Fuzzy concept

Origin of the concept

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 is relatively easy to identify. For example, the boiling point of water (where it turns into steam) is 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 establish. The Nordic myth of Loki's wager suggested that concepts which lack a precise meaning or precise boundaries of application cannot be usefully discussed at all. However, the 20th century idea of "fuzzy concepts" proposes that "somewhat vague terms" can be operated with, since we can explicate and define the variability of their application, by assigning numbers to it.

The intellectual origins of the idea of fuzzy concepts have been traced back to a diversity of famous and less well known thinkers including Eubulides, Plato, Cicero, Georg Wilhelm Friedrich Hegel, Karl Marx, Friedrich Engels, Friedrich Nietzsche, Max Black,Jan Łukasiewicz, Alfred Tarski, Stanisław Jaśkowski and Donald Knuth. This suggests that awareness of the existence of concepts with fuzzy characteristics, in one form or another, has a very long history in human thought.

However, usually the Iranian born, American computer scientist Lotfi A. Zadeh is credited with inventing the specific idea of a "fuzzy concept" in his seminal 1965 paper on fuzzy sets, because he gave a formal mathematical presentation of the phenomenon which 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.

In fact, the German scholar Dieter Klaua also published a German-language paper on fuzzy sets in 1965, but he used a different terminology (he referred to "many-valued sets"). An earlier attempt in the post-war era to create a theory of sets where set membership is a matter of degree was made by Abraham Kaplan and Hermann Schott in 1951. They intended to apply the idea to empirical research. Kaplan and Schott measured 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".

Zadeh's seminal 1965 paper is acknowledged to be one of the most-cited scholarly articles in the 20th century.

Contemporary definition

The normal scholarly definition of a concept as "fuzzy" has been in use from the 1970s onward. 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:

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 is 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. 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 the greatest precision in stating the degree of applicability of a logical rule.

Petr Hájek, writing about the foundations of fuzzy logic, 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.

Applications

In philosophical logic, fuzzy concepts are often regarded as concepts which in their application, or formally speaking, are neither completely true nor completely false, or which are partly true and partly false; they are ideas which require further elaboration, specification or qualification to understand their applicability (the conditions under which they truly make sense).

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 quantitative limits or parameters, and which can be usefully described with imprecise categories (such as "high", "medium" or "low") using some kind of qualitative scale.

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 statements are "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. Fuzzy logic in principle 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, all kinds of control systems, and various kinds of electronic equipment used for pattern recognition, surveying and monitoring (including radars, alarm systems and surveillance systems). Fuzzy reasoning is also used in artificial intelligence, virtual intelligence and soft computing research. "Fuzzy risk scores" are used by project managers and portfolio managers to express risk assessments. Fuzzy logic has even 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.

A lot of research on fuzzy logic was done by Japanese researchers inventing new machinery, electronic equipment and appliances (see also Fuzzy control system). The North American Fuzzy Information Processing Society (NAFIPS) was founded in 1981. There also exists an International Fuzzy Systems Association (IFSA). In Europe, there is a European Society for Fuzzy Logic and Technology (EUSFLAT).

Lotfi Zadeh estimates there are more than 50,000 fuzzy logic-related, patented inventions. He lists 28 journals dealing with fuzzy reasoning, and 21 journal titles on soft computing. There are now close to 100,000 publications with the word "fuzzy" in their titles, or maybe even 300,000.

Fuzzy concept lattices and big data

According to the computer scientist Andrei Popescu at Middlesex University London, a concept can be operationally defined to consist of (1) an intent, which is a description or specification stated in a language, (2) an extent, which is the collection of all the objects to which the description refers, and (3) a context, which is stated by (i) the universe of all possible objects, (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.

However, 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 object or attribute along a scale, and the results are placed in a table which links each assigned value within the given range to a numerical value 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. 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 within a large population.

Fuzzy lattices can be applied, 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; patterns of variation 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 which represents a characteristic). However, NBC News reported that the Anglo-American firm Cambridge Analytica which profiled voters for Donald Trump (Steve Bannon has been a board member) has not 400, but 4,000 data points for each of 230 million US adults. Cambridge Analytica's own website claims that "up to 5,000 data points" are collected for each of 220 million Americans, a data set of more than 1 trillion bits of formatted data. Harvard University Professor Latanya Sweeney calculated, that if a 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 4,000-5,000 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.

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, 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" A traditional objection to big data is, that it cannot cope with rapid change, but the technology now exists for corporations like Amazon, Google 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 increasingly it can be done very fast.

Definitional controversies

Some scientists claimed that in reality fuzzy concepts do not exist. For example, 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 a concept, to qualify as a concept, must be clear and precise. A vague notion would be at best a prologue to formulating a concept. However, there is no general agreement how the notion of a "concept", or a scientific concept in particular, should be defined, and of course scientists also quite often do use imprecise analogies in their models to help understanding an issue. A concept can be clear, but not (or not sufficiently) precise.

Susan Haack once claimed that a many-valued logic requires neither intermediate terms between true and false, nor a rejection of bivalence. Her suggestion was, 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, but even if it was correct, the ability to assign a numerical value to the applicability of a statement is often enormously more efficient than a long sequence of if-then statements that would have the same meaning. That point is obviously of great importance to computer programmers seeking to code a process or operation according to logical rules.

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 interpretive 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. A concept may also be deliberately created 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.

Philosophers 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". However, Lotfi Zadeh claims that "vagueness connotes insufficient specificity, whereas fuzziness connotes unsharpness of class boundaries". Thus, he argues, 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 is that fuzziness and vagueness are logically quite different qualities, rather than fuzziness being a type or subcategory of vagueness. Zadeh claims that "inappropriate use of the term 'vague' is still a common practice in the literature of philosophy".

The definitional disputes remain unresolved, 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.

For example, cosmologist Max Tegmark argues 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 that exists 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 this 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 such an exhaustive mathematical expression or reduction is feasible, and (3) 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.

Sociology and journalism

The idea of fuzzy concepts has also been applied in the philosophical, sociological and linguistic analysis of human behaviour. In a 1973 paper, George Lakoff for example analyzed hedges in the interpretation of the meaning of categories. Charles Ragin and others have applied the idea to sociological analysis.

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:

The main reason why the term 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, 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".

The use of fuzzy logic in the social sciences and humanities has remained limited until recently. Lotfi 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 are being widely applied in big data analysis of social, commercial and psychological phenomena.

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. However Lotfi 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, 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.

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 or imply?"); reducing fuzziness, perhaps by applying fuzzy logic, would generate more certainty.

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.

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.

In physics, the observer effect and Heisenberg's uncertainty principle indicate that that there is a physical limit to the amount of precision that is knowable, with regard to the movement of subatomic particles and waves. That is, features of physical reality exist, where we can know that they vary in magnitude, but can never know 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.

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 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. Bertrand Russell regarded language as intrinsically vague.

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".

This may be done instinctively, habitually or unconsciously, but it usually involves a choice of terms, assumptions or symbols whose meanings may often not be completely fixed, but which depend among other things on how the receiver of the message responds 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.

But 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. David Hilbert concluded that the existence of such 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

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).

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.

Some 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.

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").

In psychophysics, it was discovered that the perceptual distinctions we draw in the mind are often more sharply defined than they are in the real world.Thus, the brain actually tends to "sharpen up" 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"). If there are more gradations and transitions in reality, than our conceptual or perceptual distinctions can capture, then it could be argued that how those distinctions will actually apply, must necessarily become vaguer at some point. If, for example, one wants to count and quantify distinct objects using numbers, one needs to be able to distinguish between those separate objects, but 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 cannot be observed or measured directly.

Finally, 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, "Corporate governance and crisis management", in: Directors & Boards, Winter 2002

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, 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.

It could be argued that many concepts used fairly universally in daily life (e.g. "love", "God", "health", "social", "tolerance" etc.) are inherently or intrinsically fuzzy concepts, to the extent that their meaning can never be completely and exactly specified with logical operators or objective terms, and can have multiple interpretations, which are in part exclusively subjective. Yet despite this limitation, such concepts are not meaningless. People keep using the concepts, even if they are difficult to define precisely.

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.

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 could be a prologue to a more exact formulation of a concept, or a better understanding.

Economy of distinctions

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. 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. 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.

Lotfi Zadeh has picked up this point, and draws 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". He cites 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.

Different approaches to the analysis of fuzzy concepts

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. More generally, clarification techniques can be used such as:

In this way, we can obtain a more exact understanding of the 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 (the relationships between) fuzzy sets, which makes it possible to define variations in the meaning or applicability of concepts as quantities. Effectively, qualitative differences may then be described more precisely as quantitative variations or quantitative variability (assigning a numerical value then denotes the magnitude of variation).

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.