A fuzzy concept is an idea of which the boundaries of application can vary considerably according to context or conditions, instead of being fixed once and for all.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. That means the idea is somewhat vague or imprecise.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). Yet it is not unclear or meaningless. It has a definite meaning, which can often be made more exact with further elaboration and specification — including a closer definition of the context in which the concept is used."A fuzzy concept is a concept of which the content or boundaries of application vary according to content or conditions. It does have a meaning, or multiple meanings, which however can become clearer only through further elaboration and specification." – Cornelis Hoede and Xin Wang, "On fuzzy concepts". Memorandum 1814, Enschede: Dept. of Applied Mathematics, University of Twente [The Netherlands], December 2006, p. 26. The inverse of a "fuzzy concept" is a "crisp concept" (i.e. a precise concept). Fuzzy concepts are often used to navigate imprecision in the real world, when precise information is not available and an approximate indication is sufficient to be helpful.
Although the linguist George Philip Lakoff already defined the semantics of a fuzzy concept in 1973 (inspired by an unpublished 1971 paper by Eleanor Rosch,George Lakoff, "Hedges: A Study in Meaning Criteria and the Logic of Fuzzy Concepts."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 Unpublished paper, Psychology Department, UC Berkeley, 1971 (as cited by Lakoff).) the term "fuzzy concept" rarely received a standalone entry in dictionaries,The online Collins dictionary gives a separate entry for "fuzzy concept" in English and Chinese. handbooks and encyclopedias. Sometimes it was defined in encyclopedia articles on fuzzy logic,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). or it was simply equated with a mathematical “fuzzy set”. A fuzzy concept can be "fuzzy" for many different reasons in different contexts. This makes it harder to provide a precise definition that covers all cases. Paradoxically, the definition of fuzzy concepts may itself be somewhat "fuzzy". Lotfi A. Zadeh, known as "the father of fuzzy logic", claimed that "vagueness connotes insufficient specificity, whereas fuzziness connotes unsharpness of class boundaries".Lotfi A. Zadeh, "What is fuzzy logic?". IFSA Newsletter (International Fuzzy Systems Association), Vol. 10, No. 1, March 2013, pp. 5–6 (emphasis added). Not all scholars agree.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]. In: Linguistics and Philosophy, Vol. 5, issue 2, June 1982, pp. 295-297.
With increasing academic literature on the subject, the term "fuzzy concept" is now more widely recognized as a philosophical, linguistic or scientific category, and the study of the characteristics of fuzzy concepts and fuzzy language is known as fuzzy semantics.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. “Fuzzy logic” has become a generic term for many different kinds of many-valued logics, and is applied in many different areas of research, computer programming and industrial design.
For engineers, "Fuzziness is imprecision or vagueness of definition."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. For computer scientists, a fuzzy concept is an idea which is "to an extent applicable" in a situation. It means that the concept can have gradations of significance or unsharp (variable) boundaries of application — a "fuzzy statement" is a statement which is true "to some extent", and that extent can often be represented by a scaled value (a score).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. For mathematicians, a "fuzzy concept" is usually a fuzzy set or a combination of such sets (see fuzzy mathematics and fuzzy set theory). In cognitive linguistics, the things that belong to a "fuzzy category" exhibit gradations of family resemblance, and the borders of the category are not clearly defined.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.
Through most of the 20th century, the idea of reasoning with fuzzy concepts faced considerable resistance from Western academic elites.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. They did not want to endorse the use of imprecise concepts in research or argumentation, and they often regarded fuzzy logic with suspicion, derision or even hostility.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. That may partly explain why the idea of a "fuzzy concept" did not get a separate entry in encyclopedias, handbooks and dictionaries.
Yet although people might not be aware of it, the use of fuzzy concepts has risen gigantically in all walks of life from the 1970s onward.Bart Kosko, "Fuzzy logic". In: Scientific American, July 1993, pp. 76-81 ; 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.; 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. Didier Dubois, Henri Prade, Articles written on the occasion of the 50th anniversary of fuzzy set theory. Institut de Recherche Informatique de Toulouse, 2015. That is mainly due to advances in electronic engineering, fuzzy mathematics and digital computer programming. The new technology allows very complex inferences about "variations on a theme" to be anticipated and fixed in a program.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. 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] The Perseverance Mars rover, a driverless NASA vehicle used to explore the Jezero crater on the planet Mars, features fuzzy logic programming that steers it through rough terrain.Katyanna Quach, "Fuzzy logic makes a comeback – in picking where Earth sticks its probes into alien worlds". The Register, 27 Sep 2018. Jennifer Chu, "Exploring a Planet with 'Fuzzy Logic'". MIT News Office, MIT Resources Laboratory site, September 26, 2018. Similarly, to the North, the Chinese Mars rover Zhurong used fuzzy logic algorithms to calculate its travel route in Utopia Planitia from sensor data.Liwei Yang et al., "Path Planning Technique for Mobile Robots: A Review". Machines, Vol. 11, 2023, pp. 980-1026.
New neuro-fuzzy computational methodsLotfi 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. make it possible for machines to identify, measure, adjust and respond to fine gradations of significance with great precision.A useful technical overview is provided in: Enrique Ruspini et al. Handbook of fuzzy computation. Bristol & Philadelphia: Institute of Physics Publishing, 1998. It means that practically useful concepts can be coded, sharply defined, and applied to all kinds of tasks, even if ordinarily these concepts are never exactly defined. Nowadays engineers, statisticians and programmers often represent fuzzy concepts mathematically, using fuzzy logic, fuzzy values, fuzzy variables and fuzzy setsRadim Bělohlávek & George J. Klir (eds.), Concepts and fuzzy logic. Cambridge, Mass.: MIT Press, 2011. (see also fuzzy set theory). Fuzzy logic is not "woolly thinking", but a "precise logic of imprecision" which reasons with graded concepts and gradations of truth.Adriana Dumitras; George Moschytz, "Understanding Fuzzy Logic: An Interview with Lotfi Zadeh". IEEE Signal Processing Magazine, Volume 24, Issue 3, May 2007, pp. 102-105, at p. 103. Fuzzy concepts and fuzzy logic often play a significant role in artificial intelligence programming, for example because they can model human cognitive processes more easily than other methods.Edy Portmann, Fuzzy humanist. Wiesbaden: Springer, 2019; Mahdi Eftekhari et al., How fuzzy concepts contribute to machine learning. Cham (Switzerland): Springer, 2022.
Origins