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A fuzzy concept is an idea of which the boundaries of application can vary considerably according to context or conditions, instead of being fixed once and for all. This means the idea is somewhat vague or imprecise. Yet it is not unclear or meaningless. It has a definite meaning, which can be made more exact only through further elaboration and specification - including a closer definition of the context in which the concept is used. The study of the characteristics of fuzzy concepts and fuzzy language is called fuzzy semantics. The inverse of a "fuzzy concept" is a "crisp concept" (i.e. a precise concept).
For engineers, "Fuzziness is imprecision or vagueness of definition." For scientists, a fuzzy concept is an idea which is "to an extent applicable" in a situation. It means that the concept can have gradations of significance or unsharp (variable) boundaries of application; a fuzzy statement is a statement which is true "to some extent", and that extent can often be represented by a scaled value (a score). For mathematicians, a "fuzzy concept" is usually a fuzzy set or a combination of such sets (see fuzzy mathematics and fuzzy set theory). In cognitive linguistics, the things that belong to a "fuzzy category" exhibit gradations of family resemblance, and the borders of the category are not clearly defined. In a more general, popular sense – contrasting with its technical meanings – a "fuzzy concept" refers to an imprecise idea which is "somewhat vague" for any kind of reason, or which is "approximately true". Fuzzy concepts are often used to navigate imprecision in the real world, when exact information is not available.
In the past, the very idea of reasoning with fuzzy concepts faced considerable resistance from academic elites. They did not want to endorse the use of imprecise concepts in research or argumentation, and regarded fuzzy logic with suspicion or even hostility. Yet although people might not be aware of it, the use of fuzzy concepts has risen gigantically in all walks of life from the 1970s onward. That is mainly due to advances in electronic engineering, fuzzy mathematics and digital computer programming. The new technology allows very complex inferences about "variations on a theme" to be anticipated and fixed in a program. The Perseverance Mars rover, a driverless NASA vehicle used to explore the Jezero crater on the planet Mars, features fuzzy logic programming that steers it through rough terrain. Similarly, to the North, the Chinese Mars rover Zhurong used fuzzy logic algorithms to calculate its travel route in Utopia Planitia from sensor data.
New neuro-fuzzy computational methods make it possible for machines to identify, measure, correct/adjust for and respond to fine gradations of significance with great precision. It means that practically useful concepts can be coded and applied to all kinds of tasks, even if ordinarily these concepts are never precisely defined. Nowadays engineers, statisticians and programmers often represent fuzzy concepts mathematically, using fuzzy logic, fuzzy values, fuzzy variables and fuzzy sets (see also fuzzy set theory). Fuzzy logic can play a significant role in artificial intelligence programming, for example because it can model human cognitive processes more easily than other methods.
Origins
Problems of vagueness and fuzziness have probably always existed in human experience. In the West, ancient texts show that philosophers and scientists were already thinking about those kinds of problems in classical antiquity. Kit Fine states that "when a philosopher talks of vagueness he has in mind a certain kind of indeterminacy in the relation of something to the world". According to the Daoist thought of Laozi and Zhuang Zhou in ancient China, "vagueness is not regarded with suspicion, but is simply an acknowledged characteristic of the world around us" - a subject for meditation and a source of insight.
Sorites paradox
The ancient Sorites paradox first raised the logical problem of how we could exactly define the threshold at which a change in quantitative gradation turns into a qualitative or categorical difference. With some physical processes this threshold is relatively easy to identify. For example, water turns into steam at 100 °C or 212 °F (the boiling point depends partly on atmospheric pressure, which decreases at higher altitudes).
With many other processes and gradations, however, the point of change is much more difficult to locate, and remains somewhat vague. Thus, the boundaries between qualitatively different things may be unsharp: we know that there are boundaries, but we cannot define them exactly.
According to the modern idea of the continuum fallacy, the fact that a statement is to an extent vague, does not automatically mean that it has no validity. The question then arises of how we could ascertain and define the validity that the fuzzy statement does have.
Loki's wager
The Nordic myth of Loki's wager suggested that concepts that lack precise meanings or lack precise boundaries of application cannot be usefully discussed at all, because they evade any clear definition. However, the 20th-century idea of "fuzzy concepts" proposes that "somewhat vague terms" can be operated with, because we can explicate and define the variability of their application - by assigning numbers to gradations of applicability. This idea sounds simple enough, but it had large implications.
Precursors
In Western civilization, the intellectual recognition of fuzzy concepts has been traced back to a diversity of famous and less well-known thinkers, including (among many others) Eubulides,Epicurus,Plato,Cicero,Georg Wilhelm Friedrich Hegel,Karl Marx and Friedrich Engels,Friedrich Nietzsche,William James,Hugh MacColl,Charles S. Peirce,Carl Gustav Hempel,Max Black,Arto Salomaa,Ludwig Wittgenstein,Jan Łukasiewicz,Emil Leon Post,Alfred Tarski,Georg Cantor,Nicolai A. Vasiliev,Kurt Gödel,Stanisław Jaśkowski,Willard Van Orman Quine,Petr Hájek,Joseph Goguen, Jan Pavelka,George J. Klir,Didier Dubois, and Donald Knuth.
Across at least two and a half millennia, all of them had something to say about graded concepts with unsharp boundaries. This suggests at least that the awareness of the existence of concepts with "fuzzy" characteristics, in one form or another, has a very long history in human thought. Quite a few 20th century logicians, mathematicians and philosophers also tried to analyze the characteristics of fuzzy concepts as a recognized species, sometimes with the aid of some kind of many-valued logic or substructural logic.
An early attempt in the post-WW2 era to create a mathematical theory of sets with gradations of set membership was made by Abraham Kaplan and Hermann F. Schott in 1951. They intended to apply the idea to empirical research. Kaplan and Schott expressed the degree of membership of empirical classes using real numbers between 0 and 1, and they defined corresponding notions of intersection, union, complementation and subset. However, at the time, their idea "fell on stony ground".J. Barkley Rosser Sr. published a treatise on many-valued logics in 1952, anticipating "many-valued sets". Another treatise was published in 1963 by Alexander Zinoviev and others.
In 1964, the American philosopher William Alston introduced the term "degree vagueness" to describe vagueness in an idea that results from the absence of a definite cut-off point along an implied scale (in contrast to "combinatory vagueness" caused by a term that has a number of logically independent conditions of application).
The German mathematician Dieter Klaua published a German-language paper on fuzzy sets in 1965, but he used a different terminology (he referred to "many-valued sets", not "fuzzy sets").
In the late 1960s, two popular introductions to many-valued logic were published by Robert J. Ackermann and Nicholas Rescher. Rescher's book includes a bibliography on fuzzy theory up to 1965, which was extended by Robert Wolf and Joseph De Kerf for 1966–1975. Haack provides references to significant works after 1974. In 1980, Didier Dubois and Henri Prade published a detailed annotated bibliography on the field of fuzzy set theory. George J. Klir and Bo Yuan provided an overview of the subject in Fuzzy sets and fuzzy logic during the mid-1990s. Merrie Bergmann provides a more recent (2008) introduction to fuzzy reasoning. A standard modern reference work is Fuzzy Logic and Mathematics: A Historical Perspective (2017) by Radim Bělohlávek, Joseph W. Dauben and George J. Klir.
Lotfi Zadeh
The Iranian-born American computer scientist Lotfi A. Zadeh (1921–2017) is usually credited with inventing the specific idea of a "fuzzy concept" in his seminal 1965 paper on fuzzy sets, because he presented a mathematical formalization of the phenomenon that was widely accepted by scholars. It was also Zadeh who played a decisive role in developing the field of fuzzy logic, fuzzy sets and fuzzy systems, with a large number of scholarly papers. Unlike most philosophical theories of vagueness, Zadeh's engineering approach had the advantage that it could be directly applied to computer programming. Zadeh's seminal 1965 paper is acknowledged to be one of the most-cited scholarly articles in the 20th century. In 2014, it was placed 46th in the list of the world's 100 most-cited research papers of all time. Since the mid-1960s, many scholars have contributed to elaborating the theory of reasoning with graded concepts, and the research field continues to expand.
Definition
The ordinary scholarly definition of a concept as "fuzzy" has been in use from the 1970s onward.
Criteria
Radim Bělohlávek explains:
"There exists strong evidence, established in the 1970s in the psychology of concepts... that human concepts have a graded structure in that whether or not a concept applies to a given object is a matter of degree, rather than a yes-or-no question, and that people are capable of working with the degrees in a consistent way. This finding is intuitively quite appealing, because people say "this product is more or less good" or "to a certain degree, he is a good athlete", implying the graded structure of concepts. In his classic paper, Zadeh called the concepts with a graded structure fuzzy concepts and argued that these concepts are a rule rather than an exception when it comes to how people communicate knowledge. Moreover, he argued that to model such concepts mathematically is important for the tasks of control, decision making, pattern recognition, and the like. Zadeh proposed the notion of a fuzzy set that gave birth to the field of fuzzy logic..."
Hence, a concept is generally regarded as "fuzzy" in a logical sense if:
- defining characteristics of the concept apply to it "to a certain degree or extent" (or, more unusually, "with a certain magnitude of likelihood").
- or, the boundaries of applicability (the truth-value) of a concept can vary in degrees, according to different conditions.
- or, the fuzzy concept itself straightforwardly consists of a fuzzy set, or a combination of such sets.
The fact that a concept is fuzzy does not prevent its use in logical reasoning; it merely affects the type of reasoning which can be applied (see fuzzy logic). If the concept has gradations of meaningful significance, it may be necessary to specify and formalize what those gradations are, if they can make an important difference. Not all fuzzy concepts have the same logical structure, but they can often be formally described or reconstructed using fuzzy logic or other substructural logics. The advantage of this approach is, that numerical notation enables a potentially infinite number of truth-values between complete truth and complete falsehood, and thus it enables - in theory, at least - the greatest precision in stating the degree of applicability of a logical rule.
Fuzziness versus uncertainty
One of the first scholars who pointed out the need to distinguish the theory of fuzzy sets from probability theory was Zadeh's pupil Joseph Goguen.Petr Hájek, writing about the foundations of fuzzy logic, likewise sharply distinguished between "fuzziness" and "uncertainty":
"The sentence "The patient is young" is true to some degree – the lower the age of the patient (measured e.g. in years), the more the sentence is true. Truth of a fuzzy proposition is a matter of degree. I recommend to everybody interested in fuzzy logic that they sharply distinguish fuzziness from uncertainty as a degree of belief (e.g. probability). Compare the last proposition with the proposition "The patient will survive next week". This may well be considered as a crisp proposition which is either (absolutely) true or (absolutely) false; but we do not know which is the case. We may have some probability (chance, degree of belief) that the sentence is true; but probability is not a degree of truth.
In metrology (the science of measurement), it is acknowledged that for any measure we care to make, there exists an amount of uncertainty about its accuracy, but this degree of uncertainty is conventionally expressed with a magnitude of likelihood, and not as a degree of truth. In 1975, Lotfi A. Zadeh introduced a distinction between "Type 1 fuzzy sets" without uncertainty and "Type 2 fuzzy sets" with uncertainty, which has been widely accepted. Simply put, in the former case, each fuzzy number is linked to a non-fuzzy (natural) number, while in the latter case, each fuzzy number is linked to another fuzzy number.
Applications
Philosophy
In philosophical logic and linguistics, fuzzy concepts are often regarded as vague or imprecise ideas which in their application, or strictly speaking, are neither completely true nor completely false. Such ideas require further elaboration, specification or qualification to understand their applicability (the conditions under which they truly make sense). The "fuzzy area" can also refer simply to a residual number of cases which cannot be allocated to a known and identifiable group, class or set if strict criteria are used.
The French thinkers Gilles Deleuze and Félix Guattari referred occasionally to fuzzy sets in connection with their phenomenological concept of multiplicities. In A Thousand Plateaus, they state that "a set is fuzzy if its elements belong to it only by virtue of specific operations of consistency and consolidation, which themselves follow a special logic", In their book What Is Philosophy?, which deals with the functions of concepts, they suggest that all philosophical concepts could be regarded as "vague or fuzzy sets, simple aggregates of perceptions and affections, which form within the lived as immanent to a subject, to a consciousness [and which] are qualitative or intensive multiplicities, like "redness" or "baldness," where we cannot decide whether certain elements do or do not belong to the set."
Sciences
In mathematics and statistics, a fuzzy variable (such as "the temperature", "hot" or "cold") is a value which could lie in a probable range defined by some quantitative limits or parameters, and which can be usefully described with imprecise categories (such as "high", "medium" or "low") using some kind of scale or conceptual hierarchy.
Fuzzy logic
In mathematics and computer science, the gradations of applicable meaning of a fuzzy concept are described in terms of quantitative relationships defined by logical operators. Such an approach is sometimes called "degree-theoretic semantics" by logicians and philosophers, but the more usual term is fuzzy logic or many-valued logic. The novelty of fuzzy logic is, that it "breaks with the traditional principle that formalisation should correct and avoid, but not compromise with, vagueness". The basic idea of fuzzy logic is that a real number is assigned to each statement written in a language, within a range from 0 to 1, where 1 means that the statement is completely true, and 0 means that the statement is completely false, while values less than 1 but greater than 0 represent that the statement is "partly true", to a given, quantifiable extent. Susan Haack comments:
"Whereas in classical set theory an object either is or is not a member of a given set, in fuzzy set theory membership is a matter of degree; the degree of membership of an object in a fuzzy set is represented by some real number between 0 and 1, with 0 denoting no membership and 1 full membership."
"Truth" in this mathematical context usually means simply that "something is the case", or that "something is applicable". This makes it possible to analyze a distribution of statements for their truth-content, identify data patterns, make inferences and predictions, and model how processes operate. Petr Hájek claimed that "fuzzy logic is not just some "applied logic", but may bring "new light to classical logical problems", and therefore might be well classified as a distinct branch of "philosophical logic" similar to e.g. modal logics.
Machinery and analytics
Fuzzy logic offers computationally-oriented systems of concepts and methods, to formalize types of reasoning which are ordinarily approximate only, and not exact. In principle, this allows us to give a definite, precise answer to the question, "To what extent is something the case?", or, "To what extent is something applicable?". Via a series of switches, this kind of reasoning can be built into electronic devices. That was already happening before fuzzy logic was invented, but using fuzzy logic in modelling has become an important aid in design, which creates many new technical possibilities. Fuzzy reasoning (i.e., reasoning with graded concepts) turns out to have many practical uses. It is nowadays widely used in:
- The programming of vehicle and transport electronics, household appliances, video games, language filters, robotics, and driverless vehicles. Fuzzy logic washing machines are gaining popularity.
- All kinds of control systems that regulate access, traffic, movement, balance, conditions, temperature, pressure, routers etc.
- Electronic equipment used for pattern recognition, surveying and monitoring (including radars, satellites, alarm systems and surveillance systems).
- Cybernetics research, artificial intelligence,virtual intelligence, machine learning, database design and soft computing research.
- "Fuzzy risk scores" are used by project managers and portfolio managers to express financial risk assessments.
- Fuzzy logic has been applied to the problem of predicting cement strength.
It looks like fuzzy logic will eventually be applied in almost every aspect of life, even if people are not aware of it, and in that sense fuzzy logic is an astonishingly successful invention. The scientific and engineering literature on the subject is constantly increasing.
Community
Originally lot of research on fuzzy logic was done by Japanese pioneers inventing new machinery, electronic equipment and appliances (see also Fuzzy control system). The idea became so popular in Japan, that the English word entered Japanese language (ファジィ概念). "Fuzzy theory" (ファジー理論) is a recognized field in Japanese scientific research.
Since that time, the movement has spread worldwide; nearly every country nowadays has its own fuzzy systems association, although some are larger and more developed than others. In some cases, the local body is a branch of an international one. In other cases, the fuzzy systems program falls under artificial intelligence or soft computing. There are also some emerging networks of researchers which do not yet have their own website. The following list is only provisional and illustrative - many more groups could possibly be added:
- The main international body is the International Fuzzy Systems Association (IFSA).
- The Computational Intelligence Society of the Institute of Electrical and Electronics Engineers, Inc. (IEEE) has an international membership and deals with fuzzy logic, neural networks and evolutionary computing. It publishes the journal IEEE Transactions on Fuzzy Systems and holds international conferences. At the end of 2024, there were 238 chapters of IEEE/CIS across the world.
- The conference on Fuzzy Systems and Data Mining (FSDM) has its 11th International Conference (FSDM2025) in Hanshan Normal University, Chaozhou City, Guangdong Province, China.
- The Asia Pacific Neural Network Society, founded in 1993, has board members from 13 countries: Australia, China, Hong Kong, India, Japan, Malaysia, New Zealand, Singapore, South Korea, Qatar, Taiwan, Thailand, and Turkey.
- The International Association for fuzzy-set management and economy (SIGEF)[115] is based in Spain and publishes the Fuzzy Economic Review[116] since 1996.
- Intelligent and Fuzzy Systems (INFUS) is an international research forum to advance the foundations and applications of intelligent and fuzzy systems, computational intelligence, soft computing for applied research in general, complex engineering and decision support systems.
- The interdisciplinary Japan Society for Fuzzy Theory and Intelligent Informatics (SOFT) traces its origin back to 1972 and publishes two journals.
- The original Korea Fuzzy System Society founded in 1991 is now known as the Korean Institute of Intelligent Systems (KIIS).
- In mainland China, there is the Fuzzy Mathematics and Systems Association of China (FMSAC) based at the School of Mathematics, Sichuan University in Chengdu, and there exists also an important Taiwan Fuzzy Systems Association.
- The North American Fuzzy Information Processing Society (NAFIPS) was founded in 1981. There exists also a Hispanic-American Fuzzy System Association (HAFSA) based in Mexico.
- In Europe, there is a European Society for Fuzzy Logic and Technology (EUSFLAT) which includes the Working Group on Mathematical Fuzzy Logic. The North European Society of Adaptive and Intelligent Systems (NSAIS) is based in Finland.
- In 2002, the Iran Fuzzy Systems Society (nowadays merged into the Iranian Coalition on Soft Computing) was approved as an affiliate of the Statistics Association of Iran, and in 2005 registered as a non-commercial scientific institute. When Lotfi A. Zadeh received an honorary doctorate from the University of Teheran on 9 March 2017, a member of Iran's parliament stated that Iran now ranks third in the world with regard to the output of scientific research about fuzzy systems.
- In 2005, Russia's Association for Fuzzy Systems (founded in January 1990) became the Russian Association for Fuzzy Systems and Soft Computing (RAFSSoftCom). Zadeh's seminal paper on fuzzy sets was translated into Russian in 1974, and subsequently Russian fuzzy research began to take off - increasingly overcoming official skepticism.
- In 2009, the Brazilian Applied Mathematical Society (SBMAC) created the Thematic Committee on Fuzzy Systems which inspired the First Brazilian Congress on Fuzzy Systems (CBSF I) in 2010. CBSF VI was held at São Paulo State University in 2021. There also exists a Brazilian Society of Automatics (SBA).
- In India, the Center for Soft Computing Research at the Indian Statistical Institute (Kolkata) organizes and publishes research on fuzzy sets, rough sets, and applications of fuzzy logic.
- The Sri Lanka Association for Artificial Intelligence is a non-profit scientific association devoted to understanding the mechanisms underlying thoughts and intelligent behaviour, and their emulation in machines.
- Other national scientific bodies include the Hungarian Fuzzy Association (HFA), the Fuzzy Systems Association of Turkey (FSAT), the Indonesian Soft Computing Society (SC-INA), and the Vietnamese Fuzzy Systems Society (VFSS).
Achievements
Lotfi A. Zadeh estimated around 2014 that there were more than 50,000 fuzzy logic–related, patented inventions. He listed 28 journals at that time dealing with fuzzy reasoning, and 21 journal titles on soft computing. His searches found close to 100,000 publications with the word "fuzzy" in their titles, but perhaps there are even 300,000. In March 2018, Google Scholar found 2,870,000 titles which included the word "fuzzy". When he died on 11 September 2017 at age 96, Professor Zadeh had received more than 50 engineering and academic awards, in recognition of his work.
Lattices and big data sets
The technique of fuzzy concept lattices is increasingly used in programming for the formatting, relating and analysis of fuzzy data sets.
Concept formalization
According to the computer scientist Andrei Popescu at Middlesex University London, a concept can be operationally defined to consist of:
- an intent, which is a description or specification stated in a language,
- an extent, which is the collection of all the objects to which the description refers,
- a context, which is stated by: (i) the universe of all possible objects within the scope of the concept, (ii) the universe of all possible attributes of objects, and (iii) the logical definition of the relation whereby an object possesses an attribute.
Once the context is defined, we can specify relationships of sets of objects with sets of attributes which they do, or do not share.
Fuzzy concept lattice
Whether an object belongs to a concept, and whether an object does, or does not have an attribute, can often be a matter of degree. Thus, for example, "many attributes are fuzzy rather than crisp". To overcome this issue, a numerical value is assigned to each attribute along a scale, and the results are placed in a table which links each assigned object-value within the given range to a numerical value (a score) denoting a given degree of applicability.
This is the basic idea of a "fuzzy concept lattice", which can also be graphed; different fuzzy concept lattices can be connected to each other as well (for example, in "fuzzy conceptual clustering" techniques used to group data, originally invented by Enrique H. Ruspini). Fuzzy concept lattices are a useful programming tool for the exploratory analysis of big data, for example in cases where sets of linked behavioural responses are broadly similar, but can nevertheless vary in important ways, within certain limits. It can help to find out what the structure and dimensions are, of a behaviour that occurs with an important but limited amount of variation in a large population.
Big data
Coding with fuzzy lattices can be useful, for instance, in the psephological analysis of big data about voter behaviour, where researchers want to explore the characteristics and associations involved in "somewhat vague" opinions; gradations in voter attitudes; and variability in voter behaviour (or personal characteristics) within a set of parameters. The basic programming techniques for this kind of fuzzy concept mapping and deep learning are by now well-established and big data analytics had a strong influence on the US elections of 2016. A US study concluded in 2015 that for 20% of undecided voters, Google's secret search algorithm had the power to change the way they voted.
Very large quantities of data can now be explored using computers with fuzzy logic programming and open-source architectures such as Apache Hadoop, Apache Spark, and MongoDB. One author claimed in 2016 that it is now possible to obtain, link and analyze "400 data points" for each voter in a population, using Oracle systems (a "data point" is a number linked to one or more categories, which represents a characteristic).
However, NBC News reported in 2016 that the Anglo-American firm Cambridge Analytica which profiled voters for Donald Trump (Steve Bannon was a board member) did not have 400, but 4,000 data points for each of 230 million US adults. Cambridge Analytica's own website claimed that "up to 5,000 data points" were collected for each of 220 million Americans, a data set of more than 1 trillion bits of formatted data.The Guardian later claimed that Cambridge Analytica in fact had, according to its own company information, "up to 7,000 data points" on 240 million American voters.
Harvard University Professor Latanya Sweeney calculated, that if a U.S. company knows just your date of birth, your ZIP code and sex, the company has an 87% chance to identify you by name – simply by using linked data sets from various sources. With 4,000–7,000 data points instead of three, a very comprehensive personal profile becomes possible for almost every voter, and many behavioural patterns can be inferred by linking together different data sets. It also becomes possible to identify and measure gradations in personal characteristics which, in aggregate, have very large effects.
Human judgement
Some researchers argue that this kind of big data analysis has severe limitations, and that the analytical results can only be regarded as indicative, and not as definitive. This was confirmed by Kellyanne Conway, Donald Trump's campaign advisor and counselor in 2016, who emphasized the importance of human judgement and common sense in drawing conclusions from fuzzy data. Conway candidly admitted that much of her own research would "never see the light of day", because it was client confidential. Another Trump adviser criticized Conway, claiming that she "produces an analysis that buries every terrible number and highlights every positive number"
Propaganda machine
In a video interview published by The Guardian in March 2018, whistleblower Christopher Wylie called Cambridge Analytica a "full-service propaganda machine" rather than a bona fide data science company. Its own site revealed with "case studies" that it has been active in political campaigns in numerous different countries, influencing attitudes and opinions. Wylie explained, that "we spent a million dollars harvesting tens of millions of Facebook profiles, and those profiles were used as the basis of the algorithms that became the foundation of Cambridge Analytica itself. The company itself was founded on using Facebook data".
Audit
On 19 March 2018, Facebook announced it had hired the digital forensics firm Stroz Friedberg to conduct a "comprehensive audit" of Cambridge Analytica, while Facebook shares plummeted 7 percent overnight (erasing roughly $40 billion in market capitalization).Cambridge Analytica had not just used the profiles of Facebook users to compile data sets. According to Christopher Wylie's testimony, the company also harvested the data of each user's network of friends, leveraging the original data set. It then converted, combined and migrated its results into new data sets, which can in principle survive in some format, even if the original data sources are destroyed. It created and applied algorithms using data to which - critics argue - it could not have been entitled. This was denied by Cambridge Analytica, which stated on its website that it legitimately "uses data to change audience behavior" among customers and voters (who choose to view and provide information). If advertisers can do that, why not a data company? Where should the line be drawn? Legally, it remained a "fuzzy" area.
Legal issue
The tricky legal issue then became, what kind of data Cambridge Analytica (or any similar company) is actually allowed to have and keep.Facebook itself became the subject of another U.S. Federal Trade Commission inquiry, to establish whether Facebook violated the terms of a 2011 consent decree governing its handing of user data (data which was allegedly transferred to Cambridge Analytica without Facebook's and user's knowledge).Wired journalist Jessi Hempel commented in a CBNC panel discussion that "Now there is this fuzziness from the top of the company [i.e. Facebook] that I have never seen in the fifteen years that I have covered it."
Data privacy
Interrogating Facebook's CEO Mark Zuckerberg before the U.S. House Energy and Commerce Committee in April 2018, New Mexico Congressman Rep. Ben Ray Luján put it to him that the Facebook corporation might well have "29,000 data points" on each Facebook user. Zuckerberg claimed that he "did not really know". Lujan's figure was based on ProPublica research, which in fact suggested that Facebook may even have 52,000 data points for many Facebook users. When Zuckerberg replied to his critics, he stated that because the revolutionary technology of Facebook (with 2.2 billion users worldwide, at that time) had ventured into previously unknown territory, it was unavoidable that mistakes would be made, despite the best of intentions. He justified himself saying that:
"For the first ten or twelve years of the company, I viewed our responsibility primarily as building tools, that if we could put those tools in people's hands, then that would empower people to do good things. What we have learnt now... is that we need to take a more proactive role and a broader view of our responsibility."
In July 2018, Facebook and Instagram barred access from Crimson Hexagon, a company that advises corporations and governments using one trillion scraped social media posts, which it mined and processed with artificial intelligence and image analysis.
Integrity
It remained "fuzzy" what was more important to Zuckerberg: making money from user's information, or real corporate integrity in the use of personal information. Zuckerberg implied, that he believed that, on balance, Facebook had done more good than harm, and that, if he had believed that wasn't the case, he would never have persevered with the business. Thus, "the good" was itself a fuzzy concept, because it was a matter of degree ("more good than bad"). He had to sell stuff, to keep the business growing. If people do not like Facebook, then they simply should not join it, or opt out, they have the choice. Many critics however feel that people really are in no position to make an informed choice, because they have no idea of how exactly their information will or might be used by third parties contracting with Facebook; because the company legally owns the information that users provide online, they have no control over that either, except to restrict themselves in what they write online (the same applies to many other online services).
After the New York Times broke the news on 17 March 2018, that copies of the Facebook data set scraped by Cambridge Analytica could still be downloaded from the Internet, Facebook was severely criticized by government representatives. When questioned, Zuckerberg admitted that "In general we collect data on people who are not signed up for Facebook for security purposes" with the aim "to help prevent malicious actors from collecting public information from Facebook users, such as names". From 2018 onward, Facebook faced a lot more lawsuits brought against the company, alleging data breaches, security breaches and misuse of personal information (see Lawsuits involving Meta Platforms and Facebook Federal Litigation Filings). There still exists no standard international regulatory framework for social network information, and it is often unclear what happens to the stored information, after a provider company closes down, or is taken over by another company. Zuckerberg's Meta company also reports its own legal actions.
On 2 May 2018, it was reported that the Cambridge Analytica company was shutting down and was starting bankruptcy proceedings, after losing clients and facing escalating legal costs. The reputational damage which the company had suffered or caused, had become too great.
Speed
A traditional objection to big data is, that it cannot cope with rapid change: events move faster that the statistics can keep up with. Yet the technology now exists for corporations like Amazon, Google, Apple Inc. and Microsoft to pump cloud-based data streams from app-users straight into big data analytics programmes, in real time. Provided that the right kinds of analytical concepts are used, it is now technically possible to draw definite and important conclusions about gradations of human and natural behaviour using very large fuzzy data sets and fuzzy programming – and increasingly it can be done very fast.
This achievement has become highly topical in military technology, in areas such as cybersecurity; tracking and monitoring systems; guidance systems (for firearms, explosive launchers, vehicles, planes, vessels, artillery, missiles, satellites, drones and bombs); threat identification/evaluation systems; risk and strategy appraisal; arms transfer and arms race impact assessments; and targeting methods. The identification of a threat and the response to it often have to happen very fast, with a high degree of accuracy, for which comprehensive artificial intelligence is essential. Dr Tal Mimran, a lecturer at Hebrew University in Jerusalem and a former legal adviser to the Israeli Defence Force (IDF) stated:
"During the period in which I served in the target room [between 2010 and 2015], you needed a team of around 20 intelligence officers to work for around 250 days to gather something between 200 to 250 targets. Today, the AI will do that in a week.”
Although no comprehensive overviews appear to be publicly available, a large amount of scientific research on fuzzy systems was funded or sponsored by the military. However, military uses of fuzzy systems research can also have spin-offs for medical applications.
Academic debates
There have been many academic debates about the meaning, relevance and utility of fuzzy concepts, as well as their appropriate use.Rudolf E. Kálmán stated in 1972 that "there is no such thing as a fuzzy concept... We do talk about fuzzy things but they are not scientific concepts". The suggestion is that to qualify as a concept, the concept must always be clear and precise, without any fuzziness. A vague notion would be at best a prologue to formulating a concept. In 2011, three Chinese engineers alleged that "Fuzzy set, its t-norm, s-norm and fuzzy supplement theories have already become the academic virus in the world".
"Fuzzy" label
Lotfi A. Zadeh himself confessed that:
"I knew that just by choosing the label fuzzy I was going to find myself in the midst of a controversy... If it weren't called fuzzy logic, there probably wouldn't be articles on it on the front page of the New York Times. So let us say it has a certain publicity value. Of course, many people don't like that publicity value, and when they see it in the New York Times, it doesn't sit well with them."
However, the impact of the invention of fuzzy reasoning went far beyond names and labels. When Zadeh gave his acceptance speech in Japan for the 1989 Honda Foundation prize, which he received for inventing fuzzy theory, he stated that "The concept of a fuzzy set has had an upsetting effect on the established order."
Frege and Wittgenstein
According to The Foundations of Arithmetic by the logician Gottlob Frege,
"A definition of a concept... must be complete; it must unambiguously determine, as regards any object, whether or not it falls under the concept... the concept must have a sharp boundary... a concept that is not sharply defined is wrongly termed a concept. Such quasi-conceptual constructions cannot be recognized as concepts by logic. The law of the excluded middle is really just another form of the requirement that the concept should have a sharp boundary."
In his notes on language games, Ludwig Wittgenstein replied to Frege's argument as follows:
"One can say that the concept of a game is a concept with blurred edges. “But is a blurred concept a concept at all?” Is a photograph that is not sharp a picture of a person at all? Is it even always an advantage to replace a picture that is not sharp by one that is? Isn’t one that isn’t sharp often just what we need? Frege compares a concept to a region, and says that a region without clear boundaries can’t be called a region at all. This presumably means that we can’t do anything with it. But is it senseless to say “Stay roughly here”? Imagine that I were standing with someone in a city square and said that. As I say it, I do not bother drawing any boundary, but just make a pointing gesture as if I were indicating a particular spot. And this is just how one might explain what a game is."
The categorical status of concepts
There is no general agreement among philosophers and scientists about how the notion of a "concept" (and in particular, a scientific concept), should be defined. A concept could be defined as a mental representation, as a cognitive capacity, as an abstract object, as a cluster of linked phenomena etc. Edward E. Smith & Douglas L. Medin stated that "there will likely be no crucial experiments or analyses that will establish one view of concepts as correct and rule out all others irrevocably." Of course, scientists also quite often do use imprecise analogies in their models to help understanding an issue. A concept can be clear enough, but not (or not sufficiently) precise.
Rather uniquely, terminology scientists at the German National Standards Institute (Deutsches Institut für Normung) provided an official standard definition of what a concept is (under the terminology standards DIN 2330 of 1957, completely revised in 1974 and last revised in 2022; and DIN 2342 of 1986, also last revised in 2022). According to the official German definition, a concept is a unit of thought which is created through abstraction for a set of objects, and which identifies shared (or related) characteristics of those objects.
The subsequent ISO definition is very similar. Under the ISO 1087 terminology standard of the International Standards Organization (first published in October 2000, reviewed in 2005 and revised in 2019), a concept is defined as a unit of thought or an idea constituted through abstraction on the basis of properties common to a set of objects. It is acknowledged that although a concept usually has one definition or one meaning, it may have multiple designations, terms of expression, symbolizations or representations. Thus, for example, the same concept can have different names in different languages. Both verbs and nouns can express concepts. A concept can also be thought of as "a way of looking at the world".
The official terminological standards are useful for many practical purposes. But for more complex concepts the standards may not be so helpful. The reason is that complex concepts do not necessarily denote only a collection of objects which have something in common. A complex concept may for example express a Gestalt, i.e. it may express a totality which is more, means more, and does more than the sum of its parts (as recognized in Aristotle's Metaphysics). It may be that the parts cannot exist other than within the totality. The totality could also be a "totality of totalities". In such cases, the definition of the complex concept is not (or not fully) reducible to what its parts have in common. Modelling such a concept requires more than identifying and enumerating the parts that are included in (and excluded from) the concept. It requires also a specification of what all the parts together "add up to", or what they constitute collectively. In some respects at least, the totality differs qualitatively from any of its parts. The Gestalt could be a fuzzy object, figure or shape.
Potential corruption
Reasoning with fuzzy concepts is often viewed as a kind of "logical corruption" or scientific perversion because, it is claimed, fuzzy reasoning rarely reaches a definite "yes" or a definite "no". A clear, precise and logically rigorous conceptualization is no longer a necessary prerequisite, for carrying out a procedure, a project, or an inquiry, since "somewhat vague ideas" can always be accommodated, formalized and programmed with the aid of fuzzy expressions. The purist idea is, that either a rule applies, or it does not apply. When a rule is said to apply only "to some extent", then in truth the rule does not apply. Thus, a compromise with vagueness or indefiniteness is, on this view, effectively a compromise with error - an error of conceptualization, an error in the inferential system, or an error in physically carrying out a task.
Kahan's criticism
The computer scientist William Kahan argued in 1975 that "the danger of fuzzy theory is that it will encourage the sort of imprecise thinking that has brought us so much trouble." He said subsequently,
"With traditional logic there is no guaranteed way to find that something is contradictory, but once it is found, you'd be obliged to do something. But with fuzzy sets, the existence of contradictory sets can't cause things to malfunction. Contradictory information doesn't lead to a clash. You just keep computing. (...) Life affords many instances of getting the right answer for the wrong reasons... It is in the nature of logic to confirm or deny. The fuzzy calculus blurs that. (...) Logic isn't following the rules of Aristotle blindly. It takes the kind of pain known to the runner. He knows he is doing something. When you are thinking about something hard, you'll feel a similar sort of pain. Fuzzy logic is marvellous. It insulates you from pain. It's the cocaine of science."
According to Kahan, statements of a degree of probability are usually verifiable. There are standard tests one can do. By contrast, there is no conclusive procedure which can decide the validity of assigning particular fuzzy truth values to a data set in the first instance. It is just assumed that a model or program will work, "if" particular fuzzy values are accepted and used, perhaps based on some statistical comparisons or try-outs.
Bad design
In programming, a problem can usually be solved in several different ways, not just one way, but an important issue is, which solution works best in the short term, and in the long term. Kahan implies, that fuzzy solutions may create more problems in the long term, than they solve in the short term. For example, if one starts off designing a procedure, not with well thought-out, precise concepts, but rather by using fuzzy or approximate expressions which conveniently patch up (or compensate for) badly formulated ideas, the ultimate result could be a complicated, malformed mess, that does not achieve the intended goal.
Had the reasoning and conceptualization been much sharper at the start, then the design of the procedure might have been much simpler, more efficient and effective - and fuzzy expressions or approximations would not be necessary, or required much less. Thus, by allowing the use of fuzzy or approximate expressions, one might actually foreclose more rigorous thinking about design, and one might build something that ultimately does not meet expectations.
If (say) an entity X turns out to belong for 65% to category Y, and for 35% to category Z, how should X be allocated? One could plausibly decide to allocate X to Y, making a rule that, if an entity belongs for 65% or more to Y, it is to be treated as an instance of category Y, and never as an instance of category Z. One could, however, alternatively decide to change the definitions of the categorization system, to ensure that all entities such as X fall 100% in one category only.
This kind of argument claims, that boundary problems can be resolved (or vastly reduced) simply by using better categorization or conceptualization methods. If we treat X "as if" it belongs 100% to Y, while in truth it only belongs 65% to Y, then arguably we are really misrepresenting things. If we keep doing that with a lot of related variables, we can greatly distort the true situation, and make it look like something that it isn't.
In a "fuzzy permissive" environment, it might become far too easy, to formalize and use a concept which is itself badly defined, and which could have been defined much better. In that environment, there is always a quantitative way out, for concepts that do not quite fit, or which don't quite do the job for which they are intended. The cumulative adverse effect of the discrepancies might, in the end, be much larger than ever anticipated.
Counter-argument
A typical reply to Kahan's objections is, that fuzzy reasoning never "rules out" ordinary binary logic, but instead presupposes ordinary true-or-false logic. Lotfi Zadeh stated that "fuzzy logic is not fuzzy. In large measure, fuzzy logic is precise." It is a precise logic of imprecision. Fuzzy logic is not a replacement of, or substitute for ordinary logic, but an enhancement of it, with many practical uses. Fuzzy thinking does oblige action, but primarily in response to a change in quantitative gradation, not in response to a contradiction.
One could say, for example, that ultimately one is either "alive" or "dead", which is perfectly true. Meantime though one is "living", which is also a significant truth - yet "living" is a fuzzy concept. It is true that fuzzy logic by itself usually cannot eliminate inadequate conceptualization or bad design. Yet it can at least make explicit, what exactly the variations are in the applicability of a concept which has unsharp boundaries.
If one always had perfectly crisp concepts available, perhaps no fuzzy expressions would be necessary. In reality though, one often does not have all the crisp concepts to start off with. One might not have them yet for a long time, or ever - or, several successive "fuzzy" approximations might be needed, to get there. A "fuzzy permissive" environment may be appropriate and useful, precisely because it permits things to be actioned, that would never have been achieved, if there had been crystal clarity about all the consequences from the start, or if people insisted on absolute precision prior to doing anything. Scientists often try things out on the basis of "hunches", and processes like serendipity can play a role.
Learning something new, or trying to create something new, is rarely a completely formal-logical or linear process. There are not only "knowns" and "unknowns" involved, but also "partly known" phenomena, i.e., things which are known or unknown "to some degree". Even if, ideally, we would prefer to eliminate fuzzy ideas, we might need them initially to get there, further down the track. Any method of reasoning is a tool. If its application has bad results, it is not the tool itself that is to blame, but its inappropriate use. It would be better to educate people in the best use of the tool, if necessary with appropriate authorization, than to ban the tool pre-emptively, on the ground that it "could" or "might" be abused. Exceptions to this rule would include things like computer viruses and illegal weapons that can only cause great harm if they are used. There is no evidence though that fuzzy concepts as a species are intrinsically harmful, even if some bad concepts can cause harm if used in inappropriate contexts.
Reducibility
Susan Haack once claimed that a many-valued logic requires neither intermediate terms between true and false, nor a rejection of bivalence. She implied that the intermediate terms (i.e. the gradations of truth) can always be restated as conditional if-then statements, and by implication, that fuzzy logic is fully reducible to binary true-or-false logic.
This interpretation is disputed (it assumes that the knowledge already exists to fit the intermediate terms to a logical sequence), but even if it was correct, assigning a number to the applicability of a statement is often enormously more efficient than a long string of if-then statements that would have the same intended meaning. That point is obviously of great importance to computer programmers, educators and administrators seeking to code a process, activity, message or operation as simply as possible, according to logically consistent rules. Prof. Haack is, of course, quite correct when she argues that fuzzy logic does not do away with binary logic.
Quantification
It may be wonderful to have an unlimited number of distinctions available to define what one means, but not all scholars would agree that any concept is equal to, or reducible to, a mathematical set. Some phenomena are difficult or impossible to quantify and count, in particular if they lack discrete boundaries (for example, clouds). George Lakoff emphasized that it is not true that fuzzy-set theory is the only or necessarily the most appropriate way to start modelling concepts.
Formalization
Qualities may not be fully reducible to quantities – if there are no qualities, it may become impossible to say what the numbers are numbers of, or what they refer to, except that they refer to other numbers or numerical expressions such as algebraic equations. A measure requires a counting unit defined by a category, but the definition of that category is essentially qualitative; a language which is used to communicate data is difficult to operate, without any qualitative distinctions and categories. We may, for example, transmit a text in binary code, but the binary code does not tell us directly what the text intends. It has to be translated, decoded or converted first, before it becomes comprehensible.
In creating a formalization or formal specification of a concept, for example for the purpose of measurement, administrative procedure or programming, part of the meaning of the concept may be changed or lost. For example, if we deliberately program an event according to a concept, it might kill off the spontaneity, spirit, authenticity and motivational pattern which is ordinarily associated with that type of event.
Quantification is not an unproblematic process. To quantify a phenomenon, we may have to introduce special assumptions and definitions which disregard part of totality of the phenomenon.
- The economist John Maynard Keynes concluded that formalization "runs the risk of leaving behind the subjectmatter we are interested in" and "also runs the risk of increasing rather than decreasing the muddle."
- Friedrich Hayek stated that "it is certainly not scientific to insist on measurement where you don't know what your measurements mean. There are cases where measurements are not relevant."
- The Hayekian big data guru Viktor Mayer-Schönberger states that "A system based on money and price solved a problem of too much information and not enough processing power, but in the process of distilling information down to price, many details get lost."
- Michael Polanyi stated that "the process of formalizing all knowledge to the exclusion of any tacit knowing is self-defeating", since to mathematize a concept we need to be able to identify it in the first instance without mathematization.
Measurement
Programmers, statisticians or logicians are concerned in their work with the main operational or technical significance of a concept which is specifiable in objective, quantifiable terms. They are not primarily concerned with all kinds of imaginative frameworks associated with the concept, or with those aspects of the concept which seem to have no particular functional purpose – however entertaining they might be. However, some of the qualitative characteristics of the concept may not be quantifiable or measurable at all, at least not directly. The temptation exists to ignore them, or try to infer them from data results.
If, for example, we want to count the number of trees in a forest area with any precision, we have to define what counts as one tree, and perhaps distinguish them from saplings, split trees, dead trees, fallen trees etc. Soon enough it becomes apparent that the quantification of trees involves a degree of abstraction – we decide to disregard some timber, dead or alive, from the population of trees, in order to count those trees that conform to our chosen concept of a tree. We operate in fact with an abstract concept of what a tree is, which diverges to some extent from the true diversity of trees there are.
Even so, there may be some trees, of which it is not very clear, whether they should be counted as a tree or not. It may be difficult to define the exact boundary where the forest begins and ends. The forest boundary might also change somewhat in the course of time. A certain amount of "fuzziness" in the definition of a tree and of the forest may therefore remain. The implication is, that the seemingly "exact" number offered for the total quantity of trees in the forest may be much less exact than one might think - it is probably more an estimate or indication of magnitude, rather than an exact description. Yet - and this is the point - the imprecise measure can be very useful and sufficient for all intended purposes.
It is tempting to think, that if something can be measured, it must exist, and that if we cannot measure it, it does not exist. Neither might be true. Researchers try to measure such things as intelligence or gross domestic product, without much scientific agreement about what these things actually are, how they exist, and what the correct measures might be.
When one wants to count and quantify distinct objects using numbers, one needs to be able to distinguish between all of those separate objects as countable units. If this is difficult or impossible, then, although this may not invalidate a quantitative procedure as such, quantification is not really possible in practice. At best, we may be able to assume or infer indirectly a certain distribution of quantities that must be there. In this sense, scientists often use proxy variables to substitute as measures for variables which are known (or thought) to be there, but which themselves cannot be observed or measured directly.
Vague or fuzzy
The exact relationship between vagueness and fuzziness is disputed.
Philosophical interpretation
Philosophers often regard fuzziness as a particular kind of vagueness, and consider that "no specific assignment of semantic values to vague predicates, not even a fuzzy one, can fully satisfy our conception of what the extensions of vague predicates are like". Surveying recent literature on how to characterize vagueness, Matti Eklund states that appeal to lack of sharp boundaries, borderline cases and "sorites-susceptible" predicates are the three informal characterizations of vagueness which are most common in the literature.
Zadeh's argument
However, Lotfi A. Zadeh claimed that "vagueness connotes insufficient specificity, whereas fuzziness connotes unsharpness of class boundaries". Thus, he argued, a sentence like "I will be back in a few minutes" is fuzzy but not vague, whereas a sentence such as "I will be back sometime", is fuzzy and vague. His suggestion was that fuzziness and vagueness are logically quite different qualities, rather than fuzziness being a type or subcategory of vagueness. Zadeh claimed that "inappropriate use of the term 'vague' is still a common practice in the literature of philosophy".
Ethics and law
In the scholarly inquiry about ethics and meta-ethics, vague or fuzzy concepts and borderline cases are standard topics of controversy. Central to ethics are theories of "value", what is "good" or "bad" for people and why that is, and the idea of "rule following" as a condition for moral integrity, consistency and non-arbitrary behaviour.
Yet, if human valuations or moral rules are only vague or fuzzy, then they may not be able to orient or guide behaviour. It may become impossible to operationalize rules. Evaluations may not permit definite moral judgements, in that case. Hence, clarifying fuzzy moral notions is usually considered to be critical for the ethical endeavour as a whole.
Excessive precision in rule-making
Nevertheless, Scott Soames has made the case that vagueness or fuzziness can be valuable to rule-makers, because "their use of it is valuable to the people to whom rules are addressed". It may be more practical and effective to allow for some leeway (and personal responsibility) in the interpretation of how a rule should be applied - bearing in mind the overall purpose which the rule intends to achieve.
If a rule or procedure is stipulated too exactly, it can sometimes have a result which is contrary to the aim which it was intended to help achieve. For example, "The Children and Young Persons Act could have specified a precise age below which a child may not be left unsupervised. But doing so would have incurred quite substantial forms of arbitrariness (for various reasons, and particularly because of the different capacities of children of the same age)".
Conflicting rules
A related sort of problem is, that if the application of a legal concept is pursued too exactly and rigorously, it may have consequences that cause a serious conflict with another legal concept. This is not necessarily a matter of bad law-making. When a law is made, it may not be possible to anticipate all the cases and events to which it will apply later (even if 95% of possible cases are predictable). The longer a law is in force, the more likely it is, that people will run into problems with it, that were not foreseen when the law was made.
So, the further implications of one rule may conflict with another rule. "Common sense" might not be able to resolve things. In that scenario, too much precision can get in the way of justice. Very likely a special court ruling wil have to set a norm. The general problem for jurists is, whether "the arbitrariness resulting from precision is worse than the arbitrariness resulting from the application of a vague standard". David Lanius has examined nine arguments for the "value of vagueness" in different contexts.
Mathematical ontology
The definitional disputes about fuzziness remain unresolved so far, mainly because, as anthropologists and psychologists have documented, different languages (or symbol systems) that have been created by people to signal meanings suggest different ontologies. Put simply: it is not merely that describing "what is there" involves symbolic representations of some kind. How distinctions are drawn, influences perceptions of "what is there", and vice versa, perceptions of "what is there" influence how distinctions are drawn. This is an important reason why, as Alfred Korzybski noted, people frequently confuse the symbolic representation of reality, conveyed by languages and signs, with reality itself.
Fuzziness implies, that there exists a potentially infinite number of truth values between complete truth and complete falsehood. If that is the case, it creates the foundational issue of what, in the case, can justify or prove the existence of the categorical absolutes which are assumed by logical or quantitative inference. If there is an infinite number of shades of grey, how do we know what is totally black and white, and how could we identify that?
Tegmark's mathematical universe
To illustrate the ontological issues, cosmologist Max Tegmark argues boldly that the universe consists of math: "If you accept the idea that both space itself, and all the stuff in space, have no properties at all except mathematical properties," then the idea that everything is mathematical "starts to sound a little bit less insane."
Tegmark moves from the epistemic claim that mathematics is the only known symbol system which can in principle express absolutely everything, to the methodological claim that everything is reducible to mathematical relationships, and then to the ontological claim, that ultimately everything that exists is mathematical (the mathematical universe hypothesis). The argument is then reversed, so that because everything is mathematical in reality, mathematics is necessarily the ultimate universal symbol system.
The main criticisms of Tegmark's approach are that (1) the steps in this argument do not necessarily follow, (2) no conclusive proof or test is possible for the claim that a total reduction of everything to mathematics is feasible, among other things because qualitative categories remain indispensable to understand and navigate what quantities mean, and (3) it may be that a complete reduction to mathematics cannot be accomplished, without at least partly altering, negating or deleting a non-mathematical significance of phenomena, experienced perhaps as qualia.
Zalta's metaphysics
In his meta-mathematical metaphysics, Edward N. Zalta has claimed that for every set of properties of a concrete object, there always exists exactly one abstract object that encodes exactly that set of properties and no others - a foundational assumption or axiom for his ontology of abstract objects By implication, for every fuzzy object there exists always at least one defuzzified concept which encodes it exactly. It is a modern interpretation of Plato's metaphysics of knowledge, which expresses confidence in the ability of science to conceptualize the world exactly.
Platonism versus cognitive realism
The Platonic-style interpretation was critiqued by Hartry H. Field. Mark Balaguer argues that we do not really know whether mind-independent abstract objects exist or not; so far, we cannot prove whether Platonic realism is definitely true or false. Defending a cognitive realism, Scott Soames argues that the reason why this unsolvable conundrum has persisted, is because the ultimate constitution of the meaning of concepts and propositions was misconceived.
Traditionally, it was thought that concepts can be truly representational, because ultimately they are related to intrinsically representational Platonic complexes of universals and particulars (see theory of forms). However, once concepts and propositions are regarded as cognitive-event types, it is possible to claim that they are able to be representational, because they are constitutively related to intrinsically representational cognitive acts in the real world. As another philosopher put it,
"The question of how we can know the world around us is not entirely unlike the question of how it is that the food our environment provides happens to agree with our stomachs. Either can become a mystery if we forget that minds, like stomachs, originated in and have been conditioned by a pre-existent natural order."
Along these lines, it could be argued that reality, and the human cognition of reality, will inevitably contain some fuzzy characteristics, which can perhaps be represented only by concepts which are themselves fuzzy to some or other extent.
Social science and the media
The idea of fuzzy concepts has also been applied in the philosophical, sociological and linguistic analysis of human behaviour.
Sociology and linguistics
In a 1973 paper, George Lakoff analyzed hedges in the interpretation of the meaning of categories.Charles Ragin and others have applied the idea to sociological analysis. For example, fuzzy set qualitative comparative analysis ("fsQCA") has been used by German researchers to study problems posed by ethnic diversity in Latin America. In New Zealand, Taiwan, Iran, Malaysia, the European Union and Croatia, economists have used fuzzy concepts to model and measure the underground economy of their country. Kofi Kissi Dompere applied methods of fuzzy decision, approximate reasoning, negotiation games and fuzzy mathematics to analyze the role of money, information and resources in a "political economy of rent-seeking", viewed as a game played between powerful corporations and the government. The German researcher Thomas Kron has used fuzzy methods to model sociological theory, creating an integral action-theoretical model with the aid of fuzzy logic. With Lars Winter, Kron developed the system theory of Niklas Luhmann further, using the so-called "Kosko-Cube". Kron studies transnational terrorism and other contemporary phenomena using fuzzy logic, to understand conditions involving uncertainty, hybridity, violence and cultural systems.
A concept may be deliberately created by sociologists as an ideal type to understand something imaginatively, without any strong claim that it is a "true and complete description" or a "true and complete reflection" of whatever is being conceptualized. In a more general sociological or journalistic sense, a "fuzzy concept" has come to mean a concept which is meaningful but inexact, implying that it does not exhaustively or completely define the meaning of the phenomenon to which it refers – often because it is too abstract. In this context, it is said that fuzzy concepts "lack clarity and are difficult to test or operationalize". To specify the relevant meaning more precisely, additional distinctions, conditions and/or qualifiers would be required.
A few examples can illustrate this kind of usage:
- a handbook of sociology states that "The theory of interaction rituals contains some gaps that need to be filled and some fuzzy concepts that need to be differentiated." The idea is, that if finer distinctions are introduced, then the fuzziness or vagueness would be eliminated.
- a book on youth culture describes ethnicity as "a fuzzy concept that overlaps at times with concepts of race, minority, nationality and tribe". In this case, part of the fuzziness consists in the inability to distinguish precisely between a concept and a different, but closely related concept.
- a book on sociological theory argues that the Critical Theory of domination faces the problem that "reality itself has become a rather meaningless, fuzzy concept." The suggestion here is, that the variations in how theoretical concepts are applied have become so large, that the concepts could mean all kinds of things, and therefore are crucially vague (with the implication, that they are not useful any longer for that very reason).
- A history book states: "Sodomy was a vague and fuzzy concept in medieval and early modern Europe, and was often associated with a variety of supposedly related moral and criminal offenses, including heresy, witchcraft, sedition, and treason. St Thomas Aquinas... categorized sodomy with an assortment of sexual behaviours "from which generation [i.e. procreation] cannot follow". In this case, because a concept is defined by what it excludes, it remains somewhat vague what items of activity it would specifically include.
Mass media
The main reason why the term "fuzzy concept" is now often used in describing human behaviour, is that human interaction has many characteristics which are difficult to quantify and measure precisely (although we know that they have magnitudes and proportions), among other things because they are interactive and reflexive (the observers and the observed mutually influence the meaning of events). Those human characteristics can be usefully expressed only in an approximate way (see reflexivity (social theory)).
Newspaper stories frequently contain fuzzy concepts, which are readily understood and used, even although they are far from exact. Thus, many of the meanings which people ordinarily use to negotiate their way through life in reality turn out to be "fuzzy concepts". While people often do need to be exact about some things (e.g. money or time), many areas of their lives involve expressions which are far from exact.
Sometimes the term is also used in a pejorative sense. For example, a New York Times journalist wrote that Prince Sihanouk "seems unable to differentiate between friends and enemies, a disturbing trait since it suggests that he stands for nothing beyond the fuzzy concept of peace and prosperity in Cambodia".
Applied social science
The use of fuzzy logic in the social sciences and humanities has remained limited until recently. Lotfi A. Zadeh said in a 1994 interview that:
"I expected people in the social sciences – economics, psychology, philosophy, linguistics, politics, sociology, religion and numerous other areas to pick up on it. It's been somewhat of a mystery to me why even to this day, so few social scientists have discovered how useful it could be."
Two decades later, after a digital information explosion due to the growing use of the internet and mobile phones worldwide, fuzzy concepts and fuzzy logic were increasingly being applied in big data analysis of social, commercial and psychological phenomena. Many sociometric and psychometric indicators are based partly on fuzzy concepts and fuzzy variables.
Jaakko Hintikka once claimed that "the logic of natural language we are in effect already using can serve as a 'fuzzy logic' better than its trade name variant without any additional assumptions or constructions." That might help to explain why fuzzy logic has not been used much to formalize concepts in the "soft" social sciences.
Lotfi A. Zadeh rejected such an interpretation, on the ground that in many human endeavours as well as technologies it is highly important to define more exactly "to what extent" something is applicable or true, when it is known that its applicability can vary to some important extent among large populations. Reasoning which accepts and uses fuzzy concepts can be shown to be perfectly valid with the aid of fuzzy logic, because the degrees of applicability of a concept can be more precisely and efficiently defined with the aid of numerical notation.
Another possible explanation for the traditional lack of use of fuzzy logic by social scientists is simply that, beyond basic statistical analysis (using programs such as SPSS and Excel) the mathematical knowledge of social scientists is often rather limited; they may not know how to formalize and code a fuzzy concept using the conventions of fuzzy logic. The standard software packages used provide only a limited capacity to analyze fuzzy data sets, if at all, and considerable skills are required.
Yet Jaakko Hintikka may be correct, in the sense that it can be much more efficient to use natural language to denote a complex idea, than to formalize it in logical terms. The quest for formalization might introduce much more complexity, which is not wanted, and which detracts from communicating the relevant issue. Some concepts used in social science may be impossible to formalize exactly, even though they are quite useful and people understand their appropriate application quite well.
Uncertainty
Fuzzy concepts can generate uncertainty because they are imprecise (especially if they refer to a process in motion, or a process of transformation where something is "in the process of turning into something else"). In that case, they do not provide a clear orientation for action or decision-making ("what does X really mean, intend or imply?"); reducing fuzziness, perhaps by applying fuzzy logic, might generate more certainty.
Precision versus relevance
However, this is not necessarily always so. A concept, even although it is not fuzzy at all, and even though it is very exact, could equally well fail to capture the meaning of something adequately. That is, a concept can be very precise and exact, but not – or insufficiently – applicable or relevant in the situation to which it refers. In this sense, a definition can be "very precise", but "miss the point" altogether.
Security
A fuzzy concept may indeed provide more security, because it provides a meaning for something when an exact concept is unavailable – which is better than not being able to denote it at all. A concept such as God, although not easily definable, for instance can provide security to the believer.
Observer effect
In physics, the observer effect and Heisenberg's uncertainty principle indicate that there is a physical limit to the amount of precision that is knowable, with regard to the movements of subatomic particles and waves. That is, features of physical reality exist, where we can know that they vary in magnitude, but of which we can never know or predict exactly how big or small the variations are. This insight suggests that, in some areas of our experience of the physical world, fuzziness is inevitable and can never be totally removed. Since the physical universe itself is incredibly large and diverse, it is not easy to imagine it, grasp it or describe it without using fuzzy concepts.
Language
Ordinary language, which uses symbolic conventions and associations which are often not logical, inherently contains many fuzzy concepts – "knowing what you mean" in this case depends partly on knowing the context (or being familiar with the way in which a term is normally used, or what it is associated with).
This can be easily verified for instance by consulting a dictionary, a thesaurus or an encyclopedia which show the multiple meanings of words, or by observing the behaviours involved in ordinary relationships which rely on mutually understood meanings (see also Imprecise language). Bertrand Russell regarded ordinary language (in contrast to logic) as intrinsically vague.
Implicature
To communicate, receive or convey a message, an individual somehow has to bridge his own intended meaning and the meanings which are understood by others, i.e., the message has to be conveyed in a way that it will be socially understood, preferably in the intended manner. Thus, people might state: "you have to say it in a way that I understand". Even if the message is clear and precise, it may nevertheless not be received in the way it was intended.
Bridging meanings may be done instinctively, habitually or unconsciously, but it usually involves a choice of terms, assumptions or symbols whose meanings are not completely fixed, but which depend among other things on how the receivers of the message respond to it, or the context. In this sense, meaning is often "negotiated" or "interactive" (or, more cynically, manipulated). This gives rise to many fuzzy concepts.
The semantic challenge of conveying meanings to an audience was explored in detail, and analyzed logically, by the British philosopher Paul Grice - using, among other things, the concept of implicature. Implicature refers to what is suggested by a message to the recipient, without being either explicitly expressed or logically entailed by its content. The suggestion could be very clear to the recipient (perhaps a sort of code), but it could also be vague or fuzzy.
Paradoxes
Even using ordinary set theory and binary logic to reason something out, logicians have discovered that it is possible to generate statements which are logically speaking not completely true or imply a paradox, even although in other respects they conform to logical rules (see Russell's paradox). If a margin of indeterminacy therefore persists, then binary logic cannot totally remove fuzziness. David Hilbert concluded that the existence of logical paradoxes tells us "that we must develop a meta-mathematical analysis of the notions of proof and of the axiomatic method; their importance is methodological as well as epistemological".
Psychology
Various different aspects of human experience commonly generate concepts with fuzzy characteristics.
Human vs. computer
The formation of fuzzy concepts is partly due to the fact that the human brain does not operate like a computer (see also Chinese room).
- While ordinary computers use strict binary logic gates, the brain does not; i.e., it is capable of making all kinds of neural associations according to all kinds of ordering principles (or fairly chaotically) in associative patterns which are not logical but nevertheless meaningful. For example, a work of art can be meaningful without being logical.
- A pattern can be observably regular, ordered and/or non-arbitrary, hence meaningful, without it being possible to describe it completely or exhaustively in formal-logical terms.
- Something can be meaningful although we cannot name it, or we might only be able to name it and nothing else.
- Human brains can also interpret the same phenomenon in several different but interacting frames of reference, at the same time, or in quick succession, without there necessarily being an explicit logical connection between the frames (see also framing effect).
According to fuzzy-trace theory, partly inspired by Gestalt psychology, human intuition is a non-arbitrary, reasonable and rational process of cognition; it literally "makes sense" (see also: Problem of multiple generality).
Transitions in learning and consciousness
In part, fuzzy concepts arise also because learning or the growth of understanding involves a transition from a vague awareness, which cannot orient behaviour greatly, to clearer insight, which can orient behaviour. At the first encounter with an idea, the sense of the idea may be rather hazy. When more experience with the idea has occurred, a clearer and more precise grasp of the idea results, as well as a better understanding of how and when to use the idea (or not).
In his study of implicit learning, Arthur S. Reber affirms that there does not exist a very sharp boundary between the conscious and the unconscious, and "there are always going to be lots of fuzzy borderline cases of material that is marginally conscious and lots of elusive instances of functions and processes that seem to slip in and out of personal awareness".
Thus, an inevitable component of fuzziness exists and persists in human consciousness, because of continual variation of gradations in awareness, along a continuum from the conscious, the preconscious, and the subconscious to the unconscious. The hypnotherapist Milton H. Erickson similarly noted that the conscious mind and the unconscious normally interact.
Limits of distinctions and generalizations
Some psychologists and logicians argue that fuzzy concepts are a necessary consequence of the reality that any kind of distinction we might like to draw has limits of application. At a certain level of generality, a distinction works fine. But if we pursued its application in a very exact and rigorous manner, or overextend its application, it appears that the distinction simply does not apply in some areas or contexts, or that we cannot fully specify how it should be drawn. An analogy might be, that zooming a telescope, camera, or microscope in and out, reveals that a pattern which is sharply focused at a certain distance becomes blurry at another distance, or disappears altogether.
Complexity and imprecision
Faced with any large, complex and continually changing phenomenon, any short statement made about that phenomenon is likely to be "fuzzy", i.e., it is meaningful, but – strictly speaking – incorrect and imprecise. It will not really do full justice to the reality of what is happening with the phenomenon. A correct, precise statement would require a lot of elaborations and qualifiers. Nevertheless, the "fuzzy" description turns out to be a useful shorthand that saves a lot of time in communicating what is going on ("you know what I mean").
Cognition and perceptual limits
In psychophysics, it was discovered that the perceptual distinctions we draw in the mind are often more definite than they are in the real world. Thus, the brain actually tends to "sharpen up" or "enhance" our perceptions of differences in the external world.
- Between black and white, we are able to detect only a limited number of shades of gray, or colour gradations (there are "detection thresholds").
- Motion blur refers to the loss of detail when a person looks at a fast-moving object, or is moving fast while the eyes are focused on something stationary. In a movie reel, the human eye can detect a sequence of up to 10 or 12 still images per second. At around 18 to 26 frames per second, the brain will "see" the sequence of individual images as a moving scene.
If there are more gradations and transitions in reality, than our conceptual or perceptual distinctions can capture in our minds, then it could be argued that how those distinctions will actually apply, must necessarily become vaguer at some point.
Imprecision of novelty
In interacting with the external world, the human mind may often encounter new, or partly new phenomena or relationships which cannot (yet) be sharply defined given the background knowledge available, and by known distinctions, associations or generalizations.
"Crisis management plans cannot be put 'on the fly' after the crisis occurs. At the outset, information is often vague, even contradictory. Events move so quickly that decision makers experience a sense of loss of control. Often denial sets in, and managers unintentionally cut off information flow about the situation" - L. Paul Bremer.
Fuzziness and chaos
It also can be argued that fuzzy concepts are generated by a certain sort of lifestyle or way of working which evades definite distinctions, makes them impossible or inoperable, or which is in some way chaotic. To obtain concepts which are not fuzzy, it must be possible to test out their application in some way. But in the absence of any relevant clear distinctions, lacking an orderly environment, or when everything is "in a state of flux" or in transition, it may not be possible to do so, so that the amount of fuzziness increases.
Everyday occurrence
Fuzzy concepts often play a role in the creative process of forming new concepts to understand something. In the most primitive sense, this can be observed in infants who, through practical experience, learn to identify, distinguish and generalise the correct application of a concept, and relate it to other concepts. However, fuzzy concepts may also occur in scientific, journalistic, programming and philosophical activity, when a thinker is in the process of clarifying and defining a newly emerging concept which is based on distinctions which, for one reason or another, cannot (yet) be more exactly specified or validated. Fuzzy concepts are often used to denote complex phenomena, or to describe something which is developing and changing, which might involve shedding some old meanings and acquiring new ones.
Uses in different areas
- In meteorology, where changes and effects of complex interactions in the atmosphere are studied, the weather reports often use fuzzy expressions indicating a broad trend, likelihood, approximation or level. The main reason is that the forecast can rarely be totally exact for any given location. Nevertheless the information is often useful to orient behaviour (e.g. "I must not forget to take my coat, or an umbrella, just in case").
- In biology, protein complexes with multiple structural forms are called fuzzy complexes. The different conformations can result in different, even opposite functions. The conformational ensemble is modulated by the environmental conditions. Post-translational modifications or alternative splicing can also impact the ensemble and thereby the affinity or specificity of interactions. Genetic fuzzy systems use algorithms or genetic programming which simulate natural evolutionary processes, in order to understand their structures and parameters.
- In medical diagnosis, the assessment of what the symptoms of a patient are often cannot be very exactly specified, since there are many possible qualitative and quantitative gradations in severity, incidence or frequency that could occur. Different symptoms may also overlap to some extent. These gradations can be difficult to measure, it may cost a lot of time and money, and so the medical professionals might use approximate "fuzzy" categories in their judgement of a medical condition or a patient's condition. Although it may not be exact, the diagnosis is often useful enough for treatment purposes. Fuzzy logic is increasingly employed in diagnostic and medical equipment capable of measuring gradations of a condition.
- In information services, fuzzy concepts are frequently encountered because a customer or client asks a question about something which could be interpreted in different ways, or, a document is transmitted of a type or meaning which cannot be easily allocated to a known type or category, or to a known procedure. It might take considerable inquiry to "place" the information, or establish in what framework it should be understood.
- In phenomenology, which aims to study the structure of subjective experience without preconceptions, an important insight is that how someone experiences something can be shaped both by the influence of the thing being experienced itself, but also by how the person responds to it. Thus, the actual experience the person has, is shaped by an "interactive object-subject relationship". To describe this experience, fuzzy categories are often necessary, since it is often impossible to predict or describe with great exactitude what the interaction will be, and how it is experienced.
- In translation work, fuzzy concepts are analyzed for the purpose of good translation. A concept in one language may not have quite the same meaning or significance in another language, or it may not be feasible to translate it literally, or at all. Some languages have concepts which do not exist in another language, raising the problem of how one would most easily render their meaning. In computer-assisted translation, a technique called fuzzy matching is used to find the most likely translation of a piece of text, using previous translated texts as a basis.
- In hypnotherapy, fuzzy language is deliberately used for the purpose of trance induction. Hypnotic suggestions are often couched in a somewhat vague, general or ambiguous language requiring interpretation by the subject. The intention is to distract and shift the conscious awareness of the subject away from external reality to her own internal state. In response to the somewhat confusing signals she gets, the awareness of the subject spontaneously tends to withdraw inward, in search of understanding or escape.
- In business and economics, it was discovered that "we are guided less by a correct exact knowledge of our self-interest than by a socially learned, evolved, intuitive grasp derived from mental shortcuts (frames, reference points, envy, addiction, temptation, fairness)". Thus, economic preferences are often fuzzy preferences, a highly important point for suppliers of products and services. Fuzzy set empirical methodologies are increasingly used by economic analysts to analyze the extent to which members of a population belong to a specific market category, because that can make a big difference to business results.
- In sexology, sex and gender are conceptualized by gender pluralists as a spectrum or continuum, or a set of scaled characteristics. Thus, the idea that people are either heterosexual men, heterosexual women, gay, lesbian, bisexual or transsexual is far too simplistic; gender identity is a matter of degree, a graded concept, which for that very reason is a fuzzy concept with unsharp boundaries. For example, somebody who is "mainly" heterosexual, may occasionally have had non-heterosexual contacts, without this warranting a definite "bisexual" label. A great variety of sexual orientations are possible and can co-exist. In the course of history, typical male or female gender roles and gender characteristics can also gradually change, so that the extent to which they express "masculine" or "feminine" traits is, at any time, a matter of degree, i.e. fuzzy.
- In politics, it can be highly important and problematic how exactly a conceptual distinction is drawn, or indeed whether a distinction is drawn at all; distinctions used in administration may be deliberately sharpened, or kept fuzzy, due to some political motive or power relationship. Politicians may be deliberately vague about some things, and very clear and explicit about others; if there is information that proves their case, they become very precise, but if the information doesn't prove their case, they become vague or say nothing.
- In statistical research, it is an aim to measure the magnitudes of phenomena. For this purpose, phenomena have to be grouped and categorized, so that distinct and discrete counting units can be defined. It must be possible to allocate all observations to mutually exclusive categories, so that they are properly quantifiable. Survey observations do not spontaneously transform themselves into countable data; they have to be identified, categorized and classified in such a way, that identical observations can be grouped together, and that observations are not counted twice or more. A well-designed questionnaire ensures that the questions are interpreted in the same way by all respondents, and that the respondents are really able to answer them within the formats provided. Again, for this purpose, it is a requirement that the concepts being used are exactly and comprehensibly defined for all concerned, and not fuzzy. There could be a margin of measurement error, but the amount of error must be kept within tolerable limits, and preferably its magnitude should be known.
- In theology an attempt is made to define more precisely the meaning of spiritual concepts, which refer to how human beings construct the meaning of human existence, and, often, the relationship people have with a supernatural world. Many spiritual concepts and beliefs are fuzzy, to the extent that, although abstract, they often have a highly personalized meaning, or involve personal interpretation of a type that is not easy to define in a cut-and-dried way. A similar situation occurs in psychotherapy. The Dutch theologian Kees de Groot has explored the imprecise notion that psychotherapy is like an "implicit religion", defined as a "fuzzy concept" (it all depends on what one means by "psychotherapy" and "religion"). The philosopher of spirituality Ken Wilber argued that "nothing is 100% right or wrong", things merely "vary in their degree of incompleteness and dysfunction"; no one and nothing is 100% good or evil, each just varies "in their degree of ignorance and disconnection". This insight suggests, that all human valuations can be considered as graded concepts, where each qualitative judgement has at least implicitly a sense of quantitative proportion attached to it.
- In the legal system, it is essential that rules are interpreted and applied in a standard way, so that the same sorts of cases and the same sorts of circumstances are treated equally. Otherwise one would be accused of arbitrariness, which would not serve the interests of justice. Consequently, lawmakers aim to devise definitions and categories which are sufficiently precise, so that they are not open to different interpretations. For this purpose, it is critically important to remove fuzziness, and differences of interpretation are typically resolved through a court ruling based on evidence. Alternatively, some other procedure is devised which permits the correct distinction to be discovered and made.
- In administration, archiving and accounting, fuzziness problems in interpretation and boundary problems can arise, because it is not clear to what category exactly a case, item, document, transaction or piece of data belongs. In principle, each case, event or item must be allocated to the correct category in a procedure, but it may be, that it is difficult to make the appropriate or relevant distinctions.
Generalities
Many concepts which are used fairly universally in daily life (such as "love", "God", "health", "social", "sustainability" "tolerance" etc.) are considered to be intrinsically fuzzy concepts, to the extent that their meaning usually cannot be completely and exactly specified with logical operators or objective terms, and can have multiple interpretations and personal (subjective) meanings. Yet such concepts are not at all meaningless. People keep using the concepts, even if they are difficult to define precisely.
Multiple meanings
It may also be possible to specify one personal meaning for the concept, without however placing restrictions on a different use of the concept in other contexts (as when, for example, one says "this is what I mean by X" in contrast to other possible meanings). In ordinary speech, concepts may sometimes also be uttered purely randomly; for example a child may repeat the same idea in completely unrelated contexts, or an expletive term may be uttered arbitrarily. A feeling or sense is conveyed, without it being fully clear what it is about.
Happiness may be an example of a word with variable meanings depending on context or timing.
Ambiguities
Fuzzy concepts can be used deliberately to create ambiguity and vagueness, as an evasive tactic, or to bridge what would otherwise be immediately recognized as a contradiction of terms. They might be used to indicate that there is definitely a connection between two things, without giving a complete specification of what the connection is, for some or other reason. This could be due to a failure or refusal to be more precise. But it could also be a prologue to a more exact formulation of a concept, or to a better understanding of it.
Efficiency
Fuzzy concepts can be used as a practical method to describe something of which a complete description would be an unmanageably large undertaking, or very time-consuming; thus, a simplified indication of what is at issue is regarded as sufficient, although it is not exact.
Popper
There is also such a thing as an "economy of distinctions", meaning that it is not helpful or efficient to use more detailed definitions than are really necessary for a given purpose. In this sense, Karl Popper rejected pedantry and commented that:
"...it is always undesirable to make an effort to increase precision for its own sake – especially linguistic precision – since this usually leads to loss of clarity, and to a waste of time and effort on preliminaries which often turn out to be useless, because they are bypassed by the real advance of the subject: one should never try to be more precise than the problem situation demands. I might perhaps state my position as follows. Every increase in clarity is of intellectual value in itself; an increase in precision or exactness has only a pragmatic value as a means to some definite end..."
The provision of "too many details" could be disorienting and confusing, instead of being enlightening, while a fuzzy term might be sufficient to provide an orientation. The reason for using fuzzy concepts can therefore be purely pragmatic, if it is not feasible or desirable (for practical purposes) to provide "all the details" about the meaning of a shared symbol or sign. Thus people might say "I realize this is not exact, but you know what I mean" – they assume practically that stating all the details is not required for the purpose of the communication.
Fuzzy logic gambit
Lotfi A. Zadeh picked up this point, and drew attention to a "major misunderstanding" about applying fuzzy logic. It is true that the basic aim of fuzzy logic is to make what is imprecise more precise. Yet in many cases, fuzzy logic is used paradoxically to "imprecisiate what is precise", meaning that there is a deliberate tolerance for imprecision for the sake of simplicity of procedure and economy of expression.
In such uses, there is a tolerance for imprecision, because making ideas more precise would be unnecessary and costly, while "imprecisiation reduces cost and enhances tractability" (tractability means "being easy to manage or operationalize"). Zadeh calls this approach the "Fuzzy Logic Gambit" (a gambit means giving up something now, to achieve a better position later).
In the Fuzzy Logic Gambit, "what is sacrificed is precision in [quantitative] value, but not precision in meaning", and more concretely, "imprecisiation in value is followed by precisiation in meaning". Zadeh cited as example Takeshi Yamakawa's programming for an inverted pendulum, where differential equations are replaced by fuzzy if-then rules in which words are used in place of numbers.
Fuzzy vs. Boolean
Common use of this sort of approach (combining words and numbers in programming), has led some logicians to regard fuzzy logic merely as an extension of Boolean logic (a two-valued logic or binary logic is simply replaced with a many-valued logic).
However, Boolean concepts have a logical structure which differs from fuzzy concepts. An important feature in Boolean logic is, that an element of a set can also belong to any number of other sets; even so, the element either does, or does not belong to a set (or sets). By contrast, whether an element belongs to a fuzzy set is a matter of degree, and not always a definite yes-or-no question.
All the same, the Greek mathematician Costas Drossos suggests in various papers that, using a "non-standard" mathematical approach, we could also construct fuzzy sets with Boolean characteristics and Boolean sets with fuzzy characteristics. This would imply, that in practice the boundary between fuzzy sets and Boolean sets is itself fuzzy, rather than absolute. For a simplified example, we might be able to state, that a concept X is definitely applicable to a finite set of phenomena, and definitely not applicable to all other phenomena. Yet, within the finite set of relevant items, X might be fully applicable to one subset of the included phenomena, while it is applicable only "to some varying extent or degree" to another subset of phenomena which are also included in the set. Following ordinary set theory, this generates logical problems, if e.g. overlapping subsets within sets are related to other overlapping subsets within other sets.
Clarifying methods
In mathematical logic, computer programming, philosophy and linguistics fuzzy concepts can be analyzed and defined more accurately or comprehensively, by describing or modelling the concepts using the terms of fuzzy logic or other substructural logics.
With the accelerating development of computer programming languages and computer processing capacity since the 1970s, it has become more acceptable in the sciences that there isn't just one "correct" way to formalize items of knowledge. It was realized by innovators that concepts and processes can be formalized and represented using many different kinds of tools, methods and systems - according to what is most useful, effective or efficient for a given purpose. Using software and artificial intelligence, many traditional and new sorts of techniques can be applied to clarify ideas, such as:
- 1. Contextualizing the concept by defining the setting or situation in which the concept is used, or how it is used appropriately (context).
- 2. Identifying the intention, purpose, aim or goal associated with the concept (teleology and design).
- 3. Comparing and contrasting the concept with related ideas in the present or the past (comparative and comparative research).
- 4. Creating a model, likeness, analogy, metaphor, prototype or narrative which shows what the concept is about or how it is applied (isomorphism, simulation or successive approximation [117]).
- 5. Probing the assumptions on which a concept is based, or which are associated with its use (critical thought, tacit assumption).
- 6. Mapping or graphing the applications of the concept using some basic parameters, or using some diagrams or flow charts to understand the relationships between elements involved (visualization and concept map).
- 7. Examining how likely it is that the concept applies, statistically or intuitively (probability theory).
- 8. Specifying relevant conditions to which the concept applies, as a procedure (computer programming, formal concept analysis).
- 9. Concretizing the concept – finding specific examples, illustrations, details or cases to which it applies (exemplar, exemplification).
- 10. Reducing or restating fuzzy concepts in terms which are simpler or similar, and which are not fuzzy or less fuzzy (simplification, dimensionality reduction, plain language, KISS principle or concision).
- 11. Trying out a concept, by using it in interactions, practical work or in communication, and assessing the feedback to understand how the boundaries and distinctions of the concept are being drawn (trial and error or pilot experiment).
- 12. Engaging in a structured dialogue or repeated discussion, to exchange ideas about how to get specific about what it means and how to clear it up (scrum method).
- 13. Allocating different applications of the concept to different but related sets (Boolean logic).
- 14. Identifying operational rules defining the use of the concept, which can be stated in a language and which cover all or most cases (material conditional).
- 15. Classifying, categorizing, grouping, or inventorizing all or most cases or uses to which the concept applies (taxonomy, cluster analysis and typology).
- 16. Applying a meta-language which includes fuzzy concepts in a more inclusive categorical system which is not fuzzy (meta).
- 17. Creating a measure or scale of the degree to which the concept applies (metrology).
- 18. Examining the distribution patterns or distributional frequency of (possibly different) uses of the concept (statistics).
- 19. Specifying a series of logical operators or inferential system which captures all or most cases to which the concept applies (algorithm).
- 20. Relating the fuzzy concept to other concepts which are not fuzzy or less fuzzy, or simply by replacing the fuzzy concept altogether with another, alternative concept which is not fuzzy yet "works the same way" (proxy)
- 21. Engaging in meditation, taking a pause to relax, or taking the proverbial "run around the block" to clarify the mind, and thus improve precision of thought about the definitional issue (self-care).
In this way, we can obtain a more exact understanding of the meaning and use of a fuzzy concept, and possibly decrease the amount of fuzziness. It may not be possible to specify all the possible meanings or applications of a concept completely and exhaustively, but if it is possible to capture the majority of them, statistically or otherwise, this may be useful enough for practical purposes.
Defuzzification
A process of defuzzification is said to occur, when fuzzy concepts can be logically described in terms of fuzzy sets, or the relationships between fuzzy sets, which makes it possible to define variations in the meaning or applicability of concepts as quantities. Effectively, qualitative differences are in that case described more precisely as quantitative variations, or quantitative variability. Assigning a numerical value then denotes the magnitude of variation along a scale from zero to one.
The difficulty that can occur in judging the fuzziness of a concept can be illustrated with the question "Is this one of those?". If it is not possible to clearly answer this question, that could be because "this" (the object) is itself fuzzy and evades definition, or because "one of those" (the concept of the object) is fuzzy and inadequately defined.
Thus, the source of fuzziness may be in (1) the nature of the reality being dealt with, (2) the concepts used to interpret it, or (3) the way in which the two are being related by a person. It may be that the personal meanings which people attach to something are quite clear to the persons themselves, but that it is not possible to communicate those meanings to others except as fuzzy concepts.
See also
- Alternative set theory
- Approximate measures
- Classical logic
- Defuzzification
- Detection theory
- Deviant logic
- Dialectic
- European Society for Fuzzy Logic and Technology
- Fuzzy subalgebra
- Fuzzy logic
- Fuzzy set theory
- Fuzzy clustering
- Fuzzy mathematics
- Fuzzy measure theory
- Fuzzy set operations
- George Klir
- Identity (Philosophy)
- Interval finite element
- Jakobson's functions of language
- Linear partial information
- Many-valued logic
- Multiset
- Neuro-fuzzy
- Non-well-founded set theory
- Obfuscation
- Opaque context
- Paraconsistent logic
- Phenomenology (psychology)
- Precision
- Referential transparency
- reflexivity (social theory)
- Post-normal science
- Rough fuzzy hybridization
- Rough set
- Semiset
- Sørensen similarity index
- Synchronicity
- Type-2 Fuzzy Sets and Systems
- Uncertainty
- Vague set
References
- Radim Behlohlavek & George J. Klir (eds.), Concepts and fuzzy logic. Cambridge, Mass.: MIT Press, 2011; Susan Haack, Deviant logic, fuzzy logic: beyond the formalism. Chicago: University of Chicago Press, 1996.
- Richard Dietz & Sebastiano Moruzzi (eds.), Cuts and clouds. Vagueness, Its Nature, and Its Logic. Oxford University Press, 2009; Delia Graff & Timothy Williamson (eds.), Vagueness. London: Routledge, 2002. For some ordinary scientific definitions of precision and imprecision, see the articles on significant figures, accuracy and precision, precision (statistics), and precision (computer science).
- Timothy Williamson, Vagueness. London: Routledge, 1994, p. 124f; Lotfi A. Zadeh, "Quantitative fuzzy semantics". Information Sciences, Vol. 3, No. 2, April 1971, pp. 159-176.
- D. Blockley, "Earthquake risk management of civil infrastructure: integrating soft and hard risks", in: Handbook of Seismic Risk Analysis and Management of Civil Infrastructure Systems. Sawston, Cambridge: Woodhead Publishing, 2013, chapter 9, pp. 229-254, at p. 238.
- Vyvyan Evans, A glossary of cognitive linguistics. Salt Lake City: University of Utah Press, 2007, p. 88.
- Susan Haack, "Do we need ‘‘fuzzy logic’’?", in: International Journal of Man-Machine Studies, Volume 11, Issue 4, 1979, pp. 437–45; Lotfi A. Zadeh, "Is there a need for fuzzy logic?", Information Sciences, Volume 178, Issue 13, July 2008, pp. 2751-2779.
- Bart Kosko, "Fuzzy logic". In: Scientific American, July 1993, pp. 76-81[1]; Bart Kosko, Fuzzy Thinking: The New Science of Fuzzy Logic. New York: Hyperion, 1993; Bart Kosko, Heaven in a chip: fuzzy visions of society and science in the digital age. New York: Three Rivers Press, 1999; Daniel McNeill & Paul Freiberger, Fuzzy Logic: The Revolutionary Computer Technology that Is Changing Our World. New York: Simon & Schuster, 1994. Charles Elkan, "The paradoxical success of fuzzy logic." IEEE Expert, August 1994.[2]; Didier Dubois et al., "Fuzzy-set based logics - an history-oriented presentation of their main developments", in: Handbook of the history of logic. Volume 8, The many valued and non-monotonic turn in logic. Amsterdam: Elsevier- North Holland, 2007, pp. 3-125.[3] Didier Dubois, Henri Prade, Articles written on the occasion of the 50th anniversary of fuzzy set theory. Institut de Recherche Informatique de Toulouse. 2015.[4]
- Radim Bělohlávek, Joseph W. Dauben & George J. Klir, Fuzzy Logic and Mathematics: A Historical Perspective. Oxford University Press, 2017; Didier Dubois et al., "Fuzzy-set based logics - an history-oriented presentation of their main developments", in: Handbook of the history of logic. Volume 8, The many valued and non-monotonic turn in logic. Amsterdam: Elsevier- North Holland, 2007, pp. 3-125.[5] Didier Dubois, Henri Prade, Articles written on the occasion of the 50th anniversary of fuzzy set theory . Institut de Recherche Informatique de Toulouse, 2015.https://hal.science/hal-03198270v1/file/Fuzzy-sets-50.pdf]
- Katyanna Quach, "Fuzzy logic makes a comeback – in picking where Earth sticks its probes into alien worlds". The Register, 27 Sep 2018.[6]
- Liwei Yang et al., "Path Planning Technique for Mobile Robots: A Review". Machines, Vol. 11, 2023, pp. 980-1026.[7]
- Lotfi A. Zadeh, "Fuzzy logic, neural networks, and soft computing". In: Communications of the ACM, Volume 37, Issue 3, March 1994, pp. 77-84; "Artificial neural networks: an overview", in: George J. Klir & Bo Yuan, Fuzzy sets and fuzzy logic. Theory and applications. Upper Saddle River (NJ.): Prentice Hall, 1995, pp. 467-475.
- A useful technical overview is provided in: Enrique Ruspini et al. Handbook of fuzzy computation. Bristol & Philadelphia: Institute of Physics Publishing, 1998.
- Radim Behlohlavek & George J. Klir (eds.), Concepts and fuzzy logic. Cambridge, Mass.: MIT Press, 2011.
- Edy Portmann, Fuzzy humanist. Wiesbaden: Springer, 2019; Mahdi Eftekhari et al., How fuzzy concepts contribute to machine learning. Cham (Switzerland): Springer, 2022.
- Rudolf Seising et al., On fuzziness: homage to Lotfi A. Zadeh, Vol. 2. Heidelberg: Springer, 2013, p. 656; Ellen Christiaanse, "1.5 million years of information systems; from hunters-gatherers to the domestication of the networked computer". In: David Avison et al., The past and future of information systems: 1976-2006 and beyond. New York: IFIP/Springer, 2006, pp. 165-176.
- Kit Fine, Vagueness: a global approach. New York: Oxford University Press, 2020, chapter 1.
- Steve Coutinho, Zhuangzi and Early Chinese Philosophy. Vagueness, Transformation and Paradox. Abingdon: Routledge, 2016, p. 17.
- Rosanna Keefe & Peter Smith, Vagueness: a reader. Cambridge, Mass.: MIT Press, 1996; Petr Hájek and Vilém Novák, "The sorites paradox and fuzzy logic". International Journal of General Systems, Vol. 32, No. 4, 2003, pp. 373—383.
- "High Altitude Cooking" webpage of the USDA Food Safety and Inspection Service.[8]
- Massimo Pigliucci & Maarten Boudry (eds.), Philosophy of Pseudoscience: Reconsidering the Demarcation Problem. University of Chicago Press, 2013, p. 95.
- Nicholas Rescher, Many-Valued Logic. New York: McGraw-Hill, 1969.
- "The paradox of the heap", in: John L. Bell, Oppositions and Paradoxes: Philosophical Perplexities in Science and Mathematics. Peterborough (Ontario): Broadview Press, 2016, pp. 158-160.[9]
- Julia Andrina Greig,The Vagueness of Dying in Epicurean Thought: A Stoic Remedy? Masters thesis, Graduate School of Arts and Sciences, Brandeis University, May 2021.
- See for example Plato’s version of the puzzle of temporal boundaries (in: Parmenides, 156c–e): When an object begins to move, or a moving object comes to rest, does the transitional moment belong to the motion interval, or to the rest interval? (as noted in "Boundary", article in Stanford Encyclopedia of Philosophy online, 2023.[10])
- Marcus Tullius Cicero, Academica, Book 2. [written 45 BC] in: H. Rackham (transl.), Cicero: De Natura Deorum and Academica (Cambridge, Massachusetts: Harvard University Press, 1933 [11];Lisa Cordes, "Who speaks? – Ambiguity and Vagueness in the Design of Cicero’s Dialogue Speakers". In: Martin Vöhler et al. (eds.), Strategies of Ambiguity in Ancient Literature. Berlin/Boston: De Gruyter, 2021, pp. 297-314.; Robert L. Cioffi, "Fuzzy Math: The Place of Numerical Evidence in Cicero In Verrem 3.116". Mnemosyne, Vol. 64 No. 4, 2011, pp. 645–652.
- Angelica Nuzzo, "Vagueness and Meaning Variance in Hegel's Logic". In: Angelica Nuzzo, Hegel and the analytical tradition. New York: Continuum International Publishing Group, 2010, pp. 61-82.
- Robert L. Carneiro, "The transition from quantity to quality; a neglected causal mechanism in accounting for social evolution". Proceedings of the National Academy of Sciences of the United States of America (PNAS), Vol. 97 No. 23, 7 November 2000, pp. 12926-12931.[12]
- Eric Steinhart, "Nietzsche on identity". Revista di Estetica, Vol. 28, No. 1, 2005, pp. 241-256;[13] Steven D. Hales, "Nietzsche on Logic". Philosophy and Phenomenological Research. Vol. 56, No. 4, December 1996, pp. 819-835; Wilhelm Magnus, "The Significance of Mathematics; The Mathematicians Share in the General Human Condition". The American Mathematical Monthly, Vol. 104, No. 3 March 1997, pp. 261-269, at p. 263.
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- Ludwig Wittgenstein, Philosophical Investigations. Oxford University Press, 1953, Part 1, sections 65-88.
- Jan Łukasiewicz, "On three-valued logic". In: Jan Łukasiewicz, Selected Works. Amsterdam: North Holland Publishing Company, 1970, pp. 87-88.
- Emil Leon Post, "Introduction to a general theory of elementary propositions". American Journal of Mathematics, Vol. 43, No. 3, July 1921, p. 163-185.
- Alfred Tarski, Logic, semantics, metamathematics. Oxford: Oxford University Press, 1956.
- James F. Peters and Sankar K. Pal, "Cantor, Fuzzy, Near, and Rough Sets in Image Analysis". In: Sankar K. Pal and James F. Peters (eds.), Rough Fuzzy Image Analysis Foundations and Methodologies. Routledge, 2017, chapter 1.[14]
- Valentine Bazhanov, "The fate of one forgotten idea: N. A. Vasiliev and his imaginary logic." Studies in Soviet Thought, Vol.39 No. 3, 1990, pp.333-341.[15] Archived 2006-07-19 at the Wayback Machine
- Tim Lethen, "Gödel on many-valued logic". The review of Symbolic Logic, Vol. 16, issue 3, September 2023, pp. 655-671.
- Susan Haack notes that Stanisław Jaśkowski provided axiomatizations of many-valued logics in: Jaśkowski, "On the rules of supposition in formal logic". Studia Logica No. 1, 1934.[16] See Susan Haack, Philosophy of Logics. Cambridge University Press, 1978, p. 205
- W. V. Quine, "Speaking of Objects". Proceedings and Addresses of the American Philosophical Association, Vol. 31, 1957/1958, pp. 5-22, at p. 20.
- Petr Hájek, Metamathematics of fuzzy logic. Dordrecht: Springer, 1998.
- Joseph Goguen, "The logic of inexact concepts". Synthese, Vol. 19, No. 3/4), 1969, pp. 325–373.
- Radim Belohlavek, "Pavelka-style fuzzy logic in retrospect and prospect". Fuzzy Sets and Systems, Vol. 281, 15 December 2015, pp. 61-72.[17]
- Radim Belohlavek, "Systems, uncertainty, and information: A legacy of George J. Klir." International Journal of General Systems, Volume 46, Issue 8, 2017, pp. 792-823.
- Didier Dubois and Henri Prade, Fuzzy sets and systems. Theory and applications. New York: Academic Press, New York, 1980.
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- Abraham Kaplan and Hermann F. Schott, "A calculus for empirical classes", Methodos, Vol. 3, 1951, pp. 165–188.
- Timothy Williamson, Vagueness. London: Routledge, 1996, p. 120.
- J. Barkley Rosser Sr. and Atwell R. Turquette, Many-valued logics. Amsterdam: North-Holland Publishing Company, 1952, p. 109.
- Aleksandr A. Zinov'ev, David Dinsmore Comey and Guido Küng, Philosophical problems of many-valued logic. Dordrecht: D. Reidel, 1963.
- William P. Alston, Philosophy of Language. Englewood Cliffs, N.J.: Prentice Hall, 1964, p. 87; William P. Alston, "Vagueness," in Paul Edwards (ed.), Encyclopedia of Philosophy, vol. 8. New York: Macmillan, first edition 1967, pp. 218–221; William P. Alston, A Realist Conception of Truth. Ithaca: Cornell University Press, 1996 p. 62.
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- Robert John Ackermann, An introduction to many-valued logics. London, Routledge & Kegan Paul, 1967; Nicholas Rescher, Many-Valued Logic. New York: McGraw-Hill, 1969.
- Robert G. Wolf, "A survey of many-valued logic (1966–1974)", in: J. Michael Dunn and George Epstein (eds.), Modern Uses of Multiple-Valued Logic. Dordrecht: D. Reidel, 1977, 167–323; Joseph L. F. De Kerf, "A bibliography on fuzzy sets". In: Journal of Computational and Applied mathematics, vol. 1, no. 3, 1975, pp. 206-212.
- Susan Haack, Deviant logic, fuzzy logic: beyond the formalism. Chicago: University of Chicago Press, 1996.
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- Radim Bělohlávek, Joseph W. Dauben & George J. Klir, Fuzzy Logic and Mathematics: A Historical Perspective. Oxford University Press, 2017.
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- Radim Bělohlávek, Joseph W. Dauben & George J. Klir, Fuzzy Logic and Mathematics: A Historical Perspective. Oxford University Press, 2017. Lotfi A Zadeh with George J. Klir and Bo Yua, Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems: Selected Papers. Singapore and River Edge (N.J.): World Scientific Publishing Company, 1996. This last title includes a bibliography of Zadeh's writings up to 1996.
- Rudolf Seising, "Cybernetics, system(s) theory, information theory and Fuzzy Sets and Systems in the 1950s and 1960s". Information Sciences No. 180, 2010, pp. 4459-4476.
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- Radim Bělohlávek, Joseph W. Dauben & George J. Klir, Fuzzy Logic and Mathematics: A Historical Perspective. Oxford University Press, 2017.
- Radim Bělohlávek, "What is a fuzzy concept lattice? II", in: Sergei O. Kuznetsov et al. (eds.), Rough sets, fuzzy sets, data mining and granular computing. Berlin: Springer Verlag, 2011, pp. 19–20.[25]
- The vast majority of scientific or scholarly users of the idea of fuzzy concepts refer to scaled (graded) characteristics, and not to the variations in the likelihoods of their applicability. A probabilistic interpretation of concepts is discussed in Edward E. Smith & Douglas L. Medin, Categories and concepts. Cambridge: Harvard University Press, 1981.
- Nikolaos Galatos, Peter Jipsen, Tomasz Kowalski & Hiroakira Ono, Residuated lattices: an algebraic glimpse at substructural logics. Elsevier Science, 2007.
- Joseph A. Goguen, “L-fuzzy Sets.” Journal of Mathematical Analysis and Applications, Vol. 18, 1967, pp. 145–174; Radim Belohlavek, "Goguen's contributions to fuzzy logic in retrospect". International Journal of General Systems, Volume 48, issue 8, 2019, pp. 811-824 at p. 817.[26]
- Petr Hájek, Metamathematics of fuzzy logic. Dordrecht: Springer, 1998, p. 2.
- Lotfi A. Zadeh, "The Concept of a Linguistic Variable and Its Application to Approximate Reasoning–1", Information Sciences, Vol. 8, pp. 199–249, 1975. Jerry M. Mendel and Robert I. Bob John, "Type-2 Fuzzy Sets Made Simple." IEEE transactions on fuzzy systems, Vol. 10, No. 2, April 2002, pp. 117-127. Jerry M. Mendel, "Advances in type-2 fuzzy sets and systems". In: Information Sciences 177, 2007, pp. 84–110.[27]
- Timothy Williamson, Vagueness. New York, London: Routledge, 1994.
- Delia Graff Fara, "Shifting Sands: An Interest Relative Theory of Vagueness". Philosophical Topics, Vol. 28 No. 1, 2000, pp. 45-81.
- Deleuze and Guattari, A Thousand Plateaus (1988, 551).
- Deleuze and Guattari, What Is Philosophy? (New York: Columbia University Press, 1994, p. 141).
- Roy T. Cook, A dictionary of philosophical logic. Edinburgh University Press, 2009, p. 84.
- Nicholas Rescher, Many-valued logic. New York: McGraw-Hill, 1969; Alasdair Urquhart, "Basic Many-Valued Logic". In: D. M. Gabbay, F. Guenthner (eds.), Handbook of Philosophical Logic [HALO], Vol. 2., Heidelberg: Springer, 2001, pp 249–295.
- Susan Haack, Philosophy of Logics. Cambridge University Press, 1978, p. xii.
- Susan Haack, Philosophy of Logics. Cambridge University Press, 1978, p. 165.
- Petr Hájek, "Ten questions and one problem on fuzzy logic". Annals of Pure and Applied Logic, Vol. 96, Issues 1-3, March 1999, 157–165, at p. 162.
- Kazuo Tanaka, An Introduction to Fuzzy Logic for Practical Applications. Springer, 1996; Constantin Zopounidis, Panos M. Pardalos & George Baourakis, Fuzzy Sets in Management, Economics and Marketing. Singapore; World Scientific Publishing Co. 2001. Humberto Bustince et al. (eds.), Fuzzy Sets and Their Extensions: Representation, Aggregation and Models. Intelligent Systems from Decision Making to Data Mining, Web Intelligence and Computer Vision. Berlin: Springer, 2008.
- Samsung support information page [28].
- Lotfi Zadeh, "Coping with the imprecision of the real world" (interview). Communications of the ACM, Vol.27, No. 4, 1 April 1984, pp.304-311.
- Stosberg, Mark (16 December 1996). "The Role of Fuzziness in Artifical [sic] Intelligence". Minds and Machines. Archived from the original on 20 May 2013. Retrieved 19 April 2013.
- Irem Dikmen, M. Talat Birgonal and Sedat Han, "Using fuzzy risk assessment to rate cost overrun risk in international construction projects." International Journal of Project Management, Vol. 25 No. 5, July 2007, pp. 494–505.
- Fa-Liang Gao, "A new way of predicting cement strength — Fuzzy logic". Cement and Concrete Research, Volume 27, Issue 6, June 1997, Pages 883–888.
- ""2017 Golden Goose Awardee: Fuzzy Logic, Clear Impact"". Archived from the original on 2019-12-13. Retrieved 2018-03-12.
- Michio Sugeno (ed.), Industrial applications of fuzzy control. Amsterdam: North Holland, 1992; Andrew Pollack, "Technology; Fuzzy Logic For Computers". New York Times, 11 October 1984; Andrew Pollack, "Fuzzy Computer Theory: How to Mimic the Mind?" New York Times, 2 April 1989.
- The IFSA URL is: https://fuzzysystems.org/
- Cf. IEEE CIS website 2018 [29] Archived 2018-04-02 at the Wayback Machine.
- The URL is https://www.fsdmconf.org/
- See the APNNS website [30].
- INFUS URL
- See the SOFT website [31].
- KIIS website [32].
- Yingming Liu, Guoqing Chen and Mingshen Ying (eds.), Fuzzy logic, soft computing and computational intelligence. Eleventh International Fuzzy Systems Association World Congress July 28–31, 2005, Beijing, China. Volume III. Beijing: Tsinghua University Press/Springer Verlag, 2005, p. viii. and IFSA website data
- The TFSA publishes the International Journal of Fuzzy Systems
- The NAFIPS website URL is https://sites.ualberta.ca/~reformat/nafips/index.html
- The EUSFLAT URL is: http://www.eusflat.org/. Mathfuzzlog url is: https://www.eusflat.org/society-wg-mathfuzzlog.html
- The NSAIS URL is https://nsais.org/ .
- IFSS website [33].
- "Iran ranks 3rd in producing fuzzy systems related science: Official". The Iran Project, 9 March 2017.[34]
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- SLAAI website [37].
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- Indonesian society for soft computing repository site
- Vietnam Fuzzy Systems Society
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- Cade Metz, "Lotfi Zadeh, Father of Mathematical 'Fuzzy Logic,' Dies at 96." New York Times, 11 September 2017. For an autobiographical sketch, see: Lotfi Zadeh, "My life and work - a retrospective view". Applied Computational and Mathematics, Vol. 10 No. 1 (special issue on fuzzy set theory and applications), 2011, pp. 4-9.
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- Daniel Kreiss, Prototype Politics: Technology-Intensive Campaigning and the Data of Democracy. Oxford University Press, 2016.
- E.g. Mikael Collan, Mario Fedrizzi, Janusz Kacprzyk, Fuzzy Technology: Present Applications and Future Challenges. Heidelberg: Springer, 2016, p. 65f.; Daniel J. Lewis and Trevor P. Martin, "Managing Vagueness with Fuzzy in Hierarchical Big Data". Procedia Computer Science, Volume 53, 2015, pages 19–28.[41]
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- For example, Kyle C. Longest and Stephen Vaisey, "Fuzzy: A program for Performing Qualitative Comparative Analyses (QCA) in Stata." Stata Journal, Vol. 8 No. 1, 2008: pp. 79–104. Gregory Viot, "Fuzzy logic in C". Dr Dobb's journal, 1 February 1993.[50]
- Chris Preimesberger, "Big-Data Analytics Plays Big Role in 2016 Election Campaigns". eWeek, 24 September 2016.
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- Ryan Lizza, "Kellyanne Conway's political machinations", The New Yorker, 17 October 2016.[59](see also alternative facts)
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- Jonathan Shieber, "Facebook hired a forensics firm to investigate Cambridge Analytica as stock falls 7%." TC Techcrunch.com, 19 March 2018.[62]
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- Sara Salinas, "Facebook hires firm to conduct a 'comprehensive audit' of Cambridge Analytica". CNBC news, 19 March 2018 and squawkbox panel video [65]
- Christopher Carbone, "Facebook might have 29,000 data points on you, but Mark Zuckerberg doesn't really know." Fox News, 11 April 2018.[66] [67] Julia Angwin, Surya Mattu and Terry Parris Jr., "Facebook Doesn't Tell Users Everything It Really Knows About Them." ProPublica, 27 December 2016.[68]
- Zuckerberg's Senate hearings highlights video, 10 April 2018
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- Richard Waters, "Is Facebook a victim of rapid growth or an abuser of user data?" Financial Times, 20 December 2018.
- Matthew Rosenberg, Nicholas Confessore and Carole Cadwalladr, "How Trump Consultants Exploited the Facebook Data of Millions". New York Times, 17 March 2018.
- Sarah Frier and Todd Shields, "Zuckerberg Says Facebook Collects Internet Data on Non-Users", Bloomberg, 11 April 2018.
- Gabriel J.X. Dance, Michael LaForgia and Nicholas Confessore, "As Facebook Raised a Privacy Wall, It Carved an Opening for Tech Giants". New York Times, 18 December 2018.[70]
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- See the Meta "Legal Action" website.
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- Y.S. [=Yossi Sariel], The human-machine team: how to create synergy between human & artificial intelligence that will revolutionize our world. Tel Aviv: eBookPro Publishing, 2021.
- Yasmeen Serhan, "How Israel Uses AI in Gaza—And What It Might Mean for the Future of Warfare". In: Time Magazine, 18 December 2024.[74] See also: AI-assisted targeting in the Gaza Strip.
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- Michael O'Hagan, "From military to medical and commercial applications of neural networks and fuzzy logic: a modern "swords-into-plowshares" play." Proceedings of IEEE WESCON '93 conference, San Francisco, 28-30 Sept. 1993. Republished in IEEE Xplore, 6 August 2002.
- Radim Bělohlávek, George J. Klir, Harold W. Lewis III, Eileen C. Way, "Concepts and fuzzy sets: Misunderstandings, misconceptions, and oversights". International Journal of Approximate Reasoning, Vol. 51, July 2009), pp. 23–34.[76][permanent dead link ] Angel Garrido & Piedad Yuste, "controversies about the introduction of non-classical logics". Brain, Vol. 5, No. 1-4, 2014.[77] Bob Pease, "What's All This Fuzzy Logic Stuff, Anyhow?" Electronicdesign.com, May 13, 1993 - November 2020 (five parts).[78]
- Lotfi A. Zadeh, "Is there a need for fuzzy logic?", Information Sciences, No. 178, 2008, p. 2753.
- For the debate between Zadeh and Kálmán, see: Lotfi A. Zadeh, "The birth and evolution of fuzzy logic". International Journal of General Systems, Vol. 17, No. 2-3, 1990, pp. 95-105. See also: Yücel Yüksel, "On Zadeh's 'The Birth and Evolution of Fuzzy Logic'". In: Eyke Hüllermeier, Rudolf Kruse & Frank Hoffmann (eds.), Information Processing and Management of Uncertainty in Knowledge-Based Systems. Applications. Proceedings of the 13th International Conference, IPMU 2010, Dortmund, Germany, June 28–July 2, 2010 (Communications in Computer and Information Science, vol 81), Part II. Berlin: Springer, 2010, pp 350-355.
- Dong Yu-Zhen, Chen Huan and Wu He-Qinc [Hebei University of Engineering], "What is Wrong with Fuzzy Logic". Procedia Engineering, Vol. 15, August 2011, pp. 1727–1731, at p. 1731. For most of the extensions of fuzzy set theory, see: Wasim Akram Mandal & Sahidul Islam, Fundamentals on Extensions of Fuzzy Sets. Independently published, 2024.
- Daniel McNeill & Paul Freiberger, Fuzzy Logic: The Revolutionary Computer Technology that Is Changing Our World. New York: Simon & Schuster, 1994, p. 49.
- Daniel McNeill & Paul Freiberger, Fuzzy Logic: The Revolutionary Computer Technology that Is Changing Our World. New York: Simon & Schuster, 1994, p. 50. The Honda Foundation judged that Zadeh had taken an "active role in making the future of information society a more humane civilization", with a broad range of contributions in applied logic.
- P. Geach and M. Black (eds.), Translations from the Philosophical Writings of Gottlob Frege, 3rd edition. Blackwell, 1980, p. 159.
- Ludwig Wittgenstein, Philosophical investigations, Part 1, paragraph 71 (revised 4th Anscombe edition, 2009, p. 38).
- Jerry A. Fodor, Concepts: Where Cognitive Science Went Wrong. New York: Oxford University Press, 1998; Susan Carey, The origin of concepts. New York: Oxford University Press, 2009, chapter 13; Henry Cohen & Claire Lefebvre, Handbook of categorization in cognitive science. Amsterdam: Elsevier, 2005; Wolfgang G. Stock, "Concepts and Semantic Relations in Information Science". In: Journal of the American Society for Information Science and Technology Vol. 61 No. 10, October 2010, pp. 1951–1969 [79]; Eric Margolis & Stephen Laurence, "Concepts". In: Stanford Encyclopedia of Philosophy, 2011.[80]
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- Go to Standards Library for info.
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- Lotfi A. Zadeh, "The birth and evolution of fuzzy logic". International Journal of General Systems, Vol. 17, No. 2-3, 1990, pp. 95-105, at p. 98.
- Daniel McNeill & Paul Freiberger, Fuzzy Logic: The Revolutionary Computer Technology that Is Changing Our World. New York: Simon & Schuster, 1994, pp. 47-48.
- A. Dumitras, & G. Moschytz, "Understanding Fuzzy Logic: An Interview with Lotfi Zadeh". IEEE signal processing magazine, May 2007, pp. 102–105, at p. 103.
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- Susan Haack, Philosophy of Logics. Cambridge University Press, 1978, p. 213.
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- Susan L. Epstein, "Memory and concepts in reactive learning". Proceedings of the Canadian Workshop on Machine Learning 1992[87].
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- A standard definition of a forest and a tree is provided by the Food and Agriculture Organization of the United Nations. FAO provides a system of standard definitions for forestry objects.
- What statisticians then often try to do, is to create a model which can predict the magnitude of the difference between the true (accurate and exact) number and the computed number obtained, in this case the true number of trees. Such a model however still relies on imperfect or fallible definitions. Even if fuzzy values are used instead, it is likely that a definite and exact number can never be reached. At most one can say that the number is correct, if the definitions are accepted.
- Mark Benisz et al., "Abuses and misuses of intelligence tests: Facts and misconceptions". In: A. B. Kaufman & J. C. Kaufman (eds.), Pseudoscience: The conspiracy against science. Cambridge: MIT Press, 2018, pp. 351–373; Briefing "The trouble with GDP". The Economist (London), 30 April 2016.
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- Scott Soames, "The Value of Vagueness." Chapter 2 in: Andrei Marmor & Scott Soames, Philosophical Foundations of Language in the Law. Oxford: Oxford University Press, 2013, pp. 26-43, at p. 26.
- Scott Soames, "The Value of Vagueness." Chapter 2 in: Andrei Marmor & Scott Soames, Philosophical Foundations of Language in the Law. Oxford: Oxford University Press, 2013, pp. 26-43, at p. 33.
- Scott Soames, "The Value of Vagueness." Chapter 2 in: Andrei Marmor & Scott Soames, Philosophical Foundations of Language in the Law. Oxford: Oxford University Press, 2013, pp. 26-43, at p. 34.
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- "Unlike the platonic epistemology required by the classic Frege-Russell account... the epistemology of naturalized propositions sees acquaintance with, and knowledge of, propositions as rooted in acquaintance with, and knowledge of, acts and events that make up one's cognitive life" - Scott Soames, What is meaning?. Princeton: Princeton University Press, 2010, p. 106.
- William Ashley, Marxism and moral concepts. New York: Monthly Review Press, 1964, pp. 4-5. Similarly, Paul Lafargue had written in his essay "The Origin of Abstract Ideas" (1900) that "The brain has the property of thinking as the stomach has that of digesting. It cannot think but by the aid of ideas, which it fabricates with the materials furnished it by the natural environment and the social or artificial environment in which man evolves." [97]
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- Kofi Kissi Dompere, Fuzziness and approximate reasoning; epistemics on uncertainty, expectation and risk in rational behaviour. Berlin: Springer, 2009.
- Masao Mukaidono, Fuzzy logic for beginners. Singapore: World Scientific Publishing, 2001.
- Karen Armstrong, The Case for God. New York: Anchor, 2010.
- David Bohm, Wholeness and the implicate order. London: Routledge & Kegan Paul ARK paperback edition, 1983, p. 86f.
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- Paul Grice, Studies in the Way of Words. Harvard University Press, Cambridge, Mass., 1989. A critique of Grice is provided by Wayne A. Davis, Implicature: intention, convention, and principle in the failure of Gricean theory. Cambridge: Cambridge University Press, 1998. An example of a specific application of Gricean theory is: Penelope Brown & Stephen C. Levinson, Politeness: some universals in language use. Cambridge: Cambridge University Press, 1987.
- Patrick Hughes & George Brecht, Vicious Circles and Infinity. An anthology of Paradoxes. Penguin Books, 1978. Nicholas Rescher, Epistemological Studies. Frankfurt: Ontos Verlag, 2009, chapter 3. John L. Bell, Oppositions and Paradoxes: Philosophical Perplexities in Science and Mathematics. Peterborough (Ontario): Broadview Press, 2016.
- Andrea Cantini, "Paradoxes and Contemporary Logic", Stanford Encyclopedia of Philosophy online 30 April 2012.
- See further Radim Bělohlávek & George J. Klir (eds.) Concepts and Fuzzy Logic. MIT Press, 2011. John R. Searle, "Minds, brains and programs". The behavioral and brain sciences, Vol. 3, No. 3, 1980, pp. 417–457. Robert Epstein, "The empty brain", Aeon, 18 May 2016.[104]
- For non-binary computing, see Unconventional computing.
- Karl Aschenbrenner, The concept of coherence in art. Dordrecht: D. Reidel Publishing Company, 1985.
- Harry Collins, Tacit & explicit knowledge. Chicago: University of Chicago Press, 2013.
- Amos Tversky and Daniel Kahneman, "The framing of decisions and psychology of choice". Science, Vol. 211, No. 4481, January 1981, pp. 453-458.
- C. J. Brainerd and V. F. Reyna, "Gist is the grist: fuzzy-trace theory and the new intuitionism". Developmental Review, Vol. 10, No. 1, March 1990, pp. 3-47, at p. 39.
- Arthur S. Reber, Implicit learning and tacit knowledge. An essay on the cognitive unconscious. Oxford: Oxford University Press, 1993, pp. 137-138.
- Ronald A. Havens (ed.), The wisdom of Milton H. Erickson, Volume II: human behavior & psychotherapy. New York: Irvington Publishers, 1992, chapter 3.
- A. Cornelius Benjamin, "Science and vagueness". In: Philosophy of science, Vol. 6 No. 4, 1939, pp. 422-431.
- Kenneth Knoblauch, "Color Vision", in: Steven's handbook of experimental psychology, Vol 1: sensation and perception (3rd ed.). New York: John Wiley & Sons, 2002, p. 48.
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- Jean Piaget & Bärbel Inhelder, The Growth of Logical Thinking from Childhood to Adolescence. New York: Basic Books, 1958; Susan Carey, Conceptual Change in Childhood. Cambridge (Mass.): MIT Press, 1985; Philip J. Kelman & Martha E. Arterberry, The cradle of knowledge: development of perception in infancy. Cambridge, Mass.: The MIT Press, 2000.
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- Rudolf Seising, "On the absence of strict boundaries — Vagueness, haziness, and fuzziness in philosophy, science, and medicine". Applied Soft Computing, Vol 8, 2008, pp. 1232–1242, at p. 1235.
- Kazem Sadegh-Zadeh "The Fuzzy Revolution: Goodbye to the Aristotelian Weltanschauung". In: Artificial Intelligence in Medicine, 21, 2001, pp. 18–19.[107]
- Stephen Priest, Theories of the mind. London: Penguin Books, 1991, p. 183.
- Michael Hammond, Jane Howarth and Russell Keat, Understanding Phenomenology. Oxford: Blackwell, 1991.
- Cornelia Griebel, "Fuzzy concepts in translators' minds". In: Valérie Dullion, Between specialised texts and institutional contexts – competence and choice in legal translation. Amsterdam: John Benjamins Publishing Company, 2017. (Special issue of Translation and Translanguaging in Multilingual Contexts, Vol. 3, No. 1, 2017).
- Ronald A. Havens (ed.), The wisdom of Milton H. Erickson, Volume I: hypnosis and hypnotherapy. New York: Irvington Publishers, 1992, p. 106. Joseph O'Connor & John Seymour (ed.), Introducing neuro-linguistic programming. London: Thorsons, 1995, p. 116f.
- Francese Trillas, "Fuzzy logic and modern economics." In: Rudolf Seising, Enric Trillas & Janusz Kacprzyk (eds.), Towards the future of fuzzy logic. Basel: Springer International Publishing, 2015, p. 56.
- Surya Monro, Bisexuality. Houndmills, Basingstoke: Palgrave Macmillan, 2015, p. 49.
- Bart Kosko, "Yes, Candidates, There Is a Fuzzy Math". New York Times, 7 November 2000.
- Michelle Nichols, "Israel defends Gaza hospital raid, UN rights chief says its explanation is vague". Reuters, 3 January 2025.
- Russel Gordon & David Bendien, "Standard classifications". New Zealand Statistics Review, September 1993, p. 20.
- Paul C. Bauer et al., "Vague concepts in survey questions. A general problem illustrated with the left-right scale." SSRN Electronic Journal, April 2014.[108]
- C.N. de Groot, "Sociology of religion looks at psychotherapy." Recherches sociologiques (Louvain-la-Neuve, Belgium), Vol. 29, No. 2, 1998, pp. 3–17 at p. 4.[109] Archived 2013-05-23 at the Wayback Machine
- Mark Manson, "The rise and fall of Ken Wilber", markmanson.net, 4 June 2012.[110]
- That is, in applying rules, the rules are not consistently followed in all cases, so that in judging exactly the same situations different legal outcomes result.
- For more information see e.g. Ralf Posche, "Ambiguity And Vagueness In Legal Interpretation", in: Lawrence M. Solan & Peter M. Tiersma (eds.), The Oxford Handbook of Language and Law. Oxford University Press, 2012, pp. 128-144.
- David Henry, "Fuzzy Numbers', Bloomberg Businessweek, 3 October 2004.
- See e.g. Brian J.Zinnbauer, "Religion and Spirituality: Unfuzzying the Fuzzy". In: Journal for the scientific study of religion, Vol. 36, No. 4, December 1997, pp. 549-564.[112]
- K Sadegh-Zadeh, "Fuzzy health, illness, and disease". In: Journal of medical philosophy, Vol. 25, No. 5, October 2000, pp. 605-638.[113]
- In 2025, the Oxford English Dictionary lists 33 meanings for the word social, of which six are labelled obsolete.
- Lisa Mathis Butler Harrington, "Sustainability Theory and Conceptual Considerations: A Review of Key Ideas for Sustainability, and the Rural Context." Papers in Applied Geography, Vol. 2, Issue 4, November 2016; M. J. Ducey and B. C. Larson, "A fuzzy set approach to the problem of sustainability". Forest Ecology and Management, 111, 1999, pp. 29-40.
- Stuart McCready (ed.), The discovery of happiness. Naperville (illinois): Sourcebooks Inc., 2001; Desmond Morris, The nature of happiness. London: Little Books Ltd, 2004.
- Mihaela Kelemen, "Ambiguity as organisational practice: an American pragmatist perspective", in: Lindsay Hamilton et al., Contemporary issues in management, 2nd edition. Cheltenham, UK: Edward Elgar, 2019, pp. 91-103; Philip Ball, "The surprising benefits of ambiguous language". London: BBC.com, 13 March 2014.[114]
- Karl Popper, Unended quest: an intellectual autobiography. London: Routledge, 2002, p. 22.
- Lotfi A. Zadeh, "What is fuzzy logic?". IFSA Newsletter (International Fuzzy Systems Association), Vol. 10, No. 1, March 2013. Takeshi Yamakawa "Stabilization of an Inverted Pendulum by a High-speed Fuzzy Logic Controller Hardware System". Fuzzy Sets and Systems, Vol.32, pp. 161–180, 1989.
- See e.g. C. A. Drossos, "Foundations of fuzzy sets: A nonstandard approach". Fuzzy Sets and Systems, Volume 37, Issue 3, 28 September 1990, pp. 287-307.
- Nick Cercone & Gordon McCalla (eds.), The knowledge frontier: essays in the representation of knowledge. New York: Springer, 1987.
- Guy W. Mineau et al. (eds.), Conceptual graphs for knowledge representation. Berlin: Springer, 1993. Tru Hoang Cao, Conceptual graphs and fuzzy logic. Berlin: Springer, 2010.
- V. Rahmati et al. (eds.), A Novel Low Complexity Fast Response Time Fuzzy PID Controller for Antenna Adjusting Using Two Direct Current Motors.
- cf. Timothy Williamson, Vagueness. London: Routledge, 1996, p. 258.
External links
- James F. Brule, Fuzzy systems tutorial
- "Fuzzy Logic", Stanford Encyclopedia of Philosophy
- "Vagueness", Stanford Encyclopedia of Philosophy
- Calvin College Engineering Department, Getting Started with Fuzzy Logic Archived 2018-02-21 at the Wayback Machine
- 2009 Benjamin Franklin Medal Winner: Lotfi A. Zadeh
- EUSFLAT, In memory of Lotfi A. Zadeh
- RAFSoft, Remembering Lotfi Zadeh
- Lin Shang, Lecture on fuzzy and rough sets, Nanjing University
- Rudolf Kruse and Christian Moewes on fuzzy set theory
- Fuzzy Logic for "Just Plain Folks" by Thomas Sowell
- Podcast (with text) Fuzzy logic: The origin and future of non-Aristotelian thinking (20 March 2023)
This article may need to be rewritten to comply with Wikipedia s quality standards You can help The talk page may contain suggestions September 2022 A fuzzy concept is an idea of which the boundaries of application can vary considerably according to context or conditions instead of being fixed once and for all This means the idea is somewhat vague or imprecise Yet it is not unclear or meaningless It has a definite meaning which can be made more exact only through further elaboration and specification including a closer definition of the context in which the concept is used The study of the characteristics of fuzzy concepts and fuzzy language is called fuzzy semantics The inverse of a fuzzy concept is a crisp concept i e a precise concept For engineers Fuzziness is imprecision or vagueness of definition For scientists a fuzzy concept is an idea which is to an extent applicable in a situation It means that the concept can have gradations of significance or unsharp variable boundaries of application a fuzzy statement is a statement which is true to some extent and that extent can often be represented by a scaled value a score For mathematicians a fuzzy concept is usually a fuzzy set or a combination of such sets see fuzzy mathematics and fuzzy set theory In cognitive linguistics the things that belong to a fuzzy category exhibit gradations of family resemblance and the borders of the category are not clearly defined In a more general popular sense contrasting with its technical meanings a fuzzy concept refers to an imprecise idea which is somewhat vague for any kind of reason or which is approximately true Fuzzy concepts are often used to navigate imprecision in the real world when exact information is not available In the past the very idea of reasoning with fuzzy concepts faced considerable resistance from academic elites They did not want to endorse the use of imprecise concepts in research or argumentation and regarded fuzzy logic with suspicion or even hostility Yet although people might not be aware of it the use of fuzzy concepts has risen gigantically in all walks of life from the 1970s onward That is mainly due to advances in electronic engineering fuzzy mathematics and digital computer programming The new technology allows very complex inferences about variations on a theme to be anticipated and fixed in a program The Perseverance Mars rover a driverless NASA vehicle used to explore the Jezero crater on the planet Mars features fuzzy logic programming that steers it through rough terrain Similarly to the North the Chinese Mars rover Zhurong used fuzzy logic algorithms to calculate its travel route in Utopia Planitia from sensor data New neuro fuzzy computational methods make it possible for machines to identify measure correct adjust for and respond to fine gradations of significance with great precision It means that practically useful concepts can be coded and applied to all kinds of tasks even if ordinarily these concepts are never precisely defined Nowadays engineers statisticians and programmers often represent fuzzy concepts mathematically using fuzzy logic fuzzy values fuzzy variables and fuzzy sets see also fuzzy set theory Fuzzy logic can play a significant role in artificial intelligence programming for example because it can model human cognitive processes more easily than other methods OriginsProblems of vagueness and fuzziness have probably always existed in human experience In the West ancient texts show that philosophers and scientists were already thinking about those kinds of problems in classical antiquity Kit Fine states that when a philosopher talks of vagueness he has in mind a certain kind of indeterminacy in the relation of something to the world According to the Daoist thought of Laozi and Zhuang Zhou in ancient China vagueness is not regarded with suspicion but is simply an acknowledged characteristic of the world around us a subject for meditation and a source of insight Sorites paradox The ancient Sorites paradox first raised the logical problem of how we could exactly define the threshold at which a change in quantitative gradation turns into a qualitative or categorical difference With some physical processes this threshold is relatively easy to identify For example water turns into steam at 100 C or 212 F the boiling point depends partly on atmospheric pressure which decreases at higher altitudes With many other processes and gradations however the point of change is much more difficult to locate and remains somewhat vague Thus the boundaries between qualitatively different things may be unsharp we know that there are boundaries but we cannot define them exactly According to the modern idea of the continuum fallacy the fact that a statement is to an extent vague does not automatically mean that it has no validity The question then arises of how we could ascertain and define the validity that the fuzzy statement does have Loki s wager The Nordic myth of Loki s wager suggested that concepts that lack precise meanings or lack precise boundaries of application cannot be usefully discussed at all because they evade any clear definition However the 20th century idea of fuzzy concepts proposes that somewhat vague terms can be operated with because we can explicate and define the variability of their application by assigning numbers to gradations of applicability This idea sounds simple enough but it had large implications Precursors In Western civilization the intellectual recognition of fuzzy concepts has been traced back to a diversity of famous and less well known thinkers including among many others Eubulides Epicurus Plato Cicero Georg Wilhelm Friedrich Hegel Karl Marx and Friedrich Engels Friedrich Nietzsche William James Hugh MacColl Charles S Peirce Carl Gustav Hempel Max Black Arto Salomaa Ludwig Wittgenstein Jan Lukasiewicz Emil Leon Post Alfred Tarski Georg Cantor Nicolai A Vasiliev Kurt Godel Stanislaw Jaskowski Willard Van Orman Quine Petr Hajek Joseph Goguen Jan Pavelka George J Klir Didier Dubois and Donald Knuth Across at least two and a half millennia all of them had something to say about graded concepts with unsharp boundaries This suggests at least that the awareness of the existence of concepts with fuzzy characteristics in one form or another has a very long history in human thought Quite a few 20th century logicians mathematicians and philosophers also tried to analyze the characteristics of fuzzy concepts as a recognized species sometimes with the aid of some kind of many valued logic or substructural logic An early attempt in the post WW2 era to create a mathematical theory of sets with gradations of set membership was made by Abraham Kaplan and Hermann F Schott in 1951 They intended to apply the idea to empirical research Kaplan and Schott expressed the degree of membership of empirical classes using real numbers between 0 and 1 and they defined corresponding notions of intersection union complementation and subset However at the time their idea fell on stony ground J Barkley Rosser Sr published a treatise on many valued logics in 1952 anticipating many valued sets Another treatise was published in 1963 by Alexander Zinoviev and others In 1964 the American philosopher William Alston introduced the term degree vagueness to describe vagueness in an idea that results from the absence of a definite cut off point along an implied scale in contrast to combinatory vagueness caused by a term that has a number of logically independent conditions of application The German mathematician Dieter Klaua published a German language paper on fuzzy sets in 1965 but he used a different terminology he referred to many valued sets not fuzzy sets In the late 1960s two popular introductions to many valued logic were published by Robert J Ackermann and Nicholas Rescher Rescher s book includes a bibliography on fuzzy theory up to 1965 which was extended by Robert Wolf and Joseph De Kerf for 1966 1975 Haack provides references to significant works after 1974 In 1980 Didier Dubois and Henri Prade published a detailed annotated bibliography on the field of fuzzy set theory George J Klir and Bo Yuan provided an overview of the subject in Fuzzy sets and fuzzy logic during the mid 1990s Merrie Bergmann provides a more recent 2008 introduction to fuzzy reasoning A standard modern reference work is Fuzzy Logic and Mathematics A Historical Perspective 2017 by Radim Belohlavek Joseph W Dauben and George J Klir Lotfi Zadeh The Iranian born American computer scientist Lotfi A Zadeh 1921 2017 is usually credited with inventing the specific idea of a fuzzy concept in his seminal 1965 paper on fuzzy sets because he presented a mathematical formalization of the phenomenon that was widely accepted by scholars It was also Zadeh who played a decisive role in developing the field of fuzzy logic fuzzy sets and fuzzy systems with a large number of scholarly papers Unlike most philosophical theories of vagueness Zadeh s engineering approach had the advantage that it could be directly applied to computer programming Zadeh s seminal 1965 paper is acknowledged to be one of the most cited scholarly articles in the 20th century In 2014 it was placed 46th in the list of the world s 100 most cited research papers of all time Since the mid 1960s many scholars have contributed to elaborating the theory of reasoning with graded concepts and the research field continues to expand DefinitionThe ordinary scholarly definition of a concept as fuzzy has been in use from the 1970s onward Criteria Radim Belohlavek explains There exists strong evidence established in the 1970s in the psychology of concepts that human concepts have a graded structure in that whether or not a concept applies to a given object is a matter of degree rather than a yes or no question and that people are capable of working with the degrees in a consistent way This finding is intuitively quite appealing because people say this product is more or less good or to a certain degree he is a good athlete implying the graded structure of concepts In his classic paper Zadeh called the concepts with a graded structure fuzzy concepts and argued that these concepts are a rule rather than an exception when it comes to how people communicate knowledge Moreover he argued that to model such concepts mathematically is important for the tasks of control decision making pattern recognition and the like Zadeh proposed the notion of a fuzzy set that gave birth to the field of fuzzy logic Hence a concept is generally regarded as fuzzy in a logical sense if defining characteristics of the concept apply to it to a certain degree or extent or more unusually with a certain magnitude of likelihood or the boundaries of applicability the truth value of a concept can vary in degrees according to different conditions or the fuzzy concept itself straightforwardly consists of a fuzzy set or a combination of such sets The fact that a concept is fuzzy does not prevent its use in logical reasoning it merely affects the type of reasoning which can be applied see fuzzy logic If the concept has gradations of meaningful significance it may be necessary to specify and formalize what those gradations are if they can make an important difference Not all fuzzy concepts have the same logical structure but they can often be formally described or reconstructed using fuzzy logic or other substructural logics The advantage of this approach is that numerical notation enables a potentially infinite number of truth values between complete truth and complete falsehood and thus it enables in theory at least the greatest precision in stating the degree of applicability of a logical rule Fuzziness versus uncertainty One of the first scholars who pointed out the need to distinguish the theory of fuzzy sets from probability theory was Zadeh s pupil Joseph Goguen Petr Hajek writing about the foundations of fuzzy logic likewise sharply distinguished between fuzziness and uncertainty The sentence The patient is young is true to some degree the lower the age of the patient measured e g in years the more the sentence is true Truth of a fuzzy proposition is a matter of degree I recommend to everybody interested in fuzzy logic that they sharply distinguish fuzziness from uncertainty as a degree of belief e g probability Compare the last proposition with the proposition The patient will survive next week This may well be considered as a crisp proposition which is either absolutely true or absolutely false but we do not know which is the case We may have some probability chance degree of belief that the sentence is true but probability is not a degree of truth In metrology the science of measurement it is acknowledged that for any measure we care to make there exists an amount of uncertainty about its accuracy but this degree of uncertainty is conventionally expressed with a magnitude of likelihood and not as a degree of truth In 1975 Lotfi A Zadeh introduced a distinction between Type 1 fuzzy sets without uncertainty and Type 2 fuzzy sets with uncertainty which has been widely accepted Simply put in the former case each fuzzy number is linked to a non fuzzy natural number while in the latter case each fuzzy number is linked to another fuzzy number ApplicationsPhilosophy In philosophical logic and linguistics fuzzy concepts are often regarded as vague or imprecise ideas which in their application or strictly speaking are neither completely true nor completely false Such ideas require further elaboration specification or qualification to understand their applicability the conditions under which they truly make sense The fuzzy area can also refer simply to a residual number of cases which cannot be allocated to a known and identifiable group class or set if strict criteria are used The French thinkers Gilles Deleuze and Felix Guattari referred occasionally to fuzzy sets in connection with their phenomenological concept of multiplicities In A Thousand Plateaus they state that a set is fuzzy if its elements belong to it only by virtue of specific operations of consistency and consolidation which themselves follow a special logic In their book What Is Philosophy which deals with the functions of concepts they suggest that all philosophical concepts could be regarded as vague or fuzzy sets simple aggregates of perceptions and affections which form within the lived as immanent to a subject to a consciousness and which are qualitative or intensive multiplicities like redness or baldness where we cannot decide whether certain elements do or do not belong to the set Sciences In mathematics and statistics a fuzzy variable such as the temperature hot or cold is a value which could lie in a probable range defined by some quantitative limits or parameters and which can be usefully described with imprecise categories such as high medium or low using some kind of scale or conceptual hierarchy Fuzzy logic In mathematics and computer science the gradations of applicable meaning of a fuzzy concept are described in terms of quantitative relationships defined by logical operators Such an approach is sometimes called degree theoretic semantics by logicians and philosophers but the more usual term is fuzzy logic or many valued logic The novelty of fuzzy logic is that it breaks with the traditional principle that formalisation should correct and avoid but not compromise with vagueness The basic idea of fuzzy logic is that a real number is assigned to each statement written in a language within a range from 0 to 1 where 1 means that the statement is completely true and 0 means that the statement is completely false while values less than 1 but greater than 0 represent that the statement is partly true to a given quantifiable extent Susan Haack comments Whereas in classical set theory an object either is or is not a member of a given set in fuzzy set theory membership is a matter of degree the degree of membership of an object in a fuzzy set is represented by some real number between 0 and 1 with 0 denoting no membership and 1 full membership Truth in this mathematical context usually means simply that something is the case or that something is applicable This makes it possible to analyze a distribution of statements for their truth content identify data patterns make inferences and predictions and model how processes operate Petr Hajek claimed that fuzzy logic is not just some applied logic but may bring new light to classical logical problems and therefore might be well classified as a distinct branch of philosophical logic similar to e g modal logics Machinery and analytics Fuzzy logic offers computationally oriented systems of concepts and methods to formalize types of reasoning which are ordinarily approximate only and not exact In principle this allows us to give a definite precise answer to the question To what extent is something the case or To what extent is something applicable Via a series of switches this kind of reasoning can be built into electronic devices That was already happening before fuzzy logic was invented but using fuzzy logic in modelling has become an important aid in design which creates many new technical possibilities Fuzzy reasoning i e reasoning with graded concepts turns out to have many practical uses It is nowadays widely used in The programming of vehicle and transport electronics household appliances video games language filters robotics and driverless vehicles Fuzzy logic washing machines are gaining popularity All kinds of control systems that regulate access traffic movement balance conditions temperature pressure routers etc Electronic equipment used for pattern recognition surveying and monitoring including radars satellites alarm systems and surveillance systems Cybernetics research artificial intelligence virtual intelligence machine learning database design and soft computing research Fuzzy risk scores are used by project managers and portfolio managers to express financial risk assessments Fuzzy logic has been applied to the problem of predicting cement strength It looks like fuzzy logic will eventually be applied in almost every aspect of life even if people are not aware of it and in that sense fuzzy logic is an astonishingly successful invention The scientific and engineering literature on the subject is constantly increasing Community Originally lot of research on fuzzy logic was done by Japanese pioneers inventing new machinery electronic equipment and appliances see also Fuzzy control system The idea became so popular in Japan that the English word entered Japanese language ファジィ概念 Fuzzy theory ファジー理論 is a recognized field in Japanese scientific research Since that time the movement has spread worldwide nearly every country nowadays has its own fuzzy systems association although some are larger and more developed than others In some cases the local body is a branch of an international one In other cases the fuzzy systems program falls under artificial intelligence or soft computing There are also some emerging networks of researchers which do not yet have their own website The following list is only provisional and illustrative many more groups could possibly be added The main international body is the International Fuzzy Systems Association IFSA The Computational Intelligence Society of the Institute of Electrical and Electronics Engineers Inc IEEE has an international membership and deals with fuzzy logic neural networks and evolutionary computing It publishes the journal IEEE Transactions on Fuzzy Systems and holds international conferences At the end of 2024 there were 238 chapters of IEEE CIS across the world The conference on Fuzzy Systems and Data Mining FSDM has its 11th International Conference FSDM2025 in Hanshan Normal University Chaozhou City Guangdong Province China The Asia Pacific Neural Network Society founded in 1993 has board members from 13 countries Australia China Hong Kong India Japan Malaysia New Zealand Singapore South Korea Qatar Taiwan Thailand and Turkey The International Association for fuzzy set management and economy SIGEF 115 is based in Spain and publishes the Fuzzy Economic Review 116 since 1996 Intelligent and Fuzzy Systems INFUS is an international research forum to advance the foundations and applications of intelligent and fuzzy systems computational intelligence soft computing for applied research in general complex engineering and decision support systems The interdisciplinary Japan Society for Fuzzy Theory and Intelligent Informatics SOFT traces its origin back to 1972 and publishes two journals The original Korea Fuzzy System Society founded in 1991 is now known as the Korean Institute of Intelligent Systems KIIS In mainland China there is the Fuzzy Mathematics and Systems Association of China FMSAC based at the School of Mathematics Sichuan University in Chengdu and there exists also an important Taiwan Fuzzy Systems Association The North American Fuzzy Information Processing Society NAFIPS was founded in 1981 There exists also a Hispanic American Fuzzy System Association HAFSA based in Mexico In Europe there is a European Society for Fuzzy Logic and Technology EUSFLAT which includes the Working Group on Mathematical Fuzzy Logic The North European Society of Adaptive and Intelligent Systems NSAIS is based in Finland In 2002 the Iran Fuzzy Systems Society nowadays merged into the Iranian Coalition on Soft Computing was approved as an affiliate of the Statistics Association of Iran and in 2005 registered as a non commercial scientific institute When Lotfi A Zadeh received an honorary doctorate from the University of Teheran on 9 March 2017 a member of Iran s parliament stated that Iran now ranks third in the world with regard to the output of scientific research about fuzzy systems In 2005 Russia s Association for Fuzzy Systems founded in January 1990 became the Russian Association for Fuzzy Systems and Soft Computing RAFSSoftCom Zadeh s seminal paper on fuzzy sets was translated into Russian in 1974 and subsequently Russian fuzzy research began to take off increasingly overcoming official skepticism In 2009 the Brazilian Applied Mathematical Society SBMAC created the Thematic Committee on Fuzzy Systems which inspired the First Brazilian Congress on Fuzzy Systems CBSF I in 2010 CBSF VI was held at Sao Paulo State University in 2021 There also exists a Brazilian Society of Automatics SBA In India the Center for Soft Computing Research at the Indian Statistical Institute Kolkata organizes and publishes research on fuzzy sets rough sets and applications of fuzzy logic The Sri Lanka Association for Artificial Intelligence is a non profit scientific association devoted to understanding the mechanisms underlying thoughts and intelligent behaviour and their emulation in machines Other national scientific bodies include the Hungarian Fuzzy Association HFA the Fuzzy Systems Association of Turkey FSAT the Indonesian Soft Computing Society SC INA and the Vietnamese Fuzzy Systems Society VFSS Achievements Lotfi A Zadeh estimated around 2014 that there were more than 50 000 fuzzy logic related patented inventions He listed 28 journals at that time dealing with fuzzy reasoning and 21 journal titles on soft computing His searches found close to 100 000 publications with the word fuzzy in their titles but perhaps there are even 300 000 In March 2018 Google Scholar found 2 870 000 titles which included the word fuzzy When he died on 11 September 2017 at age 96 Professor Zadeh had received more than 50 engineering and academic awards in recognition of his work Lattices and big data setsThe technique of fuzzy concept lattices is increasingly used in programming for the formatting relating and analysis of fuzzy data sets Concept formalization According to the computer scientist Andrei Popescu at Middlesex University London a concept can be operationally defined to consist of an intent which is a description or specification stated in a language an extent which is the collection of all the objects to which the description refers a context which is stated by i the universe of all possible objects within the scope of the concept ii the universe of all possible attributes of objects and iii the logical definition of the relation whereby an object possesses an attribute Once the context is defined we can specify relationships of sets of objects with sets of attributes which they do or do not share Fuzzy concept lattice Whether an object belongs to a concept and whether an object does or does not have an attribute can often be a matter of degree Thus for example many attributes are fuzzy rather than crisp To overcome this issue a numerical value is assigned to each attribute along a scale and the results are placed in a table which links each assigned object value within the given range to a numerical value a score denoting a given degree of applicability This is the basic idea of a fuzzy concept lattice which can also be graphed different fuzzy concept lattices can be connected to each other as well for example in fuzzy conceptual clustering techniques used to group data originally invented by Enrique H Ruspini Fuzzy concept lattices are a useful programming tool for the exploratory analysis of big data for example in cases where sets of linked behavioural responses are broadly similar but can nevertheless vary in important ways within certain limits It can help to find out what the structure and dimensions are of a behaviour that occurs with an important but limited amount of variation in a large population Big data Coding with fuzzy lattices can be useful for instance in the psephological analysis of big data about voter behaviour where researchers want to explore the characteristics and associations involved in somewhat vague opinions gradations in voter attitudes and variability in voter behaviour or personal characteristics within a set of parameters The basic programming techniques for this kind of fuzzy concept mapping and deep learning are by now well established and big data analytics had a strong influence on the US elections of 2016 A US study concluded in 2015 that for 20 of undecided voters Google s secret search algorithm had the power to change the way they voted Very large quantities of data can now be explored using computers with fuzzy logic programming and open source architectures such as Apache Hadoop Apache Spark and MongoDB One author claimed in 2016 that it is now possible to obtain link and analyze 400 data points for each voter in a population using Oracle systems a data point is a number linked to one or more categories which represents a characteristic However NBC News reported in 2016 that the Anglo American firm Cambridge Analytica which profiled voters for Donald Trump Steve Bannon was a board member did not have 400 but 4 000 data points for each of 230 million US adults Cambridge Analytica s own website claimed that up to 5 000 data points were collected for each of 220 million Americans a data set of more than 1 trillion bits of formatted data The Guardian later claimed that Cambridge Analytica in fact had according to its own company information up to 7 000 data points on 240 million American voters Harvard University Professor Latanya Sweeney calculated that if a U S company knows just your date of birth your ZIP code and sex the company has an 87 chance to identify you by name simply by using linked data sets from various sources With 4 000 7 000 data points instead of three a very comprehensive personal profile becomes possible for almost every voter and many behavioural patterns can be inferred by linking together different data sets It also becomes possible to identify and measure gradations in personal characteristics which in aggregate have very large effects Human judgement Some researchers argue that this kind of big data analysis has severe limitations and that the analytical results can only be regarded as indicative and not as definitive This was confirmed by Kellyanne Conway Donald Trump s campaign advisor and counselor in 2016 who emphasized the importance of human judgement and common sense in drawing conclusions from fuzzy data Conway candidly admitted that much of her own research would never see the light of day because it was client confidential Another Trump adviser criticized Conway claiming that she produces an analysis that buries every terrible number and highlights every positive number Propaganda machine In a video interview published by The Guardian in March 2018 whistleblower Christopher Wylie called Cambridge Analytica a full service propaganda machine rather than a bona fide data science company Its own site revealed with case studies that it has been active in political campaigns in numerous different countries influencing attitudes and opinions Wylie explained that we spent a million dollars harvesting tens of millions of Facebook profiles and those profiles were used as the basis of the algorithms that became the foundation of Cambridge Analytica itself The company itself was founded on using Facebook data Audit On 19 March 2018 Facebook announced it had hired the digital forensics firm Stroz Friedberg to conduct a comprehensive audit of Cambridge Analytica while Facebook shares plummeted 7 percent overnight erasing roughly 40 billion in market capitalization Cambridge Analytica had not just used the profiles of Facebook users to compile data sets According to Christopher Wylie s testimony the company also harvested the data of each user s network of friends leveraging the original data set It then converted combined and migrated its results into new data sets which can in principle survive in some format even if the original data sources are destroyed It created and applied algorithms using data to which critics argue it could not have been entitled This was denied by Cambridge Analytica which stated on its website that it legitimately uses data to change audience behavior among customers and voters who choose to view and provide information If advertisers can do that why not a data company Where should the line be drawn Legally it remained a fuzzy area Legal issue The tricky legal issue then became what kind of data Cambridge Analytica or any similar company is actually allowed to have and keep Facebook itself became the subject of another U S Federal Trade Commission inquiry to establish whether Facebook violated the terms of a 2011 consent decree governing its handing of user data data which was allegedly transferred to Cambridge Analytica without Facebook s and user s knowledge Wired journalist Jessi Hempel commented in a CBNC panel discussion that Now there is this fuzziness from the top of the company i e Facebook that I have never seen in the fifteen years that I have covered it Data privacy Interrogating Facebook s CEO Mark Zuckerberg before the U S House Energy and Commerce Committee in April 2018 New Mexico Congressman Rep Ben Ray Lujan put it to him that the Facebook corporation might well have 29 000 data points on each Facebook user Zuckerberg claimed that he did not really know Lujan s figure was based on ProPublica research which in fact suggested that Facebook may even have 52 000 data points for many Facebook users When Zuckerberg replied to his critics he stated that because the revolutionary technology of Facebook with 2 2 billion users worldwide at that time had ventured into previously unknown territory it was unavoidable that mistakes would be made despite the best of intentions He justified himself saying that For the first ten or twelve years of the company I viewed our responsibility primarily as building tools that if we could put those tools in people s hands then that would empower people to do good things What we have learnt now is that we need to take a more proactive role and a broader view of our responsibility In July 2018 Facebook and Instagram barred access from Crimson Hexagon a company that advises corporations and governments using one trillion scraped social media posts which it mined and processed with artificial intelligence and image analysis Integrity It remained fuzzy what was more important to Zuckerberg making money from user s information or real corporate integrity in the use of personal information Zuckerberg implied that he believed that on balance Facebook had done more good than harm and that if he had believed that wasn t the case he would never have persevered with the business Thus the good was itself a fuzzy concept because it was a matter of degree more good than bad He had to sell stuff to keep the business growing If people do not like Facebook then they simply should not join it or opt out they have the choice Many critics however feel that people really are in no position to make an informed choice because they have no idea of how exactly their information will or might be used by third parties contracting with Facebook because the company legally owns the information that users provide online they have no control over that either except to restrict themselves in what they write online the same applies to many other online services After the New York Times broke the news on 17 March 2018 that copies of the Facebook data set scraped by Cambridge Analytica could still be downloaded from the Internet Facebook was severely criticized by government representatives When questioned Zuckerberg admitted that In general we collect data on people who are not signed up for Facebook for security purposes with the aim to help prevent malicious actors from collecting public information from Facebook users such as names From 2018 onward Facebook faced a lot more lawsuits brought against the company alleging data breaches security breaches and misuse of personal information see Lawsuits involving Meta Platforms and Facebook Federal Litigation Filings There still exists no standard international regulatory framework for social network information and it is often unclear what happens to the stored information after a provider company closes down or is taken over by another company Zuckerberg s Meta company also reports its own legal actions On 2 May 2018 it was reported that the Cambridge Analytica company was shutting down and was starting bankruptcy proceedings after losing clients and facing escalating legal costs The reputational damage which the company had suffered or caused had become too great Speed A traditional objection to big data is that it cannot cope with rapid change events move faster that the statistics can keep up with Yet the technology now exists for corporations like Amazon Google Apple Inc and Microsoft to pump cloud based data streams from app users straight into big data analytics programmes in real time Provided that the right kinds of analytical concepts are used it is now technically possible to draw definite and important conclusions about gradations of human and natural behaviour using very large fuzzy data sets and fuzzy programming and increasingly it can be done very fast This achievement has become highly topical in military technology in areas such as cybersecurity tracking and monitoring systems guidance systems for firearms explosive launchers vehicles planes vessels artillery missiles satellites drones and bombs threat identification evaluation systems risk and strategy appraisal arms transfer and arms race impact assessments and targeting methods The identification of a threat and the response to it often have to happen very fast with a high degree of accuracy for which comprehensive artificial intelligence is essential Dr Tal Mimran a lecturer at Hebrew University in Jerusalem and a former legal adviser to the Israeli Defence Force IDF stated During the period in which I served in the target room between 2010 and 2015 you needed a team of around 20 intelligence officers to work for around 250 days to gather something between 200 to 250 targets Today the AI will do that in a week Although no comprehensive overviews appear to be publicly available a large amount of scientific research on fuzzy systems was funded or sponsored by the military However military uses of fuzzy systems research can also have spin offs for medical applications Academic debatesThere have been many academic debates about the meaning relevance and utility of fuzzy concepts as well as their appropriate use Rudolf E Kalman stated in 1972 that there is no such thing as a fuzzy concept We do talk about fuzzy things but they are not scientific concepts The suggestion is that to qualify as a concept the concept must always be clear and precise without any fuzziness A vague notion would be at best a prologue to formulating a concept In 2011 three Chinese engineers alleged that Fuzzy set its t norm s norm and fuzzy supplement theories have already become the academic virus in the world Fuzzy label Lotfi A Zadeh himself confessed that I knew that just by choosing the label fuzzy I was going to find myself in the midst of a controversy If it weren t called fuzzy logic there probably wouldn t be articles on it on the front page of the New York Times So let us say it has a certain publicity value Of course many people don t like that publicity value and when they see it in the New York Times it doesn t sit well with them However the impact of the invention of fuzzy reasoning went far beyond names and labels When Zadeh gave his acceptance speech in Japan for the 1989 Honda Foundation prize which he received for inventing fuzzy theory he stated that The concept of a fuzzy set has had an upsetting effect on the established order Frege and Wittgenstein According to The Foundations of Arithmetic by the logician Gottlob Frege A definition of a concept must be complete it must unambiguously determine as regards any object whether or not it falls under the concept the concept must have a sharp boundary a concept that is not sharply defined is wrongly termed a concept Such quasi conceptual constructions cannot be recognized as concepts by logic The law of the excluded middle is really just another form of the requirement that the concept should have a sharp boundary In his notes on language games Ludwig Wittgenstein replied to Frege s argument as follows One can say that the concept of a game is a concept with blurred edges But is a blurred concept a concept at all Is a photograph that is not sharp a picture of a person at all Is it even always an advantage to replace a picture that is not sharp by one that is Isn t one that isn t sharp often just what we need Frege compares a concept to a region and says that a region without clear boundaries can t be called a region at all This presumably means that we can t do anything with it But is it senseless to say Stay roughly here Imagine that I were standing with someone in a city square and said that As I say it I do not bother drawing any boundary but just make a pointing gesture as if I were indicating a particular spot And this is just how one might explain what a game is The categorical status of concepts There is no general agreement among philosophers and scientists about how the notion of a concept and in particular a scientific concept should be defined A concept could be defined as a mental representation as a cognitive capacity as an abstract object as a cluster of linked phenomena etc Edward E Smith amp Douglas L Medin stated that there will likely be no crucial experiments or analyses that will establish one view of concepts as correct and rule out all others irrevocably Of course scientists also quite often do use imprecise analogies in their models to help understanding an issue A concept can be clear enough but not or not sufficiently precise Rather uniquely terminology scientists at the German National Standards Institute Deutsches Institut fur Normung provided an official standard definition of what a concept is under the terminology standards DIN 2330 of 1957 completely revised in 1974 and last revised in 2022 and DIN 2342 of 1986 also last revised in 2022 According to the official German definition a concept is a unit of thought which is created through abstraction for a set of objects and which identifies shared or related characteristics of those objects The subsequent ISO definition is very similar Under the ISO 1087 terminology standard of the International Standards Organization first published in October 2000 reviewed in 2005 and revised in 2019 a concept is defined as a unit of thought or an idea constituted through abstraction on the basis of properties common to a set of objects It is acknowledged that although a concept usually has one definition or one meaning it may have multiple designations terms of expression symbolizations or representations Thus for example the same concept can have different names in different languages Both verbs and nouns can express concepts A concept can also be thought of as a way of looking at the world The official terminological standards are useful for many practical purposes But for more complex concepts the standards may not be so helpful The reason is that complex concepts do not necessarily denote only a collection of objects which have something in common A complex concept may for example express a Gestalt i e it may express a totality which is more means more and does more than the sum of its parts as recognized in Aristotle s Metaphysics It may be that the parts cannot exist other than within the totality The totality could also be a totality of totalities In such cases the definition of the complex concept is not or not fully reducible to what its parts have in common Modelling such a concept requires more than identifying and enumerating the parts that are included in and excluded from the concept It requires also a specification of what all the parts together add up to or what they constitute collectively In some respects at least the totality differs qualitatively from any of its parts The Gestalt could be a fuzzy object figure or shape Potential corruption Reasoning with fuzzy concepts is often viewed as a kind of logical corruption or scientific perversion because it is claimed fuzzy reasoning rarely reaches a definite yes or a definite no A clear precise and logically rigorous conceptualization is no longer a necessary prerequisite for carrying out a procedure a project or an inquiry since somewhat vague ideas can always be accommodated formalized and programmed with the aid of fuzzy expressions The purist idea is that either a rule applies or it does not apply When a rule is said to apply only to some extent then in truth the rule does not apply Thus a compromise with vagueness or indefiniteness is on this view effectively a compromise with error an error of conceptualization an error in the inferential system or an error in physically carrying out a task Kahan s criticism The computer scientist William Kahan argued in 1975 that the danger of fuzzy theory is that it will encourage the sort of imprecise thinking that has brought us so much trouble He said subsequently With traditional logic there is no guaranteed way to find that something is contradictory but once it is found you d be obliged to do something But with fuzzy sets the existence of contradictory sets can t cause things to malfunction Contradictory information doesn t lead to a clash You just keep computing Life affords many instances of getting the right answer for the wrong reasons It is in the nature of logic to confirm or deny The fuzzy calculus blurs that Logic isn t following the rules of Aristotle blindly It takes the kind of pain known to the runner He knows he is doing something When you are thinking about something hard you ll feel a similar sort of pain Fuzzy logic is marvellous It insulates you from pain It s the cocaine of science According to Kahan statements of a degree of probability are usually verifiable There are standard tests one can do By contrast there is no conclusive procedure which can decide the validity of assigning particular fuzzy truth values to a data set in the first instance It is just assumed that a model or program will work if particular fuzzy values are accepted and used perhaps based on some statistical comparisons or try outs Bad design In programming a problem can usually be solved in several different ways not just one way but an important issue is which solution works best in the short term and in the long term Kahan implies that fuzzy solutions may create more problems in the long term than they solve in the short term For example if one starts off designing a procedure not with well thought out precise concepts but rather by using fuzzy or approximate expressions which conveniently patch up or compensate for badly formulated ideas the ultimate result could be a complicated malformed mess that does not achieve the intended goal Had the reasoning and conceptualization been much sharper at the start then the design of the procedure might have been much simpler more efficient and effective and fuzzy expressions or approximations would not be necessary or required much less Thus by allowing the use of fuzzy or approximate expressions one might actually foreclose more rigorous thinking about design and one might build something that ultimately does not meet expectations If say an entity X turns out to belong for 65 to category Y and for 35 to category Z how should X be allocated One could plausibly decide to allocate X to Y making a rule that if an entity belongs for 65 or more to Y it is to be treated as an instance of category Y and never as an instance of category Z One could however alternatively decide to change the definitions of the categorization system to ensure that all entities such as X fall 100 in one category only This kind of argument claims that boundary problems can be resolved or vastly reduced simply by using better categorization or conceptualization methods If we treat X as if it belongs 100 to Y while in truth it only belongs 65 to Y then arguably we are really misrepresenting things If we keep doing that with a lot of related variables we can greatly distort the true situation and make it look like something that it isn t In a fuzzy permissive environment it might become far too easy to formalize and use a concept which is itself badly defined and which could have been defined much better In that environment there is always a quantitative way out for concepts that do not quite fit or which don t quite do the job for which they are intended The cumulative adverse effect of the discrepancies might in the end be much larger than ever anticipated Counter argument A typical reply to Kahan s objections is that fuzzy reasoning never rules out ordinary binary logic but instead presupposes ordinary true or false logic Lotfi Zadeh stated that fuzzy logic is not fuzzy In large measure fuzzy logic is precise It is a precise logic of imprecision Fuzzy logic is not a replacement of or substitute for ordinary logic but an enhancement of it with many practical uses Fuzzy thinking does oblige action but primarily in response to a change in quantitative gradation not in response to a contradiction One could say for example that ultimately one is either alive or dead which is perfectly true Meantime though one is living which is also a significant truth yet living is a fuzzy concept It is true that fuzzy logic by itself usually cannot eliminate inadequate conceptualization or bad design Yet it can at least make explicit what exactly the variations are in the applicability of a concept which has unsharp boundaries If one always had perfectly crisp concepts available perhaps no fuzzy expressions would be necessary In reality though one often does not have all the crisp concepts to start off with One might not have them yet for a long time or ever or several successive fuzzy approximations might be needed to get there A fuzzy permissive environment may be appropriate and useful precisely because it permits things to be actioned that would never have been achieved if there had been crystal clarity about all the consequences from the start or if people insisted on absolute precision prior to doing anything Scientists often try things out on the basis of hunches and processes like serendipity can play a role Learning something new or trying to create something new is rarely a completely formal logical or linear process There are not only knowns and unknowns involved but also partly known phenomena i e things which are known or unknown to some degree Even if ideally we would prefer to eliminate fuzzy ideas we might need them initially to get there further down the track Any method of reasoning is a tool If its application has bad results it is not the tool itself that is to blame but its inappropriate use It would be better to educate people in the best use of the tool if necessary with appropriate authorization than to ban the tool pre emptively on the ground that it could or might be abused Exceptions to this rule would include things like computer viruses and illegal weapons that can only cause great harm if they are used There is no evidence though that fuzzy concepts as a species are intrinsically harmful even if some bad concepts can cause harm if used in inappropriate contexts Reducibility Susan Haack once claimed that a many valued logic requires neither intermediate terms between true and false nor a rejection of bivalence She implied that the intermediate terms i e the gradations of truth can always be restated as conditional if then statements and by implication that fuzzy logic is fully reducible to binary true or false logic This interpretation is disputed it assumes that the knowledge already exists to fit the intermediate terms to a logical sequence but even if it was correct assigning a number to the applicability of a statement is often enormously more efficient than a long string of if then statements that would have the same intended meaning That point is obviously of great importance to computer programmers educators and administrators seeking to code a process activity message or operation as simply as possible according to logically consistent rules Prof Haack is of course quite correct when she argues that fuzzy logic does not do away with binary logic Quantification It may be wonderful to have an unlimited number of distinctions available to define what one means but not all scholars would agree that any concept is equal to or reducible to a mathematical set Some phenomena are difficult or impossible to quantify and count in particular if they lack discrete boundaries for example clouds George Lakoff emphasized that it is not true that fuzzy set theory is the only or necessarily the most appropriate way to start modelling concepts Formalization Qualities may not be fully reducible to quantities if there are no qualities it may become impossible to say what the numbers are numbers of or what they refer to except that they refer to other numbers or numerical expressions such as algebraic equations A measure requires a counting unit defined by a category but the definition of that category is essentially qualitative a language which is used to communicate data is difficult to operate without any qualitative distinctions and categories We may for example transmit a text in binary code but the binary code does not tell us directly what the text intends It has to be translated decoded or converted first before it becomes comprehensible In creating a formalization or formal specification of a concept for example for the purpose of measurement administrative procedure or programming part of the meaning of the concept may be changed or lost For example if we deliberately program an event according to a concept it might kill off the spontaneity spirit authenticity and motivational pattern which is ordinarily associated with that type of event Quantification is not an unproblematic process To quantify a phenomenon we may have to introduce special assumptions and definitions which disregard part of totality of the phenomenon The economist John Maynard Keynes concluded that formalization runs the risk of leaving behind the subjectmatter we are interested in and also runs the risk of increasing rather than decreasing the muddle Friedrich Hayek stated that it is certainly not scientific to insist on measurement where you don t know what your measurements mean There are cases where measurements are not relevant The Hayekian big data guru Viktor Mayer Schonberger states that A system based on money and price solved a problem of too much information and not enough processing power but in the process of distilling information down to price many details get lost Michael Polanyi stated that the process of formalizing all knowledge to the exclusion of any tacit knowing is self defeating since to mathematize a concept we need to be able to identify it in the first instance without mathematization Measurement Programmers statisticians or logicians are concerned in their work with the main operational or technical significance of a concept which is specifiable in objective quantifiable terms They are not primarily concerned with all kinds of imaginative frameworks associated with the concept or with those aspects of the concept which seem to have no particular functional purpose however entertaining they might be However some of the qualitative characteristics of the concept may not be quantifiable or measurable at all at least not directly The temptation exists to ignore them or try to infer them from data results If for example we want to count the number of trees in a forest area with any precision we have to define what counts as one tree and perhaps distinguish them from saplings split trees dead trees fallen trees etc Soon enough it becomes apparent that the quantification of trees involves a degree of abstraction we decide to disregard some timber dead or alive from the population of trees in order to count those trees that conform to our chosen concept of a tree We operate in fact with an abstract concept of what a tree is which diverges to some extent from the true diversity of trees there are Even so there may be some trees of which it is not very clear whether they should be counted as a tree or not It may be difficult to define the exact boundary where the forest begins and ends The forest boundary might also change somewhat in the course of time A certain amount of fuzziness in the definition of a tree and of the forest may therefore remain The implication is that the seemingly exact number offered for the total quantity of trees in the forest may be much less exact than one might think it is probably more an estimate or indication of magnitude rather than an exact description Yet and this is the point the imprecise measure can be very useful and sufficient for all intended purposes It is tempting to think that if something can be measured it must exist and that if we cannot measure it it does not exist Neither might be true Researchers try to measure such things as intelligence or gross domestic product without much scientific agreement about what these things actually are how they exist and what the correct measures might be When one wants to count and quantify distinct objects using numbers one needs to be able to distinguish between all of those separate objects as countable units If this is difficult or impossible then although this may not invalidate a quantitative procedure as such quantification is not really possible in practice At best we may be able to assume or infer indirectly a certain distribution of quantities that must be there In this sense scientists often use proxy variables to substitute as measures for variables which are known or thought to be there but which themselves cannot be observed or measured directly Vague or fuzzy The exact relationship between vagueness and fuzziness is disputed Philosophical interpretation Philosophers often regard fuzziness as a particular kind of vagueness and consider that no specific assignment of semantic values to vague predicates not even a fuzzy one can fully satisfy our conception of what the extensions of vague predicates are like Surveying recent literature on how to characterize vagueness Matti Eklund states that appeal to lack of sharp boundaries borderline cases and sorites susceptible predicates are the three informal characterizations of vagueness which are most common in the literature Zadeh s argument However Lotfi A Zadeh claimed that vagueness connotes insufficient specificity whereas fuzziness connotes unsharpness of class boundaries Thus he argued a sentence like I will be back in a few minutes is fuzzy but not vague whereas a sentence such as I will be back sometime is fuzzy and vague His suggestion was that fuzziness and vagueness are logically quite different qualities rather than fuzziness being a type or subcategory of vagueness Zadeh claimed that inappropriate use of the term vague is still a common practice in the literature of philosophy Ethics and law In the scholarly inquiry about ethics and meta ethics vague or fuzzy concepts and borderline cases are standard topics of controversy Central to ethics are theories of value what is good or bad for people and why that is and the idea of rule following as a condition for moral integrity consistency and non arbitrary behaviour Yet if human valuations or moral rules are only vague or fuzzy then they may not be able to orient or guide behaviour It may become impossible to operationalize rules Evaluations may not permit definite moral judgements in that case Hence clarifying fuzzy moral notions is usually considered to be critical for the ethical endeavour as a whole Excessive precision in rule making Nevertheless Scott Soames has made the case that vagueness or fuzziness can be valuable to rule makers because their use of it is valuable to the people to whom rules are addressed It may be more practical and effective to allow for some leeway and personal responsibility in the interpretation of how a rule should be applied bearing in mind the overall purpose which the rule intends to achieve If a rule or procedure is stipulated too exactly it can sometimes have a result which is contrary to the aim which it was intended to help achieve For example The Children and Young Persons Act could have specified a precise age below which a child may not be left unsupervised But doing so would have incurred quite substantial forms of arbitrariness for various reasons and particularly because of the different capacities of children of the same age Conflicting rules A related sort of problem is that if the application of a legal concept is pursued too exactly and rigorously it may have consequences that cause a serious conflict with another legal concept This is not necessarily a matter of bad law making When a law is made it may not be possible to anticipate all the cases and events to which it will apply later even if 95 of possible cases are predictable The longer a law is in force the more likely it is that people will run into problems with it that were not foreseen when the law was made So the further implications of one rule may conflict with another rule Common sense might not be able to resolve things In that scenario too much precision can get in the way of justice Very likely a special court ruling wil have to set a norm The general problem for jurists is whether the arbitrariness resulting from precision is worse than the arbitrariness resulting from the application of a vague standard David Lanius has examined nine arguments for the value of vagueness in different contexts Mathematical ontology The definitional disputes about fuzziness remain unresolved so far mainly because as anthropologists and psychologists have documented different languages or symbol systems that have been created by people to signal meanings suggest different ontologies Put simply it is not merely that describing what is there involves symbolic representations of some kind How distinctions are drawn influences perceptions of what is there and vice versa perceptions of what is there influence how distinctions are drawn This is an important reason why as Alfred Korzybski noted people frequently confuse the symbolic representation of reality conveyed by languages and signs with reality itself Fuzziness implies that there exists a potentially infinite number of truth values between complete truth and complete falsehood If that is the case it creates the foundational issue of what in the case can justify or prove the existence of the categorical absolutes which are assumed by logical or quantitative inference If there is an infinite number of shades of grey how do we know what is totally black and white and how could we identify that Tegmark s mathematical universe To illustrate the ontological issues cosmologist Max Tegmark argues boldly that the universe consists of math If you accept the idea that both space itself and all the stuff in space have no properties at all except mathematical properties then the idea that everything is mathematical starts to sound a little bit less insane Tegmark moves from the epistemic claim that mathematics is the only known symbol system which can in principle express absolutely everything to the methodological claim that everything is reducible to mathematical relationships and then to the ontological claim that ultimately everything that exists is mathematical the mathematical universe hypothesis The argument is then reversed so that because everything is mathematical in reality mathematics is necessarily the ultimate universal symbol system The main criticisms of Tegmark s approach are that 1 the steps in this argument do not necessarily follow 2 no conclusive proof or test is possible for the claim that a total reduction of everything to mathematics is feasible among other things because qualitative categories remain indispensable to understand and navigate what quantities mean and 3 it may be that a complete reduction to mathematics cannot be accomplished without at least partly altering negating or deleting a non mathematical significance of phenomena experienced perhaps as qualia Zalta s metaphysics In his meta mathematical metaphysics Edward N Zalta has claimed that for every set of properties of a concrete object there always exists exactly one abstract object that encodes exactly that set of properties and no others a foundational assumption or axiom for his ontology of abstract objects By implication for every fuzzy object there exists always at least one defuzzified concept which encodes it exactly It is a modern interpretation of Plato s metaphysics of knowledge which expresses confidence in the ability of science to conceptualize the world exactly Platonism versus cognitive realism The Platonic style interpretation was critiqued by Hartry H Field Mark Balaguer argues that we do not really know whether mind independent abstract objects exist or not so far we cannot prove whether Platonic realism is definitely true or false Defending a cognitive realism Scott Soames argues that the reason why this unsolvable conundrum has persisted is because the ultimate constitution of the meaning of concepts and propositions was misconceived Traditionally it was thought that concepts can be truly representational because ultimately they are related to intrinsically representational Platonic complexes of universals and particulars see theory of forms However once concepts and propositions are regarded as cognitive event types it is possible to claim that they are able to be representational because they are constitutively related to intrinsically representational cognitive acts in the real world As another philosopher put it The question of how we can know the world around us is not entirely unlike the question of how it is that the food our environment provides happens to agree with our stomachs Either can become a mystery if we forget that minds like stomachs originated in and have been conditioned by a pre existent natural order Along these lines it could be argued that reality and the human cognition of reality will inevitably contain some fuzzy characteristics which can perhaps be represented only by concepts which are themselves fuzzy to some or other extent Social science and the mediaThe idea of fuzzy concepts has also been applied in the philosophical sociological and linguistic analysis of human behaviour Sociology and linguistics In a 1973 paper George Lakoff analyzed hedges in the interpretation of the meaning of categories Charles Ragin and others have applied the idea to sociological analysis For example fuzzy set qualitative comparative analysis fsQCA has been used by German researchers to study problems posed by ethnic diversity in Latin America In New Zealand Taiwan Iran Malaysia the European Union and Croatia economists have used fuzzy concepts to model and measure the underground economy of their country Kofi Kissi Dompere applied methods of fuzzy decision approximate reasoning negotiation games and fuzzy mathematics to analyze the role of money information and resources in a political economy of rent seeking viewed as a game played between powerful corporations and the government The German researcher Thomas Kron has used fuzzy methods to model sociological theory creating an integral action theoretical model with the aid of fuzzy logic With Lars Winter Kron developed the system theory of Niklas Luhmann further using the so called Kosko Cube Kron studies transnational terrorism and other contemporary phenomena using fuzzy logic to understand conditions involving uncertainty hybridity violence and cultural systems A concept may be deliberately created by sociologists as an ideal type to understand something imaginatively without any strong claim that it is a true and complete description or a true and complete reflection of whatever is being conceptualized In a more general sociological or journalistic sense a fuzzy concept has come to mean a concept which is meaningful but inexact implying that it does not exhaustively or completely define the meaning of the phenomenon to which it refers often because it is too abstract In this context it is said that fuzzy concepts lack clarity and are difficult to test or operationalize To specify the relevant meaning more precisely additional distinctions conditions and or qualifiers would be required A few examples can illustrate this kind of usage a handbook of sociology states that The theory of interaction rituals contains some gaps that need to be filled and some fuzzy concepts that need to be differentiated The idea is that if finer distinctions are introduced then the fuzziness or vagueness would be eliminated a book on youth culture describes ethnicity as a fuzzy concept that overlaps at times with concepts of race minority nationality and tribe In this case part of the fuzziness consists in the inability to distinguish precisely between a concept and a different but closely related concept a book on sociological theory argues that the Critical Theory of domination faces the problem that reality itself has become a rather meaningless fuzzy concept The suggestion here is that the variations in how theoretical concepts are applied have become so large that the concepts could mean all kinds of things and therefore are crucially vague with the implication that they are not useful any longer for that very reason A history book states Sodomy was a vague and fuzzy concept in medieval and early modern Europe and was often associated with a variety of supposedly related moral and criminal offenses including heresy witchcraft sedition and treason St Thomas Aquinas categorized sodomy with an assortment of sexual behaviours from which generation i e procreation cannot follow In this case because a concept is defined by what it excludes it remains somewhat vague what items of activity it would specifically include Mass media The main reason why the term fuzzy concept is now often used in describing human behaviour is that human interaction has many characteristics which are difficult to quantify and measure precisely although we know that they have magnitudes and proportions among other things because they are interactive and reflexive the observers and the observed mutually influence the meaning of events Those human characteristics can be usefully expressed only in an approximate way see reflexivity social theory Newspaper stories frequently contain fuzzy concepts which are readily understood and used even although they are far from exact Thus many of the meanings which people ordinarily use to negotiate their way through life in reality turn out to be fuzzy concepts While people often do need to be exact about some things e g money or time many areas of their lives involve expressions which are far from exact Sometimes the term is also used in a pejorative sense For example a New York Times journalist wrote that Prince Sihanouk seems unable to differentiate between friends and enemies a disturbing trait since it suggests that he stands for nothing beyond the fuzzy concept of peace and prosperity in Cambodia Applied social science The use of fuzzy logic in the social sciences and humanities has remained limited until recently Lotfi A Zadeh said in a 1994 interview that I expected people in the social sciences economics psychology philosophy linguistics politics sociology religion and numerous other areas to pick up on it It s been somewhat of a mystery to me why even to this day so few social scientists have discovered how useful it could be Two decades later after a digital information explosion due to the growing use of the internet and mobile phones worldwide fuzzy concepts and fuzzy logic were increasingly being applied in big data analysis of social commercial and psychological phenomena Many sociometric and psychometric indicators are based partly on fuzzy concepts and fuzzy variables Jaakko Hintikka once claimed that the logic of natural language we are in effect already using can serve as a fuzzy logic better than its trade name variant without any additional assumptions or constructions That might help to explain why fuzzy logic has not been used much to formalize concepts in the soft social sciences Lotfi A Zadeh rejected such an interpretation on the ground that in many human endeavours as well as technologies it is highly important to define more exactly to what extent something is applicable or true when it is known that its applicability can vary to some important extent among large populations Reasoning which accepts and uses fuzzy concepts can be shown to be perfectly valid with the aid of fuzzy logic because the degrees of applicability of a concept can be more precisely and efficiently defined with the aid of numerical notation Another possible explanation for the traditional lack of use of fuzzy logic by social scientists is simply that beyond basic statistical analysis using programs such as SPSS and Excel the mathematical knowledge of social scientists is often rather limited they may not know how to formalize and code a fuzzy concept using the conventions of fuzzy logic The standard software packages used provide only a limited capacity to analyze fuzzy data sets if at all and considerable skills are required Yet Jaakko Hintikka may be correct in the sense that it can be much more efficient to use natural language to denote a complex idea than to formalize it in logical terms The quest for formalization might introduce much more complexity which is not wanted and which detracts from communicating the relevant issue Some concepts used in social science may be impossible to formalize exactly even though they are quite useful and people understand their appropriate application quite well UncertaintyFuzzy concepts can generate uncertainty because they are imprecise especially if they refer to a process in motion or a process of transformation where something is in the process of turning into something else In that case they do not provide a clear orientation for action or decision making what does X really mean intend or imply reducing fuzziness perhaps by applying fuzzy logic might generate more certainty Precision versus relevance However this is not necessarily always so A concept even although it is not fuzzy at all and even though it is very exact could equally well fail to capture the meaning of something adequately That is a concept can be very precise and exact but not or insufficiently applicable or relevant in the situation to which it refers In this sense a definition can be very precise but miss the point altogether Security A fuzzy concept may indeed provide more security because it provides a meaning for something when an exact concept is unavailable which is better than not being able to denote it at all A concept such as God although not easily definable for instance can provide security to the believer Observer effect In physics the observer effect and Heisenberg s uncertainty principle indicate that there is a physical limit to the amount of precision that is knowable with regard to the movements of subatomic particles and waves That is features of physical reality exist where we can know that they vary in magnitude but of which we can never know or predict exactly how big or small the variations are This insight suggests that in some areas of our experience of the physical world fuzziness is inevitable and can never be totally removed Since the physical universe itself is incredibly large and diverse it is not easy to imagine it grasp it or describe it without using fuzzy concepts LanguageOrdinary language which uses symbolic conventions and associations which are often not logical inherently contains many fuzzy concepts knowing what you mean in this case depends partly on knowing the context or being familiar with the way in which a term is normally used or what it is associated with This can be easily verified for instance by consulting a dictionary a thesaurus or an encyclopedia which show the multiple meanings of words or by observing the behaviours involved in ordinary relationships which rely on mutually understood meanings see also Imprecise language Bertrand Russell regarded ordinary language in contrast to logic as intrinsically vague Implicature To communicate receive or convey a message an individual somehow has to bridge his own intended meaning and the meanings which are understood by others i e the message has to be conveyed in a way that it will be socially understood preferably in the intended manner Thus people might state you have to say it in a way that I understand Even if the message is clear and precise it may nevertheless not be received in the way it was intended Bridging meanings may be done instinctively habitually or unconsciously but it usually involves a choice of terms assumptions or symbols whose meanings are not completely fixed but which depend among other things on how the receivers of the message respond to it or the context In this sense meaning is often negotiated or interactive or more cynically manipulated This gives rise to many fuzzy concepts The semantic challenge of conveying meanings to an audience was explored in detail and analyzed logically by the British philosopher Paul Grice using among other things the concept of implicature Implicature refers to what is suggested by a message to the recipient without being either explicitly expressed or logically entailed by its content The suggestion could be very clear to the recipient perhaps a sort of code but it could also be vague or fuzzy Paradoxes Even using ordinary set theory and binary logic to reason something out logicians have discovered that it is possible to generate statements which are logically speaking not completely true or imply a paradox even although in other respects they conform to logical rules see Russell s paradox If a margin of indeterminacy therefore persists then binary logic cannot totally remove fuzziness David Hilbert concluded that the existence of logical paradoxes tells us that we must develop a meta mathematical analysis of the notions of proof and of the axiomatic method their importance is methodological as well as epistemological PsychologyVarious different aspects of human experience commonly generate concepts with fuzzy characteristics Human vs computer The formation of fuzzy concepts is partly due to the fact that the human brain does not operate like a computer see also Chinese room While ordinary computers use strict binary logic gates the brain does not i e it is capable of making all kinds of neural associations according to all kinds of ordering principles or fairly chaotically in associative patterns which are not logical but nevertheless meaningful For example a work of art can be meaningful without being logical A pattern can be observably regular ordered and or non arbitrary hence meaningful without it being possible to describe it completely or exhaustively in formal logical terms Something can be meaningful although we cannot name it or we might only be able to name it and nothing else Human brains can also interpret the same phenomenon in several different but interacting frames of reference at the same time or in quick succession without there necessarily being an explicit logical connection between the frames see also framing effect According to fuzzy trace theory partly inspired by Gestalt psychology human intuition is a non arbitrary reasonable and rational process of cognition it literally makes sense see also Problem of multiple generality Transitions in learning and consciousness In part fuzzy concepts arise also because learning or the growth of understanding involves a transition from a vague awareness which cannot orient behaviour greatly to clearer insight which can orient behaviour At the first encounter with an idea the sense of the idea may be rather hazy When more experience with the idea has occurred a clearer and more precise grasp of the idea results as well as a better understanding of how and when to use the idea or not In his study of implicit learning Arthur S Reber affirms that there does not exist a very sharp boundary between the conscious and the unconscious and there are always going to be lots of fuzzy borderline cases of material that is marginally conscious and lots of elusive instances of functions and processes that seem to slip in and out of personal awareness Thus an inevitable component of fuzziness exists and persists in human consciousness because of continual variation of gradations in awareness along a continuum from the conscious the preconscious and the subconscious to the unconscious The hypnotherapist Milton H Erickson similarly noted that the conscious mind and the unconscious normally interact Limits of distinctions and generalizations Some psychologists and logicians argue that fuzzy concepts are a necessary consequence of the reality that any kind of distinction we might like to draw has limits of application At a certain level of generality a distinction works fine But if we pursued its application in a very exact and rigorous manner or overextend its application it appears that the distinction simply does not apply in some areas or contexts or that we cannot fully specify how it should be drawn An analogy might be that zooming a telescope camera or microscope in and out reveals that a pattern which is sharply focused at a certain distance becomes blurry at another distance or disappears altogether Complexity and imprecision Faced with any large complex and continually changing phenomenon any short statement made about that phenomenon is likely to be fuzzy i e it is meaningful but strictly speaking incorrect and imprecise It will not really do full justice to the reality of what is happening with the phenomenon A correct precise statement would require a lot of elaborations and qualifiers Nevertheless the fuzzy description turns out to be a useful shorthand that saves a lot of time in communicating what is going on you know what I mean Cognition and perceptual limits In psychophysics it was discovered that the perceptual distinctions we draw in the mind are often more definite than they are in the real world Thus the brain actually tends to sharpen up or enhance our perceptions of differences in the external world Between black and white we are able to detect only a limited number of shades of gray or colour gradations there are detection thresholds Motion blur refers to the loss of detail when a person looks at a fast moving object or is moving fast while the eyes are focused on something stationary In a movie reel the human eye can detect a sequence of up to 10 or 12 still images per second At around 18 to 26 frames per second the brain will see the sequence of individual images as a moving scene If there are more gradations and transitions in reality than our conceptual or perceptual distinctions can capture in our minds then it could be argued that how those distinctions will actually apply must necessarily become vaguer at some point Imprecision of novelty In interacting with the external world the human mind may often encounter new or partly new phenomena or relationships which cannot yet be sharply defined given the background knowledge available and by known distinctions associations or generalizations Crisis management plans cannot be put on the fly after the crisis occurs At the outset information is often vague even contradictory Events move so quickly that decision makers experience a sense of loss of control Often denial sets in and managers unintentionally cut off information flow about the situation L Paul Bremer Fuzziness and chaos It also can be argued that fuzzy concepts are generated by a certain sort of lifestyle or way of working which evades definite distinctions makes them impossible or inoperable or which is in some way chaotic To obtain concepts which are not fuzzy it must be possible to test out their application in some way But in the absence of any relevant clear distinctions lacking an orderly environment or when everything is in a state of flux or in transition it may not be possible to do so so that the amount of fuzziness increases Everyday occurrenceFuzzy concepts often play a role in the creative process of forming new concepts to understand something In the most primitive sense this can be observed in infants who through practical experience learn to identify distinguish and generalise the correct application of a concept and relate it to other concepts However fuzzy concepts may also occur in scientific journalistic programming and philosophical activity when a thinker is in the process of clarifying and defining a newly emerging concept which is based on distinctions which for one reason or another cannot yet be more exactly specified or validated Fuzzy concepts are often used to denote complex phenomena or to describe something which is developing and changing which might involve shedding some old meanings and acquiring new ones Uses in different areas In meteorology where changes and effects of complex interactions in the atmosphere are studied the weather reports often use fuzzy expressions indicating a broad trend likelihood approximation or level The main reason is that the forecast can rarely be totally exact for any given location Nevertheless the information is often useful to orient behaviour e g I must not forget to take my coat or an umbrella just in case In biology protein complexes with multiple structural forms are called fuzzy complexes The different conformations can result in different even opposite functions The conformational ensemble is modulated by the environmental conditions Post translational modifications or alternative splicing can also impact the ensemble and thereby the affinity or specificity of interactions Genetic fuzzy systems use algorithms or genetic programming which simulate natural evolutionary processes in order to understand their structures and parameters In medical diagnosis the assessment of what the symptoms of a patient are often cannot be very exactly specified since there are many possible qualitative and quantitative gradations in severity incidence or frequency that could occur Different symptoms may also overlap to some extent These gradations can be difficult to measure it may cost a lot of time and money and so the medical professionals might use approximate fuzzy categories in their judgement of a medical condition or a patient s condition Although it may not be exact the diagnosis is often useful enough for treatment purposes Fuzzy logic is increasingly employed in diagnostic and medical equipment capable of measuring gradations of a condition In information services fuzzy concepts are frequently encountered because a customer or client asks a question about something which could be interpreted in different ways or a document is transmitted of a type or meaning which cannot be easily allocated to a known type or category or to a known procedure It might take considerable inquiry to place the information or establish in what framework it should be understood In phenomenology which aims to study the structure of subjective experience without preconceptions an important insight is that how someone experiences something can be shaped both by the influence of the thing being experienced itself but also by how the person responds to it Thus the actual experience the person has is shaped by an interactive object subject relationship To describe this experience fuzzy categories are often necessary since it is often impossible to predict or describe with great exactitude what the interaction will be and how it is experienced In translation work fuzzy concepts are analyzed for the purpose of good translation A concept in one language may not have quite the same meaning or significance in another language or it may not be feasible to translate it literally or at all Some languages have concepts which do not exist in another language raising the problem of how one would most easily render their meaning In computer assisted translation a technique called fuzzy matching is used to find the most likely translation of a piece of text using previous translated texts as a basis In hypnotherapy fuzzy language is deliberately used for the purpose of trance induction Hypnotic suggestions are often couched in a somewhat vague general or ambiguous language requiring interpretation by the subject The intention is to distract and shift the conscious awareness of the subject away from external reality to her own internal state In response to the somewhat confusing signals she gets the awareness of the subject spontaneously tends to withdraw inward in search of understanding or escape In business and economics it was discovered that we are guided less by a correct exact knowledge of our self interest than by a socially learned evolved intuitive grasp derived from mental shortcuts frames reference points envy addiction temptation fairness Thus economic preferences are often fuzzy preferences a highly important point for suppliers of products and services Fuzzy set empirical methodologies are increasingly used by economic analysts to analyze the extent to which members of a population belong to a specific market category because that can make a big difference to business results In sexology sex and gender are conceptualized by gender pluralists as a spectrum or continuum or a set of scaled characteristics Thus the idea that people are either heterosexual men heterosexual women gay lesbian bisexual or transsexual is far too simplistic gender identity is a matter of degree a graded concept which for that very reason is a fuzzy concept with unsharp boundaries For example somebody who is mainly heterosexual may occasionally have had non heterosexual contacts without this warranting a definite bisexual label A great variety of sexual orientations are possible and can co exist In the course of history typical male or female gender roles and gender characteristics can also gradually change so that the extent to which they express masculine or feminine traits is at any time a matter of degree i e fuzzy In politics it can be highly important and problematic how exactly a conceptual distinction is drawn or indeed whether a distinction is drawn at all distinctions used in administration may be deliberately sharpened or kept fuzzy due to some political motive or power relationship Politicians may be deliberately vague about some things and very clear and explicit about others if there is information that proves their case they become very precise but if the information doesn t prove their case they become vague or say nothing In statistical research it is an aim to measure the magnitudes of phenomena For this purpose phenomena have to be grouped and categorized so that distinct and discrete counting units can be defined It must be possible to allocate all observations to mutually exclusive categories so that they are properly quantifiable Survey observations do not spontaneously transform themselves into countable data they have to be identified categorized and classified in such a way that identical observations can be grouped together and that observations are not counted twice or more A well designed questionnaire ensures that the questions are interpreted in the same way by all respondents and that the respondents are really able to answer them within the formats provided Again for this purpose it is a requirement that the concepts being used are exactly and comprehensibly defined for all concerned and not fuzzy There could be a margin of measurement error but the amount of error must be kept within tolerable limits and preferably its magnitude should be known In theology an attempt is made to define more precisely the meaning of spiritual concepts which refer to how human beings construct the meaning of human existence and often the relationship people have with a supernatural world Many spiritual concepts and beliefs are fuzzy to the extent that although abstract they often have a highly personalized meaning or involve personal interpretation of a type that is not easy to define in a cut and dried way A similar situation occurs in psychotherapy The Dutch theologian Kees de Groot has explored the imprecise notion that psychotherapy is like an implicit religion defined as a fuzzy concept it all depends on what one means by psychotherapy and religion The philosopher of spirituality Ken Wilber argued that nothing is 100 right or wrong things merely vary in their degree of incompleteness and dysfunction no one and nothing is 100 good or evil each just varies in their degree of ignorance and disconnection This insight suggests that all human valuations can be considered as graded concepts where each qualitative judgement has at least implicitly a sense of quantitative proportion attached to it In the legal system it is essential that rules are interpreted and applied in a standard way so that the same sorts of cases and the same sorts of circumstances are treated equally Otherwise one would be accused of arbitrariness which would not serve the interests of justice Consequently lawmakers aim to devise definitions and categories which are sufficiently precise so that they are not open to different interpretations For this purpose it is critically important to remove fuzziness and differences of interpretation are typically resolved through a court ruling based on evidence Alternatively some other procedure is devised which permits the correct distinction to be discovered and made In administration archiving and accounting fuzziness problems in interpretation and boundary problems can arise because it is not clear to what category exactly a case item document transaction or piece of data belongs In principle each case event or item must be allocated to the correct category in a procedure but it may be that it is difficult to make the appropriate or relevant distinctions Generalities Many concepts which are used fairly universally in daily life such as love God health social sustainability tolerance etc are considered to be intrinsically fuzzy concepts to the extent that their meaning usually cannot be completely and exactly specified with logical operators or objective terms and can have multiple interpretations and personal subjective meanings Yet such concepts are not at all meaningless People keep using the concepts even if they are difficult to define precisely Multiple meanings It may also be possible to specify one personal meaning for the concept without however placing restrictions on a different use of the concept in other contexts as when for example one says this is what I mean by X in contrast to other possible meanings In ordinary speech concepts may sometimes also be uttered purely randomly for example a child may repeat the same idea in completely unrelated contexts or an expletive term may be uttered arbitrarily A feeling or sense is conveyed without it being fully clear what it is about Happiness may be an example of a word with variable meanings depending on context or timing Ambiguities Fuzzy concepts can be used deliberately to create ambiguity and vagueness as an evasive tactic or to bridge what would otherwise be immediately recognized as a contradiction of terms They might be used to indicate that there is definitely a connection between two things without giving a complete specification of what the connection is for some or other reason This could be due to a failure or refusal to be more precise But it could also be a prologue to a more exact formulation of a concept or to a better understanding of it EfficiencyFuzzy concepts can be used as a practical method to describe something of which a complete description would be an unmanageably large undertaking or very time consuming thus a simplified indication of what is at issue is regarded as sufficient although it is not exact Popper There is also such a thing as an economy of distinctions meaning that it is not helpful or efficient to use more detailed definitions than are really necessary for a given purpose In this sense Karl Popper rejected pedantry and commented that it is always undesirable to make an effort to increase precision for its own sake especially linguistic precision since this usually leads to loss of clarity and to a waste of time and effort on preliminaries which often turn out to be useless because they are bypassed by the real advance of the subject one should never try to be more precise than the problem situation demands I might perhaps state my position as follows Every increase in clarity is of intellectual value in itself an increase in precision or exactness has only a pragmatic value as a means to some definite end The provision of too many details could be disorienting and confusing instead of being enlightening while a fuzzy term might be sufficient to provide an orientation The reason for using fuzzy concepts can therefore be purely pragmatic if it is not feasible or desirable for practical purposes to provide all the details about the meaning of a shared symbol or sign Thus people might say I realize this is not exact but you know what I mean they assume practically that stating all the details is not required for the purpose of the communication Fuzzy logic gambit Lotfi A Zadeh picked up this point and drew attention to a major misunderstanding about applying fuzzy logic It is true that the basic aim of fuzzy logic is to make what is imprecise more precise Yet in many cases fuzzy logic is used paradoxically to imprecisiate what is precise meaning that there is a deliberate tolerance for imprecision for the sake of simplicity of procedure and economy of expression In such uses there is a tolerance for imprecision because making ideas more precise would be unnecessary and costly while imprecisiation reduces cost and enhances tractability tractability means being easy to manage or operationalize Zadeh calls this approach the Fuzzy Logic Gambit a gambit means giving up something now to achieve a better position later In the Fuzzy Logic Gambit what is sacrificed is precision in quantitative value but not precision in meaning and more concretely imprecisiation in value is followed by precisiation in meaning Zadeh cited as example Takeshi Yamakawa s programming for an inverted pendulum where differential equations are replaced by fuzzy if then rules in which words are used in place of numbers Fuzzy vs BooleanCommon use of this sort of approach combining words and numbers in programming has led some logicians to regard fuzzy logic merely as an extension of Boolean logic a two valued logic or binary logic is simply replaced with a many valued logic However Boolean concepts have a logical structure which differs from fuzzy concepts An important feature in Boolean logic is that an element of a set can also belong to any number of other sets even so the element either does or does not belong to a set or sets By contrast whether an element belongs to a fuzzy set is a matter of degree and not always a definite yes or no question All the same the Greek mathematician Costas Drossos suggests in various papers that using a non standard mathematical approach we could also construct fuzzy sets with Boolean characteristics and Boolean sets with fuzzy characteristics This would imply that in practice the boundary between fuzzy sets and Boolean sets is itself fuzzy rather than absolute For a simplified example we might be able to state that a concept X is definitely applicable to a finite set of phenomena and definitely not applicable to all other phenomena Yet within the finite set of relevant items X might be fully applicable to one subset of the included phenomena while it is applicable only to some varying extent or degree to another subset of phenomena which are also included in the set Following ordinary set theory this generates logical problems if e g overlapping subsets within sets are related to other overlapping subsets within other sets Clarifying methodsIn mathematical logic computer programming philosophy and linguistics fuzzy concepts can be analyzed and defined more accurately or comprehensively by describing or modelling the concepts using the terms of fuzzy logic or other substructural logics With the accelerating development of computer programming languages and computer processing capacity since the 1970s it has become more acceptable in the sciences that there isn t just one correct way to formalize items of knowledge It was realized by innovators that concepts and processes can be formalized and represented using many different kinds of tools methods and systems according to what is most useful effective or efficient for a given purpose Using software and artificial intelligence many traditional and new sorts of techniques can be applied to clarify ideas such as 1 Contextualizing the concept by defining the setting or situation in which the concept is used or how it is used appropriately context 2 Identifying the intention purpose aim or goal associated with the concept teleology and design 3 Comparing and contrasting the concept with related ideas in the present or the past comparative and comparative research 4 Creating a model likeness analogy metaphor prototype or narrative which shows what the concept is about or how it is applied isomorphism simulation or successive approximation 117 5 Probing the assumptions on which a concept is based or which are associated with its use critical thought tacit assumption 6 Mapping or graphing the applications of the concept using some basic parameters or using some diagrams or flow charts to understand the relationships between elements involved visualization and concept map 7 Examining how likely it is that the concept applies statistically or intuitively probability theory 8 Specifying relevant conditions to which the concept applies as a procedure computer programming formal concept analysis 9 Concretizing the concept finding specific examples illustrations details or cases to which it applies exemplar exemplification 10 Reducing or restating fuzzy concepts in terms which are simpler or similar and which are not fuzzy or less fuzzy simplification dimensionality reduction plain language KISS principle or concision 11 Trying out a concept by using it in interactions practical work or in communication and assessing the feedback to understand how the boundaries and distinctions of the concept are being drawn trial and error or pilot experiment 12 Engaging in a structured dialogue or repeated discussion to exchange ideas about how to get specific about what it means and how to clear it up scrum method 13 Allocating different applications of the concept to different but related sets Boolean logic 14 Identifying operational rules defining the use of the concept which can be stated in a language and which cover all or most cases material conditional 15 Classifying categorizing grouping or inventorizing all or most cases or uses to which the concept applies taxonomy cluster analysis and typology An operationalization diagram one method of clarifying fuzzy concepts 16 Applying a meta language which includes fuzzy concepts in a more inclusive categorical system which is not fuzzy meta 17 Creating a measure or scale of the degree to which the concept applies metrology 18 Examining the distribution patterns or distributional frequency of possibly different uses of the concept statistics 19 Specifying a series of logical operators or inferential system which captures all or most cases to which the concept applies algorithm 20 Relating the fuzzy concept to other concepts which are not fuzzy or less fuzzy or simply by replacing the fuzzy concept altogether with another alternative concept which is not fuzzy yet works the same way proxy 21 Engaging in meditation taking a pause to relax or taking the proverbial run around the block to clarify the mind and thus improve precision of thought about the definitional issue self care In this way we can obtain a more exact understanding of the meaning and use of a fuzzy concept and possibly decrease the amount of fuzziness It may not be possible to specify all the possible meanings or applications of a concept completely and exhaustively but if it is possible to capture the majority of them statistically or otherwise this may be useful enough for practical purposes DefuzzificationA process of defuzzification is said to occur when fuzzy concepts can be logically described in terms of fuzzy sets or the relationships between fuzzy sets which makes it possible to define variations in the meaning or applicability of concepts as quantities Effectively qualitative differences are in that case described more precisely as quantitative variations or quantitative variability Assigning a numerical value then denotes the magnitude of variation along a scale from zero to one The difficulty that can occur in judging the fuzziness of a concept can be illustrated with the question Is this one of those If it is not possible to clearly answer this question that could be because this the object is itself fuzzy and evades definition or because one of those the concept of the object is fuzzy and inadequately defined Thus the source of fuzziness may be in 1 the nature of the reality being dealt with 2 the concepts used to interpret it or 3 the way in which the two are being related by a person It may be that the personal meanings which people attach to something are quite clear to the persons themselves but that it is not possible to communicate those meanings to others except as fuzzy concepts See alsoAlternative set theory Approximate measures Classical logic Defuzzification Detection theory Deviant logic Dialectic European Society for Fuzzy Logic and Technology Fuzzy subalgebra Fuzzy logic Fuzzy set theory Fuzzy clustering Fuzzy mathematics Fuzzy measure theory Fuzzy set operations George Klir Identity Philosophy Interval finite element Jakobson s functions of language Linear partial information Many valued logic Multiset Neuro fuzzy Non well founded set theory Obfuscation Opaque context Paraconsistent logic Phenomenology psychology Precision Referential transparency reflexivity social theory Post normal science Rough fuzzy hybridization Rough set Semiset Sorensen similarity index Synchronicity Type 2 Fuzzy Sets and Systems Uncertainty Vague setReferencesRadim Behlohlavek amp George J Klir eds Concepts and fuzzy logic Cambridge Mass MIT Press 2011 Susan Haack Deviant logic fuzzy logic beyond the formalism Chicago University of Chicago Press 1996 Richard Dietz amp Sebastiano Moruzzi eds Cuts and clouds Vagueness Its Nature and Its Logic Oxford University Press 2009 Delia Graff amp Timothy Williamson eds Vagueness London Routledge 2002 For some ordinary scientific definitions of precision and imprecision see the articles on significant figures accuracy and precision precision statistics and precision computer science Timothy Williamson Vagueness London Routledge 1994 p 124f Lotfi A Zadeh Quantitative fuzzy semantics Information Sciences Vol 3 No 2 April 1971 pp 159 176 D Blockley Earthquake risk management of civil infrastructure integrating soft and hard risks in Handbook of Seismic Risk Analysis and Management of Civil Infrastructure Systems Sawston Cambridge Woodhead Publishing 2013 chapter 9 pp 229 254 at p 238 Vyvyan Evans A glossary of cognitive linguistics Salt Lake City University of Utah Press 2007 p 88 Susan Haack Do we need fuzzy logic in International Journal of Man Machine Studies Volume 11 Issue 4 1979 pp 437 45 Lotfi A Zadeh Is there a need for fuzzy logic Information Sciences Volume 178 Issue 13 July 2008 pp 2751 2779 Bart Kosko Fuzzy logic In Scientific American July 1993 pp 76 81 1 Bart Kosko Fuzzy Thinking The New Science of Fuzzy Logic New York Hyperion 1993 Bart Kosko Heaven in a chip fuzzy visions of society and science in the digital age New York Three Rivers Press 1999 Daniel McNeill amp Paul Freiberger Fuzzy Logic The Revolutionary Computer Technology that Is Changing Our World New York Simon amp Schuster 1994 Charles Elkan The paradoxical success of fuzzy logic IEEE Expert August 1994 2 Didier Dubois et al Fuzzy set based logics an history oriented presentation of their main developments in Handbook of the history of logic Volume 8 The many valued and non monotonic turn in logic Amsterdam Elsevier North Holland 2007 pp 3 125 3 Didier Dubois Henri Prade Articles written on the occasion of the 50th anniversary of fuzzy set theory Institut de Recherche Informatique de Toulouse 2015 4 Radim Belohlavek Joseph W Dauben amp George J Klir Fuzzy Logic and Mathematics A Historical Perspective Oxford University Press 2017 Didier Dubois et al Fuzzy set based logics an history oriented presentation of their main developments in Handbook of the history of logic Volume 8 The many valued and non monotonic turn in logic Amsterdam Elsevier North Holland 2007 pp 3 125 5 Didier Dubois Henri Prade Articles written on the occasion of the 50th anniversary of fuzzy set theory Institut de Recherche Informatique de Toulouse 2015 https hal science hal 03198270v1 file Fuzzy sets 50 pdf Katyanna Quach Fuzzy logic makes a comeback in picking where Earth sticks its probes into alien worlds The Register 27 Sep 2018 6 Liwei Yang et al Path Planning Technique for Mobile Robots A Review Machines Vol 11 2023 pp 980 1026 7 Lotfi A Zadeh Fuzzy logic neural networks and soft computing In Communications of the ACM Volume 37 Issue 3 March 1994 pp 77 84 Artificial neural networks an overview in George J Klir amp Bo Yuan Fuzzy sets and fuzzy logic Theory and applications Upper Saddle River NJ Prentice Hall 1995 pp 467 475 A useful technical overview is provided in Enrique Ruspini et al Handbook of fuzzy computation Bristol amp Philadelphia Institute of Physics Publishing 1998 Radim Behlohlavek amp George J Klir eds Concepts and fuzzy logic Cambridge Mass MIT Press 2011 Edy Portmann Fuzzy humanist Wiesbaden Springer 2019 Mahdi Eftekhari et al How fuzzy concepts contribute to machine learning Cham Switzerland Springer 2022 Rudolf Seising et al On fuzziness homage to Lotfi A Zadeh Vol 2 Heidelberg Springer 2013 p 656 Ellen Christiaanse 1 5 million years of information systems from hunters gatherers to the domestication of the networked computer In David Avison et al The past and future of information systems 1976 2006 and beyond New York IFIP Springer 2006 pp 165 176 Kit Fine Vagueness a global approach New York Oxford University Press 2020 chapter 1 Steve Coutinho Zhuangzi and Early Chinese Philosophy Vagueness Transformation and Paradox Abingdon Routledge 2016 p 17 Rosanna Keefe amp Peter Smith Vagueness a reader Cambridge Mass MIT Press 1996 Petr Hajek and Vilem Novak The sorites paradox and fuzzy logic International Journal of General Systems Vol 32 No 4 2003 pp 373 383 High Altitude Cooking webpage of the USDA Food Safety and Inspection Service 8 Massimo Pigliucci amp Maarten Boudry eds Philosophy of Pseudoscience Reconsidering the Demarcation Problem University of Chicago Press 2013 p 95 Nicholas Rescher Many Valued Logic New York McGraw Hill 1969 The paradox of the heap in John L Bell Oppositions and Paradoxes Philosophical Perplexities in Science and Mathematics Peterborough Ontario Broadview Press 2016 pp 158 160 9 Julia Andrina Greig The Vagueness of Dying in Epicurean Thought A Stoic Remedy Masters thesis Graduate School of Arts and Sciences Brandeis University May 2021 See for example Plato s version of the puzzle of temporal boundaries in Parmenides 156c e When an object begins to move or a moving object comes to rest does the transitional moment belong to the motion interval or to the rest interval as noted in Boundary article in Stanford Encyclopedia of Philosophy online 2023 10 Marcus Tullius Cicero Academica Book 2 written 45 BC in H Rackham transl Cicero De Natura Deorum and Academica Cambridge Massachusetts Harvard University Press 1933 11 Lisa Cordes Who speaks Ambiguity and Vagueness in the Design of Cicero s Dialogue Speakers In Martin Vohler et al eds Strategies of Ambiguity in Ancient Literature Berlin Boston De Gruyter 2021 pp 297 314 Robert L Cioffi Fuzzy Math The Place of Numerical Evidence in Cicero In Verrem 3 116 Mnemosyne Vol 64 No 4 2011 pp 645 652 Angelica Nuzzo Vagueness and Meaning Variance in Hegel s Logic In Angelica Nuzzo Hegel and the analytical tradition New York Continuum International Publishing Group 2010 pp 61 82 Robert L Carneiro The transition from quantity to quality a neglected causal mechanism in accounting for social evolution Proceedings of the National Academy of Sciences of the United States of America PNAS Vol 97 No 23 7 November 2000 pp 12926 12931 12 Eric Steinhart Nietzsche on identity Revista di Estetica Vol 28 No 1 2005 pp 241 256 13 Steven D Hales Nietzsche on Logic Philosophy and Phenomenological Research Vol 56 No 4 December 1996 pp 819 835 Wilhelm Magnus The Significance of Mathematics The Mathematicians Share in the General Human Condition The American Mathematical Monthly Vol 104 No 3 March 1997 pp 261 269 at p 263 William Joseph Gavin William James and the Reinstatement of the Vague Philadelphia Temple University Press 1992 chapter 3 S Rahman amp J Redmond Hugh MacColl and the Birth of Logical Pluralism In Handbook of the History of Logic Vol 4 Elsevier 2008 pp 533 604 Mihai Nadin The logic of vagueness in Eugene Freeman ed The Relevance of Charles Peirce La Salle Ill Open Court 1983 pp 154 166 Carl Gustav Hempel Vagueness and logic Philosophy of Science Vol 6 Issue 2 April 1939 pp 163 180 Max Black Vagueness An exercise in logical analysis Philosophy of Science Vol 4 1937 pp 427 455 Max Black Reasoning with Loose Concepts In Canadian Philosophical Review Volume 2 Issue 1 June 1963 pp 1 12 Radomir S Stankovic amp Jaakko T Astola Contributions of Arto Salomaa to Multiple Valued Logic 41st IEEE International Symposium on Multiple Valued Logic 23 25 May 2011 pp 198 204 Arto Salomaa On Many valued Systems of Logic Ajatus 22 1959 pp 138 145 also published in Journal of Symbolic Logic Vol 25 Issue 3 1960 pp 291 293 Ludwig Wittgenstein Philosophical Investigations Oxford University Press 1953 Part 1 sections 65 88 Jan Lukasiewicz On three valued logic In Jan Lukasiewicz Selected Works Amsterdam North Holland Publishing Company 1970 pp 87 88 Emil Leon Post Introduction to a general theory of elementary propositions American Journal of Mathematics Vol 43 No 3 July 1921 p 163 185 Alfred Tarski Logic semantics metamathematics Oxford Oxford University Press 1956 James F Peters and Sankar K Pal Cantor Fuzzy Near and Rough Sets in Image Analysis In Sankar K Pal and James F Peters eds Rough Fuzzy Image Analysis Foundations and Methodologies Routledge 2017 chapter 1 14 Valentine Bazhanov The fate of one forgotten idea N A Vasiliev and his imaginary logic Studies in Soviet Thought Vol 39 No 3 1990 pp 333 341 15 Archived 2006 07 19 at the Wayback Machine Tim Lethen Godel on many valued logic The review of Symbolic Logic Vol 16 issue 3 September 2023 pp 655 671 Susan Haack notes that Stanislaw Jaskowski provided axiomatizations of many valued logics in Jaskowski On the rules of supposition in formal logic Studia Logica No 1 1934 16 See Susan Haack Philosophy of Logics Cambridge University Press 1978 p 205 W V Quine Speaking of Objects Proceedings and Addresses of the American Philosophical Association Vol 31 1957 1958 pp 5 22 at p 20 Petr Hajek Metamathematics of fuzzy logic Dordrecht Springer 1998 Joseph Goguen The logic of inexact concepts Synthese Vol 19 No 3 4 1969 pp 325 373 Radim Belohlavek Pavelka style fuzzy logic in retrospect and prospect Fuzzy Sets and Systems Vol 281 15 December 2015 pp 61 72 17 Radim Belohlavek Systems uncertainty and information A legacy of George J Klir International Journal of General Systems Volume 46 Issue 8 2017 pp 792 823 Didier Dubois and Henri Prade Fuzzy sets and systems Theory and applications New York Academic Press New York 1980 Priyanka Kaushal Neeraj Mohan and Parvinder S Sandhu Relevancy of Fuzzy Concept in Mathematics International Journal of Innovation Management and Technology Vol 1 No 3 August 2010 18 Didier Dubois et al Fuzzy set based logics a history oriented presentation of their main developments In Dov M Gabbay and John Woods eds Handbook of the History of Logic Volume 8 19 Abraham Kaplan and Hermann F Schott A calculus for empirical classes Methodos Vol 3 1951 pp 165 188 Timothy Williamson Vagueness London Routledge 1996 p 120 J Barkley Rosser Sr and Atwell R Turquette Many valued logics Amsterdam North Holland Publishing Company 1952 p 109 Aleksandr A Zinov ev David Dinsmore Comey and Guido Kung Philosophical problems of many valued logic Dordrecht D Reidel 1963 William P Alston Philosophy of Language Englewood Cliffs N J Prentice Hall 1964 p 87 William P Alston Vagueness in Paul Edwards ed Encyclopedia of Philosophy vol 8 New York Macmillan first edition 1967 pp 218 221 William P Alston A Realist Conception of Truth Ithaca Cornell University Press 1996 p 62 Dieter Klaua Uber einen Ansatz zur mehrwertigen Mengenlehre Monatsberichte der Deutschen Akademie der Wissenschaften Berlin Vol 7 pp 859 867 1965 Siegfried Gottwald An early approach toward graded identity and graded membership in set theory Fuzzy Sets and Systems Vol 161 Issue 18 September 2010 pp 2369 2379 Siegfried Gottwald Shaping the logic of fuzzy set theory In Cintula Petr et al eds Witnessed years Essays in honour of Petr Hajek London College Publications 2009 pp 193 208 20 Archived 2012 10 01 at the Wayback Machine Robert John Ackermann An introduction to many valued logics London Routledge amp Kegan Paul 1967 Nicholas Rescher Many Valued Logic New York McGraw Hill 1969 Robert G Wolf A survey of many valued logic 1966 1974 in J Michael Dunn and George Epstein eds Modern Uses of Multiple Valued Logic Dordrecht D Reidel 1977 167 323 Joseph L F De Kerf A bibliography on fuzzy sets In Journal of Computational and Applied mathematics vol 1 no 3 1975 pp 206 212 Susan Haack Deviant logic fuzzy logic beyond the formalism Chicago University of Chicago Press 1996 Didier Dubois and Henri Prade Fuzzy sets and systems Theory and applications New York Academic Press New York 1980 393 pp reviewed by Ernest G Manes in Bulletin of the American Mathematical Society Vol 7 No 3 November 1982 21 George J Klir amp Bo Yuan Fuzzy sets and fuzzy logic Theory and applications Upper Saddle River NJ Prentice Hall 1995 Merrie Bergmann An Introduction to Many Valued and Fuzzy Logic Semantics Algebras and Derivation Systems Cambridge University Press 2008 Radim Belohlavek Joseph W Dauben amp George J Klir Fuzzy Logic and Mathematics A Historical Perspective Oxford University Press 2017 Lotfi A Zadeh June 1965 Fuzzy sets PDF Information and Control 8 3 338 353 doi 10 1016 S0019 9958 65 90241 X Archived from the original PDF on 2007 11 27 Retrieved 2007 11 06 See also E Trillas Lotfi A Zadeh On the man and his work Scientia Iranica Volume 18 Issue 3 June 2011 pp 574 579 22 A Dumitras amp G Moschytz Understanding Fuzzy Logic An Interview with Lotfi Zadeh IEEE signal processing magazine May 2007 pp 102 105 Radim Belohlavek Joseph W Dauben amp George J Klir Fuzzy Logic and Mathematics A Historical Perspective Oxford University Press 2017 Lotfi A Zadeh with George J Klir and Bo Yua Fuzzy Sets Fuzzy Logic and Fuzzy Systems Selected Papers Singapore and River Edge N J World Scientific Publishing Company 1996 This last title includes a bibliography of Zadeh s writings up to 1996 Rudolf Seising Cybernetics system s theory information theory and Fuzzy Sets and Systems in the 1950s and 1960s Information Sciences No 180 2010 pp 4459 4476 IFSA Newsletter International Fuzzy Systems Association Vol 10 No 1 March 2013 23 Richard Van Noorden Brendan Maher amp Regina Nuzzo The top 100 papers Nature 29 October 2014 24 Radim Belohlavek Joseph W Dauben amp George J Klir Fuzzy Logic and Mathematics A Historical Perspective Oxford University Press 2017 Radim Belohlavek What is a fuzzy concept lattice II in Sergei O Kuznetsov et al eds Rough sets fuzzy sets data mining and granular computing Berlin Springer Verlag 2011 pp 19 20 25 The vast majority of scientific or scholarly users of the idea of fuzzy concepts refer to scaled graded characteristics and not to the variations in the likelihoods of their applicability A probabilistic interpretation of concepts is discussed in Edward E Smith amp Douglas L Medin Categories and concepts Cambridge Harvard University Press 1981 Nikolaos Galatos Peter Jipsen Tomasz Kowalski amp Hiroakira Ono Residuated lattices an algebraic glimpse at substructural logics Elsevier Science 2007 Joseph A Goguen L fuzzy Sets Journal of Mathematical Analysis and Applications Vol 18 1967 pp 145 174 Radim Belohlavek Goguen s contributions to fuzzy logic in retrospect International Journal of General Systems Volume 48 issue 8 2019 pp 811 824 at p 817 26 Petr Hajek Metamathematics of fuzzy logic Dordrecht Springer 1998 p 2 Lotfi A Zadeh The Concept of a Linguistic Variable and Its Application to Approximate Reasoning 1 Information Sciences Vol 8 pp 199 249 1975 Jerry M Mendel and Robert I Bob John Type 2 Fuzzy Sets Made Simple IEEE transactions on fuzzy systems Vol 10 No 2 April 2002 pp 117 127 Jerry M Mendel Advances in type 2 fuzzy sets and systems In Information Sciences 177 2007 pp 84 110 27 Timothy Williamson Vagueness New York London Routledge 1994 Delia Graff Fara Shifting Sands An Interest Relative Theory of Vagueness Philosophical Topics Vol 28 No 1 2000 pp 45 81 Deleuze and Guattari A Thousand Plateaus 1988 551 Deleuze and Guattari What Is Philosophy New York Columbia University Press 1994 p 141 Roy T Cook A dictionary of philosophical logic Edinburgh University Press 2009 p 84 Nicholas Rescher Many valued logic New York McGraw Hill 1969 Alasdair Urquhart Basic Many Valued Logic In D M Gabbay F Guenthner eds Handbook of Philosophical Logic HALO Vol 2 Heidelberg Springer 2001 pp 249 295 Susan Haack Philosophy of Logics Cambridge University Press 1978 p xii Susan Haack Philosophy of Logics Cambridge University Press 1978 p 165 Petr Hajek Ten questions and one problem on fuzzy logic Annals of Pure and Applied Logic Vol 96 Issues 1 3 March 1999 157 165 at p 162 Kazuo Tanaka An Introduction to Fuzzy Logic for Practical Applications Springer 1996 Constantin Zopounidis Panos M Pardalos amp George Baourakis Fuzzy Sets in Management Economics and Marketing Singapore World Scientific Publishing Co 2001 Humberto Bustince et al eds Fuzzy Sets and Their Extensions Representation Aggregation and Models Intelligent Systems from Decision Making to Data Mining Web Intelligence and Computer Vision Berlin Springer 2008 Samsung support information page 28 Lotfi Zadeh Coping with the imprecision of the real world interview Communications of the ACM Vol 27 No 4 1 April 1984 pp 304 311 Stosberg Mark 16 December 1996 The Role of Fuzziness in Artifical sic Intelligence Minds and Machines Archived from the original on 20 May 2013 Retrieved 19 April 2013 Irem Dikmen M Talat Birgonal and Sedat Han Using fuzzy risk assessment to rate cost overrun risk in international construction projects International Journal of Project Management Vol 25 No 5 July 2007 pp 494 505 Fa Liang Gao A new way of predicting cement strength Fuzzy logic Cement and Concrete Research Volume 27 Issue 6 June 1997 Pages 883 888 2017 Golden Goose Awardee Fuzzy Logic Clear Impact Archived from the original on 2019 12 13 Retrieved 2018 03 12 Michio Sugeno ed Industrial applications of fuzzy control Amsterdam North Holland 1992 Andrew Pollack Technology Fuzzy Logic For Computers New York Times 11 October 1984 Andrew Pollack Fuzzy Computer Theory How to Mimic the Mind New York Times 2 April 1989 The IFSA URL is https fuzzysystems org Cf IEEE CIS website 2018 29 Archived 2018 04 02 at the Wayback Machine The URL is https www fsdmconf org See the APNNS website 30 INFUS URL See the SOFT website 31 KIIS website 32 Yingming Liu Guoqing Chen and Mingshen Ying eds Fuzzy logic soft computing and computational intelligence Eleventh International Fuzzy Systems Association World Congress July 28 31 2005 Beijing China Volume III Beijing Tsinghua University Press Springer Verlag 2005 p viii and IFSA website data The TFSA publishes the International Journal of Fuzzy Systems The NAFIPS website URL is https sites ualberta ca reformat nafips index html The EUSFLAT URL is http www eusflat org Mathfuzzlog url is https www eusflat org society wg mathfuzzlog html The NSAIS URL is https nsais org IFSS website 33 Iran ranks 3rd in producing fuzzy systems related science Official The Iran Project 9 March 2017 34 RAFSSoftCom Ildar Batyrshin A retrospective glance from Russia at wonderland of fuzziness In Rudolf Seising et al eds On Fuzziness A Homage to Lotfi A Zadeh Volume 1 Berlin Springer 2013 pp 33 38 CBSF page 35 36 CSCR website SLAAI website 37 Laszlo T Koczy Fuzzy systems research in Hungary a subjective story Archives for the philosophy and history of soft computing Issue 1 2017 38 Indonesian society for soft computing repository site Vietnam Fuzzy Systems Society Lotfi A Zadeh Factual Information about the Impact of Fuzzy Logic Berkeley Initiative in Soft Computing at Electrical Engineering and Computer Sciences Department University of Berkeley California circa 2014 39 This text was taken offline after Zadeh died Cade Metz Lotfi Zadeh Father of Mathematical Fuzzy Logic Dies at 96 New York Times 11 September 2017 For an autobiographical sketch see Lotfi Zadeh My life and work a retrospective view Applied Computational and Mathematics Vol 10 No 1 special issue on fuzzy set theory and applications 2011 pp 4 9 Andrei Popescu A general approach to fuzzy concepts Mathematical Logic Quarterly Vol 50 No 3 2005 pp 265 280 Radim Belohlavek and Vilem Vychodil What is a fuzzy concept lattice Department of Computer Science Palacky University Olomouc 2005 40 See further the COMPASSS site Archived from the original on 2017 01 01 Retrieved 2016 12 31 Daniel Kreiss Prototype Politics Technology Intensive Campaigning and the Data of Democracy Oxford University Press 2016 E g Mikael Collan Mario Fedrizzi Janusz Kacprzyk Fuzzy Technology Present Applications and Future Challenges Heidelberg Springer 2016 p 65f Daniel J Lewis and Trevor P Martin Managing Vagueness with Fuzzy in Hierarchical Big Data Procedia Computer Science Volume 53 2015 pages 19 28 41 Chris Preimesberger Big Data Analytics Plays Big Role in 2016 Election Campaigns eWeek 24 September 2016 42 permanent dead link Gregory Thomas The Big Data Advantage in the Race for the White House Bemyapp Media 2 September 2016 43 Archived 2016 11 18 at the Wayback Machine Alex Woodie Why Winning Politics Is Now Tied to Big Data Analytics Datanami com 10 May 2016 44 Lisa Ragusa And the Winner of the 2016 Election Is Big Data Liaison 4 November 2016 45 John Markman Big Data And The 2016 Election Forbes Magazine 8 August 2016 46 Taylor Armerding Big Data and elections The candidates know you better than you know them CSOonline com 17 July 2016 47 Robert Epstein How Google Could Rig the 2016 Election Politico com 19 August 2015 48 Marcel Rosenbach How Google and Facebook Can Reshape Elections Der Spiegel online English edition 8 November 2016 49 For example Kyle C Longest and Stephen Vaisey Fuzzy A program for Performing Qualitative Comparative Analyses QCA in Stata Stata Journal Vol 8 No 1 2008 pp 79 104 Gregory Viot Fuzzy logic in C Dr Dobb s journal 1 February 1993 50 Chris Preimesberger Big Data Analytics Plays Big Role in 2016 Election Campaigns eWeek 24 September 2016 Kenneth P Vogel The heiress quietly shaping Trump s operation Politico com 21 November 2016 51 Kate Brannely Trump Campaign Pays Millions to Overseas Big Data Firm NBC News 4 November 2016 52 Cambridge Analytica About Us Cambridge Analytica website Archived from the original on 2016 02 16 Carole Cadwalladr UK regulator orders Cambridge Analytica to release data on US voter The Guardian 5 May 2018 53 Adam Tanner Nine Things You Don t Know About The Gathering Of Your Personal Data Forbes Magazine 4 November 2014 54 Steve Lohr and Natasha Singernov How Data Failed Us in Calling an Election New York Times 10 November 2016 55 How Trump won the presidency Interview of Gerald F Seib with Kellyanne Conway Wall Street Journal WSJ CEO Council full interview video 14 November 2016 56 See also Jonathan Vanian How Bad Polling Data Fooled Everyone Except Donald Trump Fortune 10 November 2016 57 How Trump won the presidency Interview of Gerald F Seib with Kellyanne Conway Wall Street Journal WSJ CEO Council full interview video 14 November 2016 58 Ryan Lizza Kellyanne Conway s political machinations The New Yorker 17 October 2016 59 see also alternative facts Jina Moore Cambridge Analytica Had a Role in Kenya Election Too New York Times 20 March 2018 60 Carole Cadwalladr amp Emma Graham Harrison Pressure mounts on Cambridge Analytica and Facebook over data scandal The Guardian 18 March 2018 61 Jonathan Shieber Facebook hired a forensics firm to investigate Cambridge Analytica as stock falls 7 TC Techcrunch com 19 March 2018 62 Spencer Phade Who Should Profit From Selling Your Personal Data Futures Platform 27 March 2018 63 David McLaughlin Ben Brody and Billy House FTC Probing Facebook for Use of Personal Data Source Says Bloomberg 20 March 2018 Tony Romm and Craig Timberg FTC opens investigation into Facebook after Cambridge Analytica scrapes millions of users personal information Washington Post 20 March 2018 64 Sara Salinas Facebook hires firm to conduct a comprehensive audit of Cambridge Analytica CNBC news 19 March 2018 and squawkbox panel video 65 Christopher Carbone Facebook might have 29 000 data points on you but Mark Zuckerberg doesn t really know Fox News 11 April 2018 66 67 Julia Angwin Surya Mattu and Terry Parris Jr Facebook Doesn t Tell Users Everything It Really Knows About Them ProPublica 27 December 2016 68 Zuckerberg s Senate hearings highlights video 10 April 2018 Olivia Solon and Julia Carrie Wong Facebook suspends another analytics firm amid questions over surveillance The Guardian 20 July 2018 69 Richard Waters Is Facebook a victim of rapid growth or an abuser of user data Financial Times 20 December 2018 Matthew Rosenberg Nicholas Confessore and Carole Cadwalladr How Trump Consultants Exploited the Facebook Data of Millions New York Times 17 March 2018 Sarah Frier and Todd Shields Zuckerberg Says Facebook Collects Internet Data on Non Users Bloomberg 11 April 2018 Gabriel J X Dance Michael LaForgia and Nicholas Confessore As Facebook Raised a Privacy Wall It Carved an Opening for Tech Giants New York Times 18 December 2018 70 Heather Wallace International Regulation of Social Media A Survey Pace University Social Media Legality page October 18 2023 71 Robert Gorwa The politics of platform regulation How governments shape online content moderation New York Oxford University Press 2024 72 See the Meta Legal Action website Rebecca Ballhaus and Jenny Gross Cambridge Analytica Closing Operations Following Facebook Data Controversy Wall Street Journal 2 May 2018 Munsif Vengattil Cambridge Analytica and parent SCL Elections shutting down Reuters 2 May 2018 73 Tableau com Big data the top 8 trends for 2016 Y S Yossi Sariel The human machine team how to create synergy between human amp artificial intelligence that will revolutionize our world Tel Aviv eBookPro Publishing 2021 Yasmeen Serhan How Israel Uses AI in Gaza And What It Might Mean for the Future of Warfare In Time Magazine 18 December 2024 74 See also AI assisted targeting in the Gaza Strip Muhammet Deveci et al Survey on military operations of fuzzy set theory and its applications Journal of Naval Sciences and Engineering Vol 16 No 2 2020 pp 117 141 75 Lakshya Tyagi amp Swati Singal Application of Fuzzy Logic Control Systems in Military Platforms Noida India IEEE 9th International conference on cloud computing data science amp engineering confluence 2019 Michael O Hagan From military to medical and commercial applications of neural networks and fuzzy logic a modern swords into plowshares play Proceedings of IEEE WESCON 93 conference San Francisco 28 30 Sept 1993 Republished in IEEE Xplore 6 August 2002 Radim Belohlavek George J Klir Harold W Lewis III Eileen C Way Concepts and fuzzy sets Misunderstandings misconceptions and oversights International Journal of Approximate Reasoning Vol 51 July 2009 pp 23 34 76 permanent dead link Angel Garrido amp Piedad Yuste controversies about the introduction of non classical logics Brain Vol 5 No 1 4 2014 77 Bob Pease What s All This Fuzzy Logic Stuff Anyhow Electronicdesign com May 13 1993 November 2020 five parts 78 Lotfi A Zadeh Is there a need for fuzzy logic Information Sciences No 178 2008 p 2753 For the debate between Zadeh and Kalman see Lotfi A Zadeh The birth and evolution of fuzzy logic International Journal of General Systems Vol 17 No 2 3 1990 pp 95 105 See also Yucel Yuksel On Zadeh s The Birth and Evolution of Fuzzy Logic In Eyke Hullermeier Rudolf Kruse amp Frank Hoffmann eds Information Processing and Management of Uncertainty in Knowledge Based Systems Applications Proceedings of the 13th International Conference IPMU 2010 Dortmund Germany June 28 July 2 2010 Communications in Computer and Information Science vol 81 Part II Berlin Springer 2010 pp 350 355 Dong Yu Zhen Chen Huan and Wu He Qinc Hebei University of Engineering What is Wrong with Fuzzy Logic Procedia Engineering Vol 15 August 2011 pp 1727 1731 at p 1731 For most of the extensions of fuzzy set theory see Wasim Akram Mandal amp Sahidul Islam Fundamentals on Extensions of Fuzzy Sets Independently published 2024 Daniel McNeill amp Paul Freiberger Fuzzy Logic The Revolutionary Computer Technology that Is Changing Our World New York Simon amp Schuster 1994 p 49 Daniel McNeill amp Paul Freiberger Fuzzy Logic The Revolutionary Computer Technology that Is Changing Our World New York Simon amp Schuster 1994 p 50 The Honda Foundation judged that Zadeh had taken an active role in making the future of information society a more humane civilization with a broad range of contributions in applied logic P Geach and M Black eds Translations from the Philosophical Writings of Gottlob Frege 3rd edition Blackwell 1980 p 159 Ludwig Wittgenstein Philosophical investigations Part 1 paragraph 71 revised 4th Anscombe edition 2009 p 38 Jerry A Fodor Concepts Where Cognitive Science Went Wrong New York Oxford University Press 1998 Susan Carey The origin of concepts New York Oxford University Press 2009 chapter 13 Henry Cohen amp Claire Lefebvre Handbook of categorization in cognitive science Amsterdam Elsevier 2005 Wolfgang G Stock Concepts and Semantic Relations in Information Science In Journal of the American Society for Information Science and Technology Vol 61 No 10 October 2010 pp 1951 1969 79 Eric Margolis amp Stephen Laurence Concepts In Stanford Encyclopedia of Philosophy 2011 80 Ulric Neisser ed Concepts and conceptual development ecological and intellectual factors in categorization Cambridge Cambridge University Press 1987 Stevan Harnad ed Categorical perception the groundwork of cognition Cambridge Cambridge University Press 1987 Edward E Smith amp Douglas L Medin Categories and concepts Cambridge Harvard University Press 1981 p 182 Pawel Zeidler Models and Metaphors as Research Tools in Science Zurich Lit Verlag 2013 L Magnani N J Nersessian amp P Thagard eds Model based reasoning in scientific discovery New York Kluwer Academic Plenum Publishers 1999 Jonathan Lawry Modelling and reasoning with vague concepts New York Springer 2006 Go to Standards Library for info See e g 81 82 83 Robin Rollinger amp Carlo Ierna Christian von Ehrenfels In Stanford encyclopedia of philosophy 2019 84 Lotfi A Zadeh The birth and evolution of fuzzy logic International Journal of General Systems Vol 17 No 2 3 1990 pp 95 105 at p 98 Daniel McNeill amp Paul Freiberger Fuzzy Logic The Revolutionary Computer Technology that Is Changing Our World New York Simon amp Schuster 1994 pp 47 48 A Dumitras amp G Moschytz Understanding Fuzzy Logic An Interview with Lotfi Zadeh IEEE signal processing magazine May 2007 pp 102 105 at p 103 Joseph Tulloch We just want to stay alive An appeal from Gaza Vatican News 6 November 2023 85 Susan Haack Philosophy of Logics Cambridge University Press 1978 p 213 Joseph Grunfeld Haack on fuzzy logic The Paideia Archive Twentieth World Congress of Philosophy 8 1998 pp 65 69 86 Susan L Epstein Memory and concepts in reactive learning Proceedings of the Canadian Workshop on Machine Learning 1992 87 George Lakoff Cognitive models and prototype theory In Ulric Neisser ed Concepts and conceptual development ecological and intellectual factors in categorization Cambridge Cambridge University Press 1987 pp 63 100 at pp 90 96 Stephen Mumford Quantities and Qualities University of Nottingham blog post September 30 2012 88 Robert M Wachter How Measurement Fails Doctors and Teachers New York Times 16 January 2016 89 A well known quotation usually attributed to Einstein is Not everything that can be counted counts and not everything that counts can be counted I d amend it to a less eloquent more prosaic statement unless we know how things are counted we don t know if it s wise to count on the numbers The problem isn t with statistical tests themselves but with what we do before and after we run them First we count if we can but counting depends a great deal on previous assumptions about categorization Second after we ve gathered some numbers relating to a phenomenon we must reasonably aggregate them into some sort of recommendation or ranking This is not easy By appropriate choices of criteria measurement protocols and weights almost any desired outcome can be reached John Allen Paulos Metric Mania in New York Times 10 May 2010 90 Whether Einstein really did originate the quotation which Paulos mentions is in dispute The quote is also credited to William Bruce Cameron Informal Sociology a casual introduction to sociological thinking New York Random House 1963 p 13 91 J Coates Keynes vague concepts and fuzzy logic In G C Harcourt amp P A Riach ed A second edition of the General Theory Volume 2 London Routledge 1997 pp 244 259 at p 256 F A Hayek Coping With Ignorance Imprimis Volume 7 Number 7 July 1978 92 Viktor Mayer Schonberger and Thomas Ramge Reinventing capitalism in the age of big data London John Murray 2018 p 52 Michael Polanyi The tacit dimension 1966 Chicago University of Chicago Press 2009 pp 20 21 David Harbor A fuzzy boundary EEG 260 GIS and Remote Sensing laboratory course introducing a Geographic Information System and remote sensing Lexington VA Earth and Environmental Geoscience Department at Washington and Lee University 2025 93 A standard definition of a forest and a tree is provided by the Food and Agriculture Organization of the United Nations FAO provides a system of standard definitions for forestry objects What statisticians then often try to do is to create a model which can predict the magnitude of the difference between the true accurate and exact number and the computed number obtained in this case the true number of trees Such a model however still relies on imperfect or fallible definitions Even if fuzzy values are used instead it is likely that a definite and exact number can never be reached At most one can say that the number is correct if the definitions are accepted Mark Benisz et al Abuses and misuses of intelligence tests Facts and misconceptions In A B Kaufman amp J C Kaufman eds Pseudoscience The conspiracy against science Cambridge MIT Press 2018 pp 351 373 Briefing The trouble with GDP The Economist London 30 April 2016 Susan Haack Deviant logic fuzzy logic beyond the formalism Chicago University of Chicago Press 1996 Matti Eklund Vagueness and Second Level Indeterminacy in Richard Dietz amp Sebastiano Moruzzi eds Cuts and clouds Vagueness Its Nature and Its Logic Oxford University Press 2009 p 65 Matti Eklund Characterizing Vagueness Philosophy Compass 2 2007 pp 896 909 Lotfi A Zadeh What is fuzzy logic IFSA Newsletter International Fuzzy Systems Association Vol 10 No 1 March 2013 pp 5 6 Tom Dougherty Vague value in Philosophy and phenomenological research Vol 89 No 2 September 2014 pp 352 372 94 Tom Dougherty Vagueness and Indeterminacy in Ethics In Tristram McPherson amp David Plunkett The Routledge Handbook of Metaethics Oxford Routledge 2017 Scott Soames The Value of Vagueness Chapter 2 in Andrei Marmor amp Scott Soames Philosophical Foundations of Language in the Law Oxford Oxford University Press 2013 pp 26 43 at p 26 Scott Soames The Value of Vagueness Chapter 2 in Andrei Marmor amp Scott Soames Philosophical Foundations of Language in the Law Oxford Oxford University Press 2013 pp 26 43 at p 33 Scott Soames The Value of Vagueness Chapter 2 in Andrei Marmor amp Scott Soames Philosophical Foundations of Language in the Law Oxford Oxford University Press 2013 pp 26 43 at p 34 David Lanius What is the value of vagueness In Theoria a journal of Swedish philosophy Vol 87 No 3 May 2021 pp 752 780 Alfred Korzybski Science and Sanity An Introduction to Non Aristotelian Systems and General Semantics 5th ed Forest Hills N Y Institute of General Semantics 1995 Gregory Bateson Steps to an Ecology of Mind Collected Essays in Anthropology Psychiatry Evolution and Epistemology Chicago University Of Chicago Press 1972 Vassos Argyrou Anthropology of Magic In James D Wright ed International Encyclopedia of the Social amp Behavioural Sciences Amsterdam Elsevier 2015 2nd edition Vol 14 p 438 Dominic Hyde Vagueness Logic and Ontology Aldershot Ashgate Publishing Ltd 2008 See also object oriented ontology Alfred Korzybski Science and Sanity An Introduction to Non Aristotelian Systems and General Semantics 5th ed Forest Hills N Y Institute of General Semantics 1995 Tanya Lewis What s the Universe Made Of Math Says Scientist Live Science 30 January 2014 95 See also Raphael van Riel amp Robert Van Gulick Scientific reduction In Stanford Encyclopedia of Philosophy 2014 96 Edward N Zalta Abstract Objects An introduction to axiomatic metaphysics Dordrecht D Reidel Publishing Company 1983 Norman Gulley Plato s theory of knowledge 1962 Milton Park Routledge 2013 chapter 4 Hartry H Field Science without numbers A defense of nominalism Second edition Oxford Oxford University Press 2016 Mark Balaguer Platonism and Anti Platonism in Mathematics Oxford Oxford University Press 1998 Unlike the platonic epistemology required by the classic Frege Russell account the epistemology of naturalized propositions sees acquaintance with and knowledge of propositions as rooted in acquaintance with and knowledge of acts and events that make up one s cognitive life Scott Soames What is meaning Princeton Princeton University Press 2010 p 106 William Ashley Marxism and moral concepts New York Monthly Review Press 1964 pp 4 5 Similarly Paul Lafargue had written in his essay The Origin of Abstract Ideas 1900 that The brain has the property of thinking as the stomach has that of digesting It cannot think but by the aid of ideas which it fabricates with the materials furnished it by the natural environment and the social or artificial environment in which man evolves 97 John Coates The claims of common sense Moore Wittgenstein Keynes and the social sciences Cambridge Cambridge University Press 1996 George Lakoff Hedges A Study in Meaning Criteria and the Logic of Fuzzy Concepts Journal of Philosophical Logic Vol 2 1973 pp 458 508 98 Charles Ragin Redesigning Social Inquiry Fuzzy Sets and Beyond University of Chicago Press 2008 Shaomin Li Measuring the fuzziness of human thoughts An application of fuzzy sets to sociological research The Journal of Mathematical Sociology Volume 14 Issue 1 1989 pp 67 84 Mario Quaranta Fuzzy set theory and concepts a proposal for concept formation and operationalization Comparative Sociology Vol 12 issue 6 2013 pp 785 820 Michael Stoiber Frederik Caselitz Marie Sophie Heinelt How to deal with socio ethnic conflicts in Latin America Analysing conditions on multiple levels with fsQCA Paper for the conference QCA Applications and Methodological Challenges November 22 23 2013 Goethe University Frankfurt 99 Robert Draeseke amp David E A Giles A fuzzy logic approach to modelling the New Zealand underground economy Mathematics and Computers in Simulation Vol 59 No 1 2002 pp 115 123 Tiffany Hui Kuang Yu David Han Min Wang and Su Jane Chen A fuzzy logic approach to modeling the underground economy in Taiwan Physica Vol 362 No 2 2006 pp 471 479 Mohammad Hossien Pourkazemi Mohammad Naser Sherafat and Zahra Delfan Azari Modeling Iran s Underground Economy A Fuzzy Logic Approach In Iranian Economic Review Volume 19 Issue 1 Winter 2015 Page 91 106 Kristina Marsic amp Dijana Oreski Estimation and Comparison of Underground Economy in Croatia and European Union Countries Fuzzy Logic Approach In Journal of Information and organizational Sciences Vol 49 No 1 2016 pp 83 104 Kofi Kissi Dompere Fuzziness democracy control and collective decision choice system a theory on political economy of rent seeking and profit harvesting Heidelberg Springer 2014 For the idea of the Kosko Cube see Bart Kosko Fuzziness vs probability International Journal of General Systems Vol 17 Issue 2 3 1990 pp 211 240 100 Thomas Kron Reflexiver Terrorismus Weilerswist Velbruck 2015 Thomas Kron amp Lars Winter Fuzzy Systeme und die Corona Krise In Zeitschrift fur Theoretische Soziologie Special Issue Corona Krise und Differenzierungslagen 2021 Thomas Kron and Anna Maria Weihrauch Die Vagheit der Kultur In Interculture journal online journal for intercultural studies Vol 19 Issue 34 2021 Gewalt und emotionale Energie In Andreas Braun amp Thomas Kron eds Osterreichische Zeitschrift fur Soziologie Special Issue Bestandsaufnahme soziologischer Gewaltforschung 2020 pp 113 134 Thomas Kron amp Lars Winter Die Re Produktion des Terrors Unterscheidungen und Vagheiten In Soziale Systeme Special Issue Terrorismus fuzzy logisch und formtheoretisch Issue 1 2018 pp 15 41 Autopoiesis und Hybride zur Formkatastrophe der Gegenwartsgesellschaft In Zeitschrift fur Theoretische Soziologi Issue 2 2014 pp 220 252 Integrale Akteurtheorie zur Modellierung eines Bezugsrahmens fur komplexe Akteure In Zeitschrift fur Soziologie Issue 3 2006 pp 170 192 Thomas Kron amp Lars Winter Fuzzy Systems Uberlegungen zur Vagheit sozialer Systeme In Soziale Systeme Issue 2 2005 pp 370 394 Fuzzy Logik fur die Soziologie In Osterreichische Zeitschrift fur Soziologie Issue 3 2005 pp 51 89 Thomas Kron amp Lars Winter Logik in der Soziologie In Peter Klimczak amp Thomas Zoglauer ed Logik in den Wissenschaften Munster Mentis 2017 pp 181 198 Thomas Kron Andreas Braun and Eva Heinke Terror A hybrid perpetrator in individualized terrorism warfare In Mathieu Deflem ed Terrorism and Counterterrorism Today Bingley Emerald 2015 pp 131 149 Thomas Kron and Lars Winter Fuzzy Thinking in Sociology In Rudi Seising ed Views on fuzzy sets and systems from different perspectives Philosophy and logic criticisms and applications Berlin Springer 2009 pp 301 320 Edward A Shils amp Henry A Finch eds Max Weber on the methodology of the social sciences Glencoe Ill The Free Press 1949 p 93 Ann Markusen Fuzzy Concepts Scanty Evidence Policy Distance The Case for Rigour and Policy Relevance in Critical Regional Studies In Regional Studies Volume 37 Issue 6 7 2003 pp 701 717 Jorg Rossel and Randall Collins Conflict theory and interaction rituals The microfoundations of conflict theory In Jonathan H Turner ed Handbook of Sociological Theory New York Springer 2001 p 527 Carol Jenkins Ethnicity culture drugs and sex In Peter Aggleton Andrew Ball and Purnima Mane eds Sex Drugs and Young People International Perspectives London Routledge 2006 p 48 Elizabeth Chaplin Sociology and visual representation London Routledge 1994 p 130 Stephen J Lynch ed Christopher Marlowe Edward II with related texts Indianapolis Hackett Publishing Company 2015 p xix Loic Wacquant The fuzzy logic of practical sense in Pierre Bourdieu and Loic Wacquant An invitation to reflexive sociology London Polity Press 1992 chapter I section 4 Ph Manning Fuzzy Description Discovery and Invention in Sociology In History of the Human Sciences Vol 7 No 1 1994 pp 117 23 101 Philip Shenon Their prince is back Cambodians are baffled New York Times 6 June 1993 Betty Blair Interview with Lotfi Zadeh Creator of Fuzzy Logic Azerbaijan International Winter 1994 pp 46 47 102 Johan van Benthem et al eds The age of alternative logics Assessing philosophy of logic and mathematics today Dordrecht Springer 2006 p 203 Kofi Kissi Dompere Fuzziness and approximate reasoning epistemics on uncertainty expectation and risk in rational behaviour Berlin Springer 2009 Masao Mukaidono Fuzzy logic for beginners Singapore World Scientific Publishing 2001 Karen Armstrong The Case for God New York Anchor 2010 David Bohm Wholeness and the implicate order London Routledge amp Kegan Paul ARK paperback edition 1983 p 86f Ardis Butterfield Fuzziness and Perceptions of Language in the Middle Ages Part 1 Explosive Fuzziness The Duel Common Knowledge Vol 18 No 2 pp 255 266 2012 Part 2 Collective Fuzziness Three Treaties and a Funeral Common Knowledge Vol 19 No 1 2013 pp 51 64 Part 3 Translating Fuzziness Countertexts Common Knowledge Vol 19 3 2013 pp 446 473 Paul Teller Language and the complexity of the world Department of Philosophy University of California at Davis August 2024 103 Bertrand Russell Vagueness In Australasian Journal of Psychology and Philosophy Vol 1 pp 84 92 1923 Reprinted in Bertrand Russell Papers Vol 9 pp 147 54 Nadine Faulkner Russell and vagueness Journal of Bertrand Russell Studies Summer 2003 pp 43 63 Paul Grice Studies in the Way of Words Harvard University Press Cambridge Mass 1989 A critique of Grice is provided by Wayne A Davis Implicature intention convention and principle in the failure of Gricean theory Cambridge Cambridge University Press 1998 An example of a specific application of Gricean theory is Penelope Brown amp Stephen C Levinson Politeness some universals in language use Cambridge Cambridge University Press 1987 Patrick Hughes amp George Brecht Vicious Circles and Infinity An anthology of Paradoxes Penguin Books 1978 Nicholas Rescher Epistemological Studies Frankfurt Ontos Verlag 2009 chapter 3 John L Bell Oppositions and Paradoxes Philosophical Perplexities in Science and Mathematics Peterborough Ontario Broadview Press 2016 Andrea Cantini Paradoxes and Contemporary Logic Stanford Encyclopedia of Philosophy online 30 April 2012 See further Radim Belohlavek amp George J Klir eds Concepts and Fuzzy Logic MIT Press 2011 John R Searle Minds brains and programs The behavioral and brain sciences Vol 3 No 3 1980 pp 417 457 Robert Epstein The empty brain Aeon 18 May 2016 104 For non binary computing see Unconventional computing Karl Aschenbrenner The concept of coherence in art Dordrecht D Reidel Publishing Company 1985 Harry Collins Tacit amp explicit knowledge Chicago University of Chicago Press 2013 Amos Tversky and Daniel Kahneman The framing of decisions and psychology of choice Science Vol 211 No 4481 January 1981 pp 453 458 C J Brainerd and V F Reyna Gist is the grist fuzzy trace theory and the new intuitionism Developmental Review Vol 10 No 1 March 1990 pp 3 47 at p 39 Arthur S Reber Implicit learning and tacit knowledge An essay on the cognitive unconscious Oxford Oxford University Press 1993 pp 137 138 Ronald A Havens ed The wisdom of Milton H Erickson Volume II human behavior amp psychotherapy New York Irvington Publishers 1992 chapter 3 A Cornelius Benjamin Science and vagueness In Philosophy of science Vol 6 No 4 1939 pp 422 431 Kenneth Knoblauch Color Vision in Steven s handbook of experimental psychology Vol 1 sensation and perception 3rd ed New York John Wiley amp Sons 2002 p 48 Andrew Tarantola Why Frame Rate Matters Gizmodo com 14 January 2015 105 L Paul Bremer Corporate governance and crisis management in Directors amp Boards Winter 2002 Jean Piaget amp Barbel Inhelder The Growth of Logical Thinking from Childhood to Adolescence New York Basic Books 1958 Susan Carey Conceptual Change in Childhood Cambridge Mass MIT Press 1985 Philip J Kelman amp Martha E Arterberry The cradle of knowledge development of perception in infancy Cambridge Mass The MIT Press 2000 Rashmi Sharma et al Fuzzy complexes Specific binding without complete folding FEBS Letters Volume 589 Issue 19 Part A July 2015 pp 2533 2542 106 Rudolf Seising On the absence of strict boundaries Vagueness haziness and fuzziness in philosophy science and medicine Applied Soft Computing Vol 8 2008 pp 1232 1242 at p 1235 Kazem Sadegh Zadeh The Fuzzy Revolution Goodbye to the Aristotelian Weltanschauung In Artificial Intelligence in Medicine 21 2001 pp 18 19 107 Stephen Priest Theories of the mind London Penguin Books 1991 p 183 Michael Hammond Jane Howarth and Russell Keat Understanding Phenomenology Oxford Blackwell 1991 Cornelia Griebel Fuzzy concepts in translators minds In Valerie Dullion Between specialised texts and institutional contexts competence and choice in legal translation Amsterdam John Benjamins Publishing Company 2017 Special issue of Translation and Translanguaging in Multilingual Contexts Vol 3 No 1 2017 Ronald A Havens ed The wisdom of Milton H Erickson Volume I hypnosis and hypnotherapy New York Irvington Publishers 1992 p 106 Joseph O Connor amp John Seymour ed Introducing neuro linguistic programming London Thorsons 1995 p 116f Francese Trillas Fuzzy logic and modern economics In Rudolf Seising Enric Trillas amp Janusz Kacprzyk eds Towards the future of fuzzy logic Basel Springer International Publishing 2015 p 56 Surya Monro Bisexuality Houndmills Basingstoke Palgrave Macmillan 2015 p 49 Bart Kosko Yes Candidates There Is a Fuzzy Math New York Times 7 November 2000 Michelle Nichols Israel defends Gaza hospital raid UN rights chief says its explanation is vague Reuters 3 January 2025 Russel Gordon amp David Bendien Standard classifications New Zealand Statistics Review September 1993 p 20 Paul C Bauer et al Vague concepts in survey questions A general problem illustrated with the left right scale SSRN Electronic Journal April 2014 108 C N de Groot Sociology of religion looks at psychotherapy Recherches sociologiques Louvain la Neuve Belgium Vol 29 No 2 1998 pp 3 17 at p 4 109 Archived 2013 05 23 at the Wayback Machine Mark Manson The rise and fall of Ken Wilber markmanson net 4 June 2012 110 That is in applying rules the rules are not consistently followed in all cases so that in judging exactly the same situations different legal outcomes result For more information see e g Ralf Posche Ambiguity And Vagueness In Legal Interpretation in Lawrence M Solan amp Peter M Tiersma eds The Oxford Handbook of Language and Law Oxford University Press 2012 pp 128 144 David Henry Fuzzy Numbers Bloomberg Businessweek 3 October 2004 See e g Brian J Zinnbauer Religion and Spirituality Unfuzzying the Fuzzy In Journal for the scientific study of religion Vol 36 No 4 December 1997 pp 549 564 112 K Sadegh Zadeh Fuzzy health illness and disease In Journal of medical philosophy Vol 25 No 5 October 2000 pp 605 638 113 In 2025 the Oxford English Dictionary lists 33 meanings for the word social of which six are labelled obsolete Lisa Mathis Butler Harrington Sustainability Theory and Conceptual Considerations A Review of Key Ideas for Sustainability and the Rural Context Papers in Applied Geography Vol 2 Issue 4 November 2016 M J Ducey and B C Larson A fuzzy set approach to the problem of sustainability Forest Ecology and Management 111 1999 pp 29 40 Stuart McCready ed The discovery of happiness Naperville illinois Sourcebooks Inc 2001 Desmond Morris The nature of happiness London Little Books Ltd 2004 Mihaela Kelemen Ambiguity as organisational practice an American pragmatist perspective in Lindsay Hamilton et al Contemporary issues in management 2nd edition Cheltenham UK Edward Elgar 2019 pp 91 103 Philip Ball The surprising benefits of ambiguous language London BBC com 13 March 2014 114 Karl Popper Unended quest an intellectual autobiography London Routledge 2002 p 22 Lotfi A Zadeh What is fuzzy logic IFSA Newsletter International Fuzzy Systems Association Vol 10 No 1 March 2013 Takeshi Yamakawa Stabilization of an Inverted Pendulum by a High speed Fuzzy Logic Controller Hardware System Fuzzy Sets and Systems Vol 32 pp 161 180 1989 See e g C A Drossos Foundations of fuzzy sets A nonstandard approach Fuzzy Sets and Systems Volume 37 Issue 3 28 September 1990 pp 287 307 Nick Cercone amp Gordon McCalla eds The knowledge frontier essays in the representation of knowledge New York Springer 1987 Guy W Mineau et al eds Conceptual graphs for knowledge representation Berlin Springer 1993 Tru Hoang Cao Conceptual graphs and fuzzy logic Berlin Springer 2010 V Rahmati et al eds A Novel Low Complexity Fast Response Time Fuzzy PID Controller for Antenna Adjusting Using Two Direct Current Motors cf Timothy Williamson Vagueness London Routledge 1996 p 258 External linksJames F Brule Fuzzy systems tutorial Fuzzy Logic Stanford Encyclopedia of Philosophy Vagueness Stanford Encyclopedia of Philosophy Calvin College Engineering Department Getting Started with Fuzzy Logic Archived 2018 02 21 at the Wayback Machine 2009 Benjamin Franklin Medal Winner Lotfi A Zadeh EUSFLAT In memory of Lotfi A Zadeh RAFSoft Remembering Lotfi Zadeh Lin Shang Lecture on fuzzy and rough sets Nanjing University Rudolf Kruse and Christian Moewes on fuzzy set theory Fuzzy Logic for Just Plain Folks by Thomas Sowell Podcast with text Fuzzy logic The origin and future of non Aristotelian thinking 20 March 2023