning progress.
The total knowledge gained by the protege 𝑒2at time 𝑡𝑚is:
𝐾gain(𝑒2, 𝑡𝑚)=∫𝑡𝑚
𝑡0Õ
𝑘𝑖∈𝐾
𝑃(𝑒1, 𝑘𝑖, 𝑡 ) · 𝐺(𝑘𝑖)𝑑𝑡 .
The mentoring process is considered successful if:
𝐾gain (𝑒2, 𝑡𝑚) ≥ 𝐾target,
where 𝐾target is a predened learning threshold.
Example 4.62 (Mentoring: Software Development Training).Consider a senior software engineer 𝑒1mentor-
ing a junior developer 𝑒2over 𝑇=[0,10]days. The knowledge components 𝐾include:
𝐾={Algorithms,Debugging,Coding Standards}.
The mentor transfers knowledge 𝑃at time 𝑡, such that:
𝑃(𝑒1,Algorithms, 𝑡)=0.3, 𝑃(𝑒1,Debugging, 𝑡)=0.5, 𝑃 (𝑒1,Coding Standards, 𝑡)=0.2.
The goal attainment function 𝐺assigns weights based on importance:
𝐺(Algorithms)=1.5, 𝐺 (Debugging)=2.0, 𝐺 (Coding Standards)=1.0.
The total knowledge gained by 𝑒2at 𝑡=10 is:
𝐾gain (𝑒2,10)=∫10
0
[0.3·1.5+0.5·2.0+0.2·1.0]𝑑𝑡 =10 ·1.6=16.
If 𝐾target =15, the mentoring process is successful.
70
Denition 4.63 (Neutrosophic Mentoring).Neutrosophic Mentoring extends traditional mentoring by incor-
porating uncertainty, indeterminacy, and falsity into the knowledge transfer process. It is dened as a tuple:
𝑀𝑁=(𝐸, 𝑃𝑁, 𝐾 , 𝑇 , 𝐺),
where:
•𝐸={𝑒1, 𝑒2}: A set of agents where 𝑒1is the mentor (knowledge provider) and 𝑒2is the protege (knowl-
edge receiver), with 𝑒1≠𝑒2.
•𝐾={𝑘1, 𝑘2, . . . , 𝑘 𝑛}: A nite set of knowledge components shared in the mentoring process.
•𝑃𝑁:𝐸×𝐾×𝑇→ [0,1]3: The neutrosophic knowledge transfer function, where 𝑃𝑁(𝑒1, 𝑘𝑖, 𝑡 )=
(𝑇𝑘𝑖, 𝐼𝑘𝑖, 𝐹𝑘𝑖)represents the truth (𝑇), indeterminacy (𝐼), and falsity (𝐹) degrees of knowledge 𝑘𝑖trans-
ferred at time 𝑡.
•𝑇={𝑡0, 𝑡1, . . . , 𝑡𝑚}: A nite or innite set of discrete or continuous time steps during which mentoring
occurs.
•𝐺:𝐾→R+: The goal attainment function, mapping knowledge 𝑘𝑖to a measurable value 𝑔𝑖, represent-
ing the protege’s learning progress.
The total neutrosophic knowledge gained by the protege 𝑒2at time 𝑡𝑚is:
𝐾𝑁
gain (𝑒2, 𝑡𝑚)=∫𝑡𝑚
𝑡0Õ
𝑘𝑖∈𝐾
(𝑇𝑘𝑖−𝐹𝑘𝑖) · 𝐺(𝑘𝑖)𝑑𝑡.
The mentoring process is considered successful if:
𝐾𝑁
gain (𝑒2, 𝑡𝑚) ≥ 𝐾𝑁
target,
where 𝐾𝑁
target is a predened neutrosophic learning threshold.
Remark 4.64. Fuzzy Mentoring is a special case of Neutrosophic Mentoring where both indeterminacy and
falsity are set to zero. Furthermore, Plithogenic Mentoring is notable for its ability to generalize both Neutro-
sophic and Fuzzy Mentoring.
Example 4.65 (Neutrosophic Mentoring: Uncertain Knowledge Transfer).Consider a scenario where 𝑒1men-
tors 𝑒2on the same topics 𝐾. The neutrosophic transfer function 𝑃𝑁is:
𝑃𝑁(𝑒1,Algorithms, 𝑡)=(0.7,0.2,0.1), 𝑃𝑁(𝑒1,Debugging, 𝑡)=(0.6,0.3,0.1), 𝑃 𝑁(𝑒1,Coding Standards, 𝑡)=(0.8,0.1,0.1).
The goal attainment function 𝐺remains the same:
𝐺(Algorithms)=1.5, 𝐺 (Debugging)=2.0, 𝐺 (Coding Standards)=1.0.
The neutrosophic knowledge gained by 𝑒2over 𝑇=[0,10]is:
𝐾𝑁
gain (𝑒2,10)=∫10
0
[(0.7−0.1) · 1.5+ (0.6−0.1) · 2.0+ (0.8−0.1) · 1.0]𝑑 𝑡.
Simplifying:
𝐾𝑁
gain(𝑒2,10)=10 ·[0.6·1.5+0.5·2.0+0.7·1.0]=10 ·2.95 =29.5.
If 𝐾𝑁
target =25, the mentoring process is successful despite uncertainty.
71
4.5.2 HyperMentoring
We dene Hypermentoring and Superhypermentoring as extensions of traditional mentoring by incorporating
hyperstructure and superhyperstructure frameworks. Although these concepts remain at the conceptual stage,
we anticipate that future research will advance their understanding and application.
Denition 4.66 (Hypermentoring).Hypermentoring extends traditional mentoring by incorporating higher-
order relationships and multi-level knowledge structures among agents. It is formally dened as a tuple:
𝐻𝑀=(P (𝐸),P (𝐾), 𝑃𝐻, 𝑇 , 𝐺 𝐻),
where:
•P(𝐸): The powerset of agents 𝐸, where each element represents subsets of mentors and protégés.
Higher-order mentoring involves multiple mentors or protégés simultaneously.
•P(𝐾): The powerset of knowledge 𝐾={𝑘1, 𝑘2, . . . , 𝑘 𝑛}, where subsets of knowledge components are
shared in the mentoring process.
•𝑃𝐻:P(𝐸) × P (𝐾) ×𝑇→ [0,1]: The hyper knowledge transfer function, where 𝑃𝐻(𝐸0, 𝐾 0, 𝑡)represents
the degree of knowledge transfer among subsets 𝐸0⊆𝐸and 𝐾0⊆𝐾at time 𝑡∈𝑇, satisfying:
Õ
𝐾0⊆𝐾
𝑃𝐻(𝐸0, 𝐾0, 𝑡) ≤ 1,∀𝑡∈𝑇 .
•𝑇={𝑡0, 𝑡1, . . . , 𝑡𝑚}: A discrete or continuous set of time steps.
•𝐺𝐻:P(𝐾) → R+: The hyper goal attainment function, mapping subsets of knowledge 𝐾0to measurable
values indicating cumulative learning progress.
The total knowledge gained in a hypermentoring process by a protégé subset 𝐸2⊆𝐸at time 𝑡𝑚is:
𝐾𝐻
gain(𝐸2, 𝑡𝑚)=∫𝑡𝑚
𝑡0Õ
𝐾0⊆𝐾
𝑃𝐻(𝐸1, 𝐾0, 𝑡) · 𝐺𝐻(𝐾0)𝑑 𝑡,
where 𝐸1⊆𝐸are the mentors.
The hypermentoring process is successful if:
𝐾𝐻
gain(𝐸2, 𝑡𝑚) ≥ 𝐾𝐻
target,
where 𝐾𝐻
target is a predened hypermentoring threshold.
Example 4.67 (Hypermentoring in Research Collaboration).Consider a research collaboration program in-
volving senior researchers (mentors) and junior researchers (protégés). The Hypermentoring process is struc-
tured as follows:
•𝐸={𝑒1, 𝑒2, 𝑒3, 𝑒4, 𝑒5}: A set of agents where:
–𝑒1, 𝑒2: Senior researchers (mentors).
–𝑒3, 𝑒4, 𝑒5: Junior researchers (protégés).
•P(𝐸): Powerset of 𝐸, including subsets of mentors and protégés:
P(𝐸)={{𝑒1},{𝑒2},{𝑒3, 𝑒4},{𝑒1, 𝑒2, 𝑒5}, . . . }.
•𝐾={𝑘1, 𝑘2, 𝑘 3}: Knowledge components shared during the mentoring process:
–𝑘1: Advanced research methodologies.
–𝑘2: Statistical modeling techniques.
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–𝑘3: Paper writing and publishing skills.
•P(𝐾): Powerset of 𝐾, including combinations of knowledge components:
P(𝐾)={{ 𝑘1},{𝑘2},{𝑘1, 𝑘3},{𝑘1, 𝑘2, 𝑘3}, . . . }.
•𝑃𝐻:P(𝐸) × P (𝐾) × 𝑇→ [0,1]: The hyper knowledge transfer function. For example:
𝑃𝐻({𝑒1, 𝑒2},{𝑘1, 𝑘 2}, 𝑡 )=0.6, 𝑃𝐻({𝑒3, 𝑒4},{𝑘3}, 𝑡)=0.8.
Here, mentors 𝑒1and 𝑒2transfer knowledge 𝑘1and 𝑘2to protégés with 60% eectiveness, while protégés
𝑒3and 𝑒4focus on learning 𝑘3with 80% eectiveness.
•𝐺𝐻:P(𝐾) → R+: The hyper goal attainment function. For example:
𝐺𝐻({ 𝑘1}) =10, 𝐺𝐻({𝑘1, 𝑘 2}) =25, 𝐺𝐻({𝑘1, 𝑘 2, 𝑘 3}) =40.
•𝑇={𝑡0, 𝑡1, 𝑡2, 𝑡3}: Time steps over which mentoring occurs.
The total knowledge gained by protégés {𝑒3, 𝑒4, 𝑒5}at time 𝑡3is:
𝐾𝐻
gain({𝑒3, 𝑒4, 𝑒5}, 𝑡3)=∫𝑡3
𝑡0Õ
𝐾0⊆𝐾
𝑃𝐻({𝑒1, 𝑒2}, 𝐾 0, 𝑡) · 𝐺𝐻(𝐾0)𝑑𝑡.
Substituting values:
𝐾𝐻
gain({𝑒3, 𝑒4, 𝑒5}, 𝑡3)=(0.6·25)+(0.8·15)=15 +12 =27.
If the hypermentoring threshold 𝐾𝐻
target =25, the process is successful because:
𝐾𝐻
gain =27 ≥𝐾𝐻
target.
Denition 4.68 (n-Superhypermentoring).n-Superhypermentoring generalizes hypermentoring to 𝑛-levels of
powersets and interactions, capturing higher-order complexities across agents and knowledge structures. It is
dened as a tuple:
𝑆𝐻 𝑛
𝑀=(P 𝑛(𝐸),P𝑛(𝐾), 𝑃𝑛
𝑆𝐻 , 𝑇 , 𝐺 𝑛
𝑆𝐻 ),
where:
•P𝑛(𝐸): The 𝑛-th powerset of 𝐸, representing hierarchical and multi-level subsets of agents.
•P𝑛(𝐾): The 𝑛-th powerset of 𝐾, representing higher-order groupings of knowledge components.
•𝑃𝑛
𝑆𝐻 :P𝑛(𝐸)×P 𝑛(𝐾)×𝑇→ [0,1]: The 𝑛-superhyper knowledge transfer function, where 𝑃𝑛
𝑆𝐻 (𝐸0, 𝐾 0, 𝑡)
measures the degree of knowledge transfer among 𝑛-th level subsets 𝐸0⊆ P𝑛(𝐸)and 𝐾0⊆ P 𝑛(𝐾)at
time 𝑡.
•𝑇={𝑡0, 𝑡1, . . . , 𝑡𝑚}: A set of time steps during which mentoring occurs.
•𝐺𝑛
𝑆𝐻 :P𝑛(𝐾) → R+: The 𝑛-superhyper goal attainment function, mapping higher-order subsets 𝐾0⊆
P𝑛(𝐾)to cumulative learning values.
The total knowledge gained in an 𝑛-superhypermentoring process by 𝐸𝑛
2⊆ P𝑛(𝐸)at time 𝑡𝑚is:
𝐾𝑆𝐻 𝑛
gain (𝐸𝑛
2, 𝑡𝑚)=∫𝑡𝑚
𝑡0Õ
𝐾0⊆ P𝑛(𝐾)
𝑃𝑛
𝑆𝐻 (𝐸𝑛
1, 𝐾0, 𝑡) · 𝐺𝑛
𝑆𝐻 (𝐾0)𝑑 𝑡,
where 𝐸𝑛
1⊆ P𝑛(𝐸)are the mentor subsets at 𝑛-levels.
The 𝑛-superhypermentoring process is successful if:
𝐾𝑆𝐻 𝑛
gain (𝐸𝑛
2, 𝑡𝑚) ≥ 𝐾𝑆 𝐻 𝑛
target,
where 𝐾𝑆𝐻 𝑛
target is the predened 𝑛-superhyper mentoring threshold.
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Example 4.69 (n-Superhypermentoring in Research Collaboration).Consider a collaborative research envi-
ronment with hierarchical mentoring:
•𝐸={𝑒1, 𝑒2, 𝑒3}: Senior mentor 𝑒1, mid-level mentor 𝑒2, and junior protégé 𝑒3.
•P2(𝐸)={{𝑒1, 𝑒2},{𝑒2, 𝑒3},{𝑒1, 𝑒2, 𝑒3} }.
•𝐾={𝑘1, 𝑘2}: Research knowledge components.
•P2(𝐾)={{ 𝑘1},{𝑘2},{𝑘1, 𝑘2}}.
•𝑃2
𝑆𝐻 : Knowledge transfer function for second-level subsets:
𝑃2
𝑆𝐻 ( {𝑒1, 𝑒2},{𝑘1}, 𝑡 )=0.8, 𝑃2
𝑆𝐻 ( {𝑒2, 𝑒3},{𝑘2}, 𝑡 )=0.6.
The total knowledge gained by {𝑒2, 𝑒3}at 𝑡𝑚is:
𝐾𝑆𝐻 2
gain ({𝑒2, 𝑒3}, 𝑡𝑚)=∫𝑡𝑚
𝑡00.6·𝐺2
𝑆𝐻 ( { 𝑘2}) 𝑑𝑡.
If 𝐾𝑆𝐻 2
target =1.0, the mentoring process’s success depends on achieving this cumulative threshold.
4.6 New Storytelling Denition
4.6.1 Neutrosophic Storytelling
Storytelling is the process of conveying information, values, or experiences through structured narratives, fos-
tering emotional engagement and facilitating knowledge transfer [47,111, 169, 209, 276]. This concept is ex-
tended using Neutrosophic Logic, leading to the development of Neutrosophic Storytelling. The denitions
and associated concepts are provided below.
Denition 4.70 (Storytelling).Storytelling is the process of transmitting knowledge or values through struc-
tured narratives, dened as a tuple:
𝑆=(𝑁 , 𝑅, 𝑉 , 𝐴, 𝑇 , 𝐶),
where:
•𝑁={𝑛1, 𝑛2, . . . , 𝑛𝑚}: A sequence of narrative events 𝑛𝑖, where each 𝑛𝑖represents a discrete element of
the story.
•𝑅:𝑁×𝑁→ R: The relation function, mapping pairs of events (𝑛𝑖, 𝑛 𝑗)to a set of relationships Rsuch
as causality, sequence, or thematic links.
•𝑉:𝑁→R+: The value function, assigning a positive weight 𝑣𝑖to each narrative event 𝑛𝑖, representing
its importance or impact in the story.
•𝐴:𝐸×𝑁→ [0,1]: The audience comprehension function, where 𝐴(𝑒, 𝑛𝑖)measures the degree of
understanding or emotional response of audience member 𝑒to event 𝑛𝑖.
•𝑇={𝑡1, 𝑡2, . . . , 𝑡𝑝}: A time sequence over which the narrative is delivered.
•𝐶:𝑁→𝐾: The knowledge content function, mapping each event 𝑛𝑖to a knowledge element 𝑘∈𝐾,
where 𝐾represents the set of transferable knowledge.
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The total impact 𝐼of storytelling for an audience 𝐸is dened as:
𝐼=Õ
𝑛𝑖∈𝑁Õ
𝑒∈𝐸
𝐴(𝑒, 𝑛𝑖) · 𝑉(𝑛𝑖) · 𝐶(𝑛𝑖).
The storytelling process is deemed eective if:
𝐼≥𝐼target,
where 𝐼target is the minimum desired impact threshold.
Example 4.71 (Storytelling: Leadership Training).A manager shares a story with employees about overcoming
challenges in a previous project:
•𝑁={𝑛1:Initial failure, 𝑛2:Team collaboration, 𝑛3:Successful outcome}.
•𝑅: Events are causally related, with 𝑛1→𝑛2→𝑛3.
•𝑉(𝑛1)=2.0, 𝑉 (𝑛2)=3.0, 𝑉 (𝑛3)=5.0.
•𝐴(𝑒, 𝑛𝑖): Audience comprehension for 𝑒1and 𝑒2:
𝐴(𝑒1, 𝑛1)=0.8, 𝐴(𝑒1, 𝑛2)=0.9, 𝐴(𝑒1, 𝑛3)=1.0.
•𝐶(𝑛1)=0.5, 𝐶 (𝑛2)=1.0, 𝐶 (𝑛3)=1.5.
The total impact 𝐼is:
𝐼=Õ
𝑛𝑖∈𝑁
𝐴(𝑒1, 𝑛𝑖) · 𝑉(𝑛𝑖) · 𝐶(𝑛𝑖).
Calculating:
𝐼=(0.8·2.0·0.5)+(0.9·3.0·1.0)+(1.0·5.0·1.5)=0.8+2.7+7.5=11.0.
If 𝐼target =10, the storytelling process is eective.
Denition 4.72 (Neutrosophic Storytelling).Neutrosophic Storytelling extends traditional storytelling by in-
tegrating neutrosophic logic into the narrative process, capturing uncertainty, indeterminacy, and falsity in
audience comprehension and value transmission. It is dened as a tuple:
𝑆𝑁=(𝑁 , 𝑅, 𝑉 𝑁, 𝐴 𝑁, 𝑇 , 𝐶 𝑁),
where:
•𝑁={𝑛1, 𝑛2, . . . , 𝑛𝑚}: A sequence of narrative events 𝑛𝑖, where each 𝑛𝑖represents a discrete element of
the story.
•𝑅:𝑁×𝑁→ R: The relation function, mapping pairs of events (𝑛𝑖, 𝑛 𝑗)to a set of relationships R, such
as causality, sequence, or thematic links.
•𝑉𝑁:𝑁→ [0,1]3: The neutrosophic value function, assigning a triplet 𝑉𝑁(𝑛𝑖)=(𝑇𝑛𝑖, 𝐼𝑛𝑖, 𝐹𝑛𝑖)to each
event 𝑛𝑖, representing its truth (𝑇), indeterminacy (𝐼), and falsity (𝐹).
•𝐴𝑁:𝐸×𝑁→ [0,1]3: The neutrosophic audience comprehension function, where 𝐴𝑁(𝑒, 𝑛𝑖)=
(𝑇𝑒,𝑛𝑖, 𝐼𝑒 ,𝑛𝑖, 𝐹𝑒,𝑛𝑖)measures the audience member 𝑒’s degree of understanding, uncertainty, and mis-
understanding for event 𝑛𝑖.
•𝑇={𝑡1, 𝑡2, . . . , 𝑡𝑝}: A time sequence over which the narrative is delivered.
•𝐶𝑁:𝑁→𝐾: The neutrosophic knowledge content function, mapping each event 𝑛𝑖to a knowledge
element 𝑘∈𝐾, with truth, indeterminacy, and falsity components.
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The total neutrosophic impact 𝐼𝑁of storytelling for an audience 𝐸is dened as:
𝐼𝑁=Õ
𝑛𝑖∈𝑁Õ
𝑒∈𝐸𝑇𝑒,𝑛𝑖−𝐹𝑒,𝑛𝑖·𝑉𝑁(𝑛𝑖) · 𝐶𝑁(𝑛𝑖).
The storytelling process is deemed eective if:
𝐼𝑁≥𝐼𝑁
target,
where 𝐼𝑁
target is the minimum desired neutrosophic impact threshold.
Remark 4.73. Fuzzy Storytelling is a special case of Neutrosophic Storytelling where both indeterminacy
and falsity are set to zero. Furthermore, Plithogenic Storytelling is notable for its ability to generalize both
Neutrosophic and Fuzzy Storytelling.
Example 4.74 (Neutrosophic Storytelling: Uncertain Leadership Communication).Suppose a manager nar-
rates a project story with uncertainty:
•𝑉𝑁(𝑛1)=(0.7,0.2,0.1), 𝑉 𝑁(𝑛2)=(0.6,0.3,0.1), 𝑉 𝑁(𝑛3)=(0.9,0.05,0.05).
•𝐴𝑁(𝑒1, 𝑛1)=(0.8,0.1,0.1), 𝐴𝑁(𝑒1, 𝑛2)=(0.7,0.2,0.1), 𝐴𝑁(𝑒1, 𝑛3)=(0.9,0.05,0.05).
•𝐶𝑁(𝑛1)=0.5, 𝐶 𝑁(𝑛2)=1.0, 𝐶 𝑁(𝑛3)=1.5.
The total neutrosophic impact is:
𝐼𝑁=Õ
𝑛𝑖∈𝑁𝑇𝑒1,𝑛𝑖−𝐹𝑒1,𝑛𝑖·𝑇𝑛𝑖·𝐶𝑁(𝑛𝑖).
Simplifying:
𝐼𝑁=(0.8−0.1) · 0.7·0.5+ (0.7−0.1) · 0.6·1.0+ (0.9−0.05) · 0.9·1.5.
Calculating:
𝐼𝑁=0.49 +0.36 +1.1475 =1.9975.
If 𝐼𝑁
target =1.8, the storytelling process is eective.
4.6.2 Hyper Storytelling
Hyper Storytelling and SuperHyper Storytelling are concepts extended using Hyperstructure and SuperHyper-
structure frameworks. The related denitions and concepts are outlined below.
Denition 4.75 (Hyper Storytelling).Hyper Storytelling extends traditional storytelling by incorporating higher-
order relationships and multi-level narrative structures. It is formally dened as a tuple:
𝐻𝑆=(P (𝑁),P( 𝑅), 𝑉𝐻, 𝐴𝐻, 𝑇 , 𝐶𝐻),
where:
•P(𝑁): The powerset of narrative events 𝑁={𝑛1, 𝑛2, . . . , 𝑛𝑚}, where each subset represents a higher-
level narrative structure composed of individual events 𝑛𝑖.
•P( 𝑅): The powerset of relationships 𝑅:𝑁×𝑁→ R, where Rrepresents relationships such as causality,
sequence, and thematic links between subsets of events.
•𝑉𝐻:P(𝑁) → R+: The hyper value function, assigning a positive weight to subsets of narrative events
𝑁0⊆𝑁, representing their collective importance or impact.
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•𝐴𝐻:P(𝐸)×P ( 𝑁) → [0,1]: The hyper audience comprehension function, where 𝐴𝐻(𝐸0, 𝑁 0)measures
the degree of understanding or emotional response of audience subsets 𝐸0⊆𝐸to narrative subsets
𝑁0⊆𝑁.
•𝑇={𝑡1, 𝑡2, . . . , 𝑡𝑝}: A time sequence over which the narrative is delivered.
•𝐶𝐻:P(𝑁) → P (𝐾): The hyper knowledge content function, mapping subsets of events 𝑁0⊆𝑁to
subsets of knowledge 𝐾, where 𝐾={𝑘1, 𝑘2, . . . , 𝑘𝑛}represents transferable knowledge.
The total hyper impact 𝐼𝐻of storytelling for an audience 𝐸is dened as:
𝐼𝐻=Õ
𝑁0⊆ P ( 𝑁)Õ
𝐸0⊆ P (𝐸)
𝐴𝐻(𝐸0, 𝑁0) · 𝑉𝐻(𝑁0) · 𝐶𝐻(𝑁0).
The storytelling process is deemed eective if:
𝐼𝐻≥𝐼𝐻
target,
where 𝐼𝐻
target is the minimum desired hyper impact threshold.
Example 4.76 (Hyper Storytelling in Educational Training).Consider a company implementing a multi-level
educational training program using Hyper Storytelling to transfer knowledge eectively. The elements of Hyper
Storytelling are dened as follows:
•𝑁={𝑛1, 𝑛2, 𝑛3, 𝑛4}: A set of narrative events, where:
–𝑛1: Introduction to project management principles.
–𝑛2: A real-life case study of a successful project.
–𝑛3: A failure analysis of a previous project.
–𝑛4: A simulated project task for participants.
•P(𝑁): The powerset of 𝑁, including:
P(𝑁)={{𝑛1},{𝑛2},{𝑛3},{𝑛4},{𝑛1, 𝑛2},{𝑛2, 𝑛3, 𝑛4}, . . . }.
•P( 𝑅): The powerset of relationships, where higher-level relationships represent thematic and causal
links:
–𝑅({𝑛1},{𝑛2}): The introduction (𝑛1) prepares the audience for the case study (𝑛2).
–𝑅({𝑛2},{𝑛3}): The success story (𝑛2) contrasts with the failure analysis (𝑛3).
–𝑅({𝑛1, 𝑛2},{𝑛4}): The combined knowledge from 𝑛1and 𝑛2is applied in the simulation task 𝑛4.
•𝑉𝐻:P(𝑁) → R+: The hyper value function assigns weights to subsets of narrative events:
𝑉𝐻({𝑛1}) =0.3, 𝑉𝐻({𝑛2}) =0.5, 𝑉𝐻({𝑛3}) =0.4, 𝑉𝐻({𝑛4}) =0.8.
•𝐴𝐻:P(𝐸) × P (𝑁) → [0,1]: The hyper audience comprehension function measures understanding for
subsets of the audience 𝐸:
𝐴𝐻({𝑒1, 𝑒2},{𝑛1, 𝑛2}) =0.7, 𝐴𝐻( {𝑒2, 𝑒3},{𝑛2, 𝑛3, 𝑛4}) =0.8.
•𝐶𝐻:P(𝑁) → P (𝐾): The hyper knowledge content function maps subsets of events to subsets of
knowledge:
𝐶𝐻({𝑛1, 𝑛2}) ={𝑘1, 𝑘 2}, 𝐶𝐻( {𝑛2, 𝑛3, 𝑛4}) ={𝑘2, 𝑘 3, 𝑘4}.
Here, 𝐾={𝑘1:Project Principles, 𝑘2:Case Study Insights, 𝑘3:Failure Lessons, 𝑘4:Simulation Skills}.
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The total hyper impact 𝐼𝐻is calculated as:
𝐼𝐻=Õ
𝑁0⊆ P ( 𝑁)Õ
𝐸0⊆ P (𝐸)
𝐴𝐻(𝐸0, 𝑁0) · 𝑉𝐻(𝑁0) · 𝐶𝐻(𝑁0).
For example, considering 𝑁0={𝑛2, 𝑛3, 𝑛4}and 𝐸0={𝑒2, 𝑒3}:
𝐼𝐻=𝐴𝐻({𝑒2, 𝑒3},{𝑛2, 𝑛3, 𝑛4}) · 𝑉𝐻({𝑛2, 𝑛3, 𝑛4}) · |𝐶𝐻( {𝑛2, 𝑛3, 𝑛4}) | .
Substitute values:
𝐼𝐻=0.8· (0.5+0.4+0.8) · 3=0.8·1.7·3=4.08.
If the threshold 𝐼𝐻
target =4.0, the hyper storytelling process is deemed eective.
Denition 4.77 (n-Superhyper Storytelling).n-Superhyper Storytelling generalizes hyper storytelling to 𝑛-
levels of powersets and interactions, capturing higher-order complexities across narrative structures, relation-
ships, and audience responses. It is dened as a tuple:
𝑆𝐻 𝑛
𝑆=(P 𝑛(𝑁),P𝑛(𝑅), 𝑉 𝑛
𝑆𝐻 , 𝐴𝑛
𝑆𝐻 , 𝑇 , 𝐶 𝑛
𝑆𝐻 ),
where:
•P𝑛(𝑁): The 𝑛-th powerset of 𝑁={𝑛1, 𝑛2, . . . , 𝑛𝑚}, representing 𝑛-level narrative groupings and higher-
order event structures.
•P𝑛(𝑅): The 𝑛-th powerset of relationships 𝑅:𝑁×𝑁→ R, where higher-level relationships describe
interactions among subsets of events across multiple levels.
•𝑉𝑛
𝑆𝐻 :P𝑛(𝑁) → R+: The 𝑛-superhyper value function, assigning positive weights to 𝑛-level narrative
subsets.
•𝐴𝑛
𝑆𝐻 :P𝑛(𝐸) × P 𝑛(𝑁) → [0,1]: The 𝑛-superhyper audience comprehension function, measuring
the understanding or emotional response of audience subsets 𝐸0⊆ P𝑛(𝐸)to 𝑛-level narrative subsets
𝑁0⊆ P𝑛(𝑁).
•𝑇={𝑡1, 𝑡2, . . . , 𝑡𝑝}: A time sequence over which the narrative unfolds.
•𝐶𝑛
𝑆𝐻 :P𝑛(𝑁) → P 𝑛(𝐾): The 𝑛-superhyper knowledge content function, mapping 𝑛-level narrative
subsets to 𝑛-level knowledge components.
The total 𝑛-superhyper impact 𝐼𝑛
𝑆𝐻 for an audience 𝐸is dened as:
𝐼𝑛
𝑆𝐻 =Õ
𝑁0⊆ P𝑛(𝑁)Õ
𝐸0⊆ P𝑛(𝐸)
𝐴𝑛
𝑆𝐻 (𝐸0, 𝑁 0) · 𝑉𝑛
𝑆𝐻 (𝑁0) · 𝐶𝑛
𝑆𝐻 (𝑁0).
The 𝑛-superhyper storytelling process is deemed eective if:
𝐼𝑛
𝑆𝐻 ≥𝐼𝑆 𝐻 𝑛
target,
where 𝐼𝑆𝐻 𝑛
target is the predened 𝑛-superhyper impact threshold.
Example 4.78 (n-Superhyper Storytelling in Training Programs).Consider a corporate training program that
uses multi-level storytelling to transfer knowledge:
•𝑁={𝑛1, 𝑛2, 𝑛3}: Three narrative events 𝑛1(introductory session), 𝑛2(case study), and 𝑛3(simulation
exercise).
•P2(𝑁)={{𝑛1, 𝑛2},{𝑛2, 𝑛3},{𝑛1, 𝑛2, 𝑛3}}: Second-level narrative groupings.
78
•𝑉2
𝑆𝐻 : Narrative value function:
𝑉2
𝑆𝐻 ( {𝑛1, 𝑛2}) =0.8, 𝑉 2
𝑆𝐻 ( {𝑛2, 𝑛3}) =0.9.
•𝐴2
𝑆𝐻 : Audience comprehension function:
𝐴2
𝑆𝐻 ( {𝑒1, 𝑒2},{𝑛1, 𝑛2}) =0.7, 𝐴2
𝑆𝐻 ( {𝑒2, 𝑒3},{𝑛2, 𝑛3}) =0.8.
The total second-level superhyper impact 𝐼2
𝑆𝐻 is:
𝐼2
𝑆𝐻 =Õ
𝑁0⊆ P2(𝑁)Õ
𝐸0⊆ P2(𝐸)
𝐴2
𝑆𝐻 (𝐸0, 𝑁 0) · 𝑉2
𝑆𝐻 (𝑁0) · 𝐶2
𝑆𝐻 (𝑁0).
If the desired threshold 𝐼𝑆 𝐻2
target is met, the program achieves its storytelling objectives.
4.6.3 Neutrosophic Work-Life Balance
Work-Life Balance refers to the eective management of time and energy between professional responsibilities
and personal life to ensure well-being and productivity [32, 71, 144–146, 306, 307]. When mathematically
dened and extended using Neutrosophic Logic, it is formalized as follows. Since this concept remains in the
conceptual stage, further renements and research into its applications are anticipated as necessary.
Denition 4.79 (Work-Life Balance).Work-Life Balance (WLB) is a mathematical framework that models the
allocation of time, resources, and energy between professional responsibilities (work) and personal priorities
(life) to optimize overall well-being and sustainability. It is formally dened as:
WLB =(𝑊 , 𝐿 , 𝑇, 𝑈 , 𝐶, 𝑅, S ),
where:
•𝑊={𝑤1, 𝑤2, . . . , 𝑤𝑛}: A set of work-related activities, where 𝑤𝑖represents specic professional tasks
or obligations.
•𝐿={𝑙1, 𝑙2, . . . , 𝑙𝑚}: A set of life-related activities, where 𝑙𝑗includes personal, social, or recreational
activities.
•𝑇:𝑊∪𝐿→R+: The time allocation function, where 𝑇(𝑥)represents the time allocated to activity
𝑥∈𝑊∪𝐿, subject to: Õ
𝑥∈𝑊∪𝐿
𝑇(𝑥)=𝑇total,
where 𝑇total is the total available time.
•𝑈:𝑊∪𝐿→R+: The utility function, which quanties satisfaction, productivity, or benet derived
from activity 𝑥.
•𝐶:𝑊∪𝐿→R+: The cost function, representing physical, mental, or emotional burdens associated
with activity 𝑥.
•𝑅:𝑊∪𝐿→R: The recovery function, where:
𝑅(𝑥)>0=⇒recovery (e.g., rest, relaxation), 𝑅(𝑥)<0=⇒depletion (e.g., fatigue, stress).
•S=(𝑆𝑊, 𝑆𝐿,Ω): The sustainability state, where 𝑆𝑊and 𝑆𝐿measure cumulative work and life balance,
and Ωrepresents overall equilibrium.
The work-life balance condition is achieved if:
S=𝑆𝑊+𝑆𝐿,where Ω∈ [Ωmin,Ωmax],
and:
𝑆𝑊=Õ
𝑤𝑖∈𝑊
[𝑈(𝑤𝑖) − 𝐶(𝑤𝑖)], 𝑆𝐿=Õ
𝑙𝑗∈𝐿
[𝑈(𝑙𝑗) + 𝑅(𝑙𝑗) − 𝐶(𝑙𝑗) ].
79
Work-Life Imbalance. Work-life imbalance occurs when:
S∉[Ωmin,Ωmax],
indicating that costs outweigh benets or recovery is insucient.
Optimal Work-Life Balance. The optimal balance maximizes overall utility while maintaining sustainabil-
ity:
WLB∗=arg max
{𝑇(𝑤),𝑇 (𝑙) } [𝑆𝑊+𝑆𝐿],
subject to: Õ
𝑥∈𝑊∪𝐿
𝑇(𝑥)=𝑇total,S ∈ [Ωmin ,Ωmax].
Example 4.80 (Work-Life Balance Scenario).A software engineer allocates time in a 24-hour day as follows:
• Work activities 𝑊={𝑤1:coding, 𝑤2:meetings}with 𝑇(𝑤1)=6hours and 𝑇(𝑤2)=2hours.
• Life activities 𝐿={𝑙1:exercise, 𝑙2:family time, 𝑙3:sleep}with 𝑇(𝑙1)=1hour, 𝑇(𝑙2)=2hours, and
𝑇(𝑙3)=8hours.
The recovery values 𝑅are:
𝑅(𝑙3)=10 (high recovery), 𝑅(𝑙1)=5(moderate recovery), 𝑅(𝑤1)=−3(work fatigue).
If sleep (𝑙3) is reduced to 4 hours, 𝑅(𝑙3)decreases signicantly, leading to imbalance:
S∉[Ωmin,Ωmax],
indicating increased risk of burnout.
Denition 4.81 (Neutrosophic Work-Life Balance).Neutrosophic Work-Life Balance (NWLB) is a general-
ized mathematical model for assessing work-life equilibrium by incorporating truth, indeterminacy, and falsity
components into the evaluation of time allocation, utility, and recovery. It is formally dened as:
NWLB =(𝑊 , 𝐿 , 𝑇 𝑁, 𝑈𝑁, 𝐶 𝑁, 𝑅 𝑁,S𝑁),
where:
•𝑊={𝑤1, 𝑤2, . . . , 𝑤𝑛}: The set of work activities (e.g., meetings, projects).
•𝐿={𝑙1, 𝑙2, . . . , 𝑙𝑚}: The set of life activities (e.g., family, exercise, sleep).
•𝑇𝑁:(𝑊∪𝐿) → [0,1]3: The neutrosophic time allocation function, dened as:
𝑇𝑁(𝑥)=(𝑇𝑇(𝑥), 𝑇𝐼(𝑥), 𝑇𝐹(𝑥)),
where:
–𝑇𝑇(𝑥): Truth degree of time allocated to activity 𝑥.
–𝑇𝐼(𝑥): Indeterminacy degree of time allocation for 𝑥.
–𝑇𝐹(𝑥): Falsity degree of time allocated to 𝑥.
•𝑈𝑁:(𝑊∪𝐿) → R3: The neutrosophic utility function, where:
𝑈𝑁(𝑥)=(𝑈𝑇(𝑥), 𝑈𝐼(𝑥), 𝑈𝐹(𝑥)),
representing the truth, indeterminacy, and falsity components of utility derived from activity 𝑥.
80
•𝐶𝑁:(𝑊∪𝐿) → R3: The neutrosophic cost function, quantifying the burden of activity 𝑥as:
𝐶𝑁(𝑥)=(𝐶𝑇(𝑥), 𝐶𝐼(𝑥), 𝐶𝐹(𝑥)),
where truth, indeterminacy, and falsity components reect perceived and uncertain costs.
•𝑅𝑁:(𝑊∪𝐿) → R3: The neutrosophic recovery function, representing the recovery (restoration of
mental/physical energy) as:
𝑅𝑁(𝑥)=(𝑅𝑇(𝑥), 𝑅𝐼(𝑥), 𝑅𝐹(𝑥) ).
•S𝑁=(𝑆𝑁
𝑊, 𝑆𝑁
𝐿,Ω𝑁): The neutrosophic sustainability state, where:
–𝑆𝑁
𝑊: Cumulative neutrosophic balance for work activities.
–𝑆𝑁
𝐿: Cumulative neutrosophic balance for life activities.
–Ω𝑁: Overall neutrosophic sustainability threshold.
Neutrosophic Work-Life Balance Condition. Work-life balance is achieved if the following holds:
S𝑁=𝑆𝑁
𝑊+𝑆𝑁
𝐿,where Ω𝑁∈ [Ω𝑁
min,Ω𝑁
max],
and:
𝑆𝑁
𝑊=Õ
𝑤𝑖∈𝑊
[𝑈𝑇(𝑤𝑖) − 𝐶𝑇(𝑤𝑖)], 𝑆 𝑁
𝐿=Õ
𝑙𝑗∈𝐿𝑈𝑇(𝑙𝑗) + 𝑅𝑇(𝑙𝑗) − 𝐶𝑇(𝑙𝑗).
Neutrosophic Work-Life Imbalance. Work-life imbalance occurs when:
Ω𝑁∉[Ω𝑁
min,Ω𝑁
max],
indicating that the perceived utility, time allocation, and recovery are insucient to oset work burdens.
Optimal Neutrosophic Work-Life Balance. The optimal neutrosophic balance maximizes overall neutro-
sophic utility while accounting for indeterminacy and falsity:
NWLB∗=arg max
{𝑇𝑁(𝑤),𝑇 𝑁(𝑙) } 𝑆𝑁
𝑊+𝑆𝑁
𝐿,
subject to: Õ
𝑥∈𝑊∪𝐿
𝑇𝑇(𝑥)=𝑇total and Ω𝑁∈ [Ω𝑁
min,Ω𝑁
max].
Remark 4.82 (Neutrosophic Work-Life Balance).Fuzzy Work-Life Balance is a special case of Neutrosophic
Work-Life Balance where both indeterminacy and falsity are set to zero. Furthermore, Plithogenic Work-Life
Balance is notable for its ability to generalize both Neutrosophic and Fuzzy Work-Life Balance.
Example 4.83 (Neutrosophic Work-Life Balance Scenario).A manager allocates their time as follows in a
24-hour day:
• Work activities: 𝑊={𝑤1:emails, 𝑤2:meetings}, with neutrosophic time 𝑇𝑁(𝑤1)=(0.8,0.1,0.1)
and 𝑇𝑁(𝑤2)=(0.7,0.2,0.1).
• Life activities: 𝐿={𝑙1:exercise, 𝑙2:family, 𝑙3:sleep}, with:
𝑇𝑁(𝑙1)=(0.6,0.2,0.2), 𝑇 𝑁(𝑙2)=(0.9,0.05,0.05), 𝑇 𝑁(𝑙3)=(0.95,0.03,0.02).
The recovery values 𝑅𝑁and costs 𝐶𝑁are:
𝑅𝑁(𝑙3)=(0.9,0.05,0.05), 𝐶 𝑁(𝑤1)=(0.7,0.2,0.1).
If 𝑇𝑇(𝑙3)decreases to 0.5 (e.g., reduced sleep), recovery becomes insucient, leading to imbalance:
Ω𝑁∉[Ω𝑁
min,Ω𝑁
max],
indicating stress accumulation and unsustainability.
81
Theorem 4.84. The Neutrosophic Work-Life Balance exhibits the structure of a Neutrosophic Set.
Proof. The result follows directly from the denition.
Theorem 4.85. The Neutrosophic Work-Life Balance exhibits the structure of a Classic Work-Life Balance.
Proof. The result follows directly from the denition.
Funding
This research did not receive any external funding.
Acknowledgments
We sincerely thank everyone who contributed their invaluable support to the successful completion of this
paper. We are also deeply grateful to the readers for taking the time to engage with this work and to the authors
of the references cited, whose signicant contributions have been instrumental in shaping this study. Thank
you for your dedication and insights.
Data Availability
This paper does not involve any data analysis or datasets.
Ethical Approval
This study does not include research involving human participants or animals.
Conicts of Interest
The authors declare that there are no conicts of interest related to the publication of this paper.
Disclaimer
This study primarily explores theoretical concepts, and practical applications have not yet been validated. Fu-
ture research may involve empirical testing and renement of the proposed methodologies. While every eort
has been made to ensure the accuracy of the references cited in this paper, unintentional errors or omissions may
occur. The authors assume no legal responsibility for inaccuracies in external sources and encourage readers
to independently verify the information provided. The interpretations and opinions expressed in this paper are
solely those of the authors and do not reect the views of any aliated institutions.
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