Fuzzy concept

<h2 id="origins">Origins</h2>
Vagueness and fuzziness have probably always been a part of human experience.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a> In the West, ancient texts show that philosophers and scientists were already reflecting about that in <a href="/facts/Classical_antiquity/JBq8vEX9">classical antiquity</a>. Most often they regarded vagueness as a problem: as an obstacle to clear thinking, as a source of confusion and as an evasive tactic. This got in the way of providing clear orientation, guidance, direction and leadership. Therefore, the quality of vagueness became associated with a <a href="/facts/Hermeneutics_of_suspicion/Fxp1u8HW">hermeneutic of suspicion</a> — vagueness was considered as something to avoid, as something undesirable. By contrast, in the <a href="/facts/Taoism/Bk79rd9J">Daoist</a> thought of <a href="/facts/Laozi/H5PsnoQG">Laozi</a> and <a href="/facts/Zhuang_Zhou/Lmlr1pMl">Zhuang Zhou</a> of ancient China, "vagueness is not regarded with suspicion, but is simply an acknowledged characteristic of the world around us" — a subject for <a href="/facts/Meditation/mlU2nUt7">meditation</a> and a source of insight.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a> 

<h3>Sorites paradox</h3>
The ancient <a href="/facts/Sorites_paradox/T38qQ44U">Sorites paradox</a> 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.<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a> With some physical processes, this threshold seems relatively easy to identify. For example, water turns into steam at 100 °C or 212 °F. Of course, the boiling point depends partly on atmospheric pressure, which decreases at higher altitudes; it is also affected by the level of <a href="/facts/Humidity/Nwb28VlT">humidity</a> — in that sense, the boiling point is "somewhat fuzzy", because it can vary under different conditions.<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> Nevertheless, for every <a href="/facts/Altitude/B5eNVurK">altitude</a>, level of <a href="/facts/Air_pressure/wJLMSaVG">air pressure</a> and degree of <a href="/facts/Humidity/Nwb28VlT">humidity</a>, we can predict accurately what the boiling point will be, if we know the relevant conditions.<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a>
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. For example, to identify "the oldest city in the world", we have to define what counts as a city, and at what point a growing human settlement becomes a city.<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a>
According to the modern idea of the <a href="/facts/Continuum_fallacy/T38qQ44U">continuum fallacy</a>, 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 (by what method or approach) we could ascertain and define the validity that the fuzzy statement does have.

<h3>Loki's wager</h3>
The Nordic myth of <a href="/facts/Loki%2527s_wager/ygy3NLqK">Loki's wager</a> 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.<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a> 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.

<h3>Precursors and pioneers</h3>
In Western civilization, the intellectual recognition of fuzzy concepts has been traced back to a diversity of famous and less well-known thinkers,<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a> including (among many others) <a href="/facts/Eubulides/yYtP0Kvp">Eubulides</a>,<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a> <a href="/facts/Epicurus/ww2teSaQ">Epicurus</a>,<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a> <a href="/facts/Plato/sPedMo74">Plato</a>,<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a> <a href="/facts/Cicero/A37U7jaN">Cicero</a>,<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a> <a href="/facts/Georg_Wilhelm_Friedrich_Hegel/5NXGjvSb">Georg Wilhelm Friedrich Hegel</a>,<a class="footnote-ref" id="fnref:35" href="#fn:35">35</a> <a href="/facts/Karl_Marx/CXYTK95X">Karl Marx</a> and <a href="/facts/Friedrich_Engels/bYNwAon1">Friedrich Engels</a>,<a class="footnote-ref" id="fnref:36" href="#fn:36">36</a> <a href="/facts/Friedrich_Nietzsche/u7VdQ5sg">Friedrich Nietzsche</a>,<a class="footnote-ref" id="fnref:37" href="#fn:37">37</a> <a href="/facts/William_James/IdmT5LX7">William James</a>,<a class="footnote-ref" id="fnref:38" href="#fn:38">38</a> <a href="/facts/Hugh_MacColl/Jq7rtxfr">Hugh MacColl</a>,<a class="footnote-ref" id="fnref:39" href="#fn:39">39</a> <a href="/facts/Charles_S._Peirce/z76dNqhM">Charles S. Peirce</a>,<a class="footnote-ref" id="fnref:40" href="#fn:40">40</a> <a href="/facts/Carl_Gustav_Hempel/6sWWobAl">Carl Gustav Hempel</a>,<a class="footnote-ref" id="fnref:41" href="#fn:41">41</a> <a href="/facts/Max_Black/CpkI2S2d">Max Black</a>,<a class="footnote-ref" id="fnref:42" href="#fn:42">42</a> <a href="/facts/Arto_Salomaa/kHSrrxmP">Arto Salomaa</a>,<a class="footnote-ref" id="fnref:43" href="#fn:43">43</a> <a href="/facts/Ludwig_Wittgenstein/wN9bAgMz">Ludwig Wittgenstein</a>,<a class="footnote-ref" id="fnref:44" href="#fn:44">44</a> <a href="/facts/Jan_%25C5%2581ukasiewicz/1rlu9ej8">Jan Łukasiewicz</a>,<a class="footnote-ref" id="fnref:45" href="#fn:45">45</a> <a href="/facts/Emil_Leon_Post/Hl2CKCxw">Emil Leon Post</a>,<a class="footnote-ref" id="fnref:46" href="#fn:46">46</a> <a href="/facts/Alfred_Tarski/7h6NySAj">Alfred Tarski</a>,<a class="footnote-ref" id="fnref:47" href="#fn:47">47</a> <a href="/facts/Georg_Cantor/k5v1DvW4">Georg Cantor</a>,<a class="footnote-ref" id="fnref:48" href="#fn:48">48</a> <a href="/facts/Nicolai_A._Vasiliev/P1E7RhlW">Nicolai A. Vasiliev</a>,<a class="footnote-ref" id="fnref:49" href="#fn:49">49</a> <a href="/facts/Kurt_G%25C3%25B6del/P14DqKqK">Kurt Gödel</a>,<a class="footnote-ref" id="fnref:50" href="#fn:50">50</a> <a href="/facts/Stanis%25C5%2582aw_Ja%25C5%259Bkowski/Du5tU2hA">Stanisław Jaśkowski</a>,<a class="footnote-ref" id="fnref:51" href="#fn:51">51</a> <a href="/facts/Willard_Van_Orman_Quine/2RrhATxa">Willard Van Orman Quine</a>,<a class="footnote-ref" id="fnref:52" href="#fn:52">52</a> <a href="/facts/George_Klir/R7aAM6C8">George J. Klir</a> ,<a class="footnote-ref" id="fnref:53" href="#fn:53">53</a> <a href="/facts/Petr_H%25C3%25A1jek/1utT346h">Petr Hájek</a>,<a class="footnote-ref" id="fnref:54" href="#fn:54">54</a> <a href="/facts/Joseph_Goguen/UJBDDFIM">Joseph Goguen</a>,<a class="footnote-ref" id="fnref:55" href="#fn:55">55</a> <a href="/facts/Ronald_R._Yager/07Hf4jL8">Ronald R. Yager</a>,<a class="footnote-ref" id="fnref:56" href="#fn:56">56</a> Enrique Héctor Ruspini,<a class="footnote-ref" id="fnref:57" href="#fn:57">57</a> Jan Pavelka,<a class="footnote-ref" id="fnref:58" href="#fn:58">58</a> <a href="/facts/George_J._Klir/R7aAM6C8">George J. Klir</a>,<a class="footnote-ref" id="fnref:59" href="#fn:59">59</a> <a href="/facts/Didier_Dubois_(mathematician)/D8ycxFey">Didier Dubois</a>,<a class="footnote-ref" id="fnref:60" href="#fn:60">60</a> <a href="/facts/Bernadette_Bouchon-Meunier/SepWYMb2">Bernadette Bouchon-Meunier</a>,<a class="footnote-ref" id="fnref:61" href="#fn:61">61</a> and <a href="/facts/Donald_Knuth/GozNzDM6">Donald Knuth</a>.<a class="footnote-ref" id="fnref:62" href="#fn:62">62</a>
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 <a href="/facts/Many-valued_logic/qNI26RdF">many-valued logic</a> or <a href="/facts/Substructural_logic/4blDGy92">substructural logic</a>.<a class="footnote-ref" id="fnref:63" href="#fn:63">63</a>
An early attempt in the post-WW2 era to create a mathematical theory of sets with gradations of set membership was made by <a href="/facts/Abraham_Kaplan/nvTKzULT">Abraham Kaplan</a> 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.<a class="footnote-ref" id="fnref:64" href="#fn:64">64</a> However, at the time, their idea "fell on stony ground".<a class="footnote-ref" id="fnref:65" href="#fn:65">65</a> <a href="/facts/J._Barkley_Rosser/j6NFh46d">J. Barkley Rosser Sr.</a> published a treatise on many-valued logics in 1952, anticipating "many-valued sets".<a class="footnote-ref" id="fnref:66" href="#fn:66">66</a> Another treatise was published in 1963 by <a href="/facts/Alexander_Zinoviev/ECArvUBp">Alexander Zinoviev</a> and others.<a class="footnote-ref" id="fnref:67" href="#fn:67">67</a>
In 1964, the American philosopher <a href="/facts/William_Alston/A10PRclk">William Alston</a> 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).<a class="footnote-ref" id="fnref:68" href="#fn:68">68</a>
The German mathematician Dieter Klaua published a German-language paper on fuzzy sets in 1965,<a class="footnote-ref" id="fnref:69" href="#fn:69">69</a> but he used a different terminology (he referred to "many-valued sets", not "fuzzy sets").<a class="footnote-ref" id="fnref:70" href="#fn:70">70</a>
In the late 1960s, two popular introductions to many-valued logic were published by Robert J. Ackermann and <a href="/facts/Nicholas_Rescher/R1p2wxR1">Nicholas Rescher</a>.<a class="footnote-ref" id="fnref:71" href="#fn:71">71</a> 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.<a class="footnote-ref" id="fnref:72" href="#fn:72">72</a> Haack provides references to significant works after 1974.<a class="footnote-ref" id="fnref:73" href="#fn:73">73</a> In 1980, <a href="/facts/Didier_Dubois_(mathematician)/D8ycxFey">Didier Dubois</a> and Henri Prade published a detailed annotated bibliography on the field of fuzzy set theory.<a class="footnote-ref" id="fnref:74" href="#fn:74">74</a> George J. Klir and Bo Yuan provided an overview of the subject in Fuzzy sets and fuzzy logic during the mid-1990s.<a class="footnote-ref" id="fnref:75" href="#fn:75">75</a> Merrie Bergmann provides a more recent (2008) introduction to fuzzy reasoning.<a class="footnote-ref" id="fnref:76" href="#fn:76">76</a> A standard modern reference work is Fuzzy Logic and Mathematics: A Historical Perspective (2017) by Radim Bělohlávek, <a href="/facts/Joseph_W._Dauben/8ET2t5uw">Joseph W. Dauben</a> and <a href="/facts/George_J._Klir/R7aAM6C8">George J. Klir</a>.<a class="footnote-ref" id="fnref:77" href="#fn:77">77</a>

<h3>Lotfi Zadeh</h3>
The Iranian-born American computer scientist <a href="/facts/Lotfi_Zadeh/WojTlG4Y">Lotfi A. Zadeh</a> (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.<a class="footnote-ref" id="fnref:78" href="#fn:78">78</a> In reality, Zadeh did not literally use the expression "fuzzy concept" anywhere in his paper. Instead, he explained the concept of a <a href="/facts/Fuzzy_set/BGWudHJq">fuzzy set</a> and its formalization. However, the terms "fuzzy concept" and "fuzzy set" are often treated by engineers as interchangeable expressions. Zadeh played a decisive role in developing the field of fuzzy logic, fuzzy sets and fuzzy systems, with a large number of influential scholarly papers.<a class="footnote-ref" id="fnref:79" href="#fn:79">79</a> 
Unlike most philosophical theories of <a href="/facts/Vagueness/BUlktmmr">vagueness</a> or rival theories of <a href="/facts/Many-valued_logic/qNI26RdF">many-valued logic</a>, Zadeh's engineering approach had the great advantage that it could be directly applied to computer programming.<a class="footnote-ref" id="fnref:80" href="#fn:80">80</a> Zadeh's seminal 1965 paper is acknowledged to be one of the most-cited scholarly articles in the 20th century.<a class="footnote-ref" id="fnref:81" href="#fn:81">81</a> In 2014, it was placed 46th in the list of the world's 100 most-cited research papers of all time.<a class="footnote-ref" id="fnref:82" href="#fn:82">82</a> Zadeh's peers called him "one of the most prominent computer scientists of all-time".<a class="footnote-ref" id="fnref:83" href="#fn:83">83</a> Since the mid-1960s, many scholars have contributed to elaborating the theory of reasoning with graded concepts, and the research field continues to expand; new methods and applications of fuzzy logic are being invented all the time.<a class="footnote-ref" id="fnref:84" href="#fn:84">84</a>

<h2 id="definition">Definition</h2>
The ordinary scholarly definition of a concept as "fuzzy" has been in use from the 1970s onward.

<h3>Criteria</h3>
Radim Bělohlávek explains:

<blockquote>"There exists strong evidence, established in the 1970s in the psychology of concepts<a class="footnote-ref" id="fnref:85" href="#fn:85">85</a>... 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<a class="footnote-ref" id="fnref:86" href="#fn:86">86</a> 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 <a href="/facts/Fuzzy_set/BGWudHJq">fuzzy set</a> that gave birth to the field of <a href="/facts/Fuzzy_logic/1PvN4Att">fuzzy logic</a>..."<a class="footnote-ref" id="fnref:87" href="#fn:87">87</a> </blockquote>
Hence, a concept is generally regarded as "fuzzy" in a logical sense if:

<ul><li>defining characteristics of the concept apply to it "to a certain degree or extent" (or, more unusually, "with a certain magnitude of likelihood").<a class="footnote-ref" id="fnref:88" href="#fn:88">88</a></li>
<li>or, the boundaries of applicability (the truth-value) of a concept can vary in degrees, according to different conditions.</li>
<li>or, the fuzzy concept itself straightforwardly consists of a <a href="/facts/Fuzzy_set/BGWudHJq">fuzzy set</a>, or a combination of such sets.</li></ul>
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 <a href="/facts/Fuzzy_logic/1PvN4Att">fuzzy logic</a>). 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 <a href="/facts/Substructural_logic/4blDGy92">substructural logics</a>.<a class="footnote-ref" id="fnref:89" href="#fn:89">89</a> 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.

<h3>Fuzziness versus uncertainty</h3>
One of the first scholars who pointed out the need to distinguish the theory of fuzzy sets from probability theory was Zadeh's pupil <a href="/facts/Joseph_Goguen/UJBDDFIM">Joseph Goguen</a>.<a class="footnote-ref" id="fnref:90" href="#fn:90">90</a> <a href="/facts/Petr_H%25C3%25A1jek/1utT346h">Petr Hájek</a>, writing about the foundations of fuzzy logic, likewise sharply distinguished between "fuzziness" and "uncertainty":

<blockquote>"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.<a class="footnote-ref" id="fnref:91" href="#fn:91">91</a></blockquote>
In <a href="/facts/Metrology/HC0MmLdb">metrology</a> (the science of measurement), it is acknowledged that for any measure we care to make, there exists an <a href="/facts/Measurement_uncertainty/Fhxgrufh">amount of uncertainty</a> 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, <a href="/facts/Lotfi_Zadeh/WojTlG4Y">Lotfi A. Zadeh</a> introduced a distinction between "Type 1 fuzzy sets" without uncertainty and "<a href="/facts/Type-2_fuzzy_sets_and_systems/LsqLRemx">Type 2 fuzzy sets</a>" with uncertainty, which has been widely accepted.<a class="footnote-ref" id="fnref:92" href="#fn:92">92</a> 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.

<h2 id="applications">Applications</h2>
<h3>Philosophy</h3>
In philosophical <a href="/facts/Logic/8JnpJLtt">logic</a> 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.<a class="footnote-ref" id="fnref:93" href="#fn:93">93</a> Such ideas require further elaboration, specification or qualification to understand their applicability (the conditions under which they truly make sense).<a class="footnote-ref" id="fnref:94" href="#fn:94">94</a> <a href="/facts/Kit_Fine/0g2RJBV4">Kit Fine</a> states that "when a philosopher talks of <a href="/facts/Vagueness/BUlktmmr">vagueness</a> he has in mind a certain kind of indeterminacy in the relation of something to the world".<a class="footnote-ref" id="fnref:95" href="#fn:95">95</a> 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 <a href="/facts/Gilles_Deleuze/cnkduWTX">Gilles Deleuze</a> and <a href="/facts/F%25C3%25A9lix_Guattari/Nz2kTD70">Félix Guattari</a> referred occasionally to fuzzy sets in connection with their <a href="/facts/Phenomenology_(philosophy)/AILzLkKx">phenomenological</a> concept of <a href="/facts/Multiplicity_(philosophy)/xaSrY9cT">multiplicities</a>. In <a href="/facts/A_Thousand_Plateaus/gPVbNPm0">A Thousand Plateaus</a>, 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",<a class="footnote-ref" id="fnref:96" href="#fn:96">96</a> In their book <a href="/facts/What_Is_Philosophy%253F_(Deleuze_and_Guattari_book)/IFpe96pa">What Is Philosophy?</a>, 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."<a class="footnote-ref" id="fnref:97" href="#fn:97">97</a>

<h3>Sciences</h3>
In <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and <a href="/facts/Statistics/DNPsKGYU">statistics</a>, a fuzzy variable (such as "the temperature", "hot" or "cold") is a value which could lie in a probable <a href="/facts/Interval_(mathematics)/e9se3vTe">range</a> defined by some quantitative limits or <a href="/facts/Parameters/zFvVFMv9">parameters</a>, and which can be usefully described with imprecise categories (such as "high", "medium" or "low") using some kind of <a href="/facts/Level_of_measurement/8GAxPxFv">scale</a> or conceptual hierarchy.

<h3>Fuzzy logic</h3>
Main article: <a href="/facts/Fuzzy_logic/1PvN4Att">Fuzzy logic</a>
In mathematics and <a href="/facts/Computer_science/lN3pEyPz">computer science</a>, 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,<a class="footnote-ref" id="fnref:98" href="#fn:98">98</a> but the more usual term is <a href="/facts/Fuzzy_logic/1PvN4Att">fuzzy logic</a> or <a href="/facts/Many-valued_logic/qNI26RdF">many-valued logic</a>.<a class="footnote-ref" id="fnref:99" href="#fn:99">99</a> The novelty of fuzzy logic is, that it "breaks with the traditional principle that <a href="/facts/Formal_system/KaeDAjK7">formalisation</a> should correct and avoid, but not compromise with, vagueness".<a class="footnote-ref" id="fnref:100" href="#fn:100">100</a>
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. <a href="/facts/Susan_Haack/GehqueXu">Susan Haack</a> comments:

<blockquote>"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."<a class="footnote-ref" id="fnref:101" href="#fn:101">101</a> </blockquote>
"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.
<a href="/facts/Petr_H%25C3%25A1jek/1utT346h">Petr Hájek</a> 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. <a href="/facts/Modal_logic/I7uXvYYT">modal logics</a>.<a class="footnote-ref" id="fnref:102" href="#fn:102">102</a> Fuzzy logic does not abolish the "<a href="/facts/Hard_and_soft_science/JAPKoGQt">hard and soft science</a>" distinction, but modifies it, by redefining what scientific rigour means in many fields of research.<a class="footnote-ref" id="fnref:103" href="#fn:103">103</a>

<h3>Machinery and analytics</h3>
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.<a class="footnote-ref" id="fnref:104" href="#fn:104">104</a> It is nowadays widely used in:

<ul><li>The programming of vehicle and transport electronics, household appliances, video games, language filters, robotics, and driverless vehicles. Fuzzy logic washing machines are gaining popularity.<a class="footnote-ref" id="fnref:105" href="#fn:105">105</a></li>
<li>All kinds of control systems that regulate access, traffic, movement, balance, conditions, temperature, pressure, routers etc.</li>
<li>Electronic equipment used for pattern recognition, surveying and monitoring (including <a href="/facts/Radars/jk9BTbFA">radars</a>, <a href="/facts/Satellites/PTX8c68t">satellites</a>, <a href="/facts/Alarm_systems/QUTGg07V">alarm systems</a> and <a href="/facts/Surveillance/hJhvWdSJ">surveillance</a> systems).</li>
<li><a href="/facts/Cybernetics/4qb58zbt">Cybernetics</a> research, <a href="/facts/Artificial_intelligence/lJGauQwX">artificial intelligence</a>,<a class="footnote-ref" id="fnref:106" href="#fn:106">106</a> <a href="/facts/Virtual_Intelligence/fuCARFuK">virtual intelligence</a>, <a href="/facts/Machine_learning/e0w0XJTu">machine learning</a>, <a href="/facts/Database_design/Hdb9LRFX">database design</a> and <a href="/facts/Soft_computing/Em6u4h9N">soft computing</a> research.<a class="footnote-ref" id="fnref:107" href="#fn:107">107</a></li>
<li>"Fuzzy risk scores" are used by project managers and portfolio managers to express financial risk assessments.<a class="footnote-ref" id="fnref:108" href="#fn:108">108</a></li>
<li>Fuzzy logic has been applied to the problem of predicting cement strength.<a class="footnote-ref" id="fnref:109" href="#fn:109">109</a></li></ul>
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 class="footnote-ref" id="fnref:110" href="#fn:110">110</a> The scientific and engineering literature on the subject is constantly increasing.

<h3>Community</h3>
Originally lot of research on fuzzy logic was done by Japanese pioneers inventing new machinery, electronic equipment and appliances (see also <a href="/facts/Fuzzy_control_system/SMamr5qY">Fuzzy control system</a>).<a class="footnote-ref" id="fnref:111" href="#fn:111">111</a> 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 <a href="/facts/Artificial_intelligence/lJGauQwX">artificial intelligence</a> or <a href="/facts/Soft_computing/Em6u4h9N">soft computing</a>. 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:

<ul><li>The main international body is the International Fuzzy Systems Association (IFSA).<a class="footnote-ref" id="fnref:112" href="#fn:112">112</a></li>
<li>The Computational Intelligence Society of the Institute of Electrical and Electronics Engineers, Inc. (IEEE) has an international membership and deals with <a href="/facts/Fuzzy_logic/1PvN4Att">fuzzy logic</a>, <a href="/facts/Artificial_neural_network/6V1jMlkx">neural networks</a> and <a href="/facts/Evolutionary_computing/qXyxPI8j">evolutionary computing</a>. It publishes the journal IEEE Transactions on Fuzzy Systems and holds international conferences. At the end of 2024, there were 238 <a href="https://cis.ieee.org/activities/membership-activities/chapters">chapters</a> of IEEE/CIS across the world.<a class="footnote-ref" id="fnref:113" href="#fn:113">113</a></li>
<li>The conference on Fuzzy Systems and Data Mining (FSDM)<a class="footnote-ref" id="fnref:114" href="#fn:114">114</a> has its <a href="https://www.fsdmconf.org/">11th International Conference</a> (FSDM2025) in Hanshan Normal University, Chaozhou City, Guangdong Province, China.</li>
<li>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.<a class="footnote-ref" id="fnref:115" href="#fn:115">115</a></li>
<li>The International Association for fuzzy-set management and economy (SIGEF)<a href="https://www.sigef.net/">[124]</a> is based in Spain and publishes the Fuzzy Economic Review<a href="https://www.sigef.net/2014-09-26-07-16-23/editorial">[125]</a> since 1996.</li>
<li>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.<a class="footnote-ref" id="fnref:116" href="#fn:116">116</a></li>
<li>The interdisciplinary Japan Society for Fuzzy Theory and Intelligent Informatics (SOFT) traces its origin back to 1972 and publishes two journals.<a class="footnote-ref" id="fnref:117" href="#fn:117">117</a></li>
<li>The original Korea Fuzzy System Society founded in 1991 is now known as the Korean Institute of Intelligent Systems (KIIS).<a class="footnote-ref" id="fnref:118" href="#fn:118">118</a></li>
<li>In mainland China, there is the Fuzzy Mathematics and Systems Association of China (FMSAC) based at the School of Mathematics, Sichuan University in Chengdu,<a class="footnote-ref" id="fnref:119" href="#fn:119">119</a> and there exists also an important Taiwan Fuzzy Systems Association.<a class="footnote-ref" id="fnref:120" href="#fn:120">120</a></li>
<li>The North American Fuzzy Information Processing Society (NAFIPS) was founded in 1981.<a class="footnote-ref" id="fnref:121" href="#fn:121">121</a> There exists also a Hispanic-American Fuzzy System Association (HAFSA) based in Mexico.</li>
<li>In Europe, there is a European Society for Fuzzy Logic and Technology (EUSFLAT) which includes the Working Group on Mathematical Fuzzy Logic.<a class="footnote-ref" id="fnref:122" href="#fn:122">122</a> The North European Society of Adaptive and Intelligent Systems (NSAIS) is based in Finland.<a class="footnote-ref" id="fnref:123" href="#fn:123">123</a></li>
<li>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.<a class="footnote-ref" id="fnref:124" href="#fn:124">124</a> When <a href="/facts/Lotfi_Zadeh/WojTlG4Y">Lotfi A. Zadeh</a> received an honorary doctorate from the <a href="/facts/University_of_Teheran/3NkndgMK">University of Teheran</a> on 9 March 2017, a member of <a href="/facts/Iran/24yukge8">Iran</a>'s parliament stated that Iran now ranks third in the world with regard to the output of scientific research about fuzzy systems.<a class="footnote-ref" id="fnref:125" href="#fn:125">125</a></li>
<li>In 2005, Russia's Association for Fuzzy Systems (founded in January 1990) became the Russian Association for Fuzzy Systems and Soft Computing (RAFSSoftCom).<a class="footnote-ref" id="fnref:126" href="#fn:126">126</a> 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.<a class="footnote-ref" id="fnref:127" href="#fn:127">127</a></li>
<li>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.<a class="footnote-ref" id="fnref:128" href="#fn:128">128</a> CBSF VI was held at <a href="/facts/S%25C3%25A3o_Paulo_State_University/vzwqd6SN">São Paulo State University</a> in 2021.<a class="footnote-ref" id="fnref:129" href="#fn:129">129</a> There also exists a Brazilian Society of Automatics (SBA).</li>
<li>In India, the Center for Soft Computing Research at the <a href="/facts/Indian_Statistical_Institute/T0wB4jUX">Indian Statistical Institute</a> (Kolkata) organizes and publishes research on fuzzy sets, <a href="/facts/Rough_sets/7pRxKzxx">rough sets</a>, and applications of fuzzy logic.<a class="footnote-ref" id="fnref:130" href="#fn:130">130</a></li>
<li>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.<a class="footnote-ref" id="fnref:131" href="#fn:131">131</a></li>
<li>Other national scientific bodies include the Hungarian Fuzzy Association (HFA),<a class="footnote-ref" id="fnref:132" href="#fn:132">132</a> the Fuzzy Systems Association of Turkey (FSAT), the Indonesian Soft Computing Society (SC-INA),<a class="footnote-ref" id="fnref:133" href="#fn:133">133</a> and the Vietnamese Fuzzy Systems Society (VFSS).<a class="footnote-ref" id="fnref:134" href="#fn:134">134</a></li></ul>
<h3>Achievements</h3>
<a href="/facts/Lotfi_Zadeh/WojTlG4Y">Lotfi A. Zadeh</a> 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 <a href="/facts/Soft_computing/Em6u4h9N">soft computing</a>. His searches found close to 100,000 publications with the word "fuzzy" in their titles, but perhaps there are even 300,000.<a class="footnote-ref" id="fnref:135" href="#fn:135">135</a> In March 2018, <a href="/facts/Google_Scholar/FhEwS2QM">Google Scholar</a> 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.<a class="footnote-ref" id="fnref:136" href="#fn:136">136</a>

<h2 id="lattices-and-big-data-sets">Lattices and big data sets</h2>
The technique of fuzzy concept lattices is increasingly used in programming for the formatting, relating and analysis of fuzzy data sets.

<h3>Concept formalization</h3>
According to the computer scientist Andrei Popescu at <a href="/facts/Middlesex_University_London/Fy8X9Rez">Middlesex University London</a>,<a class="footnote-ref" id="fnref:137" href="#fn:137">137</a> a concept can be operationally defined to consist of: 

<ul><li>an intent, which is a description or specification stated in a language,</li>
<li>an extent, which is the collection of all the objects to which the description refers,</li>
<li>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.<a class="footnote-ref" id="fnref:138" href="#fn:138">138</a></li></ul>
Once the context is defined, we can specify relationships of sets of objects with sets of attributes which they do, or do not share.

<h3>Fuzzy concept lattice</h3>
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".<a class="footnote-ref" id="fnref:139" href="#fn:139">139</a> 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 "<a href="/facts/Fuzzy_clustering/MEmTL2po">fuzzy conceptual clustering</a>" techniques used to group data, originally invented by <a href="http://dblp2.uni-trier.de/pers/hd/r/Ruspini:Enrique_H=">Enrique H. Ruspini</a>). Fuzzy concept lattices are a useful programming tool for the exploratory analysis of <a href="/facts/Big_data/XGtrVg4z">big data</a>, 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.<a class="footnote-ref" id="fnref:140" href="#fn:140">140</a>

<h3>Big data</h3>
Coding with fuzzy lattices can be useful, for instance, in the <a href="/facts/Psephological/r2dP2pUD">psephological</a> analysis of <a href="/facts/Big_data/XGtrVg4z">big data</a> 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.<a class="footnote-ref" id="fnref:141" href="#fn:141">141</a> The basic programming techniques for this kind of fuzzy <a href="/facts/Concept_mapping/f9Dt1LED">concept mapping</a> and <a href="/facts/Deep_learning/JLuwD3ea">deep learning</a> are by now well-established<a class="footnote-ref" id="fnref:142" href="#fn:142">142</a> and big data analytics had a strong influence on the US elections of 2016.<a class="footnote-ref" id="fnref:143" href="#fn:143">143</a> A US study concluded in 2015 that for 20% of undecided voters, <a href="/facts/Google/GT9Sugza">Google</a>'s secret search algorithm had the power to change the way they voted.<a class="footnote-ref" id="fnref:144" href="#fn:144">144</a>
Very large quantities of data can now be explored using computers with fuzzy logic programming<a class="footnote-ref" id="fnref:145" href="#fn:145">145</a> and open-source architectures such as <a href="/facts/Apache_Hadoop/oUS1G2R7">Apache Hadoop</a>, <a href="/facts/Apache_Spark/rHQVfSBc">Apache Spark</a>, and <a href="/facts/MongoDB/9myQKGoY">MongoDB</a>. 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 <a href="/facts/Oracle_Corporation/0z14FygH">Oracle</a> systems (a "data point" is a number linked to one or more categories, which represents a characteristic).<a class="footnote-ref" id="fnref:146" href="#fn:146">146</a>
However, <a href="/facts/NBC_News/hWQ2wFfx">NBC News</a> reported in 2016 that the Anglo-American firm <a href="/facts/Cambridge_Analytica/rL11SeXd">Cambridge Analytica</a> which profiled voters for <a href="/facts/Donald_Trump/pqqKY21P">Donald Trump</a> (<a href="/facts/Steve_Bannon/fUymvQQ7">Steve Bannon</a> was a board member)<a class="footnote-ref" id="fnref:147" href="#fn:147">147</a> did not have 400, but 4,000 data points for each of 230 million US adults.<a class="footnote-ref" id="fnref:148" href="#fn:148">148</a> 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.<a class="footnote-ref" id="fnref:149" href="#fn:149">149</a> <a href="/facts/The_Guardian/H84wwJYM">The Guardian</a> 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.<a class="footnote-ref" id="fnref:150" href="#fn:150">150</a>
<a href="/facts/Harvard_University/Lhy9L1cx">Harvard University</a> Professor <a href="/facts/Latanya_Sweeney/k80bUyxn">Latanya Sweeney</a> calculated, that if a U.S. company knows just your <a href="/facts/Date_of_birth/CNhXkRxb">date of birth</a>, your <a href="/facts/ZIP_code/PHtdY0d4">ZIP code</a> and <a href="/facts/Sex/6VG77YCN">sex</a>, the company has an 87% chance to identify you by name – simply by using linked data sets from various sources.<a class="footnote-ref" id="fnref:151" href="#fn:151">151</a> 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.

<h3>Human judgement</h3>
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.<a class="footnote-ref" id="fnref:152" href="#fn:152">152</a> This was confirmed by <a href="/facts/Kellyanne_Conway/4KTqGCjM">Kellyanne Conway</a>, <a href="/facts/Donald_Trump/pqqKY21P">Donald Trump</a>'s campaign advisor and counselor in 2016, who emphasized the importance of human judgement and common sense in drawing conclusions from fuzzy data.<a class="footnote-ref" id="fnref:153" href="#fn:153">153</a> Conway candidly admitted that much of her own research would "never see the light of day", because it was client confidential.<a class="footnote-ref" id="fnref:154" href="#fn:154">154</a> Another Trump adviser criticized Conway, claiming that she "produces an analysis that buries every terrible number and highlights every positive number"<a class="footnote-ref" id="fnref:155" href="#fn:155">155</a>

<h4>Propaganda machine</h4>
In a video interview published by <a href="/facts/The_Guardian/H84wwJYM">The Guardian</a> in March 2018, whistleblower <a href="/facts/Christopher_Wylie/1zkmCgKC">Christopher Wylie</a> called <a href="/facts/Cambridge_Analytica/rL11SeXd">Cambridge Analytica</a> 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.<a class="footnote-ref" id="fnref:156" href="#fn:156">156</a> Wylie explained, that "we spent a million dollars harvesting tens of millions of <a href="/facts/Facebook/0IhXFR6u">Facebook</a> 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".<a class="footnote-ref" id="fnref:157" href="#fn:157">157</a>

<h4>Audit</h4>
On 19 March 2018, <a href="/facts/Facebook/0IhXFR6u">Facebook</a> 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).<a class="footnote-ref" id="fnref:158" href="#fn:158">158</a> <a href="/facts/Cambridge_Analytica/rL11SeXd">Cambridge Analytica</a> had not just used the profiles of <a href="/facts/Facebook/0IhXFR6u">Facebook</a> users to compile data sets. According to <a href="/facts/Christopher_Wylie/1zkmCgKC">Christopher Wylie</a>'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 <a href="/facts/Cambridge_Analytica/rL11SeXd">Cambridge Analytica</a>, 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.

<h4>Legal issue</h4>
The tricky legal issue then became, what kind of data <a href="/facts/Cambridge_Analytica/rL11SeXd">Cambridge Analytica</a> (or any similar company) is actually allowed to have and keep.<a class="footnote-ref" id="fnref:159" href="#fn:159">159</a> <a href="/facts/Facebook/0IhXFR6u">Facebook</a> itself became the subject of another <a href="/facts/U.S._Federal_Trade_Commission/Hkz0PwF9">U.S. Federal Trade Commission</a> 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).<a class="footnote-ref" id="fnref:160" href="#fn:160">160</a> <a href="/facts/Wired_(magazine)/RQb6gcyf">Wired</a> 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."<a class="footnote-ref" id="fnref:161" href="#fn:161">161</a>

<h4>Data privacy</h4>
Interrogating Facebook's CEO <a href="/facts/Mark_Zuckerberg/TJpEwVE6">Mark Zuckerberg</a> before the U.S. <a href="/facts/House_Energy_and_Commerce_Committee/paBSSfjf">House Energy and Commerce Committee</a> in April 2018, New Mexico Congressman Rep. <a href="/facts/Ben_Ray_Luj%25C3%25A1n/1zGwKe08">Ben Ray Luján</a> 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 <a href="/facts/ProPublica/BKrCjSrD">ProPublica</a> research, which in fact suggested that Facebook may even have 52,000 data points for many Facebook users.<a class="footnote-ref" id="fnref:162" href="#fn:162">162</a> 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:

<blockquote>"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."<a class="footnote-ref" id="fnref:163" href="#fn:163">163</a></blockquote>
In July 2018, <a href="/facts/Facebook/0IhXFR6u">Facebook</a> and <a href="/facts/Instagram/33s91jnV">Instagram</a> barred access from <a href="/facts/Crimson_Hexagon/kg4nI39e">Crimson Hexagon</a>, 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.<a class="footnote-ref" id="fnref:164" href="#fn:164">164</a>

<h4>Integrity</h4>
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.<a class="footnote-ref" id="fnref:165" href="#fn:165">165</a> 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 <a href="/facts/New_York_Times/T9marWVX">New York Times</a> 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.<a class="footnote-ref" id="fnref:166" href="#fn:166">166</a> 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".<a class="footnote-ref" id="fnref:167" href="#fn:167">167</a> From 2018 onward, Facebook faced a lot more lawsuits brought against the company, alleging data breaches, security breaches and misuse of personal information (see <a href="/facts/Lawsuits_involving_Meta_Platforms/htrJWDyN">Lawsuits involving Meta Platforms</a> and <a href="https://companyprofiles.justia.com/company/facebook/dockets/case">Facebook Federal Litigation Filings</a>).<a class="footnote-ref" id="fnref:168" href="#fn:168">168</a> In November 2023, <a href="/facts/Meta_Platforms/ylApt16y">Meta Platforms</a> was reported to be a defendant or co-defendant in at least 2,100 lawsuits.<a class="footnote-ref" id="fnref:169" href="#fn:169">169</a> Annual lawsuit filings by and against <a href="/facts/Meta_Platforms/ylApt16y">Meta Platforms</a> end up costing millions of dollars and even more; the cumulative legal costs run into billions of dollars. This creates a powerful financial incentive to get the company systems working right and neutralize malafide online operators. There still exists no standard international regulatory framework for social network information,<a class="footnote-ref" id="fnref:170" href="#fn:170">170</a> 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 <a href="/facts/Meta_Platforms/ylApt16y">Meta</a> company also reports its own legal actions.<a class="footnote-ref" id="fnref:171" href="#fn:171">171</a>
On 2 May 2018, it was reported that the <a href="/facts/Cambridge_Analytica/rL11SeXd">Cambridge Analytica</a> company was shutting down and was starting bankruptcy proceedings, after losing clients and facing escalating legal costs.<a class="footnote-ref" id="fnref:172" href="#fn:172">172</a> The <a href="/facts/Reputational_risk/vmuNGvEU">reputational damage</a> which the company had suffered or caused, had become too great.

<h3>Speed</h3>
A traditional objection to <a href="/facts/Big_data/XGtrVg4z">big data</a> 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 <a href="/facts/Amazon.com/YzUTUVE8">Amazon</a>, <a href="/facts/Google/GT9Sugza">Google</a>, <a href="/facts/Apple_Inc./oTJYta3k">Apple Inc.</a> and <a href="/facts/Microsoft/nGIDDXdx">Microsoft</a> to pump cloud-based data streams from app-users straight into big data analytics programmes, in real time.<a class="footnote-ref" id="fnref:173" href="#fn:173">173</a> Huge corporate data centers are being built in many different locations with enormous processing power (see also <a href="/facts/High-performance_fuzzy_computing/Nv2AmLXX">high-performance fuzzy computing</a>).<a class="footnote-ref" id="fnref:174" href="#fn:174">174</a> 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; <a href="/facts/Facial_recognition_system/mtVxaDln">digital face recognition systems</a>, <a href="/facts/Computer_and_network_surveillance/kAom0Xn0">surveillance</a> and <a href="/facts/Crowd_control/rudFoeUw">crowd control</a>; and targeting methodologies. 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.<a class="footnote-ref" id="fnref:175" href="#fn:175">175</a> Dr Tal Mimran, a lecturer at <a href="/facts/Hebrew_University/aEpNukpQ">Hebrew University</a> in <a href="/facts/Jerusalem/PKJ8LT0Y">Jerusalem</a> and a former legal adviser to the <a href="/facts/Israeli_Defence_Force/9hPaqcAM">Israeli Defence Force</a> (IDF) stated:

<blockquote>"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.”<a class="footnote-ref" id="fnref:176" href="#fn:176">176</a> </blockquote>
Although no comprehensive overviews appear to be publicly available,<a class="footnote-ref" id="fnref:177" href="#fn:177">177</a> a large amount of scientific research on fuzzy systems was funded or sponsored by the military.<a class="footnote-ref" id="fnref:178" href="#fn:178">178</a> However, military uses of fuzzy systems research can also have spin-offs for medical applications.<a class="footnote-ref" id="fnref:179" href="#fn:179">179</a>

<h2 id="academic-debates">Academic debates</h2>
There have been many academic debates about the meaning, relevance and utility of fuzzy concepts, as well as their appropriate use.<a class="footnote-ref" id="fnref:180" href="#fn:180">180</a> <a href="/facts/Rudolf_E._K%25C3%25A1lm%25C3%25A1n/SAoVkxs4">Rudolf E. Kálmán</a> 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".<a class="footnote-ref" id="fnref:181" href="#fn:181">181</a> 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.<a class="footnote-ref" id="fnref:182" href="#fn:182">182</a> In 2011, three Chinese engineers alleged that "Fuzzy set, its <a href="/facts/T-norm/oYR3JgIy">t-norm</a>, <a href="/facts/T-norm/oYR3JgIy">s-norm</a>] and fuzzy supplement theories have already become the academic virus in the world".<a class="footnote-ref" id="fnref:183" href="#fn:183">183</a>

<h3>"Fuzzy" label</h3>
<a href="/facts/Lotfi_A._Zadeh/WojTlG4Y">Lotfi A. Zadeh</a> himself confessed that:

<blockquote>"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."<a class="footnote-ref" id="fnref:184" href="#fn:184">184</a> </blockquote>
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."<a class="footnote-ref" id="fnref:185" href="#fn:185">185</a>

<h3>Frege and Wittgenstein</h3>
According to <a href="/facts/The_Foundations_of_Arithmetic/NIPBKGEJ">The Foundations of Arithmetic</a> by the logician <a href="/facts/Gottlob_Frege/FXyStVYj">Gottlob Frege</a>,

<blockquote>"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 <a href="/facts/Law_of_excluded_middle/tDhbugCN">law of the excluded middle</a> is really just another form of the requirement that the concept should have a sharp boundary."<a class="footnote-ref" id="fnref:186" href="#fn:186">186</a> </blockquote>
In his notes on <a href="/facts/Language_game_(philosophy)/UlWVPDLm">language games</a>, <a href="/facts/Ludwig_Wittgenstein/wN9bAgMz">Ludwig Wittgenstein</a> replied to Frege's argument as follows:

<blockquote>"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."<a class="footnote-ref" id="fnref:187" href="#fn:187">187</a> </blockquote>
<h3>The categorical status of concepts</h3>
There is no general agreement among philosophers and scientists about how the notion of a "<a href="/facts/Concept/VWB6yhQp">concept</a>" (and in particular, a scientific concept), should be defined.<a class="footnote-ref" id="fnref:188" href="#fn:188">188</a> A concept could be defined as a mental representation, as a cognitive capacity, as an abstract object, as a cluster of linked phenomena etc.<a class="footnote-ref" id="fnref:189" href="#fn:189">189</a> 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."<a class="footnote-ref" id="fnref:190" href="#fn:190">190</a> Of course, scientists also quite often do use imprecise analogies in their models to help understanding an issue.<a class="footnote-ref" id="fnref:191" href="#fn:191">191</a> A concept can be clear enough, but not (or not sufficiently) precise.
Terminology scientists at the German National Standards Institute (Deutsches Institut für Normung) provided the first official standard definition of what a concept is (in the terminology standard DIN 2330 of 1957, revised in 1974 and last revised in 2022). According to DIN 2330, a concept is "a unit of thought formed by abstraction from a set of objects by identifying their common characteristics". According to the ISO 1087 terminology standard of the <a href="/facts/International_Organization_for_Standardization/IacYR7Yg">International Organization for Standardization</a>, a concept is defined as “a unit of knowledge created by a unique combination of characteristics”. A concept is regarded as language-independent, and exists independently of how it is symbolized or referred to in natural language. Individual concepts refer to a single object or instance. General concepts refer to a class of objects with shared characteristics. The ISO 704 standard adds that a concept as a unit of thought comprises two parts: its extent and its intent. The extent comprises all objects belonging to the concept, and the intent comprises all attributes shared by those objects. Different standard definitions and terminologies for concepts exist for various systems in cyberspace.
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 <a href="/facts/Gestalt_psychology/mCFLuhYT">Gestalt</a>, i.e. it may express a totality which is more, does more and means more, than the sum of its parts (as recognized in <a href="/facts/Aristotle/U2bDlBFW">Aristotle</a>'s <a href="/facts/Metaphysics_(Aristotle)/JMUeXs0l">Metaphysics</a>). It may be that the parts cannot exist other than within the totality.<a class="footnote-ref" id="fnref:192" href="#fn:192">192</a> 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 <a href="/facts/Gestalt_psychology/mCFLuhYT">Gestalt</a> could be a fuzzy object, figure or shape.

<h3>Potential corruption</h3>
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.

<h4>Kahan's criticism</h4>
The computer scientist <a href="/facts/William_Kahan/J1GNQDyN">William Kahan</a> 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."<a class="footnote-ref" id="fnref:193" href="#fn:193">193</a> He said subsequently,

<blockquote>"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 <a href="/facts/Cocaine/rdQP4zwF">cocaine</a> of science."<a class="footnote-ref" id="fnref:194" href="#fn:194">194</a> </blockquote>
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.

<h4>Bad design</h4>
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.

<h4>Counter-argument</h4>
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. <a href="/facts/Lotfi_Zadeh/WojTlG4Y">Lotfi Zadeh</a> stated that "fuzzy logic is not fuzzy. In large measure, fuzzy logic is precise."<a class="footnote-ref" id="fnref:195" href="#fn:195">195</a> 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.<a class="footnote-ref" id="fnref:196" href="#fn:196">196</a> 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 <a href="/facts/Serendipity/pvcf95kQ">serendipity</a> 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.

<h3>Reducibility</h3>
<a href="/facts/Susan_Haack/GehqueXu">Susan Haack</a> once claimed that a many-valued logic requires neither intermediate terms between true and false, nor a rejection of bivalence.<a class="footnote-ref" id="fnref:197" href="#fn:197">197</a> 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.<a class="footnote-ref" id="fnref:198" href="#fn:198">198</a>

<h3>Quantification</h3>
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 <a href="/facts/Set_theory/1ptfsvUP">set</a>.<a class="footnote-ref" id="fnref:199" href="#fn:199">199</a> Some phenomena are difficult or impossible to quantify and count, in particular if they lack discrete boundaries (for example, clouds). <a href="/facts/George_Lakoff/gT0I0UNd">George Lakoff</a> emphasized that it is not true that fuzzy-set theory is the only or necessarily the most appropriate way to start modelling concepts.<a class="footnote-ref" id="fnref:200" href="#fn:200">200</a>

<h4>Formalization</h4>
Qualities may not be fully reducible to quantities<a class="footnote-ref" id="fnref:201" href="#fn:201">201</a> – 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 <a href="/facts/Formal_system/KaeDAjK7">formalization</a> or <a href="/facts/Formal_specification/4q2oDkvS">formal specification</a> 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.<a class="footnote-ref" id="fnref:202" href="#fn:202">202</a> 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.
<a href="/facts/Quantification_(science)/5YoKbeTW">Quantification</a> is not an unproblematic process.<a class="footnote-ref" id="fnref:203" href="#fn:203">203</a> To quantify a phenomenon, we may have to introduce special assumptions and definitions which disregard part of totality of the phenomenon. 

<ul><li>The economist <a href="/facts/John_Maynard_Keynes/4otHjBRu">John Maynard Keynes</a> 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."<a class="footnote-ref" id="fnref:204" href="#fn:204">204</a></li>
<li><a href="/facts/Friedrich_Hayek/sRqJlzrY">Friedrich Hayek</a> 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."<a class="footnote-ref" id="fnref:205" href="#fn:205">205</a></li>
<li>The <a href="/facts/Friedrich_Hayek/sRqJlzrY">Hayekian</a> <a href="/facts/Big_data/XGtrVg4z">big data</a> <a href="/facts/Guru/FEoxB7Le">guru</a> <a href="/facts/Viktor_Mayer-Sch%25C3%25B6nberger/h8sU3nJ2">Viktor Mayer-Schönberger</a> 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."<a class="footnote-ref" id="fnref:206" href="#fn:206">206</a></li>
<li><a href="/facts/Michael_Polanyi/rNDMTXF6">Michael Polanyi</a> stated that "the process of formalizing all knowledge to the exclusion of any <a href="/facts/Tacit_knowledge/20CNaqW5">tacit knowing</a> is self-defeating", since to mathematize a concept we need to be able to identify it in the first instance without mathematization.<a class="footnote-ref" id="fnref:207" href="#fn:207">207</a></li></ul>
<h4>Measurement</h4>
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.<a class="footnote-ref" id="fnref:208" href="#fn:208">208</a> The forest boundary might also change somewhat in the course of time.<a class="footnote-ref" id="fnref:209" href="#fn:209">209</a> 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.<a class="footnote-ref" id="fnref:210" href="#fn:210">210</a> 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 <a href="/facts/Intelligence/gwEhdLnp">intelligence</a> or <a href="/facts/Gross_domestic_product/zQRgyFKc">gross domestic product</a>, without much scientific agreement about what these things actually are, how they exist, and what the correct measures might be.<a class="footnote-ref" id="fnref:211" href="#fn:211">211</a> 
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 <a href="/facts/Proxy_(statistics)/W5V2eqaT">proxy variables</a> to substitute as measures for variables which are known (or thought) to be there, but which themselves cannot be observed or measured directly.

<h3>Vague or fuzzy</h3>
The exact relationship between vagueness and fuzziness is disputed.

<h4>Philosophical interpretation</h4>
Philosophers often regard fuzziness as a particular kind of vagueness,<a class="footnote-ref" id="fnref:212" href="#fn:212">212</a> 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".<a class="footnote-ref" id="fnref:213" href="#fn:213">213</a> 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.<a class="footnote-ref" id="fnref:214" href="#fn:214">214</a>

<h4>Zadeh's argument</h4>
However, <a href="/facts/Lotfi_Zadeh/WojTlG4Y">Lotfi A. Zadeh</a> claimed that "vagueness connotes insufficient <a href="/facts/Specificity_(statistics)/pzc9qfdz">specificity</a>, whereas fuzziness connotes unsharpness of <a href="/facts/Class_(set_theory)/F5EmDd9b">class boundaries</a>". 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".<a class="footnote-ref" id="fnref:215" href="#fn:215">215</a>

<h3>Ethics and law</h3>
In the scholarly inquiry about <a href="/facts/Ethics/wNVNgL5q">ethics</a> and <a href="/facts/Meta-ethics/a8pTEKr2">meta-ethics</a>, <a href="/facts/Vagueness/BUlktmmr">vague</a> 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.<a class="footnote-ref" id="fnref:216" href="#fn:216">216</a>

<h4>Excessive precision in rule-making</h4>
Nevertheless, <a href="/facts/Scott_Soames/mvdrWokC">Scott Soames</a> 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".<a class="footnote-ref" id="fnref:217" href="#fn:217">217</a> 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 <a href="/facts/Children_and_Young_Persons_Act_2008/819N4xja">Children and Young Persons Act</a> 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)".<a class="footnote-ref" id="fnref:218" href="#fn:218">218</a>

<h4>Conflicting rules</h4>
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".<a class="footnote-ref" id="fnref:219" href="#fn:219">219</a> David Lanius has examined nine arguments for the "value of vagueness" in different contexts.<a class="footnote-ref" id="fnref:220" href="#fn:220">220</a>

<h3>Mathematical ontology</h3>
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 <a href="/facts/Ontology/QJKTsEIJ">ontologies</a>.<a class="footnote-ref" id="fnref:221" href="#fn:221">221</a> 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.<a class="footnote-ref" id="fnref:222" href="#fn:222">222</a> This is an important reason why, as <a href="/facts/Alfred_Korzybski/p60HNSLd">Alfred Korzybski</a> noted, people frequently confuse the symbolic representation of reality, conveyed by languages and signs, with reality itself.<a class="footnote-ref" id="fnref:223" href="#fn:223">223</a>
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?

<h4>Tegmark's mathematical universe</h4>
To illustrate the ontological issues, cosmologist <a href="/facts/Max_Tegmark/4Y3DefCk">Max Tegmark</a> 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."<a class="footnote-ref" id="fnref:224" href="#fn:224">224</a>
Tegmark moves from the <a href="/facts/Epistemology/NsDwpcYH">epistemic</a> claim that mathematics is the only known symbol system which can in principle express absolutely everything, to the <a href="/facts/Methodology/dk5X9Hxx">methodological</a> claim that everything is reducible to mathematical relationships, and then to the <a href="/facts/Ontology/QJKTsEIJ">ontological</a> claim, that ultimately everything that exists is mathematical (the <a href="/facts/Mathematical_universe_hypothesis/b7JIz2kF">mathematical universe hypothesis</a>). 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 <a href="/facts/Qualia/YnwIAg5v">qualia</a>.<a class="footnote-ref" id="fnref:225" href="#fn:225">225</a>

<h4>Zalta's metaphysics</h4>
In his <a href="/facts/Meta-mathematics/yeIm8vqC">meta-mathematical</a> <a href="/facts/Metaphysics/fSFt6SHe">metaphysics</a>, <a href="/facts/Edward_N._Zalta/U8JxHaaA">Edward N. Zalta</a> 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 <a href="/facts/Axiom/mLjP14Mc">axiom</a> for his <a href="/facts/Ontology/QJKTsEIJ">ontology</a> of abstract objects<a class="footnote-ref" id="fnref:226" href="#fn:226">226</a> By implication, for every fuzzy object there would always exist at least one <a href="/facts/Defuzzification/Vx1v6N4I">defuzzified</a> concept which encodes it exactly. It is a modern interpretation of <a href="/facts/Plato/sPedMo74">Plato</a>'s <a href="/facts/Platonic_epistemology/Q1D7IGYz">metaphysics of knowledge</a>,<a class="footnote-ref" id="fnref:227" href="#fn:227">227</a> which expresses confidence in the ability of science to conceptualize the world exactly. However, such a theory — like any metaphysical theory — is impossible to test definitively. According to the Dutch <a href="/facts/Computational_linguistics/XmegJ6LC">computational linguist</a> Kees van Deemter, "The fact that vagueness abounds in the presentation of mathematical results suggests that vagueness plays an important role in our thinking, even when the concepts about which we think are completely crisp."<a class="footnote-ref" id="fnref:228" href="#fn:228">228</a>

<h4>Platonism versus cognitive realism</h4>
The Platonic-style interpretation of concepts was critiqued by <a href="/facts/Hartry_H._Field/MbcfSBfF">Hartry H. Field</a>.<a class="footnote-ref" id="fnref:229" href="#fn:229">229</a> Mark Balaguer argues that we do not really know whether mind-independent abstract objects exist or not; so far, we cannot prove whether <a href="/facts/Platonic_realism/hHn471rm">Platonic realism</a> is definitely true or false.<a class="footnote-ref" id="fnref:230" href="#fn:230">230</a> Defending a cognitive realism, <a href="/facts/Scott_Soames/mvdrWokC">Scott Soames</a> 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 <a href="/facts/Problem_of_universals/QEVe2MSK">universals</a> and <a href="/facts/Particular/AwbfD8jx">particulars</a> (see <a href="/facts/Theory_of_forms/e0zj3JR2">theory of forms</a>). 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.<a class="footnote-ref" id="fnref:231" href="#fn:231">231</a> As another philosopher put it, 

<blockquote>"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."<a class="footnote-ref" id="fnref:232" href="#fn:232">232</a> </blockquote>
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. Hongxing Li et al. comment that:

<blockquote>"Fuzzy concepts derive from fuzzy phenomena that commonly occur in the natural world. The concepts formed in human brains for perceiving, recognizing, and categorizing natural phenomena are often fuzzy concepts. Boundaries of these concepts are vague. (…) For example, “rain” is a common natural phenomenon that is difficult to describe precisely since it can “rain” with varying intensity anywhere from a light sprinkle to a torrential downpour. (…) The concepts formed in human brains for perceiving, recognizing, and categorizing natural phenomena are often fuzzy. (...) If it is raining today, you can call it light rain, or moderate rain, or heavy rain based on the relative amount of rainfall: This is fuzzy judging. If you are predicting a good, fair, or poor harvest based on the results of fuzzy judging, you are using fuzzy reasoning. The human brain has the incredible ability of processing fuzzy classification, fuzzy judgment, and fuzzy reasoning. The natural languages are ingeniously permeated with inherent fuzziness so that we can express rich information content in a few words."<a class="footnote-ref" id="fnref:233" href="#fn:233">233</a></blockquote>
<h3>Paradoxes</h3>
Even using ordinary <a href="/facts/Set_theory/1ptfsvUP">set theory</a> and <a href="/facts/Principle_of_bivalence/dnkYVCod">binary logic</a> to reason something out, logicians have discovered that it is possible to generate statements which are logically speaking not completely true or imply a <a href="/facts/Paradox/3VLGjuWf">paradox</a>,<a class="footnote-ref" id="fnref:234" href="#fn:234">234</a> even although in other respects they conform to logical rules (see <a href="/facts/Russell%2527s_paradox/XupqWGYt">Russell's paradox</a>). If a margin of indeterminacy therefore persists, then binary logic cannot totally remove fuzziness. <a href="/facts/David_Hilbert/1vhlHn4b">David Hilbert</a> 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".
<a class="footnote-ref" id="fnref:235" href="#fn:235">235</a>

<h2 id="social-science-and-the-media">Social science and the media</h2>
The idea of fuzzy concepts has also been applied in the linguistic, economic and sociological analysis of human behaviour.<a class="footnote-ref" id="fnref:236" href="#fn:236">236</a>

<h3>Sociology and linguistics</h3>
In a 1973 paper, <a href="/facts/George_Lakoff/gT0I0UNd">George Lakoff</a> analyzed hedges in the interpretation of the meaning of categories.<a class="footnote-ref" id="fnref:237" href="#fn:237">237</a> <a href="/facts/Charles_Ragin/2fsfldCb">Charles Ragin</a> and others have applied the idea to sociological analysis.<a class="footnote-ref" id="fnref:238" href="#fn:238">238</a> For example, fuzzy set qualitative comparative analysis ("fsQCA") has been used by German researchers to study problems posed by ethnic diversity in Latin America.<a class="footnote-ref" id="fnref:239" href="#fn:239">239</a> In <a href="/facts/New_Zealand/4mpSlJdg">New Zealand</a>, <a href="/facts/Taiwan/VcJoUD8w">Taiwan</a>, <a href="/facts/Iran/24yukge8">Iran</a>, <a href="/facts/Malaysia/8zbUMzCP">Malaysia</a>, the <a href="/facts/European_Union/6GYcR8qM">European Union</a> and <a href="/facts/Croatia/MSS0yyvw">Croatia</a>, economists have used fuzzy concepts to model and measure the underground economy of their country.<a class="footnote-ref" id="fnref:240" href="#fn:240">240</a> 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 <a href="/facts/Rent-seeking/3AWTfHqR">rent-seeking</a>", viewed as a game played between powerful corporations and the government.<a class="footnote-ref" id="fnref:241" href="#fn:241">241</a> 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 <a href="/facts/Niklas_Luhmann/4rqPYrPH">Niklas Luhmann</a> further, using the so-called "Kosko-Cube".<a class="footnote-ref" id="fnref:242" href="#fn:242">242</a> Kron studies transnational terrorism and other contemporary phenomena using fuzzy logic, to understand conditions involving uncertainty, hybridity, violence and cultural systems.<a class="footnote-ref" id="fnref:243" href="#fn:243">243</a>
A concept may be deliberately created by sociologists as an <a href="/facts/Ideal_type/eS9enYsC">ideal type</a> 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.<a class="footnote-ref" id="fnref:244" href="#fn:244">244</a> 
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".<a class="footnote-ref" id="fnref:245" href="#fn:245">245</a> 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:

<ul><li>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."<a class="footnote-ref" id="fnref:246" href="#fn:246">246</a> The idea is, that if finer distinctions are introduced, then the fuzziness or vagueness would be eliminated.</li>
<li>a book on youth culture describes <a href="/facts/Ethnicity/XSp5l24t">ethnicity</a> as "a fuzzy concept that overlaps at times with concepts of race, minority, nationality and tribe".<a class="footnote-ref" id="fnref:247" href="#fn:247">247</a> In this case, part of the fuzziness consists in the inability to distinguish precisely between a concept and a different, but closely related concept.</li>
<li>a book on sociological theory argues that the <a href="/facts/Critical_Theory/2R9f1DNe">Critical Theory</a> of domination faces the problem that "reality itself has become a rather meaningless, fuzzy concept."<a class="footnote-ref" id="fnref:248" href="#fn:248">248</a> 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).</li>
<li>A history book states: "<a href="/facts/Sodomy/37Jgb7mr">Sodomy</a> was a vague and fuzzy concept in <a href="/facts/Medieval/RqzEKB9v">medieval</a> and <a href="/facts/Early_modern_Europe/aeYvfAqx">early modern Europe</a>, and was often associated with a variety of supposedly related moral and criminal offenses, including <a href="/facts/Heresy/1eHqC2km">heresy</a>, <a href="/facts/Witchcraft/kdbQwmBD">witchcraft</a>, <a href="/facts/Sedition/BN1jIxAb">sedition</a>, and <a href="/facts/Treason/lqT6ECtV">treason</a>. St <a href="/facts/Thomas_Aquinas/h2vN2VkX">Thomas Aquinas</a>... categorized sodomy with an assortment of sexual behaviours "from which generation [i.e. procreation] cannot follow".<a class="footnote-ref" id="fnref:249" href="#fn:249">249</a> In this case, because a concept is defined by what it excludes, it remains somewhat vague what items of activity it would specifically include.</li></ul>
<h3>Mass media</h3>
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).<a class="footnote-ref" id="fnref:250" href="#fn:250">250</a> Those human characteristics can be usefully expressed only in an approximate way (see <a href="/facts/Reflexivity_(social_theory)/1du8jj8p">reflexivity (social theory)</a>).<a class="footnote-ref" id="fnref:251" href="#fn:251">251</a>
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 <a href="/facts/Pejorative/UId8vPsx">pejorative</a> sense. For example, a <a href="/facts/New_York_Times/T9marWVX">New York Times</a> journalist wrote that <a href="/facts/Prince_Sihanouk/VG6P9300">Prince Sihanouk</a> "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".<a class="footnote-ref" id="fnref:252" href="#fn:252">252</a>

<h3>Applied social science</h3>
The use of fuzzy logic in the social sciences and humanities has remained limited until recently. <a href="/facts/Lotfi_Zadeh/WojTlG4Y">Lotfi A. Zadeh</a> said in a 1994 interview that:

<blockquote>"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."<a class="footnote-ref" id="fnref:253" href="#fn:253">253</a> </blockquote>
Two decades later, after a digital <a href="/facts/Information_explosion/VQu3Y9J5">information explosion</a> due to the growing use of the internet and mobile phones worldwide, fuzzy concepts and fuzzy logic were increasingly being applied in <a href="/facts/Big_data/XGtrVg4z">big data</a> analysis of social, commercial and psychological phenomena. Many <a href="/facts/Sociometry/2k3HHN5L">sociometric</a> and <a href="/facts/Psychometrics/56X26MDI">psychometric</a> indicators are based partly on fuzzy concepts and fuzzy variables.
<a href="/facts/Jaakko_Hintikka/LurBDWkW">Jaakko Hintikka</a> 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."<a class="footnote-ref" id="fnref:254" href="#fn:254">254</a> That might help to explain why fuzzy logic has not been used much to formalize concepts in the "<a href="/facts/Hard_and_soft_science/JAPKoGQt">soft</a>" social sciences.
<a href="/facts/Lotfi_Zadeh/WojTlG4Y">Lotfi A. Zadeh</a> 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 <a href="/facts/SPSS/6PGBVUXF">SPSS</a> and <a href="/facts/Microsoft_Excel/Ezal6G75">Excel</a>) 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.

<h2 id="uncertainty">Uncertainty</h2>
Fuzzy concepts can generate <a href="/facts/Uncertainty/VM9pnGvk">uncertainty</a> 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,<a class="footnote-ref" id="fnref:255" href="#fn:255">255</a> might generate more certainty.
However, this is not necessarily always so.<a class="footnote-ref" id="fnref:256" href="#fn:256">256</a> 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,<a class="footnote-ref" id="fnref:257" href="#fn:257">257</a> 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 <a href="/facts/God/KAykGr3G">God</a>, although not easily definable,<a class="footnote-ref" id="fnref:258" href="#fn:258">258</a> for instance can provide security to the believer.<a class="footnote-ref" id="fnref:259" href="#fn:259">259</a>

<h2 id="physics">Physics</h2>
In physics, the <a href="/facts/Observer_effect_(physics)/PCoxTxLU">observer effect</a> and <a href="/facts/Uncertainty_principle/UUSoSp4T">Heisenberg's uncertainty principle</a><a class="footnote-ref" id="fnref:260" href="#fn:260">260</a> 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 <a href="/facts/Universe/lfoqc5m6">physical universe</a> itself is incredibly large and diverse, it is not easy to imagine it, grasp it or describe it without using fuzzy concepts.

<h2 id="language">Language</h2>
Ordinary language, which uses symbolic conventions and associations which are often not logical, inherently contains many fuzzy concepts<a class="footnote-ref" id="fnref:261" href="#fn:261">261</a> – "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).<a class="footnote-ref" id="fnref:262" href="#fn:262">262</a> This can be easily verified for instance by consulting a <a href="/facts/Dictionary/klg2kL06">dictionary</a>, a <a href="/facts/Thesaurus/w7xnH66H">thesaurus</a> or an <a href="/facts/Encyclopedia/fRsmb4q6">encyclopedia</a> which show the multiple meanings of words, or by observing the behaviours involved in ordinary relationships which rely on mutually understood meanings (see also <a href="/facts/Imprecise_language/EsMPrl8l">Imprecise language</a>). <a href="/facts/Bertrand_Russell/SRpwNX2U">Bertrand Russell</a> regarded ordinary language (in contrast to logic) as intrinsically vague.<a class="footnote-ref" id="fnref:263" href="#fn:263">263</a>

<h3>Implicature</h3>
To communicate, receive or convey a <a href="/facts/Message/zaUQGfQB">message</a>, 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.<a class="footnote-ref" id="fnref:264" href="#fn:264">264</a> 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 <a href="/facts/Symbols/GS8mTn0h">symbols</a> whose meanings are not completely fixed, but which depend among other things on how the receivers of the message respond to it, or the <a href="/facts/Context_(language_use)/eJM44o7x">context</a>. 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 <a href="/facts/Paul_Grice/IYrQGQNd">Paul Grice</a> — using, among other things, the concept of <a href="/facts/Implicature/8AaB72xS">implicature</a>.<a class="footnote-ref" id="fnref:265" href="#fn:265">265</a> 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.

<h3>Prototypes and family resemblance</h3>
<h2 id="psychology">Psychology</h2>
Various different aspects of human experience commonly generate concepts with fuzzy characteristics.

<h3>Human vs. computer</h3>
The formation of fuzzy concepts is partly due to the fact that the human brain does not operate like a computer (see also <a href="/facts/Chinese_room/YeKegsxS">Chinese room</a>).<a class="footnote-ref" id="fnref:266" href="#fn:266">266</a>

<ul><li>While ordinary computers use strict binary logic gates,<a class="footnote-ref" id="fnref:267" href="#fn:267">267</a> 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 class="footnote-ref" id="fnref:268" href="#fn:268">268</a></li>
<li>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.<a class="footnote-ref" id="fnref:269" href="#fn:269">269</a></li>
<li>Something can be meaningful although we cannot name it, or we might only be able to name it and nothing else.<a class="footnote-ref" id="fnref:270" href="#fn:270">270</a></li>
<li>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 <a href="/facts/Framing_effect/YtEUJxWj">framing effect</a>).<a class="footnote-ref" id="fnref:271" href="#fn:271">271</a></li></ul>
According to <a href="/facts/Fuzzy-trace_theory/jGrbTj8j">fuzzy-trace theory</a>, partly inspired by <a href="/facts/Gestalt_psychology/mCFLuhYT">Gestalt psychology</a>, human intuition is a non-arbitrary, reasonable and rational process of cognition; it literally "makes sense" (see also: <a href="/facts/Problem_of_multiple_generality/gdQApsy0">Problem of multiple generality</a>).<a class="footnote-ref" id="fnref:272" href="#fn:272">272</a>

<h3>Transitions in learning and consciousness</h3>
In part, fuzzy concepts arise also because <a href="/facts/Learning/lehEKRjC">learning</a> or the growth of <a href="/facts/Understanding/D8UXbWuJ">understanding</a> 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 <a href="/facts/Implicit_learning/NE4qaWv8">implicit learning</a>, <a href="/facts/Arthur_S._Reber/oEKGNjhr">Arthur S. Reber</a> 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".<a class="footnote-ref" id="fnref:273" href="#fn:273">273</a>
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 <a href="/facts/Conscious/IAxmgt6c">conscious</a>, the <a href="/facts/Preconscious/AMuoPyPp">preconscious</a>, and the <a href="/facts/Subconscious/N68pC8HJ">subconscious</a> to the <a href="/facts/Unconscious_mind/BAIRTFMM">unconscious</a>. The hypnotherapist <a href="/facts/Milton_H._Erickson/tEl0PL7o">Milton H. Erickson</a> similarly noted that the conscious mind and the unconscious normally interact.<a class="footnote-ref" id="fnref:274" href="#fn:274">274</a>

<h3>Limits of distinctions and generalizations</h3>
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 <a href="/facts/Rigorous/v23FKfhw">rigorous</a> 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 <a href="/facts/Analogy/LYcGaTdm">analogy</a> might be, that zooming a <a href="/facts/Telescope/BDwYahhR">telescope</a>, <a href="/facts/Camera/j4uqsfJQ">camera</a>, or <a href="/facts/Microscope/ldJp0CI3">microscope</a> in and out, reveals that a pattern which is sharply focused at a certain distance becomes blurry at another distance, or disappears altogether.

<h3>Complexity and imprecision</h3>
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.<a class="footnote-ref" id="fnref:275" href="#fn:275">275</a> 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").

<h3>Cognition and perceptual limits</h3>
In <a href="/facts/Psychophysics/yDPmvvGf">psychophysics</a>, 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. 

<ul><li>Between black and white, we are able to detect only a limited number of shades of gray, or colour gradations (there are "<a href="/facts/Absolute_threshold/3PMAc0vt">detection thresholds</a>").<a class="footnote-ref" id="fnref:276" href="#fn:276">276</a></li>
<li><a href="/facts/Motion_blur/e6zkU22k">Motion blur</a> 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.<a class="footnote-ref" id="fnref:277" href="#fn:277">277</a></li></ul>
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. For the philosopher <a href="/facts/William_James/IdmT5LX7">William James</a>, the existence of "vagueness" ultimately expressed the fact that the content of reality is always richer than any conceptualizations can represent.<a class="footnote-ref" id="fnref:278" href="#fn:278">278</a>

<h3>Imprecision of novelty</h3>
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.

<blockquote>"Crisis management plans cannot be put 'on the fly' after the crisis occurs. At the outset, information is often <a href="/facts/Vague/BUlktmmr">vague</a>, 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" — <a href="/facts/L._Paul_Bremer/zCI7UhCh">L. Paul Bremer</a>.<a class="footnote-ref" id="fnref:279" href="#fn:279">279</a> </blockquote>
<h3>Fuzziness and chaos</h3>
It also can be argued that fuzzy concepts are generated by a certain sort of <a href="/facts/Lifestyle_(sociology)/XVb7A8lN">lifestyle</a> 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 <a href="/facts/Test_method/sGtHudUV">test</a> 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 <a href="/facts/Flux/Aw3P5Bvb">flux</a>" or in transition, it may not be possible to do so, so that the amount of fuzziness increases.

<h2 id="everyday-occurrence">Everyday occurrence</h2>
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.<a class="footnote-ref" id="fnref:280" href="#fn:280">280</a> 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 <a href="/facts/Complexity/yaw0UrRj">complex</a> phenomena, or to describe something which is developing and changing, which might involve shedding some old meanings and acquiring new ones. 

<h3>Uses in different areas</h3>
<ul><li>In <a href="/facts/Meteorology/xblFr28q">meteorology</a>, 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").</li>
<li>In <a href="/facts/Biology/1M1ee3pB">biology</a>, protein complexes with multiple structural forms are called <a href="/facts/Fuzzy_complex/J4cDSmRb">fuzzy complexes</a>.<a class="footnote-ref" id="fnref:281" href="#fn:281">281</a> 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. <a href="/facts/Genetic_fuzzy_systems/lzokrKSP">Genetic fuzzy systems</a> use <a href="/facts/Algorithms/fnl5NmRt">algorithms</a> or <a href="/facts/Genetic_programming/kGa2jr0w">genetic programming</a> which simulate natural evolutionary processes, in order to understand their structures and parameters.</li>
<li>In <a href="/facts/Medical_diagnosis/J47rvb0p">medical diagnosis</a>, 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.<a class="footnote-ref" id="fnref:282" href="#fn:282">282</a> 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.<a class="footnote-ref" id="fnref:283" href="#fn:283">283</a></li>
<li>In <a href="/facts/Information_professional/z3G3qIVn">information services</a>, 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. Fuzzy logic can be an important aid for information retrieval systems.<a class="footnote-ref" id="fnref:284" href="#fn:284">284</a></li>
<li>In <a href="/facts/Phenomenology_(philosophy)/AILzLkKx">phenomenology</a>, which aims to study the structure of subjective experience without preconceptions,<a class="footnote-ref" id="fnref:285" href="#fn:285">285</a> 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.<a class="footnote-ref" id="fnref:286" href="#fn:286">286</a> 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.</li>
<li>In <a href="/facts/Translation/2foI82so">translation</a> 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.<a class="footnote-ref" id="fnref:287" href="#fn:287">287</a> Some languages have concepts which do not exist in another language, raising the problem of how one would most easily render their meaning. In <a href="/facts/Computer-assisted_translation/JHebmnfD">computer-assisted translation</a>, a technique called <a href="/facts/Fuzzy_matching_(computer-assisted_translation)/WsV9AUuj">fuzzy matching</a> is used to find the most likely translation of a piece of text, using previous translated texts as a basis.</li>
<li>In <a href="/facts/Hypnotherapy/xyF3uoM1">hypnotherapy</a>, 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.<a class="footnote-ref" id="fnref:288" href="#fn:288">288</a></li>
<li>In <a href="/facts/Business/For1e8Nh">business</a> and <a href="/facts/Economics/HYctA2Dn">economics</a>, 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 (<a href="/facts/Framing_effect_(psychology)/xmbNdw1g">frames</a>, reference points, envy, addiction, temptation, fairness)".<a class="footnote-ref" id="fnref:289" href="#fn:289">289</a> 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.</li>
<li>In <a href="/facts/Sexology/5q7dF5Cq">sexology</a>, sex and gender are conceptualized by gender pluralists as a spectrum or continuum, or a set of scaled characteristics.<a class="footnote-ref" id="fnref:290" href="#fn:290">290</a> Thus, the idea that people are either heterosexual <a href="/facts/Men/IMDBa3kl">men</a>, heterosexual <a href="/facts/Women/AfQPA3m4">women</a>, <a href="/facts/Gay/5zNNpbbW">gay</a>, <a href="/facts/Lesbian/b8hj21I6">lesbian</a>, <a href="/facts/Bisexual/An3RSnKG">bisexual</a> or <a href="/facts/Transsexual/xm5Kn5g0">transsexual</a> is far too simplistic; <a href="/facts/Gender_identity/GhaeHYtb">gender identity</a> is a matter of degree, a graded concept, which for that very reason is a fuzzy concept with unsharp boundaries (see also <a href="/facts/Demographics_of_sexual_orientation/mPQoXIWh">demographics of sexual orientation</a>). 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.</li>
<li>In <a href="/facts/Politics/1gDNy3RP">politics</a>, 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 <a href="/facts/Political_power/Q4feTyU6">power</a> relationship.<a class="footnote-ref" id="fnref:291" href="#fn:291">291</a> 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.<a class="footnote-ref" id="fnref:292" href="#fn:292">292</a></li>
<li>In <a href="/facts/Statistical/DNPsKGYU">statistical</a> 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 class="footnote-ref" id="fnref:293" href="#fn:293">293</a> 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.<a class="footnote-ref" id="fnref:294" href="#fn:294">294</a> 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.</li>
<li>In <a href="/facts/Theology/N71Fyw2M">theology</a> an attempt is made to define more precisely the meaning of <a href="/facts/Spirtuality/o2neYDkN">spiritual</a> concepts, which refer to how human beings construct the meaning of human existence, and, often, the relationship people have with a <a href="/facts/Supernatural/qSlTK0Ss">supernatural</a> 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 <a href="/facts/Psychotherapy/n2hDjaop">psychotherapy</a>. The Dutch theologian Kees de Groot has explored the imprecise notion that <a href="/facts/Psychotherapy/n2hDjaop">psychotherapy</a> is like an "implicit <a href="/facts/Religion/cxntrD0R">religion</a>", defined as a "fuzzy concept" (it all depends on what one means by "psychotherapy" and "religion").<a class="footnote-ref" id="fnref:295" href="#fn:295">295</a> The philosopher of spirituality <a href="/facts/Ken_Wilber/QshPBpjK">Ken Wilber</a> 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.<a class="footnote-ref" id="fnref:296" href="#fn:296">296</a></li>
<li>In the <a href="/facts/Legal_system/mc4tYzf0">legal system</a>, 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,<a class="footnote-ref" id="fnref:297" href="#fn:297">297</a> 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.<a class="footnote-ref" id="fnref:298" href="#fn:298">298</a></li>
<li>In <a href="/facts/Business_administration/Sj3A56z1">administration</a>, <a href="/facts/Archive/BgCG3dxa">archiving</a> and <a href="/facts/Accounting/sHznDY3V">accounting</a>, 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.<a class="footnote-ref" id="fnref:299" href="#fn:299">299</a></li>
<li>In everyday online <a href="/facts/Internet/Xw1rVvJU">Internet</a> use, almost all search queries by employees, citizens and consumers are processed/solved with software partly based on fuzzy concepts, fuzzy logic and fuzzy ontologies. All the hi-tech corporations, companies and state-owned organizations use <a href="/facts/Approximate_string_matching/aotktvNI">fuzzy string searching</a> algorithms in the digital services they supply.</li></ul>
<h3>Generalities</h3>
Many concepts which are used fairly universally in daily life (such as "<a href="/facts/Love/H183DM4w">love</a>",<a class="footnote-ref" id="fnref:300" href="#fn:300">300</a> "<a href="/facts/God/KAykGr3G">God</a>",<a class="footnote-ref" id="fnref:301" href="#fn:301">301</a> "<a href="/facts/Health/jjTlJf6m">health</a>",<a class="footnote-ref" id="fnref:302" href="#fn:302">302</a> "<a href="/facts/Social/twVnad19">social</a>",<a class="footnote-ref" id="fnref:303" href="#fn:303">303</a> "<a href="/facts/Sustainability/9oYH9mAs">sustainability</a>"<a class="footnote-ref" id="fnref:304" href="#fn:304">304</a> "<a href="/facts/Toleration/xJWeVd0t">tolerance</a>" 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.

<h3>Multiple meanings</h3>
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 <a href="/facts/Expletive_attributive/HjGU70ub">expletive</a> term may be uttered arbitrarily. A feeling or sense is conveyed, without it being fully clear what it is about.
<a href="/facts/Happiness/bNL2Kfp1">Happiness</a> may be an example of a word with variable meanings depending on context or timing.<a class="footnote-ref" id="fnref:305" href="#fn:305">305</a>

<h3>Ambiguities</h3>
Fuzzy concepts can be used deliberately to create <a href="/facts/Ambiguity/mgn4oEoQ">ambiguity</a> and <a href="/facts/Vagueness/BUlktmmr">vagueness</a>, as an evasive tactic or a ruse,<a class="footnote-ref" id="fnref:306" href="#fn:306">306</a> or to bridge what would otherwise be immediately recognized as a <a href="/facts/Contradiction/PQHH5a2J">contradiction</a> 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. It could be academic bluff or pretense of knowledge. But it could also be a prologue to a more exact formulation of a concept, or to a better understanding of it.<a class="footnote-ref" id="fnref:307" href="#fn:307">307</a>

<h2 id="efficiency">Efficiency</h2>
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.

<h3>Popper</h3>
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, <a href="/facts/Karl_Popper/RpLqMrz5">Karl Popper</a> rejected <a href="/facts/Pedant/TcEeR953">pedantry</a> and commented that:

<blockquote> "...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..."<a class="footnote-ref" id="fnref:308" href="#fn:308">308</a> </blockquote>
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.

<h3>Fuzzy logic gambit</h3>
<a href="/facts/Lotfi_Zadeh/WojTlG4Y">Lotfi A. Zadeh</a> 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 <a href="/facts/Fuzzy_control_system/SMamr5qY">Takeshi Yamakawa</a>'s programming for an <a href="/facts/Inverted_pendulum/4knRtpQX">inverted pendulum</a>, where differential equations are replaced by fuzzy if-then rules in which words are used in place of numbers.<a class="footnote-ref" id="fnref:309" href="#fn:309">309</a>

<h2 id="fuzzy-concepts-vs-boolean-concepts">Fuzzy concepts vs. Boolean concepts</h2>
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 <a href="/facts/Boolean_logic/VkCToAmg">Boolean logic</a> (a <a href="/facts/Two-valued_logic/dnkYVCod">two-valued logic</a> or binary logic is simply replaced with a <a href="/facts/Many-valued_logic/qNI26RdF">many-valued logic</a>).
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.<a class="footnote-ref" id="fnref:310" href="#fn:310">310</a> 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 can generate logical problems, if e.g. overlapping subsets within sets are related to other overlapping subsets within other sets (it may seem a rather obscure issue of little significance, but it can cause more complex programming routines to malfunction).

<h2 id="clarifying-methods">Clarifying methods</h2>
In <a href="/facts/Mathematical_logic/SaI7Y3wN">mathematical logic</a>, <a href="/facts/Computer_programming/QmSsFnu0">computer programming</a>, <a href="/facts/Philosophy/zAH5PVD6">philosophy</a> and <a href="/facts/Linguistics/oQuamvf2">linguistics</a>, fuzzy concepts can be defined more accurately, by describing the concepts using the terms of <a href="/facts/Fuzzy_logic/1PvN4Att">fuzzy logic</a> or other <a href="/facts/Substructural_logic/4blDGy92">substructural logics</a>. With the rapid development of computer programming languages and digital processing capacity since the 1970s, it is now accepted in the sciences that there isn't just one "correct" way to formalize items of knowledge. Innovators realized that concepts and processes can be represented using many different kinds of tools, methods and systems — according to what happens to be the most useful, effective or efficient method for a given purpose.<a class="footnote-ref" id="fnref:311" href="#fn:311">311</a> Aided by new software and artificial intelligence, many traditional and new sorts of techniques can be applied to clarify ideas, such as:

<ul><li>1. Contextualizing the concept by defining the setting or situation in which the concept is used, or how it is used appropriately (<a href="/facts/Context_(language_use)/eJM44o7x">context</a>).</li>
<li>2. Identifying the <a href="/facts/Intention/ahzUpq2G">intention</a>, <a href="/facts/Motivation/uqLMfLKq">purpose</a>, aim or goal associated with the concept (<a href="/facts/Teleology/8YR2sPUf">teleology</a> and <a href="/facts/Design/kR4GNeja">design</a>).</li>
<li>3. Comparing and contrasting the concept with related ideas in the present or the past (<a href="/facts/Comparative/qS94y639">comparative</a> and <a href="/facts/Comparative_research/MlKbeRlt">comparative research</a>).</li>
<li>4. Creating a <a href="/facts/Conceptual_model/hgysE2kP">model</a>, likeness, <a href="/facts/Analogy/LYcGaTdm">analogy</a>, <a href="/facts/Metaphor/xwsLaqJI">metaphor</a>, <a href="/facts/Prototype/q0yPLYdb">prototype</a> or <a href="/facts/Narrative/RIVoxJpV">narrative</a> which shows what the concept is about or how it is applied (<a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a>, <a href="/facts/Simulation/Q4QJV1oI">simulation</a> or successive approximation <a href="https://www.thefreedictionary.com/successive+approximation">[126]</a>).</li>
<li>5. Probing the <a href="/facts/Presupposition/IsdPl2Ub">assumptions</a> on which a concept is based, or which are associated with its use (<a href="/facts/Critical_thought/x7j7f8W9">critical thought</a>, <a href="/facts/Tacit_assumption/rMjEC2GW">tacit assumption</a>).</li>
<li>6. <a href="/facts/Map_(mathematics)/FABF1l2f">Mapping</a> or <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graphing</a> the applications of the concept using some basic <a href="/facts/Parameters/zFvVFMv9">parameters</a>, or using some diagrams or flow charts to understand the relationships between elements involved (<a href="/facts/Visualization_(graphics)/xVfxo6FJ">visualization</a> and <a href="/facts/Concept_map/f9Dt1LED">concept map</a>).<a class="footnote-ref" id="fnref:312" href="#fn:312">312</a><a class="footnote-ref" id="fnref:313" href="#fn:313">313</a></li>
<li>7. Examining how likely it is that the concept applies, statistically or intuitively (<a href="/facts/Probability_theory/mFBn51rE">probability theory</a>).</li>
<li>8. Specifying relevant conditions to which the concept applies, as a procedure (<a href="/facts/Computer_programming/QmSsFnu0">computer programming</a>, <a href="/facts/Formal_concept_analysis/ztxSSxMB">formal concept analysis</a>).</li>
<li>9. <a href="/facts/Abstract_and_concrete/uNshak6S">Concretizing</a> the concept – finding specific examples, illustrations, details or cases to which it applies (<a href="/facts/Exemplification_theory/YtfHXyUl">exemplar</a>, <a href="/facts/Exemplification/07409aLN">exemplification</a>).</li>
<li>10. Reducing or restating fuzzy concepts in terms which are simpler or similar, and which are not fuzzy or less fuzzy (<a href="/facts/Computer_algebra/hjun1KAX">simplification</a>, <a href="/facts/Dimensionality_reduction/XoRpcrB0">dimensionality reduction</a>, <a href="/facts/Plain_language/8rVK3bwa">plain language</a>, <a href="/facts/KISS_principle/eSDN26hS">KISS principle</a> or <a href="/facts/Concision/KPaPwdaw">concision</a>).</li>
<li>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 (<a href="/facts/Trial_and_error/NGm2GoEd">trial and error</a> or <a href="/facts/Pilot_experiment/MlAyYHnP">pilot experiment</a>).</li>
<li>12. Engaging in a structured <a href="/facts/Dialogue/a3etwdsd">dialogue</a> or repeated <a href="/facts/Discussion/GNKD6xb3">discussion</a>, to exchange ideas about how to get specific about what it means and how to clear it up (<a href="/facts/Scrum_(software_development)/Ih3ufnVB">scrum method</a>).</li>
<li>13. Allocating different applications of the concept to different but related sets (<a href="/facts/Boolean_logic/VkCToAmg">Boolean logic</a>).</li>
<li>14. Identifying operational rules defining the use of the concept, which can be stated in a language and which cover all or most cases (<a href="/facts/Material_conditional/hYf4b7AE">material conditional</a>).</li>
<li>15. <a href="/facts/Categorization/R6rWYttP">Classifying, categorizing, grouping</a>, or inventorizing all or most cases or uses to which the concept applies (<a href="/facts/Taxonomy_(general)/QRvFI267">taxonomy</a>, <a href="/facts/Cluster_analysis/HUeDnWPt">cluster analysis</a> and <a href="/facts/Typology_(statistics)/1hX8Muxr">typology</a>).</li></ul>

<ul><li>16. Applying a <a href="/facts/Meta-language/peNXFNUV">meta-language</a> which includes fuzzy concepts in a more inclusive categorical system which is not fuzzy (<a href="/facts/Meta_(prefix)/WQu5Xe1N">meta</a>).</li>
<li>17. Creating a <a href="/facts/Measurement/rsLLbn1s">measure</a> or <a href="/facts/Scale_(social_sciences)/2Qqrjgc1">scale</a> of the degree to which the concept applies (<a href="/facts/Metrology/HC0MmLdb">metrology</a>).</li>
<li>18. Examining the distribution patterns or distributional frequency of (possibly different) uses of the concept (<a href="/facts/Statistics/DNPsKGYU">statistics</a>).</li>
<li>19. Specifying a series of <a href="/facts/Logical_operators/VkPhUs6M">logical operators</a> or inferential system which captures all or most cases to which the concept applies (<a href="/facts/Algorithm/fnl5NmRt">algorithm</a>).</li>
<li>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" (<a href="/facts/Proxy_(statistics)/W5V2eqaT">proxy</a>)</li>
<li>21. Engaging in <a href="/facts/Meditation/mlU2nUt7">meditation</a>, 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 (<a href="/facts/Self-care/1jyq3VXS">self-care</a>).</li></ul>
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.

<h2 id="defuzzification">Defuzzification</h2>
Main article: <a href="/facts/Defuzzification/Vx1v6N4I">Defuzzification</a>
A process of <a href="/facts/Defuzzification/Vx1v6N4I">defuzzification</a> is said to occur, when fuzzy concepts can be logically described in terms of <a href="/facts/Fuzzy_set/BGWudHJq">fuzzy sets</a>, 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.<a class="footnote-ref" id="fnref:314" href="#fn:314">314</a> 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.

<h2 id="see-also">See also</h2>

<ul><li><a href="/facts/Alternative_set_theory/EJoMmlQg">Alternative set theory</a></li>
<li><a href="/facts/Approximate_measures/pXhhJ9gU">Approximate measures</a></li>
<li><a href="/facts/Approximation/LvYHsXzq">Approximation</a></li>
<li><a href="/facts/Approximation_algorithm/BWlf7M32">Approximation algorithm</a></li>
<li><a href="/facts/Approximation_theory/87AaPXLp">Approximation theory</a></li>
<li><a href="/facts/Classical_logic/ls62MAms">Classical logic</a></li>
<li><a href="/facts/Defuzzification/Vx1v6N4I">Defuzzification</a></li>
<li><a href="/facts/Degree_of_truth/5h9Nzjvy">Degree of truth</a></li>
<li><a href="/facts/Detection_theory/XWSx5V1K">Detection theory</a></li>
<li><a href="/facts/Deviant_logic/TnEly3lT">Deviant logic</a></li>
<li><a href="/facts/Dialectic/NTL69ms6">Dialectic</a></li>
<li><a href="/facts/European_Society_for_Fuzzy_Logic_and_Technology/L0GGlQFB">European Society for Fuzzy Logic and Technology</a></li>
<li><a href="/facts/Fuzzy_classification/jBMVd0nG">Fuzzy classification</a></li>
<li><a href="/facts/Fuzzy_clustering/MEmTL2po">Fuzzy clustering</a></li>
<li><a href="/facts/Fuzzy_logic/1PvN4Att">Fuzzy logic</a></li>
<li><a href="/facts/Fuzzy_markup_language/43l6v0d3">Fuzzy markup language</a></li>
<li><a href="/facts/Fuzzy_mathematics/DAdIA1HM">Fuzzy mathematics</a></li>
<li><a href="/facts/Fuzzy_measure_theory/5DUI2Nh6">Fuzzy measure theory</a></li>
<li><a href="/facts/Fuzzy_number/p76dqPeY">Fuzzy number</a></li>
<li><a href="/facts/Fuzzy_set_operations/I5HFju1C">Fuzzy set operations</a></li>
<li><a href="/facts/Fuzzy_set_theory/BGWudHJq">Fuzzy set theory</a></li>
<li><a href="/facts/Fuzzy_Sets_and_Systems/N9tL5MJE">Fuzzy Sets and Systems</a></li>
<li><a href="/facts/Fuzzy_subalgebra/LUsgYHtT">Fuzzy subalgebra</a></li>
<li><a href="/facts/George_Klir/R7aAM6C8">George Klir</a></li>
<li><a href="/facts/High-performance_fuzzy_computing/Nv2AmLXX">High-performance fuzzy computing</a></li>
<li><a href="/facts/Identity_(Philosophy)/F9rr3GkA">Identity (Philosophy)</a></li>
<li><a href="/facts/Interval_finite_element/QMn7STnt">Interval finite element</a></li>
<li><a href="/facts/Jakobson%2527s_functions_of_language/nrFFURKj">Jakobson's functions of language</a></li>
<li><a href="/facts/Linear_partial_information/IEWVXDDV">Linear partial information</a></li>
<li><a href="/facts/Many-valued_logic/qNI26RdF">Many-valued logic</a></li>
<li><a href="/facts/Multiset/bPFJGMDo">Multiset</a></li>
<li><a href="/facts/Neuro-fuzzy/EY2RTys8">Neuro-fuzzy</a></li>
<li><a href="/facts/Non-well-founded_set_theory/RkmGMC70">Non-well-founded set theory</a></li>
<li><a href="/facts/Obfuscation/4HXs1ICy">Obfuscation</a></li>
<li><a href="/facts/Opaque_context/zAqj16q1">Opaque context</a></li>
<li><a href="/facts/Paraconsistent_logic/6HLVEvrr">Paraconsistent logic</a></li>
<li><a href="/facts/Phenomenology_(psychology)/Rse4u0LN">Phenomenology (psychology)</a></li>
<li><a href="/facts/Accuracy_and_precision/JQ47xXDq">Precision</a></li>
<li><a href="/facts/Referential_transparency/g0zARbAD">Referential transparency</a></li>
<li><a href="/facts/Reflexivity_(social_theory)/1du8jj8p">reflexivity (social theory)</a></li>
<li><a href="/facts/Post-normal_science/fPGQn49c">Post-normal science</a></li>
<li><a href="/facts/Rough_fuzzy_hybridization/TSEaDwum">Rough fuzzy hybridization</a></li>
<li><a href="/facts/Rough_set/7pRxKzxx">Rough set</a></li>
<li><a href="/facts/Semiset/Ewu4KjD9">Semiset</a></li>
<li><a href="/facts/S%25C3%25B8rensen_similarity_index/rwLufgya">Sørensen similarity index</a></li>
<li><a href="/facts/Synchronicity/T3EP7j2T">Synchronicity</a></li>
<li><a href="/facts/Type-2_Fuzzy_Sets_and_Systems/LsqLRemx">Type-2 Fuzzy Sets and Systems</a></li>
<li><a href="/facts/Uncertainty/VM9pnGvk">Uncertainty</a></li>
<li><a href="/facts/Vague_set/okXsdBp4">Vague set</a></li>
<li><a href="/facts/Vagueness/BUlktmmr">Vagueness</a></li>
<li><a href="/facts/Vagueness_and_Degrees_of_Truth/nq4XTVIB">Vagueness and Degrees of Truth</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.austinlinks.com/Fuzzy/tutorial.html">James F. Brule, Fuzzy systems tutorial</a></li>
<li><a href="http://plato.stanford.edu/entries/logic-fuzzy/">"Fuzzy Logic", Stanford Encyclopedia of Philosophy</a></li>
<li><a href="http://plato.stanford.edu/entries/vagueness/">"Vagueness", Stanford Encyclopedia of Philosophy</a></li>
<li><a href="https://www.calvin.edu/~pribeiro/othrlnks/Fuzzy/home.htm">Calvin College Engineering Department, Getting Started with Fuzzy Logic</a> <a href="https://web.archive.org/web/20180221194040/http://www.calvin.edu/~pribeiro/othrlnks/Fuzzy/home.htm">Archived</a> 2018-02-21 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a></li>
<li><a href="https://www.youtube.com/watch?v=2ScTwFCcXGo">2009 Benjamin Franklin Medal Winner: Lotfi A. Zadeh</a></li>
<li><a href="https://www.eusflat.org/materials/mathware-vol24n2.pdf">EUSFLAT, In memory of Lotfi A. Zadeh</a></li>
<li><a href="https://libeldoc.bsuir.by/bitstream/123456789/30388/1/Remembering%20Lot%EF%AC%81%20Zadeh.pdf">RAFSoft, Remembering Lotfi Zadeh</a></li>
<li><a href="https://cs.nju.edu.cn/shanglin/Lecture2.pdf">Lin Shang, Lecture on fuzzy and rough sets, Nanjing University</a></li>
<li><a href="http://fuzzy.cs.ovgu.de/ci/fs/fs_ch02_fst.pdf">Rudolf Kruse and Christian Moewes on fuzzy set theory</a></li>
<li><a href="https://web.fe.up.pt/~asousa/sbld/recursos/FuzzyLogicJumpStart__free%20.pdf">Fuzzy Logic for "Just Plain Folks"</a> by <a href="/facts/Thomas_Sowell/KGhBbqy0">Thomas Sowell</a></li>
<li><a href="https://researchpod.org/arts-humanities/fuzzy-logic-origin-future-non-aristotelian-thinking">Podcast (with text) Fuzzy logic: The origin and future of non-Aristotelian thinking (20 March 2023)</a></li>
<li><a href="https://dl.acm.org/profile/81375596015/publications?Role=author">ACM digital library entries for Lotfi A. Zadeh</a></li>
<li><a href="https://www2.eecs.berkeley.edu/Pubs/Faculty/zadeh.html">UC Berkeley Faculty Publications of Lotfi A. Zadeh</a></li>
<li><a href="https://dblp.org/pid/z/LAZadeh.html">dblp computer science bibliography for Lotfi A. Zadeh</a></li>
<li><a href="https://philpapers.org/rec/ZADFLT">PhilPapers catalogue for Lotfi Zadeh's contributions to Fuzzy logic theory and applications</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Radim Bělohlávek & 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. <a href="/wiki/Susan_Haack" target="_blank">/wiki/Susan_Haack</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">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). <a href="/wiki/Significant_figures" target="_blank">/wiki/Significant_figures</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">George Lakoff, "Hedges: A Study in Meaning Criteria and the Logic of Fuzzy Concepts." Journal of Philosophical Logic, Vol. 2, 1973, pp. 458–508; Eleanor Rosch Heider, "On the Internal Structure of Perceptual and Semantic Categories". Unpublished paper, Psychology Department, UC Berkeley, 1971 (as cited by Lakoff). <a href="/wiki/George_Lakoff" target="_blank">/wiki/George_Lakoff</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">The online Collins dictionary gives a separate entry for "fuzzy concept" in English and Chinese.[1] <a href="https://www.collinsdictionary.com/zh/dictionary/english/fuzzy-concept" target="_blank">https://www.collinsdictionary.com/zh/dictionary/english/fuzzy-concept</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">For example: "The term 'fuzzy' refers to concepts without precise borders. Membership in a 'fuzzy' set — the set of things to which a 'fuzzy' concept (fuzzily) applies — is to be thought of as being a matter of degree" (Charles G. Morgan, "Fuzzy logic". In: Routledge Encyclopedia of Philosophy. London: Routledge, 1998, vol. 4, p. 664). <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">E.g. 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. <a href="/wiki/Timothy_Williamson" target="_blank">/wiki/Timothy_Williamson</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Lotfi A. Zadeh, "What is fuzzy logic?". IFSA Newsletter (International Fuzzy Systems Association), Vol. 10, No. 1, March 2013, pp. 5–6. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Matti Eklund, "Vagueness and Second-Level Indeterminacy", in: Richard Dietz & Sebastiano Moruzzi (eds.), Cuts and clouds. Vagueness, Its Nature, and Its Logic. Oxford University Press, 2009, p. 65. Stephen P. Stich, Review of Israel Scheffler, Beyond the letter: a philosophical Inquiry into ambiguity, vagueness and metaphor in language [London: Routledge & Kegan Paul, 1979]. Linguistics and Philosophy, Vol. 5, issue 2, June 1982, pp. 295-297. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">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. <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">This is more or less the definition of Lotfi Zadeh who was himself a computer scientist. For a recent book, see: Rahul Kar et al., Fuzzy Logic Concepts in Computer Science and Mathematics. Palm Bay, Florida: Apple Academic Press, 2025.[2] <a href="/wiki/Lotfi_Zadeh" target="_blank">/wiki/Lotfi_Zadeh</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Vyvyan Evans, A glossary of cognitive linguistics. Salt Lake City: University of Utah Press, 2007, p. 88; Eleanor Rosch and Carolyn Mervis, "Family resemblance: studies in the internal structure of categories", in: Cognitive Psychology Vol. 7, issue 4, October 1975, pp. 573-605; Steve Pinker and Alan Prince, "The nature of human concepts: evidence from an unusual source", in: Ray Jackendoff et al., Language, logic and concepts. Cambridge, Mass.: MIT Press, 1999, pp. 221-262. <a href="/wiki/Eleanor_Rosch" target="_blank">/wiki/Eleanor_Rosch</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">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. <a href="/wiki/Susan_Haack" target="_blank">/wiki/Susan_Haack</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Charles G. Morgan, "Fuzzy logic", in: Routledge Encyclopedia of Philosophy. London: Routledge, 1998, vol. 4, p. 664; George Lakoff, "Hedges: A Study in Meaning Criteria and the Logic of Fuzzy Concepts." Journal of Philosophical Logic, Vol. 2, 1973, pp. 458–508, at p. 491. <a href="/wiki/George_Lakoff" target="_blank">/wiki/George_Lakoff</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Bart Kosko, "Fuzzy logic". In: Scientific American, July 1993, pp. 76-81[3]; 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.[4]; 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.[6]
 <a href="/wiki/Bart_Kosko" target="_blank">/wiki/Bart_Kosko</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">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.[7] 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]
 <a href="/wiki/Joseph_W._Dauben" target="_blank">/wiki/Joseph_W._Dauben</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Katyanna Quach, "Fuzzy logic makes a comeback – in picking where Earth sticks its probes into alien worlds". The Register, 27 Sep 2018.[8] <a href="https://www.theregister.com/2018/09/27/ai_spacecraft_landing/" target="_blank">https://www.theregister.com/2018/09/27/ai_spacecraft_landing/</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Liwei Yang et al., "Path Planning Technique for Mobile Robots: A Review". Machines, Vol. 11, 2023, pp. 980-1026.[9] <a href="https://pdfs.semanticscholar.org/d035/ab3693d1300d985ca8d29f4d7565c3cff0f4.pdf" target="_blank">https://pdfs.semanticscholar.org/d035/ab3693d1300d985ca8d29f4d7565c3cff0f4.pdf</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">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 href="https://dl.acm.org/doi/pdf/10.1145/175247.175255" target="_blank">https://dl.acm.org/doi/pdf/10.1145/175247.175255</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">A useful technical overview is provided in: Enrique Ruspini et al. Handbook of fuzzy computation. Bristol & Philadelphia: Institute of Physics Publishing, 1998. <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Radim Bělohlávek & George J. Klir (eds.), Concepts and fuzzy logic. Cambridge, Mass.: MIT Press, 2011. <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">A. Dumitras, & G. Moschytz, "Understanding Fuzzy Logic: An Interview with Lotfi Zadeh". IEEE signal processing magazine, May 2007, pp. 102–105, at p. 103. <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Edy Portmann, Fuzzy humanist. Wiesbaden: Springer, 2019; Mahdi Eftekhari et al., How fuzzy concepts contribute to machine learning. Cham (Switzerland): Springer, 2022. <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">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. <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Steve Coutinho, Zhuangzi and Early Chinese Philosophy. Vagueness, Transformation and Paradox. Abingdon: Routledge, 2016, p. 17.
 <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">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. <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">"High Altitude Cooking" webpage of the USDA Food Safety and Inspection Service.[10] Christopher Harris, "Boiling Water in Space: The Science Behind a Seemingly Impossible Task". Sauceandbites.com, [11] <a href="/wiki/United_States_Department_of_Agriculture" target="_blank">/wiki/United_States_Department_of_Agriculture</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Chris Hurley, "What Do Astronauts Drink In Space?" Explaining Space, July 21, 2023.[12] <a href="https://www.explainingspace.com/what-do-astronauts-drink-in-space/" target="_blank">https://www.explainingspace.com/what-do-astronauts-drink-in-space/</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">Nick Compton, "What is the oldest city in the world?" The Guardian, 16 Feb 2015.[13] <a href="https://www.theguardian.com/cities/2015/feb/16/whats-the-oldest-city-in-the-world" target="_blank">https://www.theguardian.com/cities/2015/feb/16/whats-the-oldest-city-in-the-world</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">Massimo Pigliucci & Maarten Boudry (eds.), Philosophy of Pseudoscience: Reconsidering the Demarcation Problem. University of Chicago Press, 2013, p. 95. <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">Nicholas Rescher, Many-Valued Logic. New York: McGraw-Hill, 1969. The German scholar Rudolf Seising, whose 2004 Phd dissertation was about the origins and development of fuzzy set theory, has published many books and articles (in English and German) dealing with the history of thinking about fuzzy concepts. See: Rudolf Seising, Lebenslauf und Publikationen. Ludwig-Maximilians-Universität München/Ludwig Maximilian University of Munich.[14] <a href="/wiki/Nicholas_Rescher" target="_blank">/wiki/Nicholas_Rescher</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">"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.[15] <a href="https://broadviewpress.com/paradox-heap-john-l-bell/" target="_blank">https://broadviewpress.com/paradox-heap-john-l-bell/</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">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. <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">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.[16]) <a href="https://plato.stanford.edu/entries/boundary/" target="_blank">https://plato.stanford.edu/entries/boundary/</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">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 [17];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. <a href="https://www.attalus.org/cicero/academica2c.html" target="_blank">https://www.attalus.org/cicero/academica2c.html</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
<li id="fn:35">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. <a href="/wiki/Angelica_Nuzzo" target="_blank">/wiki/Angelica_Nuzzo</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></li>
<li id="fn:36">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.[18] <a href="/wiki/Robert_L._Carneiro" target="_blank">/wiki/Robert_L._Carneiro</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></li>
<li id="fn:37">Eric Steinhart, "Nietzsche on identity". Revista di Estetica, Vol. 28, No. 1, 2005, pp. 241-256;[19] 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.
 <a href="https://philpapers.org/archive/STENOI-2.pdf" target="_blank">https://philpapers.org/archive/STENOI-2.pdf</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></li>
<li id="fn:38">William Joseph Gavin, William James and the Reinstatement of the Vague. Philadelphia: Temple University Press, 1992, chapter 3. <a href="#fnref:38" class="footnote-back-ref">↩</a></li>
<li id="fn:39">S. Rahman & J. Redmond, "Hugh MacColl and the Birth of Logical Pluralism". In: Handbook of the History of Logic, Vol. 4. Elsevier, 2008, pp. 533-604. <a href="#fnref:39" class="footnote-back-ref">↩</a></li>
<li id="fn:40">Mihai Nadin, "The logic of vagueness", in: Eugene Freeman (ed.), The Relevance of Charles Peirce. La Salle, Ill.: Open Court, 1983, pp. 154-166. <a href="#fnref:40" class="footnote-back-ref">↩</a></li>
<li id="fn:41">Carl Gustav Hempel, "Vagueness and logic". Philosophy of Science, Vol. 6, Issue 2, April 1939, pp. 163-180. <a href="/wiki/Carl_Gustav_Hempel" target="_blank">/wiki/Carl_Gustav_Hempel</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></li>
<li id="fn:42">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. <a href="#fnref:42" class="footnote-back-ref">↩</a></li>
<li id="fn:43">Radomir S. Stankovic & 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. <a href="/wiki/Arto_Salomaa" target="_blank">/wiki/Arto_Salomaa</a> <a href="#fnref:43" class="footnote-back-ref">↩</a></li>
<li id="fn:44">Ludwig Wittgenstein, Philosophical Investigations. Oxford University Press, 1953, Part 1, sections 65-88. <a href="/wiki/Ludwig_Wittgenstein" target="_blank">/wiki/Ludwig_Wittgenstein</a> <a href="#fnref:44" class="footnote-back-ref">↩</a></li>
<li id="fn:45">Jan Łukasiewicz, "On three-valued logic". In: Jan Łukasiewicz, Selected Works. Amsterdam: North Holland Publishing Company, 1970, pp. 87-88. <a href="/wiki/Jan_%C5%81ukasiewicz" target="_blank">/wiki/Jan_%C5%81ukasiewicz</a> <a href="#fnref:45" class="footnote-back-ref">↩</a></li>
<li id="fn:46">Emil Leon Post, "Introduction to a general theory of elementary propositions". American Journal of Mathematics, Vol. 43, No. 3, July 1921, p. 163-185. <a href="/wiki/Emil_Leon_Post" target="_blank">/wiki/Emil_Leon_Post</a> <a href="#fnref:46" class="footnote-back-ref">↩</a></li>
<li id="fn:47">Jan Łukasiewicz and Alfred Tarski, "Investigations into the sentential calculus", in: Alfred Tarski, Logic, semantics, metamathematics. Papers from 1923 to 1938. Oxford: Clarendon Press, 1956, pp. 38-59; Alfred Tarski, Logic, semantics, metamathematics. Oxford: Oxford University Press, 1956. <a href="/wiki/Jan_%C5%81ukasiewicz" target="_blank">/wiki/Jan_%C5%81ukasiewicz</a> <a href="#fnref:47" class="footnote-back-ref">↩</a></li>
<li id="fn:48">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.[20] <a href="https://www.researchgate.net/publication/267071144_Cantor_fuzzy_near_and_rough_sets_in_image_analysis" target="_blank">https://www.researchgate.net/publication/267071144_Cantor_fuzzy_near_and_rough_sets_in_image_analysis</a> <a href="#fnref:48" class="footnote-back-ref">↩</a></li>
<li id="fn:49">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.[21] Archived 2006-07-19 at the Wayback Machine <a href="http://staff.ulsu.ru/bazhanov/russian/fatevas1.pdf" target="_blank">http://staff.ulsu.ru/bazhanov/russian/fatevas1.pdf</a> <a href="#fnref:49" class="footnote-back-ref">↩</a></li>
<li id="fn:50">Tim Lethen, "Gödel on many-valued logic". The review of Symbolic Logic, Vol. 16, issue 3, September 2023, pp. 655-671. <a href="#fnref:50" class="footnote-back-ref">↩</a></li>
<li id="fn:51">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.[22] See Susan Haack, Philosophy of Logics. Cambridge University Press, 1978, p. 205 <a href="/wiki/Susan_Haack" target="_blank">/wiki/Susan_Haack</a> <a href="#fnref:51" class="footnote-back-ref">↩</a></li>
<li id="fn:52">W. V. Quine, "Speaking of Objects". Proceedings and Addresses of the American Philosophical Association, Vol. 31, 1957/1958, pp. 5-22, at p. 20; Shiro Ishikawa , “Fuzzy Logic in the Quantum Mechanical Worldview; Related to Zadeh, Wittgenstein, Moore, Saussure, Quine, Lewis Carroll, etc.”. Journal of Applied Mathematics and Physics, Vol. 9, 2021, pp. 1583-1610. <a href="#fnref:52" class="footnote-back-ref">↩</a></li>
<li id="fn:53">George J. Klir & Bo Yuan, Fuzzy sets and fuzzy logic. Theory and applications. Upper Saddle River (NJ.): Prentice Hall, 1995. <a href="/wiki/George_Klir" target="_blank">/wiki/George_Klir</a> <a href="#fnref:53" class="footnote-back-ref">↩</a></li>
<li id="fn:54">Petr Hájek, Metamathematics of fuzzy logic. Dordrecht: Springer, 1998; Franco Montagna (ed.), Petr Hájek on Mathematical Fuzzy Logic. Cham: Springer, 2015. <a href="/wiki/Petr_H%C3%A1jek" target="_blank">/wiki/Petr_H%C3%A1jek</a> <a href="#fnref:54" class="footnote-back-ref">↩</a></li>
<li id="fn:55">Joseph Goguen, "The logic of inexact concepts". Synthese, Vol. 19, No. 3/4), 1969, pp. 325–37; Radim Bělohlávek, "Goguen's contributions to fuzzy logic in retrospect". International Journal of General Systems, Volume 48, issue 8, 2019, pp. 811-824. <a href="/wiki/Joseph_Goguen" target="_blank">/wiki/Joseph_Goguen</a> <a href="#fnref:55" class="footnote-back-ref">↩</a></li>
<li id="fn:56">Ronald R. Yager publications list [23]. <a href="/wiki/Ronald_R._Yager" target="_blank">/wiki/Ronald_R._Yager</a> <a href="#fnref:56" class="footnote-back-ref">↩</a></li>
<li id="fn:57">Enrique Héctor Ruspini et al., "Fuzzy Clustering: A Historical Perspective". IEEE Computational Intelligence Magazine, Volume 14, Issue 1, February 2019, pp. 45-55. Rudolf Seising, “Fuzzy Memories of Enrique Hector Ruspini” (1942–2019). In: Information Processing and Management of Uncertainty in Knowledge-Based Systems. 18th International Conference, IPMU 2020, Lisbon, Portugal, June 15–19, 2020, Proceedings, Part II. Cham: Springer Nature, 2020 [24] <a href="https://www.academia.edu/65448873/Fuzzy_Memories_of_Enrique_Hector_Ruspini_1942_2019_" target="_blank">https://www.academia.edu/65448873/Fuzzy_Memories_of_Enrique_Hector_Ruspini_1942_2019_</a> <a href="#fnref:57" class="footnote-back-ref">↩</a></li>
<li id="fn:58">Radim Bělohlávek, "Pavelka-style fuzzy logic in retrospect and prospect". Fuzzy Sets and Systems, Vol. 281, 15 December 2015, pp. 61-72.[25] <a href="https://www.sciencedirect.com/science/article/abs/pii/S0165011415003218?via%3Dihub" target="_blank">https://www.sciencedirect.com/science/article/abs/pii/S0165011415003218?via%3Dihub</a> <a href="#fnref:58" class="footnote-back-ref">↩</a></li>
<li id="fn:59">Radim Bělohlávek, "Systems, uncertainty, and information: A legacy of George J. Klir." International Journal of General Systems, Volume 46, Issue 8, 2017, pp. 792-823. <a href="#fnref:59" class="footnote-back-ref">↩</a></li>
<li id="fn:60">Didier Dubois and Henri Prade, Fuzzy sets and systems. Theory and applications. New York: Academic Press, New York, 1980. <a href="/wiki/Didier_Dubois_(mathematician)" target="_blank">/wiki/Didier_Dubois_(mathematician)</a> <a href="#fnref:60" class="footnote-back-ref">↩</a></li>
<li id="fn:61">Bernadette Bouchon-Meunier, “Fuzzy sets and possibility theory in approximate and plausible reasoning”, in: James C. Bezdek, Didier Dubois & Henri Prade (eds.), Fuzzy Sets in Approximate Reasoning and Information Systems. New York: Springer, 1999, pp. 15-190. <a href="/wiki/Bernadette_Bouchon-Meunier" target="_blank">/wiki/Bernadette_Bouchon-Meunier</a> <a href="#fnref:61" class="footnote-back-ref">↩</a></li>
<li id="fn:62">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.[26] <a href="http://ijimt.org/papers/58-M450.pdf" target="_blank">http://ijimt.org/papers/58-M450.pdf</a> <a href="#fnref:62" class="footnote-back-ref">↩</a></li>
<li id="fn:63">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.[27]
 <a href="/wiki/Didier_Dubois_(mathematician)" target="_blank">/wiki/Didier_Dubois_(mathematician)</a> <a href="#fnref:63" class="footnote-back-ref">↩</a></li>
<li id="fn:64">Abraham Kaplan and Hermann F. Schott, "A calculus for empirical classes", Methodos, Vol. 3, 1951, pp. 165–188. <a href="/wiki/Abraham_Kaplan" target="_blank">/wiki/Abraham_Kaplan</a> <a href="#fnref:64" class="footnote-back-ref">↩</a></li>
<li id="fn:65">Timothy Williamson, Vagueness. London: Routledge, 1996, p. 120. <a href="#fnref:65" class="footnote-back-ref">↩</a></li>
<li id="fn:66">J. Barkley Rosser Sr. and Atwell R. Turquette, Many-valued logics. Amsterdam: North-Holland Publishing Company, 1952, p. 109. <a href="/wiki/J._Barkley_Rosser" target="_blank">/wiki/J._Barkley_Rosser</a> <a href="#fnref:66" class="footnote-back-ref">↩</a></li>
<li id="fn:67">Aleksandr A. Zinov'ev, David Dinsmore Comey and Guido Küng, Philosophical problems of many-valued logic. Dordrecht: D. Reidel, 1963. <a href="#fnref:67" class="footnote-back-ref">↩</a></li>
<li id="fn:68">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. <a href="#fnref:68" class="footnote-back-ref">↩</a></li>
<li id="fn:69">Dieter Klaua. "Über 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. <a href="#fnref:69" class="footnote-back-ref">↩</a></li>
<li id="fn:70">Siegfried Gottwald, "Shaping the logic of fuzzy set theory". In: Cintula, Petr et al. (eds.), Witnessed years. Essays in honour of Petr Hájek. London: College Publications, 2009, pp. 193–208. [28] Archived 2012-10-01 at the Wayback Machine <a href="http://www.uni-leipzig.de/~logik/gottwald/Hajek09.pdf" target="_blank">http://www.uni-leipzig.de/~logik/gottwald/Hajek09.pdf</a> <a href="#fnref:70" class="footnote-back-ref">↩</a></li>
<li id="fn:71">Robert John Ackermann, An introduction to many-valued logics. London, Routledge & Kegan Paul, 1967; Nicholas Rescher, Many-Valued Logic. New York: McGraw-Hill, 1969. <a href="/wiki/Nicholas_Rescher" target="_blank">/wiki/Nicholas_Rescher</a> <a href="#fnref:71" class="footnote-back-ref">↩</a></li>
<li id="fn:72">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.
 <a href="#fnref:72" class="footnote-back-ref">↩</a></li>
<li id="fn:73">Susan Haack, Deviant logic, fuzzy logic: beyond the formalism. Chicago: University of Chicago Press, 1996. <a href="/wiki/Susan_Haack" target="_blank">/wiki/Susan_Haack</a> <a href="#fnref:73" class="footnote-back-ref">↩</a></li>
<li id="fn:74">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.[29] <a href="/wiki/Didier_Dubois_(mathematician)" target="_blank">/wiki/Didier_Dubois_(mathematician)</a> <a href="#fnref:74" class="footnote-back-ref">↩</a></li>
<li id="fn:75">George J. Klir & Bo Yuan, Fuzzy sets and fuzzy logic. Theory and applications. Upper Saddle River (NJ.): Prentice Hall, 1995. <a href="#fnref:75" class="footnote-back-ref">↩</a></li>
<li id="fn:76">Merrie Bergmann, An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Derivation Systems. Cambridge University Press, 2008. <a href="#fnref:76" class="footnote-back-ref">↩</a></li>
<li id="fn:77">Radim Bělohlávek, Joseph W. Dauben & George J. Klir, Fuzzy Logic and Mathematics: A Historical Perspective. Oxford University Press, 2017. <a href="/wiki/Joseph_W._Dauben" target="_blank">/wiki/Joseph_W._Dauben</a> <a href="#fnref:77" class="footnote-back-ref">↩</a></li>
<li id="fn:78">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.[30]; A. Dumitras, & G. Moschytz, "Understanding Fuzzy Logic: An Interview with Lotfi Zadeh". IEEE signal processing magazine, May 2007, pp. 102-105; Lotfi A. Zadeh, "Fuzzy logic — a personal perspective". Fuzzy Sets and Systems, Vol. 281, 15 December 2015, pp. 4-20. <a href="/wiki/Lotfi_Zadeh" target="_blank">/wiki/Lotfi_Zadeh</a> <a href="#fnref:78" class="footnote-back-ref">↩</a></li>
<li id="fn:79">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. In 2022, the National Library of Azerbaijan published a comprehensive, 607-page multilingual bibliography, titled Lotfi Zadeh, which features a chronological listing of all known publications by Zadeh, indexes of his works, and authors who have written about him. See also the external links at the bottom of this article. <a href="/wiki/National_Library_of_Azerbaijan" target="_blank">/wiki/National_Library_of_Azerbaijan</a> <a href="#fnref:79" class="footnote-back-ref">↩</a></li>
<li id="fn:80">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. <a href="#fnref:80" class="footnote-back-ref">↩</a></li>
<li id="fn:81">IFSA Newsletter (International Fuzzy Systems Association), Vol. 10, No. 1, March 2013 [31] <a href="http://isdlab.ie.ntnu.edu.tw/ntust/ifsa/files/newsletter/Vol10/IFSAnewsletterV10N1.pdf" target="_blank">http://isdlab.ie.ntnu.edu.tw/ntust/ifsa/files/newsletter/Vol10/IFSAnewsletterV10N1.pdf</a> <a href="#fnref:81" class="footnote-back-ref">↩</a></li>
<li id="fn:82">Richard Van Noorden, Brendan Maher & Regina Nuzzo, "The top 100 papers". Nature, 29 October 2014.[32] <a href="/wiki/Regina_Nuzzo" target="_blank">/wiki/Regina_Nuzzo</a> <a href="#fnref:82" class="footnote-back-ref">↩</a></li>
<li id="fn:83">Jose M. Merigo, Enrique Herrera-Viedma, Ronald R. Yager, and Janusz Kacprzyk, "A Bibliometric Overview of the Research Impact of Lotfi A. Zadeh". In: IEEE Symposium Series on Computational Intelligence (SSCI), Bangalore, 2018, pp. 441-446. <a href="#fnref:83" class="footnote-back-ref">↩</a></li>
<li id="fn:84">Radim Bělohlávek, Joseph W. Dauben & George J. Klir, Fuzzy Logic and Mathematics: A Historical Perspective. Oxford University Press, 2017. <a href="/wiki/Joseph_Dauben" target="_blank">/wiki/Joseph_Dauben</a> <a href="#fnref:84" class="footnote-back-ref">↩</a></li>
<li id="fn:85">The 1970s academic literature is discussed, for example, in Stevan Harnad, Categorical perception. The groundwork of cognition. Cambridge University Press, 1987; and Edward E. Smith & Douglas L. Medin, Categories and concepts. Cambridge: Harvard University Press, 1981. <a href="/wiki/Stevan_Harnad" target="_blank">/wiki/Stevan_Harnad</a> <a href="#fnref:85" class="footnote-back-ref">↩</a></li>
<li id="fn:86">As explained in this wiki, Zadeh did not use the term "fuzzy concept" in his 1965 article, but introduced the notion of a fuzzy set and its formalization. Elsewhere he did talk about fuzzy concepts as ideas with unsharp boundaries of application. <a href="#fnref:86" class="footnote-back-ref">↩</a></li>
<li id="fn:87">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.[33] <a href="http://belohlavek.inf.upol.cz/publications/BeVy_Wifcl.pdf" target="_blank">http://belohlavek.inf.upol.cz/publications/BeVy_Wifcl.pdf</a> <a href="#fnref:87" class="footnote-back-ref">↩</a></li>
<li id="fn:88">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. <a href="#fnref:88" class="footnote-back-ref">↩</a></li>
<li id="fn:89">Nikolaos Galatos, Peter Jipsen, Tomasz Kowalski & Hiroakira Ono, Residuated lattices: an algebraic glimpse at substructural logics. Elsevier Science, 2007. <a href="/wiki/Substructural_logics" target="_blank">/wiki/Substructural_logics</a> <a href="#fnref:89" class="footnote-back-ref">↩</a></li>
<li id="fn:90">Joseph A. Goguen, “L-fuzzy Sets.” Journal of Mathematical Analysis and Applications, Vol. 18, 1967, pp. 145–174; Radim Bělohlávek, "Goguen's contributions to fuzzy logic in retrospect". International Journal of General Systems, Volume 48, issue 8, 2019, pp. 811-824 at p. 817.[34] <a href="http://belohlavek.inf.upol.cz/publications/Bel_Gcflr-IJGS-2019.pdf" target="_blank">http://belohlavek.inf.upol.cz/publications/Bel_Gcflr-IJGS-2019.pdf</a> <a href="#fnref:90" class="footnote-back-ref">↩</a></li>
<li id="fn:91">Petr Hájek, Metamathematics of fuzzy logic. Dordrecht: Springer, 1998, p. 2. <a href="/wiki/Petr_H%C3%A1jek" target="_blank">/wiki/Petr_H%C3%A1jek</a> <a href="#fnref:91" class="footnote-back-ref">↩</a></li>
<li id="fn:92">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.[35] <a href="/wiki/Lotfi_Zadeh" target="_blank">/wiki/Lotfi_Zadeh</a> <a href="#fnref:92" class="footnote-back-ref">↩</a></li>
<li id="fn:93">Timothy Williamson, Vagueness. New York, London: Routledge, 1994. <a href="#fnref:93" class="footnote-back-ref">↩</a></li>
<li id="fn:94">Delia Graff Fara, "Shifting Sands: An Interest Relative Theory of Vagueness". Philosophical Topics, Vol. 28 No. 1, 2000, pp. 45-81. <a href="/wiki/Delia_Graff_Fara" target="_blank">/wiki/Delia_Graff_Fara</a> <a href="#fnref:94" class="footnote-back-ref">↩</a></li>
<li id="fn:95">Kit Fine, Vagueness: a global approach. New York: Oxford University Press, 2020, chapter 1. <a href="/wiki/Kit_Fine" target="_blank">/wiki/Kit_Fine</a> <a href="#fnref:95" class="footnote-back-ref">↩</a></li>
<li id="fn:96">Deleuze and Guattari, A Thousand Plateaus (1988, 551). <a href="/wiki/A_Thousand_Plateaus" target="_blank">/wiki/A_Thousand_Plateaus</a> <a href="#fnref:96" class="footnote-back-ref">↩</a></li>
<li id="fn:97">Deleuze and Guattari, What Is Philosophy? (New York: Columbia University Press, 1994, p. 141). <a href="/wiki/What_Is_Philosophy%3F_(Deleuze_and_Guattari_book)" target="_blank">/wiki/What_Is_Philosophy%3F_(Deleuze_and_Guattari_book)</a> <a href="#fnref:97" class="footnote-back-ref">↩</a></li>
<li id="fn:98">Roy T. Cook, A dictionary of philosophical logic. Edinburgh University Press, 2009, p. 84. <a href="#fnref:98" class="footnote-back-ref">↩</a></li>
<li id="fn:99">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. <a href="/wiki/Nicholas_Rescher" target="_blank">/wiki/Nicholas_Rescher</a> <a href="#fnref:99" class="footnote-back-ref">↩</a></li>
<li id="fn:100">Susan Haack, Philosophy of Logics. Cambridge University Press, 1978, p. xii. <a href="/wiki/Susan_Haack" target="_blank">/wiki/Susan_Haack</a> <a href="#fnref:100" class="footnote-back-ref">↩</a></li>
<li id="fn:101">Susan Haack, Philosophy of Logics. Cambridge University Press, 1978, p. 165. <a href="/wiki/Susan_Haack" target="_blank">/wiki/Susan_Haack</a> <a href="#fnref:101" class="footnote-back-ref">↩</a></li>
<li id="fn:102">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. See also: Petr Hájek, Metamathematics of fuzzy logic. Dordrecht: Springer, 1998; Franco Montagna (ed.), Petr Hájek on Mathematical Fuzzy Logic. Cham: Springer, 2015. <a href="/wiki/Petr_H%C3%A1jek" target="_blank">/wiki/Petr_H%C3%A1jek</a> <a href="#fnref:102" class="footnote-back-ref">↩</a></li>
<li id="fn:103">Petr Cintula, Petr Hájek and Carles Noguera (eds.), Handbook of Mathematical Fuzzy Logic, Vol. 1 & 2. London: King's College, 2011.
[36] [37] <a href="https://usiena-air.unisi.it/retrieve/7d5ebe00-f9ec-4fe8-a922-279b417d5a30/Handbook%20fuzzy%20logis%20Curatela%20vol%201.pdf" target="_blank">https://usiena-air.unisi.it/retrieve/7d5ebe00-f9ec-4fe8-a922-279b417d5a30/Handbook%20fuzzy%20logis%20Curatela%20vol%201.pdf</a> <a href="#fnref:103" class="footnote-back-ref">↩</a></li>
<li id="fn:104">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. <a href="#fnref:104" class="footnote-back-ref">↩</a></li>
<li id="fn:105">Samsung support information page [38]. <a href="http://www.samsung.com/in/support/skp/faq/138486" target="_blank">http://www.samsung.com/in/support/skp/faq/138486</a> <a href="#fnref:105" class="footnote-back-ref">↩</a></li>
<li id="fn:106">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. <a href="#fnref:106" class="footnote-back-ref">↩</a></li>
<li id="fn:107">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. <a href="https://web.archive.org/web/20130520100749/http://www.stosberg.com/Tech/fuzzy/role_in_ai.html" target="_blank">https://web.archive.org/web/20130520100749/http://www.stosberg.com/Tech/fuzzy/role_in_ai.html</a> <a href="#fnref:107" class="footnote-back-ref">↩</a></li>
<li id="fn:108">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. <a href="#fnref:108" class="footnote-back-ref">↩</a></li>
<li id="fn:109">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. <a href="#fnref:109" class="footnote-back-ref">↩</a></li>
<li id="fn:110">""2017 Golden Goose Awardee: Fuzzy Logic, Clear Impact"". Archived from the original on 2019-12-13. Retrieved 2018-03-12. <a href="https://web.archive.org/web/20191213213147/https://www.goldengooseaward.org/awardees/fuzzy-logic" target="_blank">https://web.archive.org/web/20191213213147/https://www.goldengooseaward.org/awardees/fuzzy-logic</a> <a href="#fnref:110" class="footnote-back-ref">↩</a></li>
<li id="fn:111">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. <a href="#fnref:111" class="footnote-back-ref">↩</a></li>
<li id="fn:112">The IFSA URL is: https://fuzzysystems.org/ <a href="https://fuzzysystems.org/" target="_blank">https://fuzzysystems.org/</a> <a href="#fnref:112" class="footnote-back-ref">↩</a></li>
<li id="fn:113">Cf. IEEE CIS website 2018 [39] Archived 2018-04-02 at the Wayback Machine. <a href="https://cis.ieee.org/" target="_blank">https://cis.ieee.org/</a> <a href="#fnref:113" class="footnote-back-ref">↩</a></li>
<li id="fn:114">The URL is https://www.fsdmconf.org/ <a href="https://www.fsdmconf.org/" target="_blank">https://www.fsdmconf.org/</a> <a href="#fnref:114" class="footnote-back-ref">↩</a></li>
<li id="fn:115">See the APNNS website [40]. <a href="http://www.apnns.org/" target="_blank">http://www.apnns.org/</a> <a href="#fnref:115" class="footnote-back-ref">↩</a></li>
<li id="fn:116">INFUS URL <a href="https://infus.itu.edu.tr/" target="_blank">https://infus.itu.edu.tr/</a> <a href="#fnref:116" class="footnote-back-ref">↩</a></li>
<li id="fn:117">See the SOFT website [41]. <a href="https://www.j-soft.org/post/about_eng" target="_blank">https://www.j-soft.org/post/about_eng</a> <a href="#fnref:117" class="footnote-back-ref">↩</a></li>
<li id="fn:118">KIIS website [42]. <a href="http://eng.fuzzy.or.kr/main/" target="_blank">http://eng.fuzzy.or.kr/main/</a> <a href="#fnref:118" class="footnote-back-ref">↩</a></li>
<li id="fn:119">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 <a href="#fnref:119" class="footnote-back-ref">↩</a></li>
<li id="fn:120">The TFSA publishes the International Journal of Fuzzy Systems <a href="https://www.springer.com/engineering/computational+intelligence+and+complexity/journal/40815" target="_blank">https://www.springer.com/engineering/computational+intelligence+and+complexity/journal/40815</a> <a href="#fnref:120" class="footnote-back-ref">↩</a></li>
<li id="fn:121">The NAFIPS website URL is https://sites.ualberta.ca/~reformat/nafips/index.html <a href="https://sites.ualberta.ca/~reformat/nafips/index.html" target="_blank">https://sites.ualberta.ca/~reformat/nafips/index.html</a> <a href="#fnref:121" class="footnote-back-ref">↩</a></li>
<li id="fn:122">The EUSFLAT URL is: http://www.eusflat.org/. Mathfuzzlog url is: https://www.eusflat.org/society-wg-mathfuzzlog.html <a href="http://www.eusflat.org/" target="_blank">http://www.eusflat.org/</a> <a href="#fnref:122" class="footnote-back-ref">↩</a></li>
<li id="fn:123">The NSAIS URL is https://nsais.org/ . <a href="https://nsais.org/" target="_blank">https://nsais.org/</a> <a href="#fnref:123" class="footnote-back-ref">↩</a></li>
<li id="fn:124">IFSS website [43]. <a href="http://fuzzy.ir/find.php?item=1.11.67.en" target="_blank">http://fuzzy.ir/find.php?item=1.11.67.en</a> <a href="#fnref:124" class="footnote-back-ref">↩</a></li>
<li id="fn:125">"Iran ranks 3rd in producing fuzzy systems related science: Official". The Iran Project, 9 March 2017.[44] <a href="http://theiranproject.com/blog/2016/03/09/iran-ranks-3rd-in-producing-fuzzy-systems-related-science-official/" target="_blank">http://theiranproject.com/blog/2016/03/09/iran-ranks-3rd-in-producing-fuzzy-systems-related-science-official/</a> <a href="#fnref:125" class="footnote-back-ref">↩</a></li>
<li id="fn:126">(RAFSSoftCom) <a href="http://ransmv.narod.ru/eng.html" target="_blank">http://ransmv.narod.ru/eng.html</a> <a href="#fnref:126" class="footnote-back-ref">↩</a></li>
<li id="fn:127">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. <a href="#fnref:127" class="footnote-back-ref">↩</a></li>
<li id="fn:128">CBSF page [45]. <a href="https://sites.google.com/unesp.br/vicbsf-en/contacts" target="_blank">https://sites.google.com/unesp.br/vicbsf-en/contacts</a> <a href="#fnref:128" class="footnote-back-ref">↩</a></li>
<li id="fn:129">https://sites.google.com/unesp.br/vicbsf-en/home <a href="https://sites.google.com/unesp.br/vicbsf-en/home" target="_blank">https://sites.google.com/unesp.br/vicbsf-en/home</a> <a href="#fnref:129" class="footnote-back-ref">↩</a></li>
<li id="fn:130">CSCR website <a href="https://www.isical.ac.in/~scc/" target="_blank">https://www.isical.ac.in/~scc/</a> <a href="#fnref:130" class="footnote-back-ref">↩</a></li>
<li id="fn:131">SLAAI website [46]. <a href="http://slaai.lk/" target="_blank">http://slaai.lk/</a> <a href="#fnref:131" class="footnote-back-ref">↩</a></li>
<li id="fn:132">Laszlo T. Koczy, "Fuzzy systems research in Hungary – a subjective story." Archives for the philosophy and history of soft computing, Issue 1, 2017 [47] <a href="https://itf.njszt.hu/wp-content/uploads/2024/01/fuzzy_systems_research_in_hungary.pdf" target="_blank">https://itf.njszt.hu/wp-content/uploads/2024/01/fuzzy_systems_research_in_hungary.pdf</a> <a href="#fnref:132" class="footnote-back-ref">↩</a></li>
<li id="fn:133">Indonesian society for soft computing repository site <a href="https://www.neliti.com/komunitas-softcomputing-indonesia" target="_blank">https://www.neliti.com/komunitas-softcomputing-indonesia</a> <a href="#fnref:133" class="footnote-back-ref">↩</a></li>
<li id="fn:134">Vietnam Fuzzy Systems Society <a href="https://www.easychair.org/cfp/AICI2021" target="_blank">https://www.easychair.org/cfp/AICI2021</a> <a href="#fnref:134" class="footnote-back-ref">↩</a></li>
<li id="fn:135">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.[48] This text was taken offline after Zadeh died. <a href="/wiki/Lotfi_Zadeh" target="_blank">/wiki/Lotfi_Zadeh</a> <a href="#fnref:135" class="footnote-back-ref">↩</a></li>
<li id="fn:136">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. <a href="/wiki/Lotfi_Zadeh" target="_blank">/wiki/Lotfi_Zadeh</a> <a href="#fnref:136" class="footnote-back-ref">↩</a></li>
<li id="fn:137">Andrei Popescu, "A general approach to fuzzy concepts". Mathematical Logic Quarterly Vol. 50, No. 3, 2005, pp. 265–280. <a href="#fnref:137" class="footnote-back-ref">↩</a></li>
<li id="fn:138">This definition is consistent with the ISO 704 and ISO 1087 standard definitions of a concept by the International Organization for Standardization. <a href="/wiki/International_Organization_for_Standardization" target="_blank">/wiki/International_Organization_for_Standardization</a> <a href="#fnref:138" class="footnote-back-ref">↩</a></li>
<li id="fn:139">Radim Bělohlávek and Vilem Vychodil, "What is a fuzzy concept lattice?" Department of Computer Science, Palacky University, Olomouc, 2005.[49] <a href="http://ceur-ws.org/Vol-162/paper2.pdf" target="_blank">http://ceur-ws.org/Vol-162/paper2.pdf</a> <a href="#fnref:139" class="footnote-back-ref">↩</a></li>
<li id="fn:140">"See further the COMPASSS site". Archived from the original on 2017-01-01. Retrieved 2016-12-31. <a href="https://web.archive.org/web/20170101161316/http://www.compasss.org/index.htm" target="_blank">https://web.archive.org/web/20170101161316/http://www.compasss.org/index.htm</a> <a href="#fnref:140" class="footnote-back-ref">↩</a></li>
<li id="fn:141">Daniel Kreiss, Prototype Politics: Technology-Intensive Campaigning and the Data of Democracy. Oxford University Press, 2016. <a href="#fnref:141" class="footnote-back-ref">↩</a></li>
<li id="fn:142">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.[50] <a href="/wiki/Janusz_Kacprzyk" target="_blank">/wiki/Janusz_Kacprzyk</a> <a href="#fnref:142" class="footnote-back-ref">↩</a></li>
<li id="fn:143">Chris Preimesberger, "Big-Data Analytics Plays Big Role in 2016 Election Campaigns". eWeek, 24 September 2016. [51][permanent dead link];Gregory Thomas, "The Big Data Advantage in the Race for the White House." Bemyapp Media, 2 September 2016.[52] 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.[53]; Lisa Ragusa, "And the Winner of the 2016 Election Is… Big Data". Liaison, 4 November 2016.[54]; John Markman, "Big Data And The 2016 Election". Forbes Magazine, 8 August 2016.[55]; Taylor Armerding, "Big Data and elections: The candidates know you – better than you know them." CSOonline.com, 17 July 2016.[56] <a href="http://www.eweek.com/database/big-data-analytics-plays-big-role-in-2016-election-campaigns.html" target="_blank">http://www.eweek.com/database/big-data-analytics-plays-big-role-in-2016-election-campaigns.html</a> <a href="#fnref:143" class="footnote-back-ref">↩</a></li>
<li id="fn:144">Robert Epstein, "How Google Could Rig the 2016 Election". Politico.com, 19 August 2015 [57]; Marcel Rosenbach, "How Google and Facebook Can Reshape Elections", Der Spiegel online (English edition), 8 November 2016. [58] <a href="https://www.politico.com/magazine/story/2015/08/how-google-could-rig-the-2016-election-121548/" target="_blank">https://www.politico.com/magazine/story/2015/08/how-google-could-rig-the-2016-election-121548/</a> <a href="#fnref:144" class="footnote-back-ref">↩</a></li>
<li id="fn:145">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.[59] <a href="http://www.drdobbs.com/cpp/fuzzy-logic-in-c/184408940" target="_blank">http://www.drdobbs.com/cpp/fuzzy-logic-in-c/184408940</a> <a href="#fnref:145" class="footnote-back-ref">↩</a></li>
<li id="fn:146">Chris Preimesberger, "Big-Data Analytics Plays Big Role in 2016 Election Campaigns". eWeek, 24 September 2016. <a href="#fnref:146" class="footnote-back-ref">↩</a></li>
<li id="fn:147">Kenneth P. Vogel, "The heiress quietly shaping Trump's operation." Politico.com, 21 November 2016.[60] <a href="https://www.politico.com/story/2016/11/rebekah-mercer-donald-trump-231693" target="_blank">https://www.politico.com/story/2016/11/rebekah-mercer-donald-trump-231693</a> <a href="#fnref:147" class="footnote-back-ref">↩</a></li>
<li id="fn:148">Kate Brannely, "Trump Campaign Pays Millions to Overseas Big Data Firm." NBC News, 4 November 2016.[61] <a href="http://www.nbcnews.com/storyline/2016-election-day/trump-campaign-pays-millions-overseas-big-data-firm-n677321" target="_blank">http://www.nbcnews.com/storyline/2016-election-day/trump-campaign-pays-millions-overseas-big-data-firm-n677321</a> <a href="#fnref:148" class="footnote-back-ref">↩</a></li>
<li id="fn:149">"Cambridge Analytica – About Us". Cambridge Analytica website. Archived from the original on 2016-02-16. <a href="https://web.archive.org/web/20160216023554/https://cambridgeanalytica.org/about" target="_blank">https://web.archive.org/web/20160216023554/https://cambridgeanalytica.org/about</a> <a href="#fnref:149" class="footnote-back-ref">↩</a></li>
<li id="fn:150">Carole Cadwalladr, "UK regulator orders Cambridge Analytica to release data on US voter". The Guardian, 5 May 2018.[62] <a href="https://www.theguardian.com/uk-news/2018/may/05/cambridge-analytica-uk-regulator-release-data-us-voter-david-carroll" target="_blank">https://www.theguardian.com/uk-news/2018/may/05/cambridge-analytica-uk-regulator-release-data-us-voter-david-carroll</a> <a href="#fnref:150" class="footnote-back-ref">↩</a></li>
<li id="fn:151">Adam Tanner, "Nine Things You Don't Know About The Gathering Of Your Personal Data." Forbes Magazine, 4 November 2014.[63] <a href="https://www.forbes.com/sites/adamtanner/2014/11/04/nine-things-you-dont-know-about-the-gathering-of-your-personal-data/#3faf581675b7" target="_blank">https://www.forbes.com/sites/adamtanner/2014/11/04/nine-things-you-dont-know-about-the-gathering-of-your-personal-data/#3faf581675b7</a> <a href="#fnref:151" class="footnote-back-ref">↩</a></li>
<li id="fn:152">Steve Lohr and Natasha Singernov, "How Data Failed Us in Calling an Election." New York Times, 10 November 2016 [64] <a href="/wiki/New_York_Times" target="_blank">/wiki/New_York_Times</a> <a href="#fnref:152" class="footnote-back-ref">↩</a></li>
<li id="fn:153">"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.[65] See also: Jonathan Vanian, "How Bad Polling Data Fooled Everyone Except Donald Trump". Fortune, 10 November 2016.[66] <a href="/wiki/Kellyanne_Conway" target="_blank">/wiki/Kellyanne_Conway</a> <a href="#fnref:153" class="footnote-back-ref">↩</a></li>
<li id="fn:154">"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.[67] <a href="https://www.wsj.com/video/how-trump-won-the-presidency/969FC367-31B3-436F-8388-6C9CE7546AC7.html" target="_blank">https://www.wsj.com/video/how-trump-won-the-presidency/969FC367-31B3-436F-8388-6C9CE7546AC7.html</a> <a href="#fnref:154" class="footnote-back-ref">↩</a></li>
<li id="fn:155">Ryan Lizza, "Kellyanne Conway's political machinations", The New Yorker, 17 October 2016.[68](see also alternative facts) <a href="https://www.newyorker.com/magazine/2016/10/17/kellyanne-conways-political-machinations" target="_blank">https://www.newyorker.com/magazine/2016/10/17/kellyanne-conways-political-machinations</a> <a href="#fnref:155" class="footnote-back-ref">↩</a></li>
<li id="fn:156">Jina Moore, "Cambridge Analytica Had a Role in Kenya Election, Too". New York Times, 20 March 2018.[69] <a href="https://www.nytimes.com/2018/03/20/world/africa/kenya-cambridge-analytica-election.html" target="_blank">https://www.nytimes.com/2018/03/20/world/africa/kenya-cambridge-analytica-election.html</a> <a href="#fnref:156" class="footnote-back-ref">↩</a></li>
<li id="fn:157">Carole Cadwalladr & Emma Graham-Harrison, "Pressure mounts on Cambridge Analytica and Facebook over data scandal." The Guardian, 18 March 2018.[70] <a href="/wiki/The_Guardian" target="_blank">/wiki/The_Guardian</a> <a href="#fnref:157" class="footnote-back-ref">↩</a></li>
<li id="fn:158">Jonathan Shieber, "Facebook hired a forensics firm to investigate Cambridge Analytica as stock falls 7%." TC Techcrunch.com, 19 March 2018.[71] <a href="/wiki/Cambridge_Analytica" target="_blank">/wiki/Cambridge_Analytica</a> <a href="#fnref:158" class="footnote-back-ref">↩</a></li>
<li id="fn:159">Spencer Phade, "Who Should Profit From Selling Your Personal Data?". Futures Platform, 27 March 2018.[72] <a href="https://www.futuresplatform.com/blog/your-data-and-profit-data-scraping" target="_blank">https://www.futuresplatform.com/blog/your-data-and-profit-data-scraping</a> <a href="#fnref:159" class="footnote-back-ref">↩</a></li>
<li id="fn:160">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.[73] <a href="https://www.washingtonpost.com/news/the-switch/wp/2018/03/20/ftc-opens-investigation-into-facebook-after-cambridge-analytica-scrapes-millions-of-users-personal-information/" target="_blank">https://www.washingtonpost.com/news/the-switch/wp/2018/03/20/ftc-opens-investigation-into-facebook-after-cambridge-analytica-scrapes-millions-of-users-personal-information/</a> <a href="#fnref:160" class="footnote-back-ref">↩</a></li>
<li id="fn:161">Sara Salinas, "Facebook hires firm to conduct a 'comprehensive audit' of Cambridge Analytica". CNBC news, 19 March 2018 and squawkbox panel video [74] <a href="https://www.cnbc.com/2018/03/19/facebook-to-conduct-a-comprehensive-audit-of-cambridge-analytica.html" target="_blank">https://www.cnbc.com/2018/03/19/facebook-to-conduct-a-comprehensive-audit-of-cambridge-analytica.html</a> <a href="#fnref:161" class="footnote-back-ref">↩</a></li>
<li id="fn:162">Christopher Carbone, "Facebook might have 29,000 data points on you, but Mark Zuckerberg doesn't really know." Fox News, 11 April 2018.[75] [76] Julia Angwin, Surya Mattu and Terry Parris Jr., "Facebook Doesn't Tell Users Everything It Really Knows About Them." ProPublica, 27 December 2016.[77] <a href="https://www.foxnews.com/tech/facebook-might-have-29000-data-points-on-you-but-mark-zuckerberg-doesnt-really-know" target="_blank">https://www.foxnews.com/tech/facebook-might-have-29000-data-points-on-you-but-mark-zuckerberg-doesnt-really-know</a> <a href="#fnref:162" class="footnote-back-ref">↩</a></li>
<li id="fn:163">Zuckerberg's Senate hearings highlights video, 10 April 2018 <a href="https://www.youtube.com/watch?v=EgI_KAkSyCw" target="_blank">https://www.youtube.com/watch?v=EgI_KAkSyCw</a> <a href="#fnref:163" class="footnote-back-ref">↩</a></li>
<li id="fn:164">Olivia Solon and Julia Carrie Wong, "Facebook suspends another analytics firm amid questions over surveillance." The Guardian, 20 July 2018.[78] <a href="/wiki/Julia_Carrie_Wong" target="_blank">/wiki/Julia_Carrie_Wong</a> <a href="#fnref:164" class="footnote-back-ref">↩</a></li>
<li id="fn:165">Richard Waters, "Is Facebook a victim of rapid growth or an abuser of user data?" Financial Times, 20 December 2018. <a href="#fnref:165" class="footnote-back-ref">↩</a></li>
<li id="fn:166">Matthew Rosenberg, Nicholas Confessore and Carole Cadwalladr, "How Trump Consultants Exploited the Facebook Data of Millions". New York Times, 17 March 2018. <a href="/wiki/Donald_Trump" target="_blank">/wiki/Donald_Trump</a> <a href="#fnref:166" class="footnote-back-ref">↩</a></li>
<li id="fn:167">Sarah Frier and Todd Shields, "Zuckerberg Says Facebook Collects Internet Data on Non-Users", Bloomberg, 11 April 2018. <a href="#fnref:167" class="footnote-back-ref">↩</a></li>
<li id="fn:168">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.[79] <a href="https://www.nytimes.com/2018/12/18/technology/facebook-privacy.html" target="_blank">https://www.nytimes.com/2018/12/18/technology/facebook-privacy.html</a> <a href="#fnref:168" class="footnote-back-ref">↩</a></li>
<li id="fn:169">Samson Amore, "Here are the biggest lawsuits Facebook is facing". Daily Dot, 27 November 2023.[80] <a href="https://www.dailydot.com/news/facebook-lawsuits/" target="_blank">https://www.dailydot.com/news/facebook-lawsuits/</a> <a href="#fnref:169" class="footnote-back-ref">↩</a></li>
<li id="fn:170">Heather Wallace, "International Regulation of Social Media- A Survey". Pace University, Social Media Legality page, October 18, 2023 [81]; Robert Gorwa, The politics of platform regulation. How governments shape online content moderation. New York: Oxford University Press, 2024.[82] <a href="https://socialmediablawg.blogs.pace.edu/2023/10/18/international-regulation-of-social-media-a-survey/" target="_blank">https://socialmediablawg.blogs.pace.edu/2023/10/18/international-regulation-of-social-media-a-survey/</a> <a href="#fnref:170" class="footnote-back-ref">↩</a></li>
<li id="fn:171">See the Meta "Legal Action" website. <a href="https://about.fb.com/news/tag/legal-action/" target="_blank">https://about.fb.com/news/tag/legal-action/</a> <a href="#fnref:171" class="footnote-back-ref">↩</a></li>
<li id="fn:172">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.[83] <a href="https://www.reuters.com/article/us-faceboook-privacy/cambridge-analytica-and-parent-scl-elections-shutting-down-idUSKBN1I32L7" target="_blank">https://www.reuters.com/article/us-faceboook-privacy/cambridge-analytica-and-parent-scl-elections-shutting-down-idUSKBN1I32L7</a> <a href="#fnref:172" class="footnote-back-ref">↩</a></li>
<li id="fn:173">Tableau.com, Big data: the top 8 trends for 2016. <a href="#fnref:173" class="footnote-back-ref">↩</a></li>
<li id="fn:174">15 Massive Data Centers Under Construction, Top Fives channel, 8 February 2025. <a href="https://www.youtube.com/watch?v=Isr-WK1GEJc" target="_blank">https://www.youtube.com/watch?v=Isr-WK1GEJc</a> <a href="#fnref:174" class="footnote-back-ref">↩</a></li>
<li id="fn:175">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. <a href="/wiki/Yossi_Sariel" target="_blank">/wiki/Yossi_Sariel</a> <a href="#fnref:175" class="footnote-back-ref">↩</a></li>
<li id="fn:176">Yasmeen Serhan, "How Israel Uses AI in Gaza—And What It Might Mean for the Future of Warfare". In: Time Magazine, 18 December 2024.[84] See also: AI-assisted targeting in the Gaza Strip. <a href="https://time.com/7202584/gaza-ukraine-ai-warfare/" target="_blank">https://time.com/7202584/gaza-ukraine-ai-warfare/</a> <a href="#fnref:176" class="footnote-back-ref">↩</a></li>
<li id="fn:177">Funded fuzzy systems research in the defence sector is often a component of artificial intelligence projects, robotics projects, or autonomous systems projects. Military fuzzy logic projects are usually classified or proprietary; many contracts are for in-house work for which no detailed public information is available. <a href="/wiki/Classified_information" target="_blank">/wiki/Classified_information</a> <a href="#fnref:177" class="footnote-back-ref">↩</a></li>
<li id="fn:178">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.[85] Lakshya Tyagi & Swati Singal, "Application of Fuzzy Logic Control Systems in Military Platforms". Noida, India: IEEE 9th International conference on cloud computing, data science & engineering (confluence), 2019.
 <a href="https://dergipark.org.tr/en/download/article-file/1245069" target="_blank">https://dergipark.org.tr/en/download/article-file/1245069</a> <a href="#fnref:178" class="footnote-back-ref">↩</a></li>
<li id="fn:179">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. <a href="#fnref:179" class="footnote-back-ref">↩</a></li>
<li id="fn:180">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.[86][permanent dead link] Angel Garrido & Piedad Yuste, "controversies about the introduction of non-classical logics". Brain, Vol. 5, No. 1-4, 2014.[87] Bob Pease, "What's All This Fuzzy Logic Stuff, Anyhow?" Electronicdesign.com, May 13, 1993 — November 2020 (five parts).[88] <a href="http://ac.els-cdn.com/S0888613X09001327/1-s2.0-S0888613X09001327-main.pdf?_tid=7f4bada4-37bd-11e7-9bfe-00000aacb361&acdnat=1494667448_9f6a0acd8f43577a7af214d56512b12a" target="_blank">http://ac.els-cdn.com/S0888613X09001327/1-s2.0-S0888613X09001327-main.pdf?_tid=7f4bada4-37bd-11e7-9bfe-00000aacb361&acdnat=1494667448_9f6a0acd8f43577a7af214d56512b12a</a> <a href="#fnref:180" class="footnote-back-ref">↩</a></li>
<li id="fn:181">Lotfi A. Zadeh, "Is there a need for fuzzy logic?", Information Sciences, No. 178, 2008, p. 2753. <a href="/wiki/Lotfi_Zadeh" target="_blank">/wiki/Lotfi_Zadeh</a> <a href="#fnref:181" class="footnote-back-ref">↩</a></li>
<li id="fn:182">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. <a href="#fnref:182" class="footnote-back-ref">↩</a></li>
<li id="fn:183">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. <a href="#fnref:183" class="footnote-back-ref">↩</a></li>
<li id="fn:184">Daniel McNeill & Paul Freiberger, Fuzzy Logic: The Revolutionary Computer Technology that Is Changing Our World. New York: Simon & Schuster, 1994, p. 49. <a href="#fnref:184" class="footnote-back-ref">↩</a></li>
<li id="fn:185">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. <a href="#fnref:185" class="footnote-back-ref">↩</a></li>
<li id="fn:186">P. Geach and M. Black (eds.), Translations from the Philosophical Writings of Gottlob Frege, 3rd edition. Blackwell, 1980, p. 159. <a href="#fnref:186" class="footnote-back-ref">↩</a></li>
<li id="fn:187">Ludwig Wittgenstein, Philosophical investigations, Part 1, paragraph 71 (revised 4th Anscombe edition, 2009, p. 38); see also Eleanor Rosch, "Wittgenstein and categorization research in cognitive psychology", in M. Chapman & R. Dixon (eds.), Meaning and the growth of understanding. Wittgenstein's significance for developmental Psychology. Hillsdale, New Jersey: Erlbaum, 1987, pp. 151-166. <a href="/wiki/Ludwig_Wittgenstein" target="_blank">/wiki/Ludwig_Wittgenstein</a> <a href="#fnref:187" class="footnote-back-ref">↩</a></li>
<li id="fn:188">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 [89]; Eric Margolis & Stephen Laurence, "Concepts". In: Stanford Encyclopedia of Philosophy, 2011.[90] <a href="https://www.researchgate.net/publication/220433962_Concepts_and_Semantic_Relations_in_Information_Science" target="_blank">https://www.researchgate.net/publication/220433962_Concepts_and_Semantic_Relations_in_Information_Science</a> <a href="#fnref:188" class="footnote-back-ref">↩</a></li>
<li id="fn:189">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; Steve Pinker and Alan Prince, "The nature of human concepts: evidence from an unusual source", in: Ray Jackendoff et al., Language, logic and concepts. Cambridge, Mass.: MIT Press, 1999, pp. 221-262. <a href="/wiki/Ulric_Neisser" target="_blank">/wiki/Ulric_Neisser</a> <a href="#fnref:189" class="footnote-back-ref">↩</a></li>
<li id="fn:190">Edward E. Smith & Douglas L. Medin, Categories and concepts. Cambridge: Harvard University Press, 1981, p. 182. <a href="#fnref:190" class="footnote-back-ref">↩</a></li>
<li id="fn:191">Pawel Zeidler, Models and Metaphors as Research Tools in Science. Zurich: Lit Verlag, 2013; L. Magnani, N. J. Nersessian & 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. <a href="#fnref:191" class="footnote-back-ref">↩</a></li>
<li id="fn:192">Robin Rollinger & Carlo Ierna, "Christian von Ehrenfels". In Stanford encyclopedia of philosophy, 2019.[91] <a href="/wiki/Christian_von_Ehrenfels" target="_blank">/wiki/Christian_von_Ehrenfels</a> <a href="#fnref:192" class="footnote-back-ref">↩</a></li>
<li id="fn:193">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. <a href="#fnref:193" class="footnote-back-ref">↩</a></li>
<li id="fn:194">Daniel McNeill & Paul Freiberger, Fuzzy Logic: The Revolutionary Computer Technology that Is Changing Our World. New York: Simon & Schuster, 1994, pp. 47-48. <a href="#fnref:194" class="footnote-back-ref">↩</a></li>
<li id="fn:195">A. Dumitras, & G. Moschytz, "Understanding Fuzzy Logic: An Interview with Lotfi Zadeh". IEEE signal processing magazine, May 2007, pp. 102–105, at p. 103. <a href="#fnref:195" class="footnote-back-ref">↩</a></li>
<li id="fn:196">Joseph Tulloch, "‘We just want to stay alive’: An appeal from Gaza." Vatican News, 6 November 2023.[92] <a href="/wiki/Vatican_News" target="_blank">/wiki/Vatican_News</a> <a href="#fnref:196" class="footnote-back-ref">↩</a></li>
<li id="fn:197">Susan Haack, Philosophy of Logics. Cambridge University Press, 1978, p. 213. <a href="#fnref:197" class="footnote-back-ref">↩</a></li>
<li id="fn:198">Joseph Grunfeld, "Haack on fuzzy logic". The Paideia Archive: Twentieth World Congress of Philosophy, 8, 1998, pp. 65-69.[93] <a href="https://www.pdcnet.org/collection/fshow?id=wcp20-paideia_1998_0008_0065_0069&pdfname=wcp20-paideia_1998_0008_0065_0069.pdf&file_type=pdf" target="_blank">https://www.pdcnet.org/collection/fshow?id=wcp20-paideia_1998_0008_0065_0069&pdfname=wcp20-paideia_1998_0008_0065_0069.pdf&file_type=pdf</a> <a href="#fnref:198" class="footnote-back-ref">↩</a></li>
<li id="fn:199">Susan L. Epstein, "Memory and concepts in reactive learning". Proceedings of the Canadian Workshop on Machine Learning 1992[94]. <a href="http://www.cs.hunter.cuny.edu/~epstein/papers/CSCSI92.pdf" target="_blank">http://www.cs.hunter.cuny.edu/~epstein/papers/CSCSI92.pdf</a> <a href="#fnref:199" class="footnote-back-ref">↩</a></li>
<li id="fn:200">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. <a href="#fnref:200" class="footnote-back-ref">↩</a></li>
<li id="fn:201">Stephen Mumford, "Quantities and Qualities", University of Nottingham blog post, September 30, 2012.[95] <a href="https://blogs.nottingham.ac.uk/artsmatters/2012/09/30/quantities-and-qualities/" target="_blank">https://blogs.nottingham.ac.uk/artsmatters/2012/09/30/quantities-and-qualities/</a> <a href="#fnref:201" class="footnote-back-ref">↩</a></li>
<li id="fn:202">Robert M. Wachter, "How Measurement Fails Doctors and Teachers". New York Times, 16 January 2016.[96] <a href="https://www.nytimes.com/2016/01/17/opinion/sunday/how-measurement-fails-doctors-and-teachers.html" target="_blank">https://www.nytimes.com/2016/01/17/opinion/sunday/how-measurement-fails-doctors-and-teachers.html</a> <a href="#fnref:202" class="footnote-back-ref">↩</a></li>
<li id="fn:203">"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.[97] 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.[98] <a href="https://www.nytimes.com/2010/05/16/magazine/16FOB-WWLN-t.html" target="_blank">https://www.nytimes.com/2010/05/16/magazine/16FOB-WWLN-t.html</a> <a href="#fnref:203" class="footnote-back-ref">↩</a></li>
<li id="fn:204">J. Coates, "Keynes, vague concepts and fuzzy logic." In: G.C. Harcourt & P.A. Riach (ed.), A second edition of the General Theory, Volume 2. London: Routledge, 1997, pp. 244–259, at p. 256. <a href="#fnref:204" class="footnote-back-ref">↩</a></li>
<li id="fn:205">F.A. Hayek, "Coping With Ignorance". Imprimis, Volume 7, Number 7, July 1978.[99] <a href="/wiki/Friedrich_Hayek" target="_blank">/wiki/Friedrich_Hayek</a> <a href="#fnref:205" class="footnote-back-ref">↩</a></li>
<li id="fn:206">Viktor Mayer-Schönberger and Thomas Ramge, Reinventing capitalism in the age of big data. London: John Murray, 2018, p. 52. <a href="/wiki/Viktor_Mayer-Sch%C3%B6nberger" target="_blank">/wiki/Viktor_Mayer-Sch%C3%B6nberger</a> <a href="#fnref:206" class="footnote-back-ref">↩</a></li>
<li id="fn:207">Michael Polanyi, The tacit dimension [1966]. Chicago: University of Chicago Press, 2009, pp. 20-21. <a href="#fnref:207" class="footnote-back-ref">↩</a></li>
<li id="fn:208">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.[100] <a href="https://geol260.academic.wlu.edu/course-notes/fuzzy-logic-fuzzy-sets-conditional-inclusion-and-bayes-theorem/a-fuzzy-boundary/" target="_blank">https://geol260.academic.wlu.edu/course-notes/fuzzy-logic-fuzzy-sets-conditional-inclusion-and-bayes-theorem/a-fuzzy-boundary/</a> <a href="#fnref:208" class="footnote-back-ref">↩</a></li>
<li id="fn:209">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. <a href="https://www.fao.org/4/Y1997E/y1997e1m.htm" target="_blank">https://www.fao.org/4/Y1997E/y1997e1m.htm</a> <a href="#fnref:209" class="footnote-back-ref">↩</a></li>
<li id="fn:210">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. <a href="#fnref:210" class="footnote-back-ref">↩</a></li>
<li id="fn:211">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. <a href="https://www.economist.com/briefing/2016/04/30/the-trouble-with-gdp" target="_blank">https://www.economist.com/briefing/2016/04/30/the-trouble-with-gdp</a> <a href="#fnref:211" class="footnote-back-ref">↩</a></li>
<li id="fn:212">Susan Haack, Deviant logic, fuzzy logic — beyond the formalism. Chicago: University of Chicago Press, 1996. <a href="#fnref:212" class="footnote-back-ref">↩</a></li>
<li id="fn:213">Matti Eklund, "Vagueness and Second-Level Indeterminacy", in: Richard Dietz & Sebastiano Moruzzi (eds.), Cuts and clouds. Vagueness, Its Nature, and Its Logic. Oxford University Press, 2009, p. 65. <a href="#fnref:213" class="footnote-back-ref">↩</a></li>
<li id="fn:214">Matti Eklund, "Characterizing Vagueness". Philosophy Compass, 2, 2007, pp. 896-909. <a href="#fnref:214" class="footnote-back-ref">↩</a></li>
<li id="fn:215">Lotfi A. Zadeh, "What is fuzzy logic?". IFSA Newsletter (International Fuzzy Systems Association), Vol. 10, No. 1, March 2013, pp. 5–6. <a href="/wiki/Lotfi_Zadeh" target="_blank">/wiki/Lotfi_Zadeh</a> <a href="#fnref:215" class="footnote-back-ref">↩</a></li>
<li id="fn:216">Tom Dougherty, "Vague value", in: Philosophy and phenomenological research, Vol. 89, No. 2, September 2014, pp. 352–372 [101]; Tom Dougherty, "Vagueness and Indeterminacy in Ethics". In: Tristram McPherson & David Plunkett, The Routledge Handbook of Metaethics. Oxford: Routledge, 2017. <a href="https://philpapers.org/archive/DOUVV.pdf" target="_blank">https://philpapers.org/archive/DOUVV.pdf</a> <a href="#fnref:216" class="footnote-back-ref">↩</a></li>
<li id="fn:217">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. <a href="/wiki/Andrei_Marmor" target="_blank">/wiki/Andrei_Marmor</a> <a href="#fnref:217" class="footnote-back-ref">↩</a></li>
<li id="fn:218">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. <a href="#fnref:218" class="footnote-back-ref">↩</a></li>
<li id="fn:219">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. <a href="#fnref:219" class="footnote-back-ref">↩</a></li>
<li id="fn:220">David Lanius, "What is the value of vagueness?" In: Theoria: a journal of Swedish philosophy, Vol. 87, No. 3, May 2021, pp. 752-780. <a href="#fnref:220" class="footnote-back-ref">↩</a></li>
<li id="fn:221">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. <a href="/wiki/Alfred_Korzybski" target="_blank">/wiki/Alfred_Korzybski</a> <a href="#fnref:221" class="footnote-back-ref">↩</a></li>
<li id="fn:222">Vassos Argyrou, "Anthropology of Magic". In: James D. Wright (ed.), International Encyclopedia of the Social & 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. <a href="/wiki/Object-oriented_ontology" target="_blank">/wiki/Object-oriented_ontology</a> <a href="#fnref:222" class="footnote-back-ref">↩</a></li>
<li id="fn:223">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. <a href="/wiki/Alfred_Korzybski" target="_blank">/wiki/Alfred_Korzybski</a> <a href="#fnref:223" class="footnote-back-ref">↩</a></li>
<li id="fn:224">Tanya Lewis, "What's the Universe Made Of? Math, Says Scientist." Live Science, 30 January 2014.[102] <a href="http://www.livescience.com/42839-the-universe-is-math.html" target="_blank">http://www.livescience.com/42839-the-universe-is-math.html</a> <a href="#fnref:224" class="footnote-back-ref">↩</a></li>
<li id="fn:225">See also Raphael van Riel & Robert Van Gulick, "Scientific reduction". In: Stanford Encyclopedia of Philosophy, 2014.[103] <a href="https://plato.stanford.edu/entries/scientific-reduction/" target="_blank">https://plato.stanford.edu/entries/scientific-reduction/</a> <a href="#fnref:225" class="footnote-back-ref">↩</a></li>
<li id="fn:226">Edward N. Zalta, Abstract Objects. An introduction to axiomatic metaphysics. Dordrecht: D. Reidel Publishing Company, 1983. <a href="/wiki/Edward_N._Zalta" target="_blank">/wiki/Edward_N._Zalta</a> <a href="#fnref:226" class="footnote-back-ref">↩</a></li>
<li id="fn:227">Norman Gulley, Plato's theory of knowledge [1962]. Milton Park: Routledge, 2013, chapter 4. <a href="#fnref:227" class="footnote-back-ref">↩</a></li>
<li id="fn:228">Kees van Deemter, Not exactly. in praise of vagueness. Oxford: Oxford Universty Press, 2010, p. 90. <a href="https://www.uu.nl/staff/CJvanDeemter" target="_blank">https://www.uu.nl/staff/CJvanDeemter</a> <a href="#fnref:228" class="footnote-back-ref">↩</a></li>
<li id="fn:229">Hartry H. Field, Science without numbers. A defense of nominalism. Second edition, Oxford: Oxford University Press, 2016. <a href="/wiki/Hartry_H._Field" target="_blank">/wiki/Hartry_H._Field</a> <a href="#fnref:229" class="footnote-back-ref">↩</a></li>
<li id="fn:230">Mark Balaguer, Platonism and Anti-Platonism in Mathematics. Oxford: Oxford University Press, 1998. <a href="#fnref:230" class="footnote-back-ref">↩</a></li>
<li id="fn:231">"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. <a href="#fnref:231" class="footnote-back-ref">↩</a></li>
<li id="fn:232">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." [104] <a href="/wiki/Paul_Lafargue" target="_blank">/wiki/Paul_Lafargue</a> <a href="#fnref:232" class="footnote-back-ref">↩</a></li>
<li id="fn:233">Hongxing Li, C. L. Philip Chen & Han-Pang Huang, Fuzzy Neural Intelligent Systems. Mathematical Foundation and the Applications in Engineering. Boca Raton, Florida: CRC Press, 2001. <a href="#fnref:233" class="footnote-back-ref">↩</a></li>
<li id="fn:234">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. <a href="#fnref:234" class="footnote-back-ref">↩</a></li>
<li id="fn:235">Andrea Cantini, "Paradoxes and Contemporary Logic", Stanford Encyclopedia of Philosophy online 30 April 2012. <a href="https://plato.stanford.edu/entries/paradoxes-contemporary-logic/#RusCon" target="_blank">https://plato.stanford.edu/entries/paradoxes-contemporary-logic/#RusCon</a> <a href="#fnref:235" class="footnote-back-ref">↩</a></li>
<li id="fn:236">John Coates, The claims of common sense; Moore, Wittgenstein, Keynes and the social sciences. Cambridge: Cambridge University Press, 1996. <a href="#fnref:236" class="footnote-back-ref">↩</a></li>
<li id="fn:237">George Lakoff, "Hedges: A Study in Meaning Criteria and the Logic of Fuzzy Concepts." Journal of Philosophical Logic, Vol. 2, 1973, pp. 458–508.[105] <a href="/wiki/George_Lakoff" target="_blank">/wiki/George_Lakoff</a> <a href="#fnref:237" class="footnote-back-ref">↩</a></li>
<li id="fn:238">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. <a href="/wiki/Charles_Ragin" target="_blank">/wiki/Charles_Ragin</a> <a href="#fnref:238" class="footnote-back-ref">↩</a></li>
<li id="fn:239">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.[106] <a href="https://www.fernuni-hagen.de/conpar/download/stoibercaselitzheinelt.pdf" target="_blank">https://www.fernuni-hagen.de/conpar/download/stoibercaselitzheinelt.pdf</a> <a href="#fnref:239" class="footnote-back-ref">↩</a></li>
<li id="fn:240">Robert Draeseke & 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 & 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. <a href="#fnref:240" class="footnote-back-ref">↩</a></li>
<li id="fn:241">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. <a href="#fnref:241" class="footnote-back-ref">↩</a></li>
<li id="fn:242">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.[107] <a href="/wiki/Bart_Kosko" target="_blank">/wiki/Bart_Kosko</a> <a href="#fnref:242" class="footnote-back-ref">↩</a></li>
<li id="fn:243">Thomas Kron, Reflexiver Terrorismus. Weilerswist: Velbrück, 2015; Thomas Kron & Lars Winter, "Fuzzy-Systeme und die "Corona-Krise". In: Zeitschrift für 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 & Thomas Kron (eds.), Österreichische Zeitschrift für Soziologie, Special Issue "Bestandsaufnahme soziologischer Gewaltforschung", 2020, pp. 113-134; Thomas Kron & 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 Hybrid — zur Formkatastrophe der Gegenwartsgesellschaft". In: Zeitschrift für Theoretische Soziologi, Issue 2, 2014: pp. 220-252; "Integrale Akteurtheorie – zur Modellierung eines Bezugsrahmens für komplexe Akteure". In: Zeitschrift für Soziologie, Issue 3, 2006, pp. 170-192; Thomas Kron & Lars Winter, "Fuzzy Systems — Überlegungen zur Vagheit sozialer Systeme". In: Soziale Systeme, Issue 2, 2005, pp. 370–394; "Fuzzy-Logik für die Soziologie". In: Österreichische Zeitschrift für Soziologie, Issue 3, 2005, pp. 51-89; Thomas Kron & Lars Winter, "Logik in der Soziologie". In: Peter Klimczak & Thomas Zoglauer (ed.), Logik in den Wissenschaften. Münster: 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. <a href="#fnref:243" class="footnote-back-ref">↩</a></li>
<li id="fn:244">Edward A. Shils & Henry A. Finch (eds.), Max Weber on the methodology of the social sciences. Glencoe, Ill.: The Free Press, 1949, p. 93. <a href="#fnref:244" class="footnote-back-ref">↩</a></li>
<li id="fn:245">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. <a href="#fnref:245" class="footnote-back-ref">↩</a></li>
<li id="fn:246">Jörg Rössel 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. <a href="/wiki/Randall_Collins" target="_blank">/wiki/Randall_Collins</a> <a href="#fnref:246" class="footnote-back-ref">↩</a></li>
<li id="fn:247">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. <a href="#fnref:247" class="footnote-back-ref">↩</a></li>
<li id="fn:248">Elizabeth Chaplin, Sociology and visual representation. London: Routledge, 1994, p. 130. <a href="#fnref:248" class="footnote-back-ref">↩</a></li>
<li id="fn:249">Stephen J. Lynch (ed.), Christopher Marlowe: Edward II, with related texts. Indianapolis: Hackett Publishing Company, 2015, p. xix. <a href="#fnref:249" class="footnote-back-ref">↩</a></li>
<li id="fn:250">Loïc Wacquant, "The fuzzy logic of practical sense." in: Pierre Bourdieu and Loïc Wacquant, An invitation to reflexive sociology. London: Polity Press, 1992, chapter I section 4. <a href="/wiki/Lo%C3%AFc_Wacquant" target="_blank">/wiki/Lo%C3%AFc_Wacquant</a> <a href="#fnref:250" class="footnote-back-ref">↩</a></li>
<li id="fn:251">Ph. Manning "Fuzzy Description: Discovery and Invention in Sociology". In: History of the Human Sciences, Vol. 7, No. 1, 1994, pp. 117–23.[108] <a href="http://engagedscholarship.csuohio.edu/cgi/viewcontent.cgi?article=1049&context=clsoc_crim_facpub" target="_blank">http://engagedscholarship.csuohio.edu/cgi/viewcontent.cgi?article=1049&context=clsoc_crim_facpub</a> <a href="#fnref:251" class="footnote-back-ref">↩</a></li>
<li id="fn:252">Philip Shenon, "Their prince is back: Cambodians are baffled." New York Times, 6 June 1993. <a href="#fnref:252" class="footnote-back-ref">↩</a></li>
<li id="fn:253">Betty Blair, "Interview with Lotfi Zadeh, Creator of Fuzzy Logic". Azerbaijan International, Winter 1994, pp. 46–47.[109] <a href="http://www.azer.com/aiweb/categories/magazine/24_folder/24_articles/24_fuzzylogic.html" target="_blank">http://www.azer.com/aiweb/categories/magazine/24_folder/24_articles/24_fuzzylogic.html</a> <a href="#fnref:253" class="footnote-back-ref">↩</a></li>
<li id="fn:254">Johan van Benthem et al. (eds.), The age of alternative logics. Assessing philosophy of logic and mathematics today. Dordrecht: Springer, 2006, p. 203. <a href="/wiki/Johan_van_Benthem_(logician)" target="_blank">/wiki/Johan_van_Benthem_(logician)</a> <a href="#fnref:254" class="footnote-back-ref">↩</a></li>
<li id="fn:255">Kofi Kissi Dompere, Fuzziness and approximate reasoning; epistemics on uncertainty, expectation and risk in rational behaviour. Berlin: Springer, 2009. <a href="#fnref:255" class="footnote-back-ref">↩</a></li>
<li id="fn:256">Masao Mukaidono, Fuzzy logic for beginners. Singapore: World Scientific Publishing, 2001. <a href="#fnref:256" class="footnote-back-ref">↩</a></li>
<li id="fn:257">Himanshu Mishra, Arul Mishra and Baba Shiv, “In Praise of Vagueness: Malleability of Vague Information as a Performance Booster”. Psychological Science, Volume 22, Issue 6, June 2011, pp. 733-738. <a href="#fnref:257" class="footnote-back-ref">↩</a></li>
<li id="fn:258">Daniele Bertini, “The Vagueness of Religious Beliefs”. European Journal for Philosophy of Religion, Vol. 12, No. 2, June 2020, pp. 181-210. <a href="#fnref:258" class="footnote-back-ref">↩</a></li>
<li id="fn:259">Karen Armstrong, The Case for God. New York: Anchor, 2010. <a href="/wiki/Karen_Armstrong" target="_blank">/wiki/Karen_Armstrong</a> <a href="#fnref:259" class="footnote-back-ref">↩</a></li>
<li id="fn:260">David Bohm, Wholeness and the implicate order. London: Routledge & Kegan Paul ARK paperback edition, 1983, p. 86f. <a href="/wiki/David_Bohm" target="_blank">/wiki/David_Bohm</a> <a href="#fnref:260" class="footnote-back-ref">↩</a></li>
<li id="fn:261">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. <a href="#fnref:261" class="footnote-back-ref">↩</a></li>
<li id="fn:262">Paul Teller, "Language and the complexity of the world". Department of Philosophy, University of California at Davis, August 2024.[110] <a href="https://philpapers.org/archive/TELLAT-2.pdf" target="_blank">https://philpapers.org/archive/TELLAT-2.pdf</a> <a href="#fnref:262" class="footnote-back-ref">↩</a></li>
<li id="fn:263">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. <a href="#fnref:263" class="footnote-back-ref">↩</a></li>
<li id="fn:264">Bertram F. Malle & Sara D. Hodges (eds.), Other Minds: How Humans Bridge the Divide Between Self and Others. New York: The Guilford Press, 2005. <a href="#fnref:264" class="footnote-back-ref">↩</a></li>
<li id="fn:265">Paul Grice, Studies in the Way of Words. Cambridge: Harvard University Press, 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. <a href="#fnref:265" class="footnote-back-ref">↩</a></li>
<li id="fn:266">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.[111] <a href="/wiki/John_R._Searle" target="_blank">/wiki/John_R._Searle</a> <a href="#fnref:266" class="footnote-back-ref">↩</a></li>
<li id="fn:267">For non-binary computing, see Unconventional computing. <a href="/wiki/Unconventional_computing" target="_blank">/wiki/Unconventional_computing</a> <a href="#fnref:267" class="footnote-back-ref">↩</a></li>
<li id="fn:268">Karl Aschenbrenner, The concept of coherence in art. Dordrecht: D. Reidel Publishing Company, 1985. <a href="#fnref:268" class="footnote-back-ref">↩</a></li>
<li id="fn:269">A design was inscribed on a gold plate mounted on the NASA space probe Voyager 1, of which scientists hoped that alien civilizations would be able to decipher it.[112] <a href="/wiki/NASA" target="_blank">/wiki/NASA</a> <a href="#fnref:269" class="footnote-back-ref">↩</a></li>
<li id="fn:270">Harry Collins, Tacit & explicit knowledge. Chicago: University of Chicago Press, 2013. <a href="#fnref:270" class="footnote-back-ref">↩</a></li>
<li id="fn:271">Amos Tversky and Daniel Kahneman, "The framing of decisions and psychology of choice". Science, Vol. 211, No. 4481, January 1981, pp. 453-458. <a href="/wiki/Amos_Tversky" target="_blank">/wiki/Amos_Tversky</a> <a href="#fnref:271" class="footnote-back-ref">↩</a></li>
<li id="fn:272">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. <a href="#fnref:272" class="footnote-back-ref">↩</a></li>
<li id="fn:273">Arthur S. Reber, Implicit learning and tacit knowledge. An essay on the cognitive unconscious. Oxford: Oxford University Press, 1993, pp. 137-138. <a href="/wiki/Arthur_S._Reber" target="_blank">/wiki/Arthur_S._Reber</a> <a href="#fnref:273" class="footnote-back-ref">↩</a></li>
<li id="fn:274">Ronald A. Havens (ed.), The wisdom of Milton H. Erickson, Volume II: human behavior & psychotherapy. New York: Irvington Publishers, 1992, chapter 3. <a href="#fnref:274" class="footnote-back-ref">↩</a></li>
<li id="fn:275">A. Cornelius Benjamin, "Science and vagueness". In: Philosophy of science, Vol. 6 No. 4, 1939, pp. 422-431. <a href="#fnref:275" class="footnote-back-ref">↩</a></li>
<li id="fn:276">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. <a href="#fnref:276" class="footnote-back-ref">↩</a></li>
<li id="fn:277">Andrew Tarantola, "Why Frame Rate Matters". Gizmodo.com, 14 January 2015.[113] <a href="https://gizmodo.com/why-frame-rate-matters-1675153198" target="_blank">https://gizmodo.com/why-frame-rate-matters-1675153198</a> <a href="#fnref:277" class="footnote-back-ref">↩</a></li>
<li id="fn:278">João Pina-Cabral, "On embracing the vague", in: HAU: Journal of Ethnographic Theory, Volume 10, Number 3, Winter 2020; William Joseph Gavin, William James and the Reinstatement of the Vague. Philadelphia: Temple University Press, 1992. <a href="#fnref:278" class="footnote-back-ref">↩</a></li>
<li id="fn:279">L. Paul Bremer, "Corporate governance and crisis management", in: Directors & Boards, Winter 2002 <a href="/wiki/L._Paul_Bremer" target="_blank">/wiki/L._Paul_Bremer</a> <a href="#fnref:279" class="footnote-back-ref">↩</a></li>
<li id="fn:280">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. <a href="/wiki/Jean_Piaget" target="_blank">/wiki/Jean_Piaget</a> <a href="#fnref:280" class="footnote-back-ref">↩</a></li>
<li id="fn:281">Rashmi Sharma et al., "Fuzzy complexes: Specific binding without complete folding". FEBS Letters, Volume 589, Issue 19, Part A, July 2015, pp. 2533-2542.[114] <a href="https://febs.onlinelibrary.wiley.com/doi/epdf/10.1016/j.febslet.2015.07.022" target="_blank">https://febs.onlinelibrary.wiley.com/doi/epdf/10.1016/j.febslet.2015.07.022</a> <a href="#fnref:281" class="footnote-back-ref">↩</a></li>
<li id="fn:282">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. <a href="#fnref:282" class="footnote-back-ref">↩</a></li>
<li id="fn:283">Kazem Sadegh-Zadeh "The Fuzzy Revolution: Goodbye to the Aristotelian Weltanschauung". In: Artificial Intelligence in Medicine, 21, 2001, pp. 18–19.[115] <a href="http://www.iiia.csic.es/~puyol/SEIAD2001/publicacions/sadegh.pdf" target="_blank">http://www.iiia.csic.es/~puyol/SEIAD2001/publicacions/sadegh.pdf</a> <a href="#fnref:283" class="footnote-back-ref">↩</a></li>
<li id="fn:284">See e.g. the work of Prof. Gabriella Pasi at University of Milano-Bicocca. An early paper was Gloria Bordogna & Gabriella Pasi, “A fuzzy linguistic approach generalizing Boolean Information Retrieval: A model and its evaluation”. Journal of the American Society for Information Science, Volume 44, Issue 2, March 1993, pp. 70-82. <a href="#fnref:284" class="footnote-back-ref">↩</a></li>
<li id="fn:285">Stephen Priest, Theories of the mind. London: Penguin Books, 1991, p. 183. <a href="#fnref:285" class="footnote-back-ref">↩</a></li>
<li id="fn:286">Michael Hammond, Jane Howarth and Russell Keat, Understanding Phenomenology. Oxford: Blackwell, 1991. <a href="#fnref:286" class="footnote-back-ref">↩</a></li>
<li id="fn:287">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). <a href="#fnref:287" class="footnote-back-ref">↩</a></li>
<li id="fn:288">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. <a href="#fnref:288" class="footnote-back-ref">↩</a></li>
<li id="fn:289">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. <a href="/wiki/Janusz_Kacprzyk" target="_blank">/wiki/Janusz_Kacprzyk</a> <a href="#fnref:289" class="footnote-back-ref">↩</a></li>
<li id="fn:290">Surya Monro, Bisexuality. Houndmills, Basingstoke: Palgrave Macmillan, 2015, p. 49. <a href="#fnref:290" class="footnote-back-ref">↩</a></li>
<li id="fn:291">Bart Kosko, "Yes, Candidates, There Is a Fuzzy Math". New York Times, 7 November 2000. <a href="/wiki/Bart_Kosko" target="_blank">/wiki/Bart_Kosko</a> <a href="#fnref:291" class="footnote-back-ref">↩</a></li>
<li id="fn:292">Michelle Nichols, "Israel defends Gaza hospital raid, UN rights chief says its explanation is vague". Reuters, 3 January 2025. <a href="#fnref:292" class="footnote-back-ref">↩</a></li>
<li id="fn:293">Russel Gordon & David Bendien, "Standard classifications". New Zealand Statistics Review, September 1993, p. 20. <a href="#fnref:293" class="footnote-back-ref">↩</a></li>
<li id="fn:294">Paul C. Bauer et al., "Vague concepts in survey questions. A general problem illustrated with the left-right scale." SSRN Electronic Journal, April 2014.[116] <a href="http://pablobarbera.com/static/vague-concepts.pdf" target="_blank">http://pablobarbera.com/static/vague-concepts.pdf</a> <a href="#fnref:294" class="footnote-back-ref">↩</a></li>
<li id="fn:295">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.[117] Archived 2013-05-23 at the Wayback Machine <a href="http://arno.uvt.nl/show.cgi?fid=76988" target="_blank">http://arno.uvt.nl/show.cgi?fid=76988</a> <a href="#fnref:295" class="footnote-back-ref">↩</a></li>
<li id="fn:296">Mark Manson, "The rise and fall of Ken Wilber", markmanson.net, 4 June 2012.[118] <a href="https://markmanson.net/ken-wilber" target="_blank">https://markmanson.net/ken-wilber</a> <a href="#fnref:296" class="footnote-back-ref">↩</a></li>
<li id="fn:297">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. <a href="#fnref:297" class="footnote-back-ref">↩</a></li>
<li id="fn:298">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. <a href="/wiki/Lawrence_Solan" target="_blank">/wiki/Lawrence_Solan</a> <a href="#fnref:298" class="footnote-back-ref">↩</a></li>
<li id="fn:299">David Henry, "Fuzzy Numbers', Bloomberg Businessweek, 3 October 2004.[119] <a href="https://web.archive.org/web/20120817073213/http://www.businessweek.com/stories/2004-10-03/fuzzy-numbers" target="_blank">https://web.archive.org/web/20120817073213/http://www.businessweek.com/stories/2004-10-03/fuzzy-numbers</a> <a href="#fnref:299" class="footnote-back-ref">↩</a></li>
<li id="fn:300">Beverley Fehr & James A. Russell, "The Concept of Love Viewed From a Prototype Perspective". Journal of Personality and Social Psychology Vol. 60 Number 3, March 1991, pp. 425-438. <a href="#fnref:300" class="footnote-back-ref">↩</a></li>
<li id="fn:301">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.[120] <a href="https://www.psychology.hku.hk/ftbcstudies/refbase/docs/zinnbauer/1997/34_Zinnbauer_etal1997.pdf" target="_blank">https://www.psychology.hku.hk/ftbcstudies/refbase/docs/zinnbauer/1997/34_Zinnbauer_etal1997.pdf</a> <a href="#fnref:301" class="footnote-back-ref">↩</a></li>
<li id="fn:302">K. Sadegh-Zadeh, "Fuzzy health, illness, and disease". In: Journal of medical philosophy, Vol. 25, No. 5, October 2000, pp. 605-638.[121] <a href="https://pubmed.ncbi.nlm.nih.gov/11035544/" target="_blank">https://pubmed.ncbi.nlm.nih.gov/11035544/</a> <a href="#fnref:302" class="footnote-back-ref">↩</a></li>
<li id="fn:303">In 2025, the Oxford English Dictionary lists 33 meanings for the word social, of which six are labelled obsolete. <a href="/wiki/Oxford_English_Dictionary" target="_blank">/wiki/Oxford_English_Dictionary</a> <a href="#fnref:303" class="footnote-back-ref">↩</a></li>
<li id="fn:304">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, Vol. 115, Issue 1, March 1999, pp. 29-40. <a href="#fnref:304" class="footnote-back-ref">↩</a></li>
<li id="fn:305">Stuart McCready (ed.), The discovery of happiness. Naperville (illinois): Sourcebooks Inc., 2001; Desmond Morris, The nature of happiness. London: Little Books Ltd, 2004. <a href="/wiki/Desmond_Morris" target="_blank">/wiki/Desmond_Morris</a> <a href="#fnref:305" class="footnote-back-ref">↩</a></li>
<li id="fn:306">Peter McGee, "Vague language as a means of avoiding controversy". Training, Language and Culture, Vol. 2 Issue 2, pp. 40-54.[122] <a href="https://rudn.tlcjournal.org/archive/2(2)/2(2)-03.pdf" target="_blank">https://rudn.tlcjournal.org/archive/2(2)/2(2)-03.pdf</a> <a href="#fnref:306" class="footnote-back-ref">↩</a></li>
<li id="fn:307">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.[123] <a href="/wiki/Philip_Ball" target="_blank">/wiki/Philip_Ball</a> <a href="#fnref:307" class="footnote-back-ref">↩</a></li>
<li id="fn:308">Karl Popper, Unended quest: an intellectual autobiography. London: Routledge, 2002, p. 22. <a href="#fnref:308" class="footnote-back-ref">↩</a></li>
<li id="fn:309">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. <a href="/wiki/Lotfi_Zadeh" target="_blank">/wiki/Lotfi_Zadeh</a> <a href="#fnref:309" class="footnote-back-ref">↩</a></li>
<li id="fn:310">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. <a href="#fnref:310" class="footnote-back-ref">↩</a></li>
<li id="fn:311">Nick Cercone & Gordon McCalla (eds.), The knowledge frontier: essays in the representation of knowledge. New York: Springer, 1987. <a href="#fnref:311" class="footnote-back-ref">↩</a></li>
<li id="fn:312">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. <a href="#fnref:312" class="footnote-back-ref">↩</a></li>
<li id="fn:313">V. Rahmati et al. (eds.), A Novel Low Complexity Fast Response Time Fuzzy PID Controller for Antenna Adjusting Using Two Direct Current Motors. <a href="#fnref:313" class="footnote-back-ref">↩</a></li>
<li id="fn:314">cf. Timothy Williamson, Vagueness. London: Routledge, 1996, p. 258. <a href="#fnref:314" class="footnote-back-ref">↩</a></li>
</ol>

Fuzzy concept open-in-new

Fuzzy concept