oSets: Observer-Dependent Sets

Abstract

Sets play a foundational role in organizing, understanding, and interacting with the world in our daily lives. They also play a critical role in the functioning and behavior of social robots and artificial intelligence systems, which are designed to interact with humans and their environments in meaningful and socially intelligent ways. A multitude of non-classical set theories emerged during the last half-century aspiring to supplement Cantor’s set theory, allowing sets to be true to the reality of life by supporting, for example, human imprecision and uncertainty. The aim of this paper is to continue this effort of introducing oSets, which are sets depending on the perception of their observers. Our main objective is to align set theory with human cognition and perceptual diversity. In this context, an accessible set is a class of objects for which perception is passive, i.e., it is independent of perception; otherwise, it is called an oSet, which cannot be known exactly with respect to its observers, but it can only be approximated by a family of sets representing the diversity of its perception. Thus, the new introduced membership function is a three-place predicate denoted ${\in }_i$, where the expression “$x{\in }_{i}X$” indicates that the “observer” i perceives the element x as belonging to the set X. The accessibility notion is related to perception and can be best summarized as follows: “to be accessible is to be perceived”, presenting a weaker stance than Berkeley’s idealism, which asserts that “to be is to be perceived”.

1. Introduction

In the beginning of the 21st century, theories evolve faster because of high-speed computing, data collection, and interdisciplinary science stimulated by the advent of artificial intelligence and quantum computing. In fact, Artificial Intelligence (AI) has transformed from a theoretical concept to a crucial part of modern society. Its evolution has been shaped by advances in mathematics, computing power, neuroscience, and data availability. This transformation is deeply rooted in the evolution of AI theories, which have gradually aligned with the complexities of the real world. Significant research is emerging, driven by the advent of AI, that explores how science and technology align with human values and needs. For example, AI, family dynamics and cultural identity developing empathetic AI [1], human-AI coevolution, defined as a process in which humans and AI algorithms continuously influence each other [2], creation of AI technologies that are human- centered [3], Generative AI (GenAI) and Social-Emotional Learning (SEL) [4], etc.

The real world consists of all physical entities and phenomena, such as mountains, rivers, animals, and human-made objects such as cars and phones. Humans interpret and understand reality through mental representations known as concepts, which are abstract ideas that help categorize and make sense of the myriad of elements in our environment. Humans perceive the physical world through their five senses [5]: sight, hearing, touch, taste, and smell. For example, when encountering a lion on a mountain, an individual uses these senses to recognize and respond to the animal. Each observed lion is unique; no two lions are identical. However, through experience and learning, humans develop the “lion” abstract concept, which encompasses the general characteristics and attributes common to all lions. Hence, there is an abstract, non-physical concept that defines all lions, aligning with Platonic philosophy and the theory of forms, and similarly, allows for the classification and understanding of individual lions encountered in reality. Concepts [6] serve as the foundational elements of human thought, enabling processes such as classification, learning, inference, and decision making. By forming concepts, humans can group individual entities based on shared characteristics, facilitating efficient processing and understanding of information. For example, the concept of a “chair” allows people to recognize and categorize various types of chairs, despite differences in design or material, based on their common purpose and features.

In computer science, representing real-world entities and abstract concepts requires the use of appropriate data structures. Sets, as defined in set theory, are fundamental structures employed to model collections of distinct objects. A set is a unordered collection that contains unique elements, reflecting the mathematical concept of a finite set. This abstraction aligns with how concepts group individual instances sharing common properties. For example, the concept of a “lion” can be represented in a computer system as a set that contains all instances of lions; i.e., each lion is an element within this set. This representation allows for efficient data processing and retrieval, enabling machines to perform operations such as classification, search, and pattern recognition based on the defined sets. Hence, sets provide a basis for various computational operations and algorithms. They facilitate the organization of data into manageable and logical groupings, allowing machines to perform tasks such as union, intersection, and difference operations. These set operations are crucial in fields such as database management, information retrieval, and artificial intelligence, where the ability to manipulate and analyze groups of related data is essential.

Consequently, the interplay between the physical world, human cognition, and computational representation is intricate and profound. Humans perceive tangible entities through their senses and abstract commonalities into concepts, which are then utilized in thought processes. In computing, these concepts are modeled using data structures like sets, enabling machines to process and analyze information effectively. This alignment between human cognitive frameworks and computational models underscores the importance of set theory and related abstractions in both understanding the world and designing intelligent systems capable of engaging with the complexities of the real world.

In the near future, AI systems will live with humans in the real world, serving as assistant agents or collaborating in hybrid teams to perform specific tasks autonomously. The relationship between abstract concepts, their mathematical representation as sets, and their human perception is central to understanding how meaning is formed and shared, or misunderstood, across individuals and contexts. In cognitive science, concepts such as “lion” are mental abstractions that can be represented formally as sets that contain all instances that satisfy the concept. For example, the concept “lion” can be modeled as a set L, including biological lions, their behaviors, habitats, and symbolic associations.

Now consider a more context-specific abstraction: “lions of Atlas”. This refers to a concept $LA$ that is a specialization or contextual refinement of L, thus $LA\subseteq L$ in set-theoretic terms. However, this formal subset relation does not capture the full semantic divergence introduced by individual, cultural, and linguistic contexts. In practice, “lions of Atlas” can mean radically different things to different people depending on their background knowledge and interpretive context. For example, a biologist can interpret the expression “lions of Atlas” as referring to the subspecies of lions that roamed the Atlas Mountains in Morocco. However, a Moroccan football fan can immediately associate the “lions of Atlas” with the national football team.

Thus, concepts, sets, perceptions, and computing are four key cornerstones, which may play a pivotal role in the development of a future AI-dependent society, where AI is not just a tool, but a new frontier in intelligence and lifestyles. The relationship between the real world, concepts, and mathematical sets is a profound topic that bridges philosophy, cognitive science, and mathematics. In fact, perception is a fundamental issue in epistemology, as it directly influences our understanding of reality, knowledge, and truth. Different theories have emerged to explain how humans perceive the world, each with different implications for knowledge acquisition; see Section 2. Today, computing is challenged by the new roles of machines, which are going beyond their classical role, consisting of playing with numbers to perform very quickly a huge number of numerical operations. In fact, machines interact with and serve humans; they are not only connected together, but they are also connected to humans. But can machines think [7] and have human abilities to live in the physical world? Humans achieve goals during their daily life using, among other things, their ability to think; see Section 8. Thus, set theory has evolved significantly from Cantor’s classical set theory to modern extensions such as fuzzy and rough sets. These developments reflect the adaptation of mathematical theories to better model uncertainty, vagueness, and incomplete information in real-world applications; see Section 4.

This work is part of the evolution of sets that adapt to the real world inhabited by humans. The oSets are sets that depend not only on their members but also on their observers. Section 5 introduces fundamental notions and properties, whereas different classes of perception functions are studied in Section 6. We interpret sets, such as fuzzy and rough sets, through perception in Section 7. The computing with perception is introduced in Section 8, considering perception and sets, shared perceptions, perception diversity analysis, and examples of applications. Finally, a general discussion and concluding remarks are developed in Section 9 and Section 10, respectively.

2. Perception in Epistemology

Epistemology is the study of knowledge [8], where perception is a fundamental topic because it is related to how we gain knowledge about the world. In fact, perception plays a crucial role in epistemology, as it is one of the primary ways humans acquire information about the world. Philosophers debate whether perception provides direct or indirect access to reality and whether it can serve as a reliable foundation for knowledge. A fundamental question is as follows: Can we trust perception as a “source of knowledge”? This question explores whether sensory experience provides reliable knowledge about reality.

Several arguments are in favor of trusting perception; for example, most of our everyday knowledge (e.g., seeing a tree, hearing a voice) comes from perception, Moreover, our perceptions generally match up over time, i.e., if we see a car on the road, we can later touch it, reinforcing its existence. However, there are other arguments against trusting perception, for example, perception errors. In fact, optical illusions, hallucinations, and mirages show that perception can be misleading, so if it can deceive us sometimes, how do we know it is ever accurate? Also, there is subjectivity of experience, as different people perceive the same thing differently (e.g., color blindness, taste preferences). This suggests that perception is not a purely objective way of knowing.

Beyond trusting perception, the following important question is asked: does perception give us direct access to reality, or is it shaped by the Mind? This question explores whether perception is a passive reflection of the world or if the mind actively interprets sensory input. In epistemology, two main perceptions theories are developed: direct access and indirect access theories. Direct access theories (Direct Realism or Naive Realism) consider that the world exists as we see it. Objects have properties such as color, shape, and size that we perceive directly. Alternatively, indirect access theories (Constructivism and Idealism) have allowed the emergence of several perception schools [9]:

• Naive realism [10]: An Aristotelian theory, where we directly perceive the world as it is; i.e., things are what they seem and our perception of the world reflects it exactly as it is, unbiased and unfiltered. According to this school, we simply receive information about the world through our senses, and our consciousness ’mimics’ reality in some way (i.e., objects have the properties that they appear to us as they really are). Hence, perception is passive as it consists of the absorption of the external world into consciousness.
• Representative Realism (John Locke [11] and René Descartes [12]): The mind does not perceive the external world directly; instead, it perceives mental representations of objects. For example, when we see a tree, we are not experiencing the tree itself but an image processed by our brain.
• Transcendental Idealism (Immanuel Kant [13]): The mind actively organizes perception. We never perceive the world “as it is” but only through our mental filters (space and time). For example, imagine wearing colored glasses that tint everything blue. We never see things as they are, only how our mind structures them.
• Idealism (George Berkeley [14]): This school is defended by George Berkeley, who is persuaded by the thought that we only have direct access to our experiences of the world but not to the world itself, asserting that to be is to be perceived.Thus, objects are nothing more than our experiences of them, and God is constantly perceiving everything to ensure the continued existence of objects of the world.

In the 1950s, John McCarthy highlighted the epistemological problems of artificial intelligence [15] and thought that artificial intelligence has closer scientific connections to philosophy than other sciences [16]. Since then, AI researchers have sought to replicate human perception, asking, can machines perceive like humans? Today, in computer vision, AI analyzes images to recognize objects, faces, and text, with applications in self-driving cars and medical imaging. In natural language processing (NLP), AI interprets language for tasks such as simulating human conversation with an end user (chatbots) and text translation. AI also integrates data from multiple sensors, such as cameras, microphones, and radar, to create a more comprehensive perception, used in autonomous robots and drones. In doing so, AI processes data through mathematical models and pattern recognition, lacking intuitive comprehension and requiring specific training for novel scenarios. In contrast, human perception integrates sensory input with memory and experience, enabling holistic understanding and adaptability to new situations. Humans can be deceived by optical illusions due to misinterpretation of sensory information, whereas AI systems are vulnerable to adversarial attacks, where minor input alterations lead to misclassification or incorrect decisions [17]. Additionally, humans naturally grasp context and subtleties such as sarcasm, whereas AI often struggles with nuanced meanings and contextual variations.

In conclusion, while AI is powerful in data processing and pattern recognition, human perception remains superior in flexibility, intuition, and contextual awareness. AI continues to evolve, but it has yet to reach the depth of human understanding. Hence, we introduce oSets based on perception theories developed in epistemology while advocating for a computing-with-perception framework to improve Human–AI symbiosis [18].

3. Computing and Perceptions

Automation refers to the use of technology, such as software, algorithms, artificial intelligence (AI), and robotic systems, to perform tasks with less or more autonomy and human intervention. Through AI and natural language processing (NLP), it enables real-time analysis of human communication on social networks, filtering, and categorizing massive volumes of data. This improves user experience and platform regulation by detecting trends, monitoring sentiment, and identifying misinformation. However, significant challenges remain, including biases in AI models, privacy concerns, and errors in detecting misinformation.

Beyond data-driven processing, human perception significantly influences how information is understood and interpreted. Perception is shaped not only by the five senses but also by individual factors such as education, culture, personal experiences, and the real operational context in which humans act. In an increasingly connected world, people continuously share information and perspectives, even when they hold different worldviews. This growing interconnectedness fosters rich exchanges but also highlights conflicting perceptions, often leading to debate and confrontation. The expansion of digital communication platforms, social media, and transportation technologies has made the world feel smaller than ever. These advances facilitate the rapid dissemination of opinions, sentiments, sarcasm, and even misinformation. The widespread nature of online communication means that information reaches diverse audiences, each interpreting it through their perception. Consequently, AI-powered systems must evolve to account for the diversity of human perception.

In his paper [19], L.A. Zadeh pointed out that “Humans have a remarkable capability to perform a wide variety of physical and mental tasks without any measurements and any computations”. He introduces the computational theory of perceptions (CTP) by acknowledging the intrinsic imprecision of perceptions and incorporating fuzzy sets. Here, we introduce oSets and focus on the diversity of perceptions. For example, consider the short message “Edible meat becomes expensive!” being shared on social media platforms. It is generally analyzed using natural language processing tools. Let us analyze it using the CoreNLP tool (version 4.5.8), which assigns specific parts of speech to each word to understand their grammatical roles. In this case, “Edible” is tagged as an adjective (JJ), “meat” as a noun (NN), “becomes” as a verb (VBZ), and “expensive” as an adjective (JJ). Beyond identifying parts of speech, CoreNLP examines the grammatical relationships between words through dependency parsing. This process reveals how words connect to convey meaning. For instance, “edible” modifies “meat”, indicating the type of meat being discussed. “Meat” serves as the subject of the verb “becomes”, which denotes a change of state. The word “expensive” functions as the complement, describing the new state of “meat” (Figure 1).

However, understanding “Edible meat becomes expensive!” requires real-world knowledge and contextual awareness. NLP tools recognize the basic sentence structure, but we need to fully capture the meaning of the message according to the diversity of human perception. The term meat is broad, and the definition of “edible meat” varies across cultures. Although some societies consume beef and pork, others may include horse, dog, or cat meat in their diet. AI often assumes a universal definition of “meat”, applying statements indiscriminately across all cultures. However, dietary customs vary significantly. In some Asian countries, dog and cat meat is consumed, whereas in much of Europe, these animals are regarded as pets. Horse meat is eaten in certain parts of Europe but is largely avoided in England. In contrast, while beef is widely consumed in Arab countries, cows are considered sacred in India. Meanwhile, vegetarians and vegans refrain from eating meat altogether. Ultimately, the concept of “edible meat” is shaped by cultural norms, traditions, personal beliefs, and preferences. In Section 4, we show how oSets represent this diversity of perception of the concept “edible meat”.

4. From Cantorian Sets to oSets

The classical set theory was introduced by G. Cantor in late 1873, defining a set as “a collection into a whole, of definite, well-distinguished objects (called the ‘elements’) of our perception or of our thought…”. The standard definition of a classical set is “a set is a well-determined collection that is completely characterized by its elements”. Furthermore, the basic relation in the theory of sets is that of membership, denoted ∈, where $x\in X$ indicates that the element x is a member of the set X. However, the naive use of the set notion led to several inconsistencies, such as Russell’s paradox [20]. To avoid inconsistency, Zermelo and Fraenkel adopted an axiomatic approach, leading to the well-known ZFC set theory.

Beyond these initial considerations, a multitude of non-classical set theories emerged during the last half-century aspiring to supplement the classical set theory, i.e., Heyting-valued set theory, intuitionistic and constructive set theory, quantum set theory, fuzzy sets, rough sets, etc. Various factors have contributed to the development of these theories. For example, the classical membership function is Boolean, assigning values from $\{0,1\}$ and disregarding repetitions, since an element cannot appear more than once in a set. Multisets [21] lift the occurrence constraint, taking into account repeated elements in a set, and the number of times an element occurs is called its multiplicity. Hence, the binary membership function is extended to consider the multiplicity of an element in a set, and $x{\in }^{n}X$ is interpreted as “the element x belongs to the set X with a multiplicity of n”. Moreover, Lotfi A. Zadeh noted that the Boolean function is often inadequate in real-world scenarios, as objects in the physical world rarely have strictly defined membership criteria [22]. Consequently, he introduced a membership function that assigns values within the interval $[0,1]$, allowing elements to possess varying degrees of membership within a set. To maintain notational consistency, we denote the fuzzy membership relation as ${\in }_{\alpha }$, where $x{\in }_{\alpha }X$ means that x is a member of the fuzzy set X with a membership degree $\alpha \in [0,1]$. Alternatively, Z. Pawlak pointed out that the relation “belongs to” is not an absolute property but depends upon our knowledge, which is summarized by the relation R. He introduced rough sets [23] using an equivalence relation R as available knowledge, claiming that knowledge is deep and rooted in the classificatory abilities of human beings. A set X is said to be R-definable if it is the union of some R-basic categories; otherwise, it is said to be R-undefinable or rough. Hence, a rough set cannot be known exactly, with respect to our knowledge, but it can only be approximated by two sets called lower and upper approximations. Consequently, x belongs to the lower approximation of X, denoted $x{\underline{\in }}_{R}X$, which means that x “surely” belongs to X with respect to R, while x “possibly” belongs to X with respect to R when it is a member of its upper approximation, that is, $x{\overline{\in }}_{R}X$. More generally, many non-Cantorian sets theories have been put forth to grow beyond the traditional binary “belongs to” relation relying on different notions like cardinality, multiplicity, plurality, distinguishability, discernibility, definability, etc.

The aim of this paper is to continue this effort, considering that our knowledge is closely related to our perception and assuming this central assumption: the relation “belongs to” does not just involve elements and sets, but it also implies observers. Let us return to the previous message “Edible meat becomes expensive!”, which is broadcast on social networks. Its meaning depends on the semantics of the concept “Edible meat”, represented by the set X, which is ultimately private depending on the reader of the message or observer. In fact, the interpretation of a person is shaped by what they consider “edible meat”. According to this example, the set X is represented by one of the following sets: $X_{asian}=\{cat,dog,horse,pork,cow\}$, $X_{european}=\{horse,pork,cow\}$, $X_{english}=\{pork,cow\}$, $X_{arabic}=\{beef,cow\}$, $X_{hindu}=\left\{pork\right\}$, and $X_{vegetarian}=\varnothing$. Food choices are deeply tied to culture, religion, and personal beliefs, making interpretation highly subjective. For this reason, the concept of “edible meat” cannot be represented by a single set. Thus, we propose to represent the semantics of the concept “Edible meat” by a family of sets to take into account the diversity of its perception, i.e., Φ$\left(X\right)=\{X_{asian},X_{european},X_{english},X_{arabic},X_{hindu},X_{vegetarian}\}$, where Φ is the semantic function. In conclusion, the concept of “edible meat” is represented by the set Φ$\left(X\right)$, which is a family of sets that reflect the perception diversity of its observers.

Consequently, we face classes that are not only characterized by their members but also depend on perceptions of their observers. Such classes do not constitute classic sets, but form a family of sets depending on the diversity of their perceptions. When a class is reduced to only one set, regardless of its perception, it is said to be accessible. Humans seem to be familiar with non-accessible sets because their education, values, socioeconomic status, experiences, and more generally their egocentric particulars are different, which leads to the diversity of their perception of the world. In short, the diversity of perception necessarily leads to a plurality of representation.

5. Fundamental Notions and Properties

Given a set U of objects in the real world denoted by x, the set of all parts (subsets) of U is called the power set of U and is generally denoted by $\mathcal{P}\left(U\right)$, which is the collection of all subsets of U, including both the empty set ∅ and U. Each object x of U is represented by the set $\left\{x\right\}$, therefore all objects are represented by the set of parts of U such that their cardinality is equal to 1. This is the set of all single element subsets (or singleton sets) of U, denoted by:

$${\mathcal{P}}_1\left(U\right)=\{X\in \mathcal{P}\left(U\right):|X|=1\}$$

(1)

where $\left|X\right|=1$ means that X contains exactly one element from U. Concepts are subsets of U whose cardinality is greater than or equal to 2. This is the collection of all subsets of U that contain at least two elements, denoted by:

$${\mathcal{P}}_{\ge 2}\left(U\right)=\{X\in \mathcal{P}\left(U\right):|X|\ge 2\}$$

(2)

Equations (1) and (2) define objects and concepts corresponding to physical entities and abstract ideas, respectively, using subsets of the power set $\mathcal{P}\left(U\right)$. In short, a singleton set, containing exactly one element, represents a specific object in the universe, while a set with two or more elements corresponds to a concept or abstract idea that generalizes multiple instances.

In modern philosophy and cognitive science, the concept notion refers to mental representations or abstract ideas that we use to categorize and understand the world. Concepts are the building blocks of our thoughts and knowledge, allowing us to recognize and reason about objects, events, and ideas. The set theory framework enables us to formalize the relationships between physical entities, mental constructs, and symbolic representations used in logic, computation, and language.

Human perception interacts with reality through the senses, so we introduce a function $g_i$ that encodes how an individual i perceives objects in the world. Moreover, perception also extends to abstract concepts, processed through a separate function $f_i$. Thus, each individual i possesses a unique perception function $q_i=<g_i,f_i>$, using both $g_i$ and $f_i$ to perceive objects and concepts, respectively.

Definition 1 (Perception space).

A perception space is a pair $(U,I)$, where U is a non-empty set called universe, and $I\subset \mathbb{N}$ is a set of observers.

As an observer i perceives the world through his perception function $q_i$, when he observes an element X, he perceives $q_i\left(X\right)$, which may or may not be equal to X. Consequently, $q_i\left(X\right)$ replaces X in the observer’s perception space $U_i$. This perception diversity is summarized as follows: we are in the same world (U), but each lives in his own world ($(U,\{i\left\}\right)$, or $U_i$ in short).

$$U_i=\{q_i\left(X\right):X\in \mathcal{P}\left(U\right)\}$$

(3)

Definition 2 (Object perception function g).

Let U be the universe, i be an observer, then $\exists !g_i:{\mathcal{P}}_1\left(U\right)\to U_i$, such that $g_i\left(\left\{x\right\}\right)$ is the perception of the object $x\in U$ by the observer i.

In the remainder of this paper, we assume that this perception function of objects satisfies the following three minimal coherence properties:

$$\left\{\begin{array}{c}x\in g_i\left(\left\{x\right\}\right)\hfill \\ x\in g_i\left(\left\{y\right\}\right)\iff y\in g_i\left(\left\{x\right\}\right)\hfill \\ x\in g_i\left(\left\{y\right\}\right)\wedge y\in g_i\left(\left\{z\right\}\right)\Rightarrow x\in g_i\left(\left\{z\right\}\right)\hfill \end{array}\right.$$

(4)

Using these properties, note that objects of the universe are not necessarily perceived as they are, because we assume only that $x\in g_i\left(\left\{x\right\}\right)$, but not $g_i\left(\left\{x\right\}\right)=\left\{x\right\}$.

Example 1 (Object perception).

Table 1. Perception of objects $x_j$ by observers $g_i$: bee ($x_1$), eagle ($x_2$), drone ($x_3$), pigeon ($x_4$), robot ($x_5$), lizard ($x_6$), parrot ($x_7$).

	$\left\{{\mathit{x}}_1\right\}$	$\left\{{\mathit{x}}_2\right\}$	$\left\{{\mathit{x}}_3\right\}$	$\left\{{\mathit{x}}_4\right\}$	$\left\{{\mathit{x}}_5\right\}$	$\left\{{\mathit{x}}_6\right\}$	$\left\{{\mathit{x}}_7\right\}$
$g_1$	{$x_1\}$	$\left\{x_2\right\}$	$\left\{x_3\right\}$	$\left\{x_4\right\}$	$\left\{x_5\right\}$	$\left\{x_6\right\}$	$\left\{x_7\right\}$
$g_2$	{$x_1\}$	$\left\{x_2\right\}$	$\left\{x_3\right\}$	$\{x_4,x_7\}$	$\{x_5,x_6\}$	$\{x_5,x_6\}$	$\{x_4,x_7\}$
$g_3$	{$x_1,x_2,x_7\}$	$\{x_1,x_2,x_7\}$	$\{x_3,x_4\}$	$\{x_3,x_4\}$	$\{x_5,x_6\}$	$\{x_5,x_6\}$	$\{x_1,x_2,x_7\}$

shows seven objects of the real world U, i.e., a bee, an eagle, a drone, a pigeon, a robot, a lizard, and a parrot, represented by elements $x_1,x_2,x_3,x_4,x_5,x_6$, and $x_7$, respectively. These objects are perceived by three observers. Observer 1 perceives objects as they are, i.e., $\forall x\in U,g_1\left(\left\{x\right\}\right)=\left\{x\right\}$, whereas observers 2 and 3 fail to distinguish some objects from others. In fact, the observer 2 confuses the pigeon $x_4$ with the parrot $x_7$, and the robot $x_5$ with the lizard $x_6$, even if he perceives the bee $x_1$, eagle $x_2$, and drone $x_3$ as they are. However, the observer 3 does not observe any object as it is; it confuses $x_1,x_2$ and $x_7$, $x_3$ with $x_4$, and $x_5$ with $x_6$.

Beyond this individual perception of objects, an observer can also perceive categories of objects, or concepts.

Definition 3 (Concept perception function f).

Let U be the universe, i be an observer, then $\exists !f_i:{\mathcal{P}}_{\ge 2}\left(U\right)\to U_i$, such that $f_i\left(X\right)$ is the perception of X by the observer i.

The function f can be built on g using a given property $\mathbb{P}$. For example, $x\in f_i\left(X\right)$ iff $\mathbb{P}(X,g_i\left(\left\{x\right\}\right))$, where $\mathbb{P}$ is a given property, such as $\left({\mathbb{P}}_1\right)g_i\left(\left\{x\right\}\right)\subseteq X$, $\left({\mathbb{P}}_2\right)g_i\left(\left\{x\right\}\right)\cap X\ne \varnothing$, $\left({\mathbb{P}}_3\right)bel(X\cap g_i\left(\left\{x\right\}\right))⩾0.6$, $\left({\mathbb{P}}_4\right)P(X\cap g_i\left(\left\{x\right\}\right))\le 0.2$, and $\left({\mathbb{P}}_5\right)pl(X\cap g_i\left(\left\{x\right\}\right))⩾0.95$, where $bel$, P, and $pl$ are belief, probability, and plausibility measures, respectively.

Example 2 (Concept perception).

Taking into account the objects introduced in Table 1, the concept “Flying animals” is represented by the set $Z=\{x_1,x_2,x_4,x_7\}$. Considering, for example, that $\forall X\in {\mathcal{P}}_{\ge 2}\left(U\right),f_i\left(X\right)=\{x:g_i\left(x\right)\subseteq X\}$, then the perceptions of the concept Z by the three observers are $f_1\left(Z\right)=f_2\left(Z\right)=Z$, $f_3\left(Z\right)=\{x_1,x_2,x_7\}$. Thus, observer 1 perceives each object as it is and the concept Z as well, whereas observer 2 perceives Z as it is even if he confuses some objects of the universe, e.g, $g_2\left(x_4\right)=\{x_4,x_7\}$. Furthermore, the observer 3 has a partially correct perception of Z (e.g., $f_3\left(Z\right)\cap Z\ne \varnothing$).

Beyond these mathematical properties that reflect the imprecise and uncertain character of humans, we can consider other properties based on features behind the diversity of human perception, due to multiple biological, cognitive, cultural, and psychological factors, for example:

• Sensory variability: Differences in sensory organs (e.g., vision, hearing) affect perception. Color blindness alters how a person sees colors; what looks “red” to one person might appear as “brown” to another.
• Cognitive processing and prior knowledge: The brain filters and interprets information based on past experiences, education, memory, context, goal to meet, etc. Two people watching the same ambiguous image (e.g., the famous “duck–rabbit” illusion) may see different shapes depending on their mental expectations.
• Cultural background and social influences: Language, traditions, and education influence how concepts and emotions are understood. In western cultures, white symbolizes purity, while in some eastern cultures it represents mourning. The same color carries different meanings based on cultural perception.
• Emotional and psychological state: Mood and mental state shape perception and decision-making. A person feeling anxious might interpret a neutral face as threatening, whereas someone in a calm state may see it as indifferent.
• Personal interests and preferences: Subjective biases affect attention and value judgments. A musician and a mathematician listening to a piece of music may focus on different aspects, like the melody vs. the mathematical structure of the rhythm.

In short, human perception of the same reality varies due to multiple factors, and these differences shape how individuals interpret, judge, and react to the world around them. Now, we formally define the perception function $q=<g,f>$.

Definition 4 (Perception function $q_i$).

Let U be the universe, i be an observer, then $\exists !q_i=<g_i,f_i>$, which defines his perception as follows:

$$q_i\left(X\right)=\left\{\begin{array}{cc}\varnothing \hfill & ifX=\varnothing \hfill \\ g_i\left(X\right)\hfill & ifX\in {\mathcal{P}}_1\left(U\right)\hfill \\ f_i\left(X\right)\hfill & ifX\in {\mathcal{P}}_{\ge 2}\left(U\right)\hfill \end{array}\right.$$

(5)

A constructive approach to define the perception function $q_i$ is to consider that the object perception function $g_i$ is generated by a binary relation ${\mathcal{R}}_i$. Hence, the perception of objects is represented by clusters of related objects or granules, that is, granular perception. Then, construct $f_i$ based on $g_i$ and a given property $\mathbb{P}$ to define $q_i=<g_i,f_i>$, which is the perception function of the observer i.

Definition 5

(Granular perception function—see Table 1). Let U be the universe, i be a given observer, ${\mathcal{R}}_i$ be a binary relation on U and $\mathbb{P}$ a predefined property. The perception of i is granular if $g_i$ is generated by ${\mathcal{R}}_i$ on U, i.e., $\forall x\in U$, $g_i\left(\left\{x\right\}\right)=\{y\in U|x{\mathcal{R}}_{i}y\}$, and $f_i$ is based on $g_i$ according to a property $\mathbb{P}$, i.e., $\forall X\in {\mathcal{P}}_{\ge 2}\left(U\right),f_i\left(X\right)=\{x\in U:\mathbb{P}(g_i\left(\left\{x\right\}\right),X)\}$.

To extend the individual perception function considering several observers I, we use a family of sets reflecting the principle of variability of perception. This multiplicity of perception of the same reality between several observers is defined by the function $q_I$.

Definition 6 (Perception function $q_I$).

Let U be the universe of objects, I be a set of observers, $X\in \mathcal{P}\left(U\right)$ and $q_I$ be the perception function of the set of observers I, which is defined as follows:

$$\begin{array}{cc}{q}_I:\hfill & \mathcal{P}\left(U\right)\to \prod _{i\in I}\mathcal{P}\left(U_i\right)\hfill \\ \hfill & X\mapsto {\left(q_i\left(X\right)\right)}_{i\in I}\hfill \end{array}$$

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Hence, the perception $q_I\left(X\right)$ of $X\in \mathcal{P}\left(U\right)$ by a set of observers is a family of sets consisting of an index set I, and, for each observer, $i\in I$, $q_i\left(X\right)$ represents the perception of X by the observer i.

Definition 7 (Accessible set).

Given a perception space $(U,I)$ and $q_i$, the perception function of the observer $i\in I$, $X\in \mathcal{P}\left(U\right)$ is said to be accessible in $(U,I)$ if and only if,

$$\forall i\in I,q_i\left(X\right)=X$$

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If X is accessible, it means that there is a consensus among observers who perceive X as it is; otherwise, an element $x\in U$ can be a member of X for an observer, but not for another. Note that this accessibility notion is Boolean, as a set is accessible to all observers in I or not. To move beyond a binary approach to accessibility, it is possible to extend the concept by considering relative accessibility. For example, the partial accessibility function ${\psi }_i$ defines the degree to which the observer i can access X:

$${\psi }_i\left(X\right)={\displaystyle \frac{\mid q_i\left(X\right)\cap X{\mid }^2}{\mid X\mid \ast \mid q_i\left(X\right)\mid }}$$

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The function ${\psi }_i$ has the following properties: ${\psi }_i(\varnothing )=1$ as $q_i(\varnothing )=\varnothing$, and:

$${\psi }_i\left(X\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}{q}_i\left(X\right)=X\hfill \\ 0\hfill & \mathrm{if}{q}_i\left(X\right)\cap X=\varnothing \hfill \end{array}\right.$$

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The membership function not only links an element x to a set X but also establishes a ternary membership relation that incorporates observers.

Definition 8 (Ternary Membership Relation ${\in }_i$).

Given a perception space $(U,I)$ and $q_i$, the perception function of the observer $i\in I$. The element $x\in U$ is perceived by i to be a member of the set $X\in \mathcal{P}\left(U\right)$ denoted $x{\in }_{i}X$, where:

$$x{\in }_{i}X\iff x\in q_i\left(X\right).$$

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Thus, we introduce a new notion of membership, denoted ${\in }_i$, where $x{\in }_{i}X$ indicates that the observer i perceives the element x as belonging to the set X. In doing so, we do not exclude the variability of perceptions, assuming their multiplicity. In fact, the observer i has his own universe $U_i$ constructed according to his perception function and used for reasoning, decision making, interacting with outside and collaborating in teaming. Such personal universes may be identical to the real world, more or less different, or completely different, as developed in epistemology; see Section 2.

In conclusion, a set X is said to be accessible if it is independent of its observers; otherwise, it is said to be an observer-dependent set or oSet, which cannot be known exactly with respect to its observers, but it can only be approximated by a family of sets representing the diversity of its perception. Thus, accessibility is intrinsically dependent on perception, expressed as “to be accessible is to be perceived” in contrast to Berkeley’s subjective idealism or immaterialism paradigm, “to be is to be perceived”. We will now focus on some classes of perception function, considering, for example, naive, pessimistic, optimistic, and doubtful perceptions.

6. Perception Function Classes

Naive observers perceive the world as it is; therefore, the elementship relation is independent of observers, and consequently, we face an absolute truth as $x\in X$, or $x\notin X$ for all observers.

6.1. Naive or Cantorian Perception Function (NR Class)

Definition 9 (Naive Perception Function).

Given a perception space $(U,I)$, the perception function $q_i^c=<g_i^c,f_i^c>$, where $i\in I$, belongs to the class of naive or Cantorian perception if and only if:

$$\forall x\in U,\forall X\in \mathcal{P}\left(U\right),x\in q_i^c\left(X\right)\iff x\in X$$

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All naive observers have the same perception function, which is equal to the identity function. The identity function on a set X is the function that maps every element of X to itself. This means that applying the identity function to any element in X does not change its value, it simply returns the same element. Thus, $q_i^c\left(X\right)=<g_i^c\left(X\right),f_i^c\left(X\right)>$, where $g_i^c\left(X\right)=X$ if $X\in {\mathcal{P}}_1$, and on the contrary $f_i^c\left(X\right)=X$.

Proposition 1.

The following properties hold for naive perception functions:

(6.1.1) 

Equality: $f_i^c\left(X\right)=f_i^c\left(Y\right)\iff X=Y$;

(6.1.2) 

Intersection: $f_i^c(X\cap Y)=f_i^c\left(X\right)\cap f_i^c\left(Y\right)$;

(6.1.3) 

Union: $f_i^c(X\cup Y)$ = $f_i^c\left(X\right)\cup f_i^c\left(Y\right)$;

(6.1.4) 

Monotony: $X\subseteq Y\Rightarrow f_i^c\left(X\right)\subseteq f_i^c\left(Y\right)$;

(6.1.5) 

Idempotent: $f_i^c\circ f_i^c\left(X\right)=f_i^c\left(X\right)$;

(6.1.6) 

Accessibility: $f_i^c\left(X\right)=X$;

Proposition 2.

For naive observers, all sets are accessible.

Proof. 

This proposition follows from equation (6.1.6).   □

Furthermore, naive observers construct the algebraic structure $(\mathcal{P}\left(U\right),\cap ,\cup ,-,\{q_i^c:i\in I\})$, for example, Figure 2 depicts this structure considering two sets A and B.

In conclusion, naive perception corresponds to naive realism theory (see Section 2), where perception is a passive process, as objects of the world have the properties that they appear to have. Furthermore, from a mathematical viewpoint, this perception corresponds to Cantor’s set theory, which defines crisp sets using a Boolean membership relation excluding any variation of perception.

Now, we consider that objects have primary and secondary qualities as assumed by the representative realism school (see Section 2), and consequently, they can be perceived differently from one observer to another. In this Representative Realism (RR) class, we distinguish three different perception functions called Pessimistic, Optimistic, and Doubtful. For simplicity, we consider that the perception function of objects (i.e., g) is always the same for all these three perception functions; see Table 1, and then we construct the perception functions of concepts accordingly.

6.2. Pessimistic Perception Function (RR Class)

Definition 10 (Pessimistic Perception Function).

Given a perception space $(U,I)$, an observer $i\in I$, the pessimistic perception function, denoted $q_i^{\circ }=<g_i^{\circ },f_i^{\circ }>$, is defined as follows:

$$\forall x\in U,\forall X\in \mathcal{P}\left(U\right),x\in f_i^{\circ }\left(X\right)\iff g_i^{\circ }\left(\left\{x\right\}\right)\subseteq X$$

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An object x is a member of perception $f_i^{\circ }\left(X\right)$ if and only if its perception $g_i^{\circ }(\left\{x\right\}$ is included in X (i.e., $g_i^{\circ }\left(\left\{x\right\}\right)\subseteq X$).

Proposition 3.

The following properties hold for pessimistic perceptions:

(6.2.1) 

Intersection: $f_i^{\circ }(X\cap Y)=f_i^{\circ }\left(X\right)\cap f_i^{\circ }\left(Y\right)$

(6.2.2) 

Union: $f_i^{\circ }\left(X\right)\cup f_i^{\circ }\left(Y\right)\subseteq f_i^{\circ }(X\cup Y)$

(6.2.3) 

Monotony: $X\subseteq Y\Rightarrow f_i^{\circ }\left(X\right)\subseteq f_i^{\circ }\left(Y\right)$

(6.2.4) 

Idempotent: $f_i^{\circ }\circ f_i^{\circ }\left(X\right)=f_i^{\circ }\left(X\right)$

(6.2.5) 

Accessibility: $f_i^{\circ }\left(X\right)\subseteq X$

The main difference from naive perception concerns both the union and the monotony of perceptions. In fact, the union of two perceptions is not necessarily equal to the perception of their union (6.2.2). In addition, a pessimistic perception of X is only included in it but not necessarily equal to it (6.2.5). Furthermore, a pessimistic perception $q_i^{\circ }$ is not necessarily injective, for example, $g_3^{\circ }\left(\left\{x_5\right\}\right)=g_3^{\circ }\left(\left\{x_6\right\}\right)=\{x5,x6\}$ (see Table 1, $g_i^{\circ }=g_i$).

Proposition 4.

Not all sets are necessarily accessible for pessimistic perceptions.

Proof. 

This proposition follows from equation (6.2.5).    □

Proposition 5.

If $A,B$ are accessible, then $A\cap B$ and $A\cup B$ are also accessible for pessimistic perceptions.

Proof. 

• A and B are accessible, then $A\cap B$ is accessible by (6.2.1).
• $f_i^{\circ }\left(A\right)\cup f_i^{\circ }\left(B\right)\subseteq f_i^{\circ }(A\cup B)\subseteq A\cup B$ by $(6.2.2)$, and $(6.2.5)$. So, if A and B are accessible, we have $f_i^{\circ }\left(A\right)\cup f_i^{\circ }\left(B\right)=A\cup B$, then, $A\cup B\subseteq f_i^{\circ }(A\cup B)$ by $(6.2.2)$ and $f_i^{\circ }(A\cup B)\subseteq A\cup B$ by $(6.2.5)$. So, $f_i^{\circ }(A\cup B)=A\cup B$.

   □

A pessimistic perception of X is included in it, but under what conditions is it equal to it (i.e., X is accessible)?

Proposition 6.

If $X={\cup }_{x\in X}{g}_i^{\circ }\left(\left\{x\right\}\right)$, then X is accessible for pessimistic perceptions.

Proof. 

$X={\cup }_{x\in X}{g}_i^{\circ }\left(\left\{x\right\}\right)$, using (6.2.3), we obtain ${\cup }_{x\in X}{g}_i^{\circ }\left(\left\{x\right\}\right)\subseteq f_i^{\circ }\left({\cup }_{x\in X}\left\{x\right\}\right)=f_i^{\circ }\left(X\right)$, thus $X\subseteq f_i^{\circ }\left(X\right)$. So, X is accessible by (6.2.5).   □

We use this proposition to check if a given set X is accessible by computing ${\cup }_{x\in X}{g}_i^{\circ }\left(\left\{x\right\}\right)$, and the response is affirmative if this union is equal to X.

Example 3.

Now, we want to compute the perception of concepts and, as we have said before, this computation depends on the class of perception functions. The perception functions of observers 2 and 3 are pessimistic because $x\in f_i^{\circ }\left(X\right)\iff g_i^{\circ }\left(\left\{x\right\}\right)\subseteq X$, with $2\le i\le 3$. Consider the concept Flying objects, which is defined by the set $X=\{x_1,x_2,x_3,x_4,x_7\}$, then $x_4\in f_2^{\circ }\left(X\right)$ because $f_2^{\circ }\left(\left\{x_4\right\}\right)=\{x_4,x_7\}\subseteq X$. By adding the following two concepts, which are $Y=\{x_1,x_2,x_4,x_6,x_7\}$ (Animals) and $Z=\{x_1,x_2,x_4,x_7\}$ (Flying animals), we want to compare their accessibility according to the perceptions of observers 2 and 3. Using property 6, we conclude that X is accessible to observers 2 and 3, Z is accessible only to observer 2, however, Y is not accessible for observers 2 and 3 (see

Table 2. Pessimistic perceptions of X (Flying objects), Y (Animals) and Z (Flying animals)—values of ${\cup }_{x\in •}{g}_i^{\circ }\left(\left\{x\right\}\right)$, where • is X, Y, or Z.

	$\mathit{X}=\{{\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_3,{\mathit{x}}_4,{\mathit{x}}_7\}$	$\mathit{Y}=\{{\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_4,{\mathit{x}}_6,{\mathit{x}}_7\}$	$\mathit{Z}=\{{\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_4,{\mathit{x}}_7\}$
$f_1^{\circ }$	X	Y	Z
$f_2^{\circ }$	X	$\{x_1,x_2,x_4,x_5,x_6,x_7\}$	Z
$f_3^{\circ }$	X	U	$\{x_1,x_2,x_3,x_4,x_7\}$

).

A pessimistic observer constructs and uses the following algebraic structure $\left(\mathcal{P}\right(U)$, ∩, ∪, −, $\{q_i^{\circ }:i\in I\})$. The dashed lines in Figure 3 connect concepts that are not accessible and cannot be used by pessimistic observers, where their structure is only defined by solid lines (Figure 3 considers only two sets A and B).

To what extent is observer i more pessimistic than observer j? To answer this question, we define the structure $(I,{\prec }^{\circ })$, where ${\prec }^{\circ }$ is a binary relation that allows us to compare the pessimism of observers defined by I.

Definition 11 (more pessimistic).

Given a perception space $(U,I)$, we say that i is more pessimistic than j, denoted $j{\prec }^{\circ }i$, iff $\forall X\in \mathcal{P}\left(U\right),q_i^{\circ }\left(X\right)\subseteq q_j^{\circ }\left(X\right)$.

6.3. Optimistic Perception Function (RR Class)

We now introduce a second perception function of the RR class, designed to be optimistic.

Definition 12 (Optimistic Perception Function).

Given a perception space $(U,I)$, an observer $i\in I$, the optimistic perception function, denoted ${\overline{q}}_i=<{\overline{g}}_i,{\overline{f}}_i>$, is defined as follows:

$$\forall x\in U,\forall X\in \mathcal{P}\left(U\right),x\in {\overline{f}}_i\left(X\right)\iff {\overline{g}}_i\left(\left\{x\right\}\right)\cap X\ne \varnothing$$

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For an optimistic observer, an element x can belong to the perception of X (i.e., $x\in {\overline{f}}_i\left(X\right)$) without being a member of X. The properties of the optimistic perception functions are presented in the following way.

Proposition 7.

The following properties hold for optimistic perception functions:

(6.3.1) 

Intersection: ${\overline{f}}_i(X\cap Y)\subseteq {\overline{f}}_i\left(X\right)\cap {\overline{f}}_i\left(Y\right)$;

(6.3.2) 

Union: ${\overline{f}}_i(X\cup Y)={\overline{f}}_i\left(X\right)\cup {\overline{f}}_i\left(Y\right)$;

(6.3.3) 

Monotony: $X\subseteq Y\Rightarrow {\overline{f}}_i\left(X\right)\subseteq {\overline{f}}_i\left(Y\right)$;

(6.3.4) 

Idempotent: ${\overline{f}}_i\circ {\overline{f}}_i\left(X\right)={\overline{f}}_i\left(X\right)$.

(6.3.5) 

Accessibility: $X\subseteq {\overline{f}}_i\left(X\right)$;

For optimistic observers, the perception of the intersection of two sets is not equal to the intersection of their perception (6.3.1), and sets are not necessarily accessible (6.3.5). In addition, an optimistic perception ${\overline{q}}_i$ is not necessarily injective.

Proposition 8.

Not all sets are necessarily accessible for optimistic perceptions.

Proof. 

This proposition follows from equation (6.3.5).    □

Proposition 9.

If A and B are accessible, then $A\cap B$ and $A\cup B$ are accessible for optimistic perceptions.

Proof. 

• $A\cup B$ is accessible if both A and B are accessible; see $(6.3.2)$.
• $A\cap B\subseteq {\overline{f}}_i(A\cap B)$ by $(6.3.5)$ and ${\overline{f}}_i(A\cap B)\subseteq {\overline{f}}_i\left(A\right)\cap {\overline{f}}_i\left(B\right)$ by $(6.3.1)$. As A and B are accessible, then $A\cap B\subseteq {\overline{f}}_i(A\cap B)\subseteq A\cap B$. Consequently, ${\overline{f}}_i(A\cap B)=A\cap B$.

   □

Similarly to pessimistic perception, the accessibility of a set can be determined using the Proposition 10. Specifically, we compute ${\cup }_{x\in X}{\overline{g}}_i\left(\left\{x\right\}\right)$ and verify whether the result is included in or equal to X. If the computing result is a subset of or equal to X, then X is considered accessible; otherwise, X is deemed inaccessible.

Proposition 10.

If ${\cup }_{x\in X}{\overline{g}}_i\left(\left\{x\right\}\right)\subseteq X$, then X is accessible for optimistic perceptions.

Proof. 

 As $X={\cup }_{x\in X}\left\{x\right\}$, so ${\overline{f}}_i\left(X\right)={\overline{f}}_i\left({\cup }_{x\in X}\left\{x\right\}\right)$, and ${\overline{f}}_i\left({\cup }_{x\in X}\left\{x\right\}\right)={\cup }_{x\in X}{\overline{g}}_i\left(\left\{x\right\}\right)$ by (6.3.2). Taking into account ${\cup }_{x\in X}{\overline{g}}_i\left(\left\{x\right\}\right)\subseteq X$, we have ${\overline{f}}_i\left(X\right)\subseteq X$ and we conclude that X is accessible using (6.3.5).    □

Example 4.

Given the perception functions ${\overline{q}}_2$ and ${\overline{q}}_3$ for observers 2 and 3, respectively, are the concepts “Flying objects” (X), “Animals” (Y) and “Flying animals” (Z) accessible to each of them? (see

Table 3. Optimistic perceptions of X (Flying objects), Y (Animals), and Z (Flying animals)—values of ${\cup }_{x\in •}{g}_i\left(\left\{x\right\}\right)$, where • is X, Y, or Z.

	$\mathit{X}=\{{\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_3,{\mathit{x}}_4,{\mathit{x}}_7\}$	$\mathit{Y}=\{{\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_4,{\mathit{x}}_6,{\mathit{x}}_7\}$	$\mathit{Z}=\{{\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_4,{\mathit{x}}_7\}$
${\overline{f}}_1$	X	Y	Z
${\overline{f}}_2$	X	$\{x_1,x_2,x_4,x_5,x_6,x_7\}$	Z
${\overline{f}}_3^{\circ }$	X	U	$\{x_1,x_2,x_3,x_4,x_7\}$

).

More generally, optimistic perception leads to the algebraic structure $(P\left(U\right),\cap ,\cup ,-,\{{\overline{f}}_i:i\in I\})$, which is based on the power set of the universe U (Figure 4 considers only two sets A and B).

In order to compare observers according to their optimistic observers according to perception, we introduce the binary relation $\overline{\prec }$.

Definition 13 (more optimistic).

Given a perception space $(U,I)$, we say that i is more optimistic than j, denoted $j\overline{\prec }i$, iff $\forall X\in \mathcal{P}\left(U\right),{\overline{q}}_j\left(X\right)\subseteq {\overline{q}}_i\left(X\right)$.

We have introduced the class NR for which all sets are accessible, and next we have defined pessimistic and optimistic perceptions that are representatives of the class RR. Now, we focus on perceptions related to idealism, i.e., class I, where no set is accessible.

6.4. Doubtful Perception Function (I Class)

Definition 14 (Doubtful Perception Function).

Given a perception space $(U,I)$, an observer $i\in I$, the doubtful perception function, denoted $q_i^{\sim }=<g_i^{\sim },f_i^{\sim }>$, is defined as follows:

$$\forall x\in U,\forall X\in \mathcal{P}\left(U\right),x\in f_i^{\sim }\left(X\right)\iff (g_i^{\sim }\left(\left\{x\right\}\right)\cap X\ne \varnothing )\wedge (g_i^{\sim }\left(\left\{x\right\}\right)\cap -X\ne \varnothing )$$

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For a doubtful observer, an element x belongs to the perception of X (i.e., $x\in f_i^{\sim }\left(X\right)$) only if its perception intersects at the same time X and its complement (i.e., $g_i^{\sim }\left(\left\{x\right\}\right)\cap X\ne \varnothing$ and $g_i^{\sim }\left(\left\{x\right\}\right)\cap -X\ne \varnothing$). Hence, the perception of the object x is never true. Such perceptions may arise, for example, when the observer has incomplete knowledge, exhibits subjectivity, holds ambiguous preferences, is influenced by personal experience, or displays hesitation.

Proposition 11.

The doubtful perception function $f_i^{\sim }$ is not necessarily monotonic.

Proof. 

 $f^{\sim }$ is a doubtful perception function, $X=\{a,b,c\},Y=\{a,b,c,d,e\}$, considering $f^{\sim }\left(X\right)=\{b,c,d\}$ and $f^{\sim }\left(Y\right)=\{e,g,h\}$, $f^{\sim }\left(X\right)$, resp. $f^{\sim }\left(Y\right)$ have a non-empty intersection with both X and $-X$, respectively, with Y and $-Y$, however, $f^{\sim }\left(X\right)=\{b,c,d\}⊈f^{\sim }\left(Y\right)=\{e,g,h\}$, even if $X\subset Y$.    □

In the remainder of this paper, we will consider only the doubtful monotonic functions assuming the following axiom.

$$X\subseteq Y\Rightarrow f_i^{\sim }\left(X\right)\subseteq f_i^{\sim }\left(Y\right)$$

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This axiom is necessary to guarantee a minimal internal coherence of doubtful perceptions, leading to the following properties:

Proposition 12.

The following properties hold for monotonic doubtful perception functions:

(6.4.1) 

Intersection: $f_i^{\sim }(X\cap Y)\subseteq f_i^{\sim }\left(X\right)\cap f_i^{\sim }\left(Y\right)$

(6.4.2) 

Union: $f_i^{\sim }\left(X\right)\cup f_i^{\sim }\left(Y\right)\subseteq f_i^{\sim }(X\cup Y)$

(6.4.3) 

Idempotent: $f_i^{\sim }\left(X\right)\subset f_i^{\sim }\circ f_i^{\sim }\left(X\right)$

(6.4.4) 

Accessibility:$X\subseteq f_i^{\sim }\left(X\right)$

The structure used by doubtful monotonic observers is deduced from the properties $(6.4.1)–(6.4.4)$, with the monotony axiom, and drawn in Figure 5. Consequently, we cannot compute exactly the perception of concept X, we can only give its lower approximation because $f_i^{\sim }\left(X\right)=f_i^{\sim }\left({\cup }_{x\in X}\left\{x\right\}\right)$, but unfortunately, $f_i^{\sim }\left({\cup }_{x\in X}\left\{x\right\}\right)$ is not necessarily equal to ${\cup }_{x\in X}{g}_i^{\sim }\left(\left\{x\right\}\right)$, see $(6.4.2)$. Thus, we approximate $f_i^{\sim }\left(X\right)$ by its lower approximation ${\cup }_{x\in X}{g}_i^{\sim }\left(\left\{x\right\}\right)$. Moreover, the perception of an intersection $f_i^{\sim }(X\cap Y)$ can only be approximated by its upper approximation $f_i^{\sim }\left(X\right)\cap f_i^{\sim }\left(Y\right)$, see $(6.4.1)$, whereas $f_i^{\sim }(X\cup Y)$ is only approximated by its lower approximation $f_i^{\sim }\left(X\right)\cup f_i^{\sim }\left(Y\right)$, see $(6.4.2)$.

Proposition 13.

No set is accessible for doubtful perceptions; i.e., $\forall X\in \mathcal{P}\left(U\right),\forall i\in I,q_i^{\sim }\left(X\right)\ne X$;

Proof. 

 This proposition follows from equation $(6.4.4)$.    □

Example 5.

Let us compute the perception of the concept Flying Objects $\{x_1,x_2,x_3,x_4,x_7\}$ considering observers 1, 2, and 3. The three concepts X (Flying objects), Y (Animals), and Z (Flying animals): how to approximate the perception of concepts X, Y, and Z for the three observers 1, 2, and 3 (see

Table 4. Doubtful perceptions of X (Flying objects), Y (Animals), and Z (Flying animals)—values of ${\cup }_{x\in •}{g}_i\left(\left\{x\right\}\right)$, where • is X, Y, or Z.

	$\mathit{X}=\{{\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_3,{\mathit{x}}_4,{\mathit{x}}_7\}$	$\mathit{Y}=\{{\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_4,{\mathit{x}}_6,{\mathit{x}}_7\}$	$\mathit{Z}=\{{\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_4,{\mathit{x}}_7\}$
$f_1^{\sim }$	∅	∅	∅
$f_2^{\sim }$	∅	$\left\{x_6\right\}$	∅
$f_3^{\sim }$	∅	$\{x_4,x_6\}$	$\left\{x_4\right\}$

)?

Similarly, doubtful perception leads also to the algebraic structure $(P\left(U\right),\cap ,\cup ,-,\{f_i^{\sim }:i\in I\})$, which is based on the power set of universe U (Figure 5—considers only two sets A and B).

How much more doubtful is one observer than another? The relation ${\prec }^{\sim }$ allows for the comparison of two doubtful observers i and j.

Definition 15 (more doubtful).

Given a perception space $(U,I)$, we say that i is more doubtful than j, denoted $j{\prec }^{\sim }i$, iff $\forall X\in \mathcal{P}\left(U\right),q_j^{\sim }\left(X\right)\subseteq q_i^{\sim }\left(X\right)$.

7. Interpreting Sets Through Perception

For Cantorian sets, the perception is passive, adopting a Boolean approach where an element is a member of a set or not. In this context, the perception function $q^c$ is equal to the identity, i.e., $\forall X\in \mathcal{P}\left(U\right),q^c\left(X\right)=X$. It corresponds to the Naive Realism (NR) school in epistemology, where we perceive the world as it is, which can be rewritten, in terms of accessibility, as follows: all sets are accessible regarding their observers.

7.1. Zadeh’s Perception of Sets

In classical set theory, the membership function ∈ expresses a connection between an element x and a set X, allowing one to know if the element x is a member of X or not. This assumes that sets or classes of objects have precisely defined criteria of membership, which is not always the case in the real world. For example, the term “animal” seems straightforward, but it is actually ambiguous due to differences in biological classification, common language use, and conceptual boundaries. In everyday use, “animal” often refers to vertebrates such as dogs and cats, making it unclear whether things like starfish or jellyfish count. Taxonomy shows a clear definition, but perception varies, leading to confusion in non-experts. Moreover, bacteria are not animals, and despite this, some people might mistakenly group bacteria with animals because both are living organisms that move and consume energy. In addition, several concepts are subjective, such as “tall people”, “fat person”, “expensive price”, etc. Ambiguity is ignored in the classical set theory, even if it arises in the real world. For this reason, Lotfi A. Zadeh introduced fuzzy sets to handle vagueness and ambiguity [22]. Thus, the membership function is not a function connecting elements of U to sets, but each set X has its own membership function, denoted ${\mu }_X$, where ${\mu }_X\left(x\right)$ is the membership degree of x to X. In short, a fuzzy set is a pair $(X,\mu )$, where $X\subseteq U$ and ${\mu }_X:U\mapsto [0,1]$, allowing the gradual assessment of the membership of elements in a set: x is not included in X iff ${\mu }_X\left(x\right)=0$, fully included iff ${\mu }_X\left(x\right)=1$, and partially included otherwise (i.e., $0<{\mu }_X\left(x\right)<1$).

Fuzzy sets can be interpreted in terms of perception, assuming a single observer with an imprecise perception function $q^z=<g^z,f^z>$, where $g^z\left(\left\{x\right\}\right)=\left\{x\right\}$ and $f^z\left(X\right)=\{(x,{\mu }_X\left(x\right)):x\in U\}$. Hence, objects are perceived as they are, and the perception of X may be imprecise since $f^z\left(X\right)$ is a fuzzy set, and consequently all properties of fuzzy sets are inherited.

There are several generalizations of fuzzy sets that extend and complement the original theory in order to deal with more specific problems. For example, intuitionistic fuzzy sets (IFS), introduced by Atanassov [24], allow one to consider not only imprecision, but also hesitation. To do so, an IFS $X\in \mathcal{P}\left(U\right)$ has the form $X=\{<x,{\mu }_X\left(x\right),{\nu }_X\left(x\right)>:x\in U\}$, where the functions ${\mu }_X\left(x\right)$ and ${\nu }_X\left(x\right)$ define, respectively, the degree of membership and degree of non-membership of the element x to the set X. These two degrees respect the following constraints: $0\le {\mu }_A\left(x\right)\le 1$, $0\le {\nu }_A\left(x\right)\le 1$ and $0\le {\mu }_A\left(x\right)+{\nu }_A\left(x\right)\le 1$. Moreover, these two degrees are aggregated into a single degree ${\pi }_X\left(x\right)=1-{\mu }_X\left(x\right)-{\nu }_X\left(x\right)$, which is called the hesitation margin or the degree of indeterminacy of $x\in X$.

How to interpret the IFS in terms of perception? Similarly to the interpretation of fuzzy sets, we consider only one intuitionistic observer, which is imprecise and hesitant at the same time. His perception function is $q^{iz}=<g^{iz},f^{iz}>$, where $g^{iz}\left(\left\{x\right\}\right)=\left\{x\right\}$ and $f^{iz}\left(X\right)=\left\{(x,{\pi }_X\left(x\right))\right):x\in U\}$. Hence, objects are perceived as they are, and concepts are perceived with imprecision and hesitation, since $f^{iz}\left(X\right)$ is an intuitionistic fuzzy set and consequently all properties of intuitionistic fuzzy sets are inherited.

Another generalization of a fuzzy set is called a Hesitant Fuzzy Set (HFS) [25], which considers situations where there are some difficulties in determining the membership of an element to a set, caused by a doubt between a few different values. For example, in decision making, experts may have divergent opinions on alternatives due to their varying knowledge backgrounds or interests. Since they cannot easily persuade each other, achieving a consensus evaluation can be challenging. However, multiple evaluation values may exist, reflecting the diversity of expert perspectives and the complexity of aggregating judgments in a structured decision-making process. In cases where we have a set of possible values, hesitant fuzzy set (HFSs) allow the membership degree of an element to a set presented by several possible values between 0 and 1. Thus, a hesitant fuzzy set X is defined as follows: $X=\{<x,h_X\left(x\right)>:x\in U\}$, where $h_X\left(x\right)$ is a set of possible membership degrees of $x\in U$ to the set X. For example, let $U=\{x_1,x_2,x_3\}$ be the universe, $h_X\left(x_1\right)=\{0.8,0.7,0.9\}$, $h_X\left(x_2\right)=\{0.4,0.5\}$, $h_X\left(x_3\right)=\{0.1,0.2\}$, and $h_X\left(x_4\right)=\{0.1,0.9,0.8,0.3\}$, which are members degrees of elements $x_1,x_2,x_3$, and $x_4$ to X, respectively. So, X is an HFS like $X=\{<x_1,\{0.8,0.7,0.9\}>,<x_2,\{0.4,0.5\}>,<x_3,\{0.1,0.2\}>,<x_4,\{0.1,0.9,0.8,0.3\}>\}$. Unlike fuzzy and intuitionistic fuzzy sets where there is only one observer, which is imprecise and also hesitant, respectively, in the case of hesitant fuzzy sets, there is a set of imprecise observers, denoted $I_{x,X}$, perceiving the same reality (an element x, and a set X), and each one gives a member degree of elements x to X. Note that the number of observers depends on the reality they observe. In this context, the hesitant perception function is $q^{hz}=<g^{hz},f^{hz}>$, where $g^{hz}\left(\left\{x\right\}\right)=\left\{x\right\}$ and $f^{hz}\left(X\right)=\{<x,h_X\left(x\right)>:x\in U\}$, where |$h_X\left(x\right)$| is the number of observers perceiving the same reality (x, and X). In addition, the perception space is $(U,I)$, where $I={\bigcup }_{x\in U,X\in {\mathcal{P}}_{\ge 2}\left(U\right)}{I}_{x,X}$. In short, the same reality is perceived by a set of imprecise observers, which may change according to the observed reality, with disagreements among themselves. Moreover, objects are perceived as they are, and concepts are perceived by multiple imprecise observers, since $f^{hz}\left(X\right)$ is a hesitant fuzzy set and consequently, all properties of hesitant fuzzy sets are inherited.

Beyond fuzzy, intuitionistic, and hesitant sets that we have interpreted in terms of perception, there are several other extensions of fuzzy sets [26], like type-2 fuzzy set, fuzzy multiset, etc. However, interpreting all of these sets in terms of perception is beyond the scope of this paper. Nevertheless, we continue by interpreting rough sets to show how perception is related to roughness and to better understand the relationship between rough and fuzzy sets to shed different light.

7.2. Pawlak’s Perception of Sets

The rough set theory (RST), introduced by Zdzisław Pawlak in 1982, is a mathematical framework for dealing with vagueness and uncertainty in data analysis—Book Pawlak [27]. Unlike classical set theory, which assumes sharp and precise membership, rough sets allow for incomplete or imprecise knowledge by approximating a set using two definable boundaries: a lower approximation and an upper approximation. Rough set theory provides a mathematically rigorous framework for dealing with uncertainty and vagueness, particularly in decision systems, artificial intelligence, and machine learning. It differs from fuzzy set theory by focusing on indiscernibility relations rather than membership degrees. Its ability to perform data reduction, classification, and knowledge discovery makes it a powerful tool in data science and AI applications. Let U be a universal set (also called the universe of discourse) and let R be an equivalence relation on U, called an indisernibility relation as if x, $y\in U$ and $(x,y)\in R$ we say that x and y are indistinguishable in U. This means that R partitions U into equivalence classes called granules or information granules. The pair $(U,R)$ is called an approximation space. For a given target subset $X\subseteq U$, we define its rough approximation as follows:

• Lower approximation: the lower approximation of X, denoted as $\underline{R}\left(X\right)$, consists of all elements of U that definitely belong to X based on the equivalence relation R: $\underline{R}\left(X\right)=\{x\in U|{\left[x\right]}_R\subseteq X\}$, where ${\left[x\right]}_R$ is the equivalence class of x under R.
• Upper approximation: the upper approximation of X, denoted as $\overline{R}\left(X\right)$, consists of all elements of U that possibly belong to X, meaning that their equivalence class overlaps with X: $\overline{R}\left(X\right)=\{x\in U\mid {\left[x\right]}_R\cap X\ne \varnothing \}$

Using these two approximations, the boundary region of X is given by the difference between the upper and lower approximations: $BR\left(X\right)=\overline{R}\left(X\right)-\underline{R}\left(X\right)$, representing elements that cannot be definitively classified as belonging to X or not. Hence, the roughness of a set X can be quantified using the accuracy measure ${\alpha }_R\left(X\right)={\displaystyle \frac{|\underline{R}\left(X\right)|}{|\overline{R}\left(X\right)|}}$, with ${\alpha }_R\left(X\right)\in [0,1]$. So, if ${\alpha }_R\left(X\right)=1$, then X is a crisp (classical) set (i.e., its boundary region is empty), whereas, if ${\alpha }_R\left(X\right)<1$, then X is a rough set, meaning there is uncertainty in classifying some elements of U, i.e., those of the boundary of X, as an element of X or not.

To interpret rough sets in terms of perception, we have to consider the equivalence relation R, which leads to perception of objects as granules and consequently inheriting granular perceptions of objects; see Definition 5. In this context, observers do not have direct access to all objects in the universe U but perceive them through the relation R. Hence, the perception of concepts, or sets, is a consequence of the granular perception of objects, which is based on the lower and upper approximation operators. That is why we consider two observers l and u, who perceive objects using their relation $\underline{R}$ and $\overline{R}$, respectively. The first observer l is sure and expresses a necessity, whereas the second observer u is uncertain and expresses a possibility. Consequently, we define the rough perception function as $q^p=<q_l^p,q_u^p>$, where $q_l^p=<g_l^p,f_l^p>$ and $q_u^p=<g_u^p,g_u^p>$ represent perception functions of the observers l and u, respectively. For this rough perception, both observers do not perceive objects as they are because $g_u^p\left(\left\{x\right\}\right)={\left[x\right]}_{\underline{R}}$ and $g_u^p\left(\left\{x\right\}\right)={\left[x\right]}_{\overline{R}}$. In fact, objects are observed as granules (i.e., ${\left[x\right]}_{\underline{R}}$, ${\left[x\right]}_{\overline{R}}$) because of indiscernibility because both observers cannot separate some objects from each other, even if they are not identical, leading to the perception of granules. In addition, these two observers l and u perceive concepts using the functions $f_l^p\left(X\right)=\underline{R}\left(X\right)$ and $f_u^p\left(X\right)=\overline{R}\left(X\right)$, respectively. So, a set X is crisp or accessible if there is a consensus between these two observers, i.e., $q_l^p\left(X\right)=q_u^p\left(X\right)$, and rough otherwise. For more details on the relationship between roughness and accessibility, see [28], where the author shows that rough sets are $ϵ$-accessible. Note that $q_l^p$ belongs to the pessimistic perception class introduced in Section 6.2, while $q_u^p$ is an optimistic perception function (Section 6.3).

Many proposals have been made for generalizing rough sets, especially in relation with fuzzy sets, focusing on the indiscernibility relation, as granules play a pivotal role in Rough Set Theory (RST). For example, $\alpha$-RST [29] has been developed to approximate fuzzy sets using a parameterized indiscernibility relation $R_{\alpha }$ based on the classical indiscernibility relation R, a similarity function S, and a threshold $\alpha$ to control the size of granules:

$$\forall x,y\in U,(x,y)\in R_{\alpha }\iff ((x,y)\in R)\wedge (S(x,y)\ge \alpha )$$

(16)

All basic concepts of rough sets are extended using $R_{\alpha }$ instead of R. Similarly to rough sets, we also consider two observers, but here, $l^{\alpha }$ and $u^{\alpha }$, controlling the size of granules they perceive with a parameter $\alpha$. In addition, the parametrized perception function is $q_{\alpha }^p=(q_{l^{\alpha }}^p,q_{u^{\alpha }}^p)$, where $q_{l^{\alpha }}^p=<g_{l^{\alpha }}^p,f_{l^{\alpha }}^p>$ and $q_{u^{\alpha }}^p=<g_{u^{\alpha }}^p,g_{u^{\alpha }}^p>$. As for rough observers used for rough sets, these two parametrized rough observers also do not perceive objects as they are, but they perceive them at the same level $\alpha$, where $g_{l^{\alpha }}^p\left(\left\{x\right\}\right)={\left[x\right]}_{{\underline{R}}_{\alpha }}$ and $g_{u^{\alpha }}^p\left(\left\{x\right\}\right)={\left[x\right]}_{{\overline{R}}_{\alpha }}$. Moreover, they also perceive concepts at the same level $\alpha$ using their perception functions $f_{l^{\alpha }}^p\left(X\right)={\underline{R}}_{\alpha }\left(X\right)$ and $f_{u^{\alpha }}^p\left(X\right)={\overline{R}}_{\alpha }\left(X\right)$, respectively.

More generally, rough set theory is particularly useful in machine learning, data mining, pattern recognition, and artificial intelligence, where data often contain incomplete, inconsistent, or noisy information. Several other extensions of rough sets were developed; however, interpreting them in terms of perception is beyond the scope of this paper.

In conclusion, this section elucidates and reinterprets established set theories by situating them within the oSet formalism, which explicitly incorporates the role of observers. This unified perspective facilitates the comparison of different types of sets, regardless of their notational and structural heterogeneity. For instance, in fuzzy sets, elements of the universe are clearly distinguished, but their membership is uncertain. In contrast, rough sets introduce indiscernibility, where some objects cannot be distinguished from others, leading to granular representations. Thus, imprecision in fuzzy sets arises from uncertainty in membership attribution, whereas uncertainty in rough sets results from indistinguishability among elements. Additionally, intuitionistic fuzzy sets simultaneously capture both membership and non-membership uncertainties, explicitly accounting for hesitancy. In contrast, hesitant fuzzy sets primarily represent the diversity of expert opinions rather than any inherent hesitation. This distinction makes hesitant fuzzy sets particularly suitable for multi-expert decision making, where diverse yet imprecise judgments must be aggregated into a coherent evaluative framework.

Moreover, the oSet framework can serve as a foundation for constructing new set theories, such as Intuitionestic-Hesitant Sets, that integrate multiple imprecise hesitant observers. Formally, let $X=\{<x,h_X\left(x\right)>:x\in U\}$, $h_X\left(x\right)$ be a set of ${\pi }_X\left(x\right)$ representing a hesitation margin or the degree of indeterminacy of $x\in X$. In this case, we consider multiple imprecise and hesitant observers, and consequently, these sets can be viewed as a generalized framework encompassing fuzzy, intuitionistic, and hesitant sets simultaneously.

8. Computing with Perceptions

8.1. Perceptions as Sets

In mathematics and related disciplines, different kinds of set have been introduced to model various forms of knowledge, uncertainty, vagueness, and hesitation. These sets go beyond classical set theory to capture more nuanced situations in real-world applications (engineering, AI, medicine, social sciences, etc.). Several types of sets and the types of phenomena they are designed to model are summarized in

Table 5. Sets.

Set Type	Captures	Membership	Example
Classical Set	Exact knowledge	0 or 1	Set of even numbers
Fuzzy Set	Vagueness, imprecision	$[0,1]$	“Hot temperature”
Interval-Valued Fuzzy Set	Uncertainty in imprecision	Interval $[a,b]\subseteq [0,1]$	Uncertain evaluation of temperature
Intuitionistic Fuzzy Set	Imprecision, hesitation	Membership and non-membership	Student performance evaluation
Rough Set	Incompleteness, indiscernibility	Lower and upper approximations	Patient diagnosis from incomplete data
Hesitant Fuzzy Set	Multiple possible degrees	Set of values in $[0,1]$	Varying expert opinions
Type-2 Fuzzy Set	Uncertainty about fuzziness itself	Fuzzy membership function	Linguistic uncertainty across regions

. The relationship between such sets and their corresponding perception functions are discussed in Section 7.

In practice, if observers are uncertain, we can use fuzzy sets that allow partial membership of elements to a set. Consider perceptions of X by two uncertain observers, denoted $\mathcal{X}$ and $\mathcal{Y}$, respectively, and use fuzzy set operators:

• Fuzzy Union:

  $${\mu }_{\mathcal{X}\cup \mathcal{Y}}\left(x\right)=max({\mu }_{\mathcal{X}}\left(x\right),{\mu }_{\mathcal{Y}}\left(x\right))$$

  (17)
• Fuzzy Intersection:

  $${\mu }_{\mathcal{X}\cap \mathcal{Y}}\left(x\right)=min({\mu }_{\mathcal{X}}\left(x\right),{\mu }_{\mathcal{Y}}\left(x\right))$$

  (18)
• Fuzzy Complement:

  $${\mu }_{-\mathcal{X}}\left(x\right)=1-{\mu }_{\mathcal{X}}\left(x\right)$$

  (19)
• Fuzzy Difference:

  $${\mu }_{\mathcal{X}\setminus \mathcal{Y}}\left(x\right)=min({\mu }_{\mathcal{X}}\left(x\right),1-{\mu }_{\mathcal{Y}}\left(x\right))$$

  (20)

We can also account for the perceptions of different observers by using generalized operators such as t-norms and t-conorms. Moreover, when perceptions are affected by incompleteness or indiscernibility, we can employ rough sets. In contexts involving multiple experts with varying opinions, hesitant fuzzy sets provide a suitable modeling framework.

8.2. Perceptions as Family of Sets

In this section, we define operations on oSets or on families as point-wise (or element-wise) operations between two families of sets that have the same size. This idea naturally generalizes set operations to paired sets across two families. Let $\mathcal{X}=(X_1,X_2,...,X_n)$ and $\mathcal{Y}=(Y_1,Y_2,...,Y_n)$ be two perceptions of the n observers of the two sets X and Y, where $X_i$, and $X_i$ are the perceptions of the observers i and j, respectively. We now define set operations pointwise:

• Pointwise Union:

  $$\mathcal{X}\cup \mathcal{Y}=(X_1\cup Y_1,X_2\cup Y_2,...,X_n\cup Y_n)$$

  (21)
• Pointwise Intersection:

  $$\mathcal{X}\cap \mathcal{Y}=(X_1\cap Y_1,X_2\cap Y_2,...,X_n\cap Y_n)$$

  (22)
• Pointwise Difference:

  $$\mathcal{X}\setminus \mathcal{Y}=(X_1\setminus Y_1,X_2\setminus Y_2,...,X_n\setminus Y_n)$$

  (23)
• Pointwise Symmetric Difference:

  $$\mathcal{X}△\mathcal{Y}=(X_1△Y_1,X_2△Y_2,...,X_n△Y_n)$$

  (24)
• Pointwise Cartesian Product:

  $$\mathcal{X}\times \mathcal{Y}=(X_1\times Y_1,X_2\times Y_2,...,X_n\times Y_n)$$

  (25)

We can use this representation and such operations in different applications, for example, to compare knowledge or sensor readings between agents, to manage decision in multiple specialist contexts, to compare user profiles using, for example, interest sets, to match feature sets across ontology instances, to compute overlaps between user preferences, etc.

8.3. Perceptions as Multisets

We introduce now the representation of a family of sets as a multiset over the universe based on element multiplicity and then adapt classical set operations accordingly. Let us formalize a multiset-based representation of the perception of X by n observers. Thus, we count the number of times that each element of U appears in the perceptions $X_1$, $X_2$, …, $X_n$ of the n observers. Multisets provide a compressed and algebraic way to capture redundancy, prevalence, overlap, and uniqueness in perception, going beyond each observer’s personal perception.

Consider the perception of X, i.e., $q_{\{1,2,3\}}\left(X\right)$, denoted $\mathcal{X}$, by three obsevers 1, 2, and 3, i.e., $\mathcal{X}=(X_1,X_2,X_3)$, where $X_1=\{1,2\}$, $X_2=\{1,2,5\}$, and $X_3=\{1,4,5\}$. The multiset that represents the perception $\mathcal{X}$ is $M_{\mathcal{X}}=\{1:3,2:2,4:1,5:2\}$, which means that the element 1 appears in 3 perceptions, the elements 2 and 5 appear in 2 perceptions, and the element 4 appears only in one perception. Using $m_{M_{\mathcal{X}}}\left(x\right)$ to represent the multiplicity of element $x\in U$ in the multiset $M_{\mathcal{X}}$, we have $m_{M_{\mathcal{X}}}\left(1\right)=3$, $m_{M_{\mathcal{X}}}\left(2\right)=m_{M_{\mathcal{X}}}\left(5\right)=2$, and $m_{M_{\mathcal{X}}}\left(4\right)=1$. The support of perception $\mathcal{X}$ is defined as:

$$S_{\mathcal{X}}=\{x/x\in U,m_{M_{\mathcal{X}}}\left(x\right)>0\}$$

(26)

The combined supports of two perceptions $\mathcal{X}$ and $\mathcal{Y}$, i.e., $S_{\mathcal{X},\mathcal{Y}}$ is the union of their respective supports. Formally, the multiset representing the perception $\mathcal{X}$ is:

$$M_{\mathcal{X}}=\{x:m_{M_{\mathcal{X}}\left(x\right)}/x\in U,m_{M_{\mathcal{X}}\left(x\right)}=\left|\{X/X\in \mathcal{X},x\in X\}\right|\}$$

(27)

In this context, the empty multiset is the set that contains the null multiplicities for all elements in U. We define operations on perceptions as operations on multisets that represent them. We said that a perception $\mathcal{X}$ is included into $\mathcal{Y}$ whenever:

$$\forall x\in M_{\mathcal{X}},m_{M_{\mathcal{X}}}\left(x\right)\le m_{M_{\mathcal{Y}}}\left(x\right)$$

(28)

For simplicity’s sake, we will use the same inclusion symbol as for sets, i.e., $M_{\mathcal{X}}\subseteq M_{\mathcal{Y}}$. More generally, operations on perceptions $\mathcal{X}$ and $\mathcal{Y}$ are defined as operations on multisets that represent them, i.e., $M_{\mathcal{X}}$ and $M_{\mathcal{Y}}$ respectively. Hence, the union of two perceptions $\mathcal{X}$ and $\mathcal{Y}$ is defined as:

$$\begin{array}{cc}\hfill & M_{\mathcal{Z}}=M_{\mathcal{X}}\cup M_{\mathcal{Y}}=\{x:m_{M_Z}\left(x\right),x\in S_{\mathcal{X},\mathcal{Y}}\}\hfill \\ \hfill & m_{M_Z}\left(x\right)=max(m_{M_X}\left(x\right),m_{M_Y}\left(x\right))\hfill \end{array}$$

(29)

The intersection of two perceptions $\mathcal{X}$ and $\mathcal{Y}$ is defined as:

$$\begin{array}{cc}\hfill & M_{\mathcal{Z}}=M_{\mathcal{X}}\cap M_{\mathcal{Y}}=\{x:m_{M_Z}\left(x\right),x\in S_{\mathcal{X},\mathcal{Y}}\}\hfill \\ \hfill & m_{M_Z}\left(x\right)=min(m_{M_X}\left(x\right),m_{M_Y}\left(x\right))\hfill \end{array}$$

(30)

The sum of two perceptions is represented as $M_{\mathcal{X}}+M_{\mathcal{Y}}$:

$$\begin{array}{cc}\hfill & M_{\mathcal{Z}}=M_{\mathcal{X}}+M_{\mathcal{Y}}=\{x:m_{M_Z}\left(x\right),x\in S_{\mathcal{X},\mathcal{Y}}\}\hfill \\ \hfill & m_{M_Z}\left(x\right)=m_{M_X}\left(x\right)+m_{M_Y}\left(x\right)\hfill \end{array}$$

(31)

The difference of two perceptions is a subtraction between two perceptions:

$$\begin{array}{cc}\hfill & M_{\mathcal{Z}}=M_{\mathcal{X}}-M_{\mathcal{Y}}=\{x:m_{M_Z}\left(x\right),x\in S_{\mathcal{X},\mathcal{Y}}\}\hfill \\ \hfill & m_{M_Z}\left(x\right)=max(m_{M_X}\left(x\right)-m_{M_Y}\left(x\right),0)\hfill \end{array}$$

(32)

Representing perceptions as multisets offers a powerful and versatile framework for various domains. For instance, in information retrieval, term frequency can be modeled across multiple data sources; in diversity analysis, element frequency captures variation within or across populations; in probabilistic modeling, multisets reflect discrete distributions over observed categories; and in sociology, they can describe the frequency of attributes or responses across groups.

Formally, a multiset derived from multiple individual perceptions of a set X aggregates information by associating each object $x\in U$ with its multiplicity, i.e., the number of observers who consider x to belong to X. Although this multi-set representation compactly captures how frequently an object is perceived as a member of X, it abstracts away the identity of the observers. To recover this richer layer of information, we introduce a function ${\phi }_{\mathcal{X}}:U\to I$, where I denotes the set of observers. The function ${\phi }_{\mathcal{X}}$ maps each object $x\in U$ to the subset of observers in I who perceive x as a member of X. This function enables us to go beyond frequency and reconstruct the observer-specific structure of perception, which is crucial in applications requiring traceability, personalized modeling, or observer-aware reasoning.

8.4. Shared Perceptions

Unlike an elementary or individual perception $q_i\left(X\right)$ of a single observer i, a shared perception $\widehat{X}$ of X is defined, according to the set of observers I as an alternative representation that incorporates the perspectives of all observers in I.

Definition 16 (Shared perception).

Let U be the universe of objects, $X\in \mathcal{P}\left(U\right)$, I be the index set of observers, $q_i$ the elementary perception function of the observer i, $q_I\left(X\right)={\left(q_i\left(X\right)\right)}_{i\in I}$ be the perception of X. $\widehat{X}\in \mathcal{P}\left(U\right)$ is a shared perception of X iff:

$$\forall i\in I,\widehat{X}\cap q_i\left(X\right)\ne \varnothing$$

(33)

Hence, a shared perception meets the perceptions of all observers of X and aims to reflect a collective, rather than an individual, perception or understanding of X. A shared perception $\widehat{X}$ is called minimal if no proper subset ${\widehat{X}}^{\prime }\subset \widehat{X}$ is a shared perception.

Definition 17 (Minimal shared perception).

Let U be the universe of objects, $X\in \mathcal{P}\left(U\right)$, I be the index set of observers, $q_i$ the elementary perception function of the observer i, $q_I\left(X\right)={\left(q_i\left(X\right)\right)}_{i\in I}$ be the perception of X. $\widehat{X}$ is a shared perception of X iff:

$$\widehat{X}\in {\widehat{q}}_I\left(X\right)\iff (\forall i\in I,\widehat{X}\cap q_i\left(X\right)\ne \varnothing )\wedge (\forall Y\subset \widehat{X},\exists i\in I,Y\cap q_i\left(X\right)=\varnothing ).$$

(34)

To compute shared perceptions, we first compute the minimal shared perceptions and then generate the other shared perceptions of X according to the set of observers I. Then, we use them to generate the set of all shared perceptions. To achieve this goal, we change the representation of perceptions from a tuple $q_I\left(X\right)=(X_1,X_2,...,X_{\left|I\right|})$ to a hypergraph.

Definition 18 (Hypergraph $\mathcal{H}$).

A hypergraph $\mathcal{H}$ is a pair $\mathcal{H}=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}=\{v_1,v_2,...,v_n\}$ is a finite set of vertices and $\mathcal{E}={\left(e_i\right)}_{i\in 1,2,...,m}$ a set of non-empty subsets of $\mathcal{V}$ called hyperedges such that ${\cup }_{1\le i\le m}\left\{e_i\right\}=\mathcal{V}$.

In hypergraph theory [30], a vertex cover or transversal, denoted $Tr\left(\mathcal{H}\right)$, is a set of vertices such that every hyperedge $e_i\in \mathcal{E}$ of the hypergraph contains at least one vertex $v_i$ of that set. Moreover, a transversal T is called minimal if no proper subset of $T^{\prime }\subset T$ is a transversal. The set of minimal transversals of the hypergraph $\mathcal{H}$ is denoted $MinTr\left(\mathcal{H}\right)$.

The perception of X, i.e., $q_I\left(X\right)=(X_1,X_2,...,X_{\left|I\right|})$, can be represented by the hypergraph ${\mathcal{H}}_I\left(X\right)$=(${\mathcal{V}}_I\left(X\right)$, ${\mathcal{E}}_I\left(X\right)$), where the set of its vertices is ${\mathcal{V}}_I\left(X\right)={\cup }_{i\in I}\left\{q_i\left(X\right)\right\}$ and ${\mathcal{E}}_I\left(X\right)=\{q_i\left(X\right):i\in I\}$ is the set of its edges. In this context, the minimal shared perceptions $\widehat{X}$ of $q_I\left(X\right)$ are minimal transversals of the hypergraph ${\mathcal{H}}_I\left(X\right)$:

$$\widehat{X}=MinTr\left({\mathcal{H}}_I\left(X\right)\right)$$

(35)

After computing $\widehat{X}$ using, for example, Berge’s algorithm, we generate the other shared perception to define the Consistent Shared Perception Space.

Definition 19 (Consistent Shared Perception Space—CSPS).

The space of consistent shared perceptions considering the set of observers I is the sub-lattice defined by the interval $[\cap \{Y\in {\widehat{q}}_I\left(X\right)\},\cup \{Y\in {\widehat{q}}_I\left(X\right)\}]$.

The Algorithm 1 begins by modeling the perceptions of the set X by a group of observers I as a hypergraph $\mathcal{H}$ (lines 3–4). In this representation, the perception of each observer corresponds to a hyperedge connecting the elements they associate with X. Following this, the initial consistent shared perception space ($CSPS$) is initialized using the Minimal Shared Perceptions of $\mathcal{H}$ (line 6), which serves as the starting point for exploration. Subsequently, the algorithm enters an iterative phase (lines 7–12), where it systematically computes intersections and unions between all sets in $CSPS$. This process continues until closure, i.e., when no new sets can be generated from further combinations, indicating that the system has reached a convergence point in the perception space.

Algorithm 1 CSPS Algorithm

1: Input: The perception of X, i.e., $q_I\left(X\right)={\left(q_i\left(X\right)\right)}_{i\in I}$  2: Ouput: set of shared perceptions ${\widehat{q}}_I\left(X\right)$  3: and the generated consistent shared perception space, i.e., $(CSPS,⪯)$  4: $\mathcal{V}={\cup }_{i\in I}\left\{q_i\left(X\right)\right\}$  5: $\mathcal{E}=\{q_i\left(X\right):i\in I\}$  6: $\mathcal{H}=(\mathcal{V},\mathcal{E})$  7: $CSPS=MinTr\left(\mathcal{H}\right)$  8: repeat  9:    $CSPSold=CSPS$ 10:    $A=\{X\cup Y:X,Y\in CSPS\}$ 11:    $B=\{X\cap Y:X,Y\in CSPS\}$ 12:    CSPS = $CSPSold\cup A\cup B$ 13: until Convergence: $CSPS=CSPSold$ Ensure:  ${\widehat{q}}_I\left(X\right)$ and $(CSPS,⪯)$.

Shared perceptions can be useful in solving problems that represent a particularly complex decision, which are at the intersection of personal identity, cultural symbolism, aesthetic preferences, and social expectations. For example, the wedding dress problem, see Section 8.6.2, is not merely a matter of clothing choice but a multifaceted process that reflects how an individual wishes to express herself during a highly ritualized and socially significant event.

8.5. Perception Diversity Analysis

8.5.1. Diversity of Systems

Diversity is a foundational concept in analyzing complex systems, whether ecological, technological, economic, or social, because it directly influences the system’s resilience, functionality, adaptability, and fairness. Diversity analysis (DA) plays a central role across numerous disciplines, as it provides insights into the structure, stability, adaptability, and equity of complex systems. Although the contexts differ, the underlying rationale for considering diversity remains conceptually consistent: systems with greater diversity are generally more robust, innovative, and sustainable.

Alternatively, Perception Diversity Analysis (PDA) is a complementary research domain to DA. While DA focuses on analyzing diversity within a system, examining the variety of elements, structures, or behaviors that constitute it, PDA addresses the diversity of external perceptions of that system. In other words, PDA explores how different observers, shaped by their backgrounds, knowledge, values, and experiences, perceive and interpret the same system in varied ways.

For example, in ecology, DA helps quantify species richness and evenness within ecosystems. Metrics are used to assess biodiversity and assess plant diversity in a rainforest to predict resilience to climate change. In this context, consider that a garden rich in floral biodiversity may go unappreciated if its diversity is not adequately perceived by the local community. As a result, the community might vote to repurpose it, perhaps into a bicycle parking lot, highlighting the gap between ecological value and social perception.

In addition, in the social sciences, diversity analysis is employed to assess cultural, ethnic, linguistic, and ideological variation within populations. These dimensions of diversity significantly influence policy making, social cohesion, and conflict dynamics. For example, in multilingual societies, measuring linguistic diversity is crucial for designing inclusive and equitable educational systems. In this context, consider a society in which small minority groups, though statistically insignificant, are physically and culturally visible, often as descendants of immigrants. Despite the numerical dominance and cultural continuity of the majority population, some may perceive these minorities as a threat, fueling narratives of demographic replacement. As a result, they become focal points in political discourse, particularly during election periods. This perception can generate disproportionate media attention and public anxiety, even when the lived realities of these minorities, often marginalized and with limited societal influence, contrast sharply with the dominant narrative.

Another example concerns AI, machine learning, and, more particularly, ensemble learning, where diversity among classifiers improves generalization. Diverse models bring different “opinions” to a problem, reducing overfitting and improving performance. In fact, diversity increases the robustness of the models to unseen data.

In short, diversity is not merely a philosophical or ethical concept; it is a quantifiable and critical feature of system design and analysis. Whether in natural ecosystems, engineered systems, or social structures, diversity contributes to robustness, adaptability, and innovation, making it indispensable in both theoretical modeling and practical implementation. In this paper, we have introduced the perception diversity analysis as a dual approach to diversity analysis. We can summarize the relationship between them as follows: what matters most is not only the intrinsic nature of systems but also how they are perceived by observers. In this context, DA focuses on the internal diversity within a system, while PDA emphasizes the external, observer-dependent diversity of interpretations or perceptions of that system. The distinction between “what systems are” (referring to their objective structure) and “how they are perceived” (highlighting subjective and contextual factors) captures the epistemological importance of perspective and interpretation. This distinction is particularly relevant in multidisciplinary fields such as human–AI interaction, sociology, cognitive science, and the philosophy of science.

8.5.2. Perception Diversity Metrics

Diversity analysis employs a variety of quantitative metrics to measure variation within a set of elements, whether they are biological species, data points, social groups, or model components. These metrics are often drawn from disciplines such as ecology, information theory, statistics, machine learning, and sociology. The most widely used diversity metrics can be classified into three main categories: richness, evenness, and composite indices.

Diversity, measured using metrics as disagreement measures or entropy-based metrics, increases the robustness of models to unseen data and adversarial attacks. For example, an ensemble of neural networks with different architectures performs better on image classification tasks than any individual model.

The diversity metrics are adapted here to quantify and analyze the variability of human perceptions across different observers. An oSet can cannot be known exactly with respect to its observers, so it is represented by a family of sets according to the diversity of its perceptions. We introduce metrics on a family of sets to support perceptions diversity analysis.

Consider $U=\{x_1,x_2,...,x_M\}$ the universe of elements and N observers, where the observer i perceives X as $X_i\subseteq U$. All perceptions of X are represented by the family $\mathcal{X}=\{X_1,X_2,...,X_N\}$. To quantify the diversity of perceptions of X, we employ several established diversity indices, each capturing different aspects of diversity, such as richness, evenness, dominance, and disparity. To achieve this goal, we start to calculate the following quantities:

• Frequency $fre{q}_j$, which is the number of sets $Xi$ in which element $x_j$ appears.
• Total occurrences $T={\sum }_{j=1}^{M}fre{q}_j$.
• Proportion $p_j={\displaystyle \frac{fre{q}_j}{T}}$, representing the relative frequency of $x_j$ across all perceptions.

Now, we give examples of metrics to measure the perceptions diversity or the diversity of a family of sets.

• Richness ($R=|{\cup }_{i=1}^M{X}_i|$): The total number of distinct types (e.g., species, products, categories, classes) in a dataset. For example, in ecology, if a forest has 10 tree species, then species richness = 10. However, this metric does not consider relative abundances.
• Shannon Entropy or Shannon–Weaver Index ($H^{\prime }=-{\sum }_{j=1}^M{p}_{i}log\left(p_i\right)$): This metric incorporates both richness and evenness. It is used in ecology, NLP (text entropy), recommender systems, and fairness analysis. Higher $H^{\prime }$ indicates greater uncertainty or diversity.
• Pielou’s Evenness Index ($J^{\prime }={\displaystyle \frac{H^{\prime }}{log\left(R\right)}}$): Evenness captures how uniformly elements are distributed across categories. Values range from 0 (uneven distribution) to 1 (perfectly even). It is often used to compare distribution fairness in ecology or sociology.

8.6. Example of Applications

8.6.1. Socially Informed AI

There is a broader class of decision-making problems that involve high personal stakes, strong social and cultural expectations, emotional investment, and identity expression. These types of choice and decision are often multidimensional, subjective, and difficult to ‘optimize’ in a traditional rational sense. For example, choosing a wedding dress is a uniquely challenging decision, as it involves personal identity, cultural symbolism, social expectation, preferences, etc., see Section 8.6.2 for more details.

Like the wedding dress problem, others similar problems are:

• Choosing a Career Path: a career often reflects personal identity, values, and aspirations, while also being influenced by family expectations, financial constraints, and social norms.
• Naming a Child: the name must resonate personally and culturally and often reflect heritage, while also considering social perceptions, pronunciation, and uniqueness.
• Decorating a Home Interior: reflects aesthetic taste, social identity, emotional comfort, and budget constraints.
• Making a Life-Altering Medical Decision: combines bodily autonomy, family opinions, societal ethics, and long-term consequences.
• Designing a Logo: encodes the essence of the brand, balances uniqueness and clarity, and needs to appeal to diverse stakeholders.

These examples show that the wedding dress dilemma is not unique, it is a rich metaphor for a class of decisions that are ambiguous, expressive, and multi-constrained. They highlight the need for human-centered design, empathy, and interdisciplinary understanding in real-world decision support systems.

Beyond these personal matters, many domains, like industry, health, trade, public policy, environment, etc., present complex decision-making problems emotionally significant, highly visible, symbolically or economically loaded, and shaped by a network of practical and psychological constraints. Across these domains, decisions are not just about optimization—they are about negotiating meaning, emotion, identity, and constraints. Recognizing this helps design human-centered systems for decision support, AI interaction, policy making, and innovation management.

In artificial intelligence, there are several problems that cannot be reduced only to an optimization problem. For example, evaluation is subjective because success is not just functional (e.g., accuracy) but experiential, how users feel about the system. In addition, different users react differently to the same AI system output due to culture, education, or experience. Finally, many decisions invoke ethical considerations that lack universal consensus, which lead to an increasing ethical and emotional complexity of AI systems.

• Explainable AI (XAI): designing an explanation for an AI decision requires cultural and cognitive alignment because an explanation that satisfies one user may confuse or alienate another.
• Algorithmic Fairness: What are fairness criteria (e.g., equal opportunity vs. demographic parity). Different group values and different definitions of fairness lead to no one-size-fits-all solution.
• Chatbots/AI Assistants: The design of tone and personality of voice agents is a crucial problem. In fact, the same tone may be perceived as friendly or patronizing based on user background and expectations.
• Autonomous Vehicles: Programming ethical decisions in edge cases is not simple because moral dilemmas (e.g., trolley problem) depend on cultural norms and societal values.
• Content Moderation AI: detecting hate speech or misinformation is a difficult problem because what is considered “harmful” or “inappropriate” is often subjective and context-dependent.
• Recommender Systems: Such systems suggest content (e.g., news, videos, jobs). However, users may feel manipulated or stereotyped by recommendations, even if technically accurate.
• Affective Computing: detecting and responding to human emotions can only be reduced to plying with numbers, because cultural, personal, and contextual differences affect how emotions are expressed and interpreted.
• Medical Diagnosis AI: How to communicate diagnoses or risk predictions? The same medical risk output may cause fear or empowerment due to interpretation issues.
• Generative AI: What is creative in generating text, music, and producing art? Creativity is subjective; users may see AI as innovative or derivative depending on perception.

AI systems that model, quantify, and act under uncertainty are more robust, trustworthy, and capable of interacting with complex real-world environments. Combining multiple approaches is often essential to build AI systems that understand risk, adapt under ambiguity, and make context-aware decisions. Moreover, many challenges lie not in building the system but in ensuring that it aligns with human values, perceptions, and expectations across cultures, disciplines, and individual differences. These are perception-sensitive problems, and they demand interdisciplinary, human-centered design thinking.

8.6.2. Social Networks for Decision Making

The shared perceptions are experimented with using social networks for decision making and not only for broadcasting information and news. For example, choosing a wedding dress is a uniquely challenging decision that sits at the intersection of personal identity, cultural symbolism, aesthetic preference, and social expectation. This complexity can be unpacked through a multidisciplinary lens, drawing on psychology, sociology, fashion studies, and decision theory to understand both why it is difficult and how one might navigate the process more effectively. In fact, a wedding dress is not just a garment; it represents a milestone, a moment of public visibility, and often a deeply personal dream. Brides may feel intense pressure to find “the perfect dress”, a notion reinforced by media and cultural narratives. This expectation can lead to paralysis decision, perfectionism, or post-purchase regret. In addition, wedding dresses are rich with cultural meaning. In many societies, the white dress symbolizes purity, commitment, and social belonging. Deviating from these norms, e.g., choosing a red dress in a Western context, can invite criticism or confusion, even if it better suits the individual. In addition, family members, friends, and bridal consultants often weigh in. While well-intentioned, these opinions can overshadow the bride’s own preferences, leading to dresses that reflect others’ desires rather than her own. Finally, fashion norms and body ideals can lead brides to choose dresses that flatter perceived “flaws” rather than celebrate their bodies authentically. This complicates the experience with issues of self-esteem and self-presentation.

Consider a bride-to-be facing the emotionally significant and culturally symbolic task of choosing her wedding dress. Rather than relying solely on conventional e-commerce search engines, we propose an iterative and interactive decision-making process grounded in the perceptual feedback of her social network (e.g., Facebook or Instagram). We present here a socially embedded approach to decision making in a high-stakes, emotionally significant context. This approach leverages collective perception and interpersonal affinity to guide the selection process more organically and contextually.

The main steps of this process (illustrated in Figure 6) are as follows:

• Initial Selection: The bride selects a subset of dresses from online platforms or physical catalogs over which she is undecided.
• Social Sharing: These shortlisted dresses are posted to her social media profile, such as her Facebook or Instagram wall.
• Perception Collection: Friends and family provide feedback by expressing their preferences (likes, comments, votes), which reflect their individual perceptions of the dresses.
• Filtering Based on Perceptions: The bride filters the options based on the aggregated perceptions received from her network.
• Affinity-Weighted Feedback Integration: She navigates within a conceptual space of collective subjective perceptions (CSPS), wherein the feedback is weighted based on her perceived affinity with respondents. For example, the opinions of her sister or grandmother may carry more weight than those of distant acquaintances. This weighting is informed by relationship strength, personal trust, shared taste history, and emotional closeness.
• Iterative Refinement: Updated selections are reposted to gather further feedback, allowing the solution space to be refined iteratively until only a few highly preferred options remain.

After narrowing down her options based on social feedback, the bride must further refine her choice by considering objective criteria such as price, delivery time, alteration possibilities, return policies, and other logistical constraints. This stage exemplifies a multi-criteria decision-making problem where both subjective perceptions and practical considerations are involved.

Suppose two remaining dresses differ in price and delivery time. Dress A is preferred aesthetically by most close friends, but it is more expensive and requires international shipping with uncertain delivery. Dress B is slightly less favored in terms of style but is affordable and available locally. The system can display a perceptual distribution graph, highlight decision trade-offs, and simulate outcomes under various constraints—without dictating a final answer. The bride remains the ultimate decision maker, supported by an informed social-computational process.

Importantly, this human–machine interaction is not governed by a traditional optimization algorithm that seeks a single best solution based on predefined utility functions. Instead, the process is driven by an iterative perception-exchange mechanism, where human participants (e.g., friends, family) contribute their opinions and preferences, and computational systems aggregate, structure, and analyze these inputs.

The system’s role is not to prescribe a decision but to compute and represent the space of shared perceptions, effectively forming a perception landscape or preference map. This landscape helps the human decision maker to navigate complex trade-offs and social signals while still maintaining agency in the final choice.

In short, this framework exemplifies a human-centric and socially informed method of decision making under uncertainty. It integrates perception aggregation, trust-based weighting, and iterative refinement, making it relevant to fields such as recommender systems, collective intelligence, and collaborative human–AI agents. The process respects the emotional and identity-expressive nature of the decision while using computational principles such as social graph analysis and oSets-based perception modeling. This wedding dress problem illustrates a broader paradigm in human–AI systems, where AI serves as a facilitator of shared understanding rather than a solitary decision maker. It aligns with ongoing research in explainable AI (XAI), collaborative filtering, decision support systems, and human-centered computing, particularly in domains requiring emotional sensitivity, cultural awareness, and social trust.

9. Discussion

We have introduced four perception functions, the main classes of which are denoted Class NR, Class RR, and Class I corresponding, respectively, to Naive Realism, Representative Realism, and Idealism, well-known schools in epistemology. The accessibility plays a key role in bridging the gap between sets and perceptions, i.e., all sets are accessible for naive perceptions, not all sets are necessarily accessible for optimistic perceptions, whereas no set is accessible for idealist observers. Considering that each observer has his own perception function, one may induce a relationship between sets and observers. More precisely, we have induced the algebraic structure associated with each perception function; see Figure 7.

The view of the world that naive observers derive from their perception faithfully represents objects and concepts of the real world assuming a sharp distinction between them and excluding any variability of their perception. Thus, all objects and concepts are perceived as they are; i.e., all sets are accessible. Consequently, all naive observers, living in the same real world, construct the same algebraic structure (Figure 7a), which can be used for decision making and general reasoning. However, the limitations of this naive perception occur when dealing with concepts in the real world, which may be imprecisely defined. In addition to their imprecision, the boundaries of such concepts are unsharp, culminating in the problematic situation, where we can neither confirm with certainty that an element is a member of a set or not.

However, very little attention was given to the mathematical foundations of concepts perception and its variability despite its fundamental importance to human thinking and the social nature of human beings. In addition to our perception of the universe through our five senses, our age, education, gender, psychological peculiarities, and past individual experiences determine our perceptions of the world and suggest the nature of our decision and reasoning. Oriented by this need, two perception functions of the Class RR are introduced to represent the hidden nature of observers like pessimism (Figure 7b), optimism (Figure 7c), and doubt function represents the class I (Figure 7d).

Now, we present some properties of cross-perception functions:

Proposition 14.

The following properties are directly inferred from the definition of perception functions and axioms:

(9.1) 

$q^{\circ }(-X)=-\overline{q}\left(X\right)$

(9.2) 

$\overline{q}(-X)=-q^{\circ }\left(X\right)$

(9.3) 

$q^{\circ }\left(X\right)\subseteq q_i^{\sim }\left(X\right)$

(9.4) 

$q^{\circ }\left(X\right)\cap {\overline{q}}_i\left(X\right)=\varnothing$

(9.5) 

$q_i^{\sim }\left(X\right)=-q_i^{\sim }\left(X\right)$

(9.6) 

$q^{\circ }\left(X\right)\subseteq X\subseteq \overline{q}\left(X\right)$

(9.7) 

$\overline{q}\left(x\right)=q^{\circ }\left(y\right)\Rightarrow \overline{q}\circ q^{\circ }\left(X\right)=q^{\circ }\left(X\right)$

(9.8) 

$\overline{q}\left(x\right)=q^{\circ }\left(y\right)\Rightarrow q^{\circ }\circ \overline{q}\left(X\right)=\overline{q}\left(X\right)$

In real-world contexts, human perceptions are significantly more nuanced than the basic mathematical perceptual functions considered here. In practice, we can address this complexity by framing perception acquisition as an experiential process, one that unfolds during interactive exchanges between AI systems and human users. Through these shared experiences, the system can progressively uncover and formalize the human’s values, needs, and preferences. To operationalize this, a large language model (LLM)-based system can engage in structured dialogue, adapting to different roles, such as assistant, collaborator, or supervisor, depending on the context and the user’s level of expertise or authority. This conversational approach enables the system to refine its understanding of human perceptions dynamically and contextually, facilitating more aligned and ethically sensitive decision making.

Finally, let us focus on the bridge that we are attempting to build between perception theories developed in epistemology and sets to define oSets, which depend on their observer. In this spirit, we introduced different perception functions, which are discussed in the light of perception theories, mathematical structures, and the notion of accessibility. If we want to define a new perception function representing the idealism school, we need to change the perception space, replacing $(U,I)$ by $(U,V_i,I)$, under the constraint $U\cap V_i=\varnothing$, with U representing the real world and $V_i$ is the perception world of the observer i. Alternatively, with the advent of new technologies such as Virtual Reality (VR) headsets and artificial intelligence (AI), we can develop new perception functions according to representative realism and transcendental idealism approaches. For example, when wearing a VR headset, the mind does not structure reality through our mental filters. The observer is living in a virtual and/or augmented reality that evolves dynamically according to the observer’s behavior and context.

oSets are particularly useful for computing without consensus, i.e., no unique solution or decision to deal with the different alternatives. They can play a central role in the human–AI collaboration [31], because AI systems must account for the diversity of human perceptions as individuals construct their own reality based on experiences, culture, and cognitive biases, even when observing the same objective world. This diversity profoundly impacts reasoning, decision making, and problem solving. AI systems that rigidly impose ranking methods overlook the nuances of human subjectivity, potentially leading to biased or contextually inappropriate decisions. Instead, AI should incorporate perception-aware models, enabling adaptive, context-sensitive collaboration. For example, in medical diagnosis or legal reasoning, diverse expert opinions shape better outcomes than a single ranking approach, promoting inclusion, adaptability, and trust in AI. A hypergraph-based method was introduced to compute shared perceptions with respect to all observers [18].

In addition, vision-language models (VLMs) are central to the future of computer vision, as they enable more intelligent, general-purpose, and communicative AI systems. These models not only enhance traditional computer vision (CV) tasks but also broaden the scope of what machines can see/perceive and describe about the world. While conventional CV models can detect objects or segment scenes, they often lack deep semantic reasoning. By integrating language, VLMs can generate captions, answer questions, and explain visual decisions, thereby enriching the functionality of vision systems. For instance, in image captioning, a model might generate a description such as “A man riding a bicycle through a city street”, which conveys more meaning than simply detecting “man”, “bicycle”, and “street”. Beyond improved semantic understanding, VLMs also benefit from weak supervision by leveraging large-scale image–text pairs. This enables them to learn visual concepts without requiring dense pixel-level annotations. Moreover, VLMs are valuable for multimodal reasoning, human–AI interaction, and other tasks, helping manage complexity more effectively when addressing real-world problems.

Recently, researchers have begun exploring the development of Perception Language Models (PLMs) for image and video understanding [32]. Vision-language models are revolutionizing AI’s capacity to perceive, interpret, and communicate across modalities. These multi-modal, vision-centered models represent a key piece of the AI puzzle. However, to make them more effective and human-aligned, it is essential to incorporate the diversity of human perception—shaped by personal experiences, cultural contexts, ethical norms, education, values, and socioeconomic backgrounds. To advance human–AI interaction, creativity, and understanding, we propose Human Diversity Perception Language Models (HDPLMs). These models account for the egocentric and contextual nature of perception across individuals. The oSet framework provides a foundation for representing and computing with diverse perceptions, making it possible to build AI systems that are not only more robust but also more socially and culturally aware.

10. Concluding Remarks

In computing, particularly in artificial intelligence, acknowledging the variability of human perception is fundamental for designing systems that can adapt to diverse interpretations of reality. Traditional AI models operate under the assumption of an objective and well-defined world, often failing to capture the nuances of human subjectivity. However, future AI systems must go beyond rigid one-size-fits-all approaches to integrate perceptual diversity into their decision-making and modeling processes.

To achieve this, AI must incorporate frameworks that allow observer-dependent representations, accommodating uncertainty, ambiguity, preferences, cultural variability, and contexts. This requires moving from purely data-driven models to hybrid approaches that blend neural computation with symbolic reasoning, enabling AI to process and reason with multiple, sometimes conflicting, perspectives. Such models will be crucial in fields like natural language processing, human–computer interaction, and autonomous decision making, where understanding human intent and perception is vital.

Furthermore, in an increasingly interconnected world, AI systems will not operate in isolation but will coexist and interact with humans in shared environments. Future intelligent systems must be designed to dynamically adapt to human interpretations of reality rather than imposing a singular computational perspective. This calls for the development of perception-aware AI capable of recognizing when different observers interpret the same reality differently and adjusting their responses accordingly.

Ultimately, integrating human perception into AI systems will not only enhance their robustness and fairness but will also ensure their alignment with human values. As AI becomes embedded in daily life, shaping everything from personalized recommendations to autonomous decision making, its ability to recognize and respect the diversity of human experiences will define its success in creating meaningful and responsible interactions within the globally connected small world.

The oSets framework is designed to align with the complexities of the real world, taking into account the diversity of human perceptions for effective communication, AI development, and decision making in multidisciplinary fields. We also try to open new horizons for interdisciplinary research by building bridges between epistemology and more particularly theories of perception, mathematics starting with set theories, artificial intelligence, and computing.

In summary, our vision is for an AI revolution that prioritizes human-centric collaboration. We advocate for AI systems that are not only technically advanced but also sensitive to the rich diversity of human experiences, ethical standards, and cultural contexts. This approach ensures that AI becomes a powerful tool for positive transformation, enabling a fair, diverse, and respectful interaction between technology and humanity.