fuzzy-logic

Fuzzy Logic

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Fuzzy Logic is a mathematical approach addressing uncertainty. It features membership functions and linguistic rules. Fuzzy Sets use membership grades for flexibility. Applications include control systems and AI. Advantages encompass handling uncertainty, while challenges involve complexity. Future trends include fuzzy deep learning and IoT applications for adaptive control.

Characteristics of Fuzzy Logic:

  1. Membership Functions:
    • Fuzzy Logic employs membership functions to determine the degree of membership of an element in a set. These functions can take various shapes, such as triangular, trapezoidal, or sigmoidal curves, enabling flexible modeling of uncertainty.
  2. Linguistic Variables:
    • It utilizes linguistic variables to describe data using human-like terms such as “very hot,” “somewhat cold,” or “quite tall,” facilitating intuitive reasoning and decision-making.
  3. Fuzzy Rules:
    • Fuzzy Logic operates on fuzzy rules expressed in IF-THEN statements, linking linguistic variables and membership functions to derive conclusions based on fuzzy logic reasoning.
  4. Fuzzy Sets:
    • Fuzzy sets form the foundation of Fuzzy Logic, defining the boundaries of membership for elements and allowing for the representation of vague or ambiguous concepts.
  5. Operations:
    • Operations such as union, intersection, and complement are employed in Fuzzy Logic to manipulate fuzzy sets and derive meaningful insights from uncertain or incomplete data.

Applications of Fuzzy Logic:

  1. Control Systems:
    • Fuzzy Logic finds extensive use in control systems, including automotive control (e.g., anti-lock braking systems), HVAC systems, and industrial process control, where it can adapt to changing conditions and handle imprecise sensor data effectively.
  2. Pattern Recognition:
    • In pattern recognition tasks, Fuzzy Logic aids in image and speech recognition by accommodating noisy or uncertain data, enabling more robust recognition algorithms.
  3. Artificial Intelligence:
    • Fuzzy Logic is integrated into artificial intelligence systems for decision-making, allowing machines to mimic human reasoning and make informed choices in uncertain environments.

Advantages of Fuzzy Logic:

  1. Handling Uncertainty:
    • Fuzzy Logic excels in managing real-world uncertainty, making it well-suited for applications where precise data is unavailable or difficult to obtain.
  2. Linguistic Interpretability:
    • Its use of linguistic variables and simple IF-THEN rules enhances interpretability, enabling users to understand and interpret the reasoning process easily.
  3. Approximate Reasoning:
    • Fuzzy Logic enables approximate reasoning, providing solutions that are acceptable even when exact solutions are not feasible or practical.

Challenges of Fuzzy Logic:

  1. Computational Complexity:
    • Handling a large number of fuzzy rules and variables can lead to computational complexity, affecting response times in control systems and requiring efficient algorithms for implementation.
  2. Data-Intensive Learning:
    • Fuzzy systems may require substantial training data to adapt and generalize effectively, posing challenges in domains where labeled data is scarce or expensive to acquire.

Future Trends in Fuzzy Logic:

  1. Fuzzy Deep Learning:
    • Researchers are exploring the integration of Fuzzy Logic with deep learning techniques to develop Fuzzy Deep Learning models capable of leveraging the strengths of both approaches for improved performance in complex and uncertain environments.
  2. IoT Applications:
    • With the proliferation of the Internet of Things (IoT), Fuzzy Logic is increasingly applied in IoT systems for adaptive and context-aware control, enabling intelligent decision-making based on real-time sensor data.

Case Studies

1. Washing Machine:

  • Fuzzy Logic is used in washing machines to determine the appropriate wash cycle and water temperature based on the type of fabric and the degree of dirtiness. It can handle the imprecise inputs like “lightly soiled” or “delicate fabrics.”

2. Elevator Control:

  • In elevator systems, Fuzzy Logic helps optimize elevator movement by considering factors like passenger load, waiting time, and floor requests. It adapts elevator behavior to varying conditions efficiently.

3. Car Airbag Systems:

  • Fuzzy Logic is employed in airbag systems to determine the force and timing of airbag deployment during a crash. It takes into account factors like the collision speed, impact angle, and occupant position.

4. Traffic Light Control:

  • Fuzzy Logic is used to control traffic lights at intersections. It adjusts the timing of traffic signals based on traffic flow, reducing congestion and optimizing traffic patterns.

5. Camera Autofocus:

  • Fuzzy Logic assists digital cameras in achieving accurate and quick autofocus by evaluating the sharpness of an image and adjusting the focus accordingly.

6. Temperature Control in HVAC:

  • HVAC (Heating, Ventilation, and Air Conditioning) systems use Fuzzy Logic to maintain a comfortable indoor temperature by considering factors like room occupancy, outside temperature, and humidity levels.

7. Financial Forecasting:

  • Fuzzy Logic is applied in financial modeling to assess market trends and make predictions. It can handle imprecise economic data and provide more nuanced forecasts.

8. Medical Diagnosis:

  • Fuzzy Logic is used in medical diagnostic systems to interpret patient data and assess the likelihood of various medical conditions. It accounts for uncertainties in test results and symptoms.

9. Product Quality Control:

  • Fuzzy Logic plays a role in quality control processes, especially in manufacturing. It assesses the quality of products based on various parameters and can make real-time adjustments to production processes.

10. Natural Language Processing: – Fuzzy Logic enhances natural language processing tasks, such as sentiment analysis and language translation, by dealing with the inherent ambiguity and imprecision in human language.

Key Highlights

  • Uncertainty Management: Fuzzy Logic excels at handling uncertainty and imprecise data. It allows systems to work with partial truths, making it suitable for scenarios where information is not black and white.
  • Degrees of Truth: Unlike traditional binary logic, Fuzzy Logic deals with degrees of truth. It captures the nuances of information, allowing for a more human-like decision-making process.
  • Flexibility and Adaptability: Fuzzy Logic systems are highly flexible and adaptable to changing conditions. They can adjust their rules and parameters as circumstances evolve.
  • Wide Applicability: Fuzzy Logic finds applications across various domains, including control systems, artificial intelligence, image processing, and natural language understanding.
  • Interpretable Results: Fuzzy Logic provides results that are easy to interpret and understand, which is crucial in applications where decision transparency is essential.
  • Noise Tolerance: Fuzzy Logic is less sensitive to noisy or incomplete data, making it robust in scenarios where data quality is not perfect.
  • Real-World Applications: Fuzzy Logic is extensively used in real-world applications such as automotive control systems (e.g., anti-lock brakes), consumer electronics (e.g., washing machines), and financial modeling.
  • Adaptive Systems: It enables the development of adaptive systems that can learn and improve their performance over time.
  • Improved Precision: Fuzzy Logic helps improve precision in decision-making by capturing the nuances of information, leading to more accurate outcomes.
  • Multi-Industry Use: Fuzzy Logic is valuable in various industries, including engineering (process control), healthcare (medical diagnosis), and artificial intelligence (fuzzy expert systems).

Framework NameDescriptionWhen to Apply
Neuro-Fuzzy Systems– Neuro-Fuzzy Systems integrate fuzzy logic principles with neural network models to create hybrid intelligent systems capable of learning from data and handling uncertainty. They combine the learning capabilities of neural networks with the interpretability of fuzzy logic, allowing for adaptive and robust decision-making in complex and uncertain environments.– When developing intelligent systems that need to learn from data and handle uncertainty, to apply Neuro-Fuzzy Systems by combining fuzzy logic with neural network architectures, enabling adaptive decision-making in domains such as robotics, financial forecasting, pattern recognition, and control systems where both learning from data and handling uncertainty are crucial.
Evolutionary Fuzzy Systems– Evolutionary Fuzzy Systems combine fuzzy logic with evolutionary algorithms to optimize fuzzy rule sets and membership functions for improved performance in uncertain environments. They use evolutionary algorithms to search for optimal solutions to fuzzy logic problems, adapting fuzzy rule sets and membership functions to fit specific applications and datasets.– When optimizing fuzzy logic systems for specific applications or datasets, to apply Evolutionary Fuzzy Systems by combining fuzzy logic with evolutionary algorithms, enabling automatic optimization of fuzzy rule sets and membership functions in domains such as control systems, data mining, pattern recognition, and decision support systems where manual tuning of fuzzy logic parameters is challenging or impractical.
Fuzzy Cognitive Maps (FCMs)– Fuzzy Cognitive Maps are graphical representations of fuzzy logic concepts used to model and analyze complex systems, such as social networks, economic systems, and decision-making processes. They represent causal relationships between concepts as nodes connected by weighted edges, allowing for the simulation and analysis of system dynamics and the identification of feedback loops and causal dependencies.– When modeling and analyzing complex systems with interconnected components and feedback loops, to apply Fuzzy Cognitive Maps by representing causal relationships as nodes and edges in a graph, enabling the simulation and analysis of system dynamics, decision-making processes, and social networks in domains such as economics, sociology, political science, and organizational behavior.
Fuzzy Petri Nets– Fuzzy Petri Nets extend traditional Petri Nets with fuzzy logic concepts, enabling the modeling and analysis of systems with imprecise or uncertain transitions, such as manufacturing processes and workflow systems. They represent system states as places, transitions as events, and tokens as resources, allowing for the simulation and analysis of system behavior under uncertain conditions.– When modeling and analyzing systems with dynamic behavior and uncertain transitions, to apply Fuzzy Petri Nets by extending traditional Petri Nets with fuzzy logic concepts, enabling the representation and analysis of manufacturing processes, workflow systems, and other dynamic systems in domains such as logistics, supply chain management, and process automation where uncertainties play a significant role in system behavior.
Interval Type-2 Fuzzy Logic– Interval Type-2 Fuzzy Logic enhances traditional fuzzy logic by introducing intervals to represent uncertainty more effectively, allowing for more robust decision-making and control in dynamic environments. It extends Type-1 fuzzy sets with additional uncertainty information, enabling a more accurate representation of uncertain concepts and better handling of imprecise or incomplete data.– When dealing with highly uncertain or dynamic environments where traditional fuzzy logic may not capture uncertainty adequately, to apply Interval Type-2 Fuzzy Logic by representing fuzzy sets with intervals, enabling more robust decision-making and control in domains such as autonomous systems, financial modeling, medical diagnosis, and risk management where uncertainties need to be explicitly considered in the decision-making process.
Fuzzy Reinforcement Learning– Fuzzy Reinforcement Learning combines fuzzy logic with reinforcement learning algorithms to enable agents to learn optimal decision-making policies in uncertain or ambiguous environments. It integrates fuzzy logic concepts such as linguistic variables, fuzzy rules, and fuzzy inference with reinforcement learning frameworks to handle uncertainties and vagueness in state and action spaces.– When training agents to learn optimal decision-making policies in uncertain or ambiguous environments, to apply Fuzzy Reinforcement Learning by combining fuzzy logic with reinforcement learning algorithms, enabling adaptive and robust decision-making in domains such as robotics, autonomous systems, game playing, and financial trading where uncertainties and ambiguous states are prevalent and traditional reinforcement learning may struggle to converge to optimal solutions.
Fuzzy Time Series Analysis– Fuzzy Time Series Analysis applies fuzzy logic techniques to analyze time series data with imprecise or uncertain patterns, such as financial forecasting, weather prediction, and demand forecasting. It models time series data using fuzzy sets and fuzzy rules, allowing for the representation and analysis of complex temporal patterns and trends in uncertain and dynamic environments.– When analyzing time series data with imprecise or uncertain patterns and trends, to apply Fuzzy Time Series Analysis by modeling time series data with fuzzy sets and rules, enabling the representation and analysis of complex temporal patterns in domains such as financial forecasting, weather prediction, and demand forecasting where uncertainties and dynamic changes in data need to be explicitly considered in the analysis process.
Fuzzy Clustering– Fuzzy Clustering methods use fuzzy logic principles to partition data into clusters with overlapping boundaries, allowing for more flexible and interpretable cluster assignments in pattern recognition and data analysis tasks. They assign membership degrees to data points in multiple clusters, allowing for a soft assignment of data to clusters and handling data points that belong to multiple clusters simultaneously.– When clustering data points with overlapping boundaries and uncertain cluster assignments, to apply Fuzzy Clustering by assigning membership degrees to data points in multiple clusters, enabling flexible and interpretable cluster assignments in domains such as pattern recognition, data analysis, image segmentation, and customer segmentation where traditional hard clustering methods may lead to ambiguous or non-intuitive cluster assignments.
Fuzzy Decision Trees– Fuzzy Decision Trees use fuzzy logic principles to build decision trees capable of handling imprecise or uncertain data and making robust decisions in uncertain environments. They extend traditional decision trees by incorporating fuzzy sets and fuzzy rules to represent uncertainty in attribute values and decision criteria, allowing for more flexible and adaptive decision-making processes.– When building decision trees from imprecise or uncertain data and making robust decisions in uncertain environments, to apply Fuzzy Decision Trees by incorporating fuzzy sets and rules into decision tree construction, enabling more flexible and adaptive decision-making processes in domains such as data mining, classification, and decision support systems where uncertainties in attribute values and decision criteria need to be explicitly considered in the decision-making process.

Connected Thinking Frameworks

Convergent vs. Divergent Thinking

convergent-vs-divergent-thinking
Convergent thinking occurs when the solution to a problem can be found by applying established rules and logical reasoning. Whereas divergent thinking is an unstructured problem-solving method where participants are encouraged to develop many innovative ideas or solutions to a given problem. Where convergent thinking might work for larger, mature organizations where divergent thinking is more suited for startups and innovative companies.

Critical Thinking

critical-thinking
Critical thinking involves analyzing observations, facts, evidence, and arguments to form a judgment about what someone reads, hears, says, or writes.

Biases

biases
The concept of cognitive biases was introduced and popularized by the work of Amos Tversky and Daniel Kahneman in 1972. Biases are seen as systematic errors and flaws that make humans deviate from the standards of rationality, thus making us inept at making good decisions under uncertainty.

Second-Order Thinking

second-order-thinking
Second-order thinking is a means of assessing the implications of our decisions by considering future consequences. Second-order thinking is a mental model that considers all future possibilities. It encourages individuals to think outside of the box so that they can prepare for every and eventuality. It also discourages the tendency for individuals to default to the most obvious choice.

Lateral Thinking

lateral-thinking
Lateral thinking is a business strategy that involves approaching a problem from a different direction. The strategy attempts to remove traditionally formulaic and routine approaches to problem-solving by advocating creative thinking, therefore finding unconventional ways to solve a known problem. This sort of non-linear approach to problem-solving, can at times, create a big impact.

Bounded Rationality

bounded-rationality
Bounded rationality is a concept attributed to Herbert Simon, an economist and political scientist interested in decision-making and how we make decisions in the real world. In fact, he believed that rather than optimizing (which was the mainstream view in the past decades) humans follow what he called satisficing.

Dunning-Kruger Effect

dunning-kruger-effect
The Dunning-Kruger effect describes a cognitive bias where people with low ability in a task overestimate their ability to perform that task well. Consumers or businesses that do not possess the requisite knowledge make bad decisions. What’s more, knowledge gaps prevent the person or business from seeing their mistakes.

Occam’s Razor

occams-razor
Occam’s Razor states that one should not increase (beyond reason) the number of entities required to explain anything. All things being equal, the simplest solution is often the best one. The principle is attributed to 14th-century English theologian William of Ockham.

Lindy Effect

lindy-effect
The Lindy Effect is a theory about the ageing of non-perishable things, like technology or ideas. Popularized by author Nicholas Nassim Taleb, the Lindy Effect states that non-perishable things like technology age – linearly – in reverse. Therefore, the older an idea or a technology, the same will be its life expectancy.

Antifragility

antifragility
Antifragility was first coined as a term by author, and options trader Nassim Nicholas Taleb. Antifragility is a characteristic of systems that thrive as a result of stressors, volatility, and randomness. Therefore, Antifragile is the opposite of fragile. Where a fragile thing breaks up to volatility; a robust thing resists volatility. An antifragile thing gets stronger from volatility (provided the level of stressors and randomness doesn’t pass a certain threshold).

Systems Thinking

systems-thinking
Systems thinking is a holistic means of investigating the factors and interactions that could contribute to a potential outcome. It is about thinking non-linearly, and understanding the second-order consequences of actions and input into the system.

Vertical Thinking

vertical-thinking
Vertical thinking, on the other hand, is a problem-solving approach that favors a selective, analytical, structured, and sequential mindset. The focus of vertical thinking is to arrive at a reasoned, defined solution.

Maslow’s Hammer

einstellung-effect
Maslow’s Hammer, otherwise known as the law of the instrument or the Einstellung effect, is a cognitive bias causing an over-reliance on a familiar tool. This can be expressed as the tendency to overuse a known tool (perhaps a hammer) to solve issues that might require a different tool. This problem is persistent in the business world where perhaps known tools or frameworks might be used in the wrong context (like business plans used as planning tools instead of only investors’ pitches).

Peter Principle

peter-principle
The Peter Principle was first described by Canadian sociologist Lawrence J. Peter in his 1969 book The Peter Principle. The Peter Principle states that people are continually promoted within an organization until they reach their level of incompetence.

Straw Man Fallacy

straw-man-fallacy
The straw man fallacy describes an argument that misrepresents an opponent’s stance to make rebuttal more convenient. The straw man fallacy is a type of informal logical fallacy, defined as a flaw in the structure of an argument that renders it invalid.

Streisand Effect

streisand-effect
The Streisand Effect is a paradoxical phenomenon where the act of suppressing information to reduce visibility causes it to become more visible. In 2003, Streisand attempted to suppress aerial photographs of her Californian home by suing photographer Kenneth Adelman for an invasion of privacy. Adelman, who Streisand assumed was paparazzi, was instead taking photographs to document and study coastal erosion. In her quest for more privacy, Streisand’s efforts had the opposite effect.

Heuristic

heuristic
As highlighted by German psychologist Gerd Gigerenzer in the paper “Heuristic Decision Making,” the term heuristic is of Greek origin, meaning “serving to find out or discover.” More precisely, a heuristic is a fast and accurate way to make decisions in the real world, which is driven by uncertainty.

Recognition Heuristic

recognition-heuristic
The recognition heuristic is a psychological model of judgment and decision making. It is part of a suite of simple and economical heuristics proposed by psychologists Daniel Goldstein and Gerd Gigerenzer. The recognition heuristic argues that inferences are made about an object based on whether it is recognized or not.

Representativeness Heuristic

representativeness-heuristic
The representativeness heuristic was first described by psychologists Daniel Kahneman and Amos Tversky. The representativeness heuristic judges the probability of an event according to the degree to which that event resembles a broader class. When queried, most will choose the first option because the description of John matches the stereotype we may hold for an archaeologist.

Take-The-Best Heuristic

take-the-best-heuristic
The take-the-best heuristic is a decision-making shortcut that helps an individual choose between several alternatives. The take-the-best (TTB) heuristic decides between two or more alternatives based on a single good attribute, otherwise known as a cue. In the process, less desirable attributes are ignored.

Bundling Bias

bundling-bias
The bundling bias is a cognitive bias in e-commerce where a consumer tends not to use all of the products bought as a group, or bundle. Bundling occurs when individual products or services are sold together as a bundle. Common examples are tickets and experiences. The bundling bias dictates that consumers are less likely to use each item in the bundle. This means that the value of the bundle and indeed the value of each item in the bundle is decreased.

Barnum Effect

barnum-effect
The Barnum Effect is a cognitive bias where individuals believe that generic information – which applies to most people – is specifically tailored for themselves.

First-Principles Thinking

first-principles-thinking
First-principles thinking – sometimes called reasoning from first principles – is used to reverse-engineer complex problems and encourage creativity. It involves breaking down problems into basic elements and reassembling them from the ground up. Elon Musk is among the strongest proponents of this way of thinking.

Ladder Of Inference

ladder-of-inference
The ladder of inference is a conscious or subconscious thinking process where an individual moves from a fact to a decision or action. The ladder of inference was created by academic Chris Argyris to illustrate how people form and then use mental models to make decisions.

Goodhart’s Law

goodharts-law
Goodhart’s Law is named after British monetary policy theorist and economist Charles Goodhart. Speaking at a conference in Sydney in 1975, Goodhart said that “any observed statistical regularity will tend to collapse once pressure is placed upon it for control purposes.” Goodhart’s Law states that when a measure becomes a target, it ceases to be a good measure.

Six Thinking Hats Model

six-thinking-hats-model
The Six Thinking Hats model was created by psychologist Edward de Bono in 1986, who noted that personality type was a key driver of how people approached problem-solving. For example, optimists view situations differently from pessimists. Analytical individuals may generate ideas that a more emotional person would not, and vice versa.

Mandela Effect

mandela-effect
The Mandela effect is a phenomenon where a large group of people remembers an event differently from how it occurred. The Mandela effect was first described in relation to Fiona Broome, who believed that former South African President Nelson Mandela died in prison during the 1980s. While Mandela was released from prison in 1990 and died 23 years later, Broome remembered news coverage of his death in prison and even a speech from his widow. Of course, neither event occurred in reality. But Broome was later to discover that she was not the only one with the same recollection of events.

Crowding-Out Effect

crowding-out-effect
The crowding-out effect occurs when public sector spending reduces spending in the private sector.

Bandwagon Effect

bandwagon-effect
The bandwagon effect tells us that the more a belief or idea has been adopted by more people within a group, the more the individual adoption of that idea might increase within the same group. This is the psychological effect that leads to herd mentality. What in marketing can be associated with social proof.

Moore’s Law

moores-law
Moore’s law states that the number of transistors on a microchip doubles approximately every two years. This observation was made by Intel co-founder Gordon Moore in 1965 and it become a guiding principle for the semiconductor industry and has had far-reaching implications for technology as a whole.

Disruptive Innovation

disruptive-innovation
Disruptive innovation as a term was first described by Clayton M. Christensen, an American academic and business consultant whom The Economist called “the most influential management thinker of his time.” Disruptive innovation describes the process by which a product or service takes hold at the bottom of a market and eventually displaces established competitors, products, firms, or alliances.

Value Migration

value-migration
Value migration was first described by author Adrian Slywotzky in his 1996 book Value Migration – How to Think Several Moves Ahead of the Competition. Value migration is the transferal of value-creating forces from outdated business models to something better able to satisfy consumer demands.

Bye-Now Effect

bye-now-effect
The bye-now effect describes the tendency for consumers to think of the word “buy” when they read the word “bye”. In a study that tracked diners at a name-your-own-price restaurant, each diner was asked to read one of two phrases before ordering their meal. The first phrase, “so long”, resulted in diners paying an average of $32 per meal. But when diners recited the phrase “bye bye” before ordering, the average price per meal rose to $45.

Groupthink

groupthink
Groupthink occurs when well-intentioned individuals make non-optimal or irrational decisions based on a belief that dissent is impossible or on a motivation to conform. Groupthink occurs when members of a group reach a consensus without critical reasoning or evaluation of the alternatives and their consequences.

Stereotyping

stereotyping
A stereotype is a fixed and over-generalized belief about a particular group or class of people. These beliefs are based on the false assumption that certain characteristics are common to every individual residing in that group. Many stereotypes have a long and sometimes controversial history and are a direct consequence of various political, social, or economic events. Stereotyping is the process of making assumptions about a person or group of people based on various attributes, including gender, race, religion, or physical traits.

Murphy’s Law

murphys-law
Murphy’s Law states that if anything can go wrong, it will go wrong. Murphy’s Law was named after aerospace engineer Edward A. Murphy. During his time working at Edwards Air Force Base in 1949, Murphy cursed a technician who had improperly wired an electrical component and said, “If there is any way to do it wrong, he’ll find it.”

Law of Unintended Consequences

law-of-unintended-consequences
The law of unintended consequences was first mentioned by British philosopher John Locke when writing to parliament about the unintended effects of interest rate rises. However, it was popularized in 1936 by American sociologist Robert K. Merton who looked at unexpected, unanticipated, and unintended consequences and their impact on society.

Fundamental Attribution Error

fundamental-attribution-error
Fundamental attribution error is a bias people display when judging the behavior of others. The tendency is to over-emphasize personal characteristics and under-emphasize environmental and situational factors.

Outcome Bias

outcome-bias
Outcome bias describes a tendency to evaluate a decision based on its outcome and not on the process by which the decision was reached. In other words, the quality of a decision is only determined once the outcome is known. Outcome bias occurs when a decision is based on the outcome of previous events without regard for how those events developed.

Hindsight Bias

hindsight-bias
Hindsight bias is the tendency for people to perceive past events as more predictable than they actually were. The result of a presidential election, for example, seems more obvious when the winner is announced. The same can also be said for the avid sports fan who predicted the correct outcome of a match regardless of whether their team won or lost. Hindsight bias, therefore, is the tendency for an individual to convince themselves that they accurately predicted an event before it happened.

Read Next: Biases, Bounded Rationality, Mandela Effect, Dunning-Kruger Effect, Lindy Effect, Crowding Out Effect, Bandwagon Effect.

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