A fuzzy concept is a concept of which the boundaries of application can vary considerably according to context or conditions, instead of being fixed once and for all. This means the concept is vague in some way, lacking a fixed, precise meaning, without however being unclear or meaningless altogether. It has a definite meaning, which can be made more precise only through further elaboration and specification - including a closer definition of the context in which the concept is used. The study of the characteristics of fuzzy concepts and fuzzy language is called fuzzy semantics. The inverse of a "fuzzy concept" is a "crisp concept" (i.e. a precise concept).
In philosophical logic and linguistics, fuzzy concepts are often regarded as vague concepts which in their application, or formally speaking, are neither completely true nor completely false, or which are partly true and partly false; they are ideas which require further elaboration, specification or qualification to understand their applicability (the conditions under which they truly make sense). 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.
In mathematics and computer science, the gradations of applicable meaning of a fuzzy concept are described in terms of quantitative relationships defined by logical operators. Such an approach is sometimes called "degree-theoretic semantics" by logicians and philosophers, but the more usual term is fuzzy logic or many-valued logic. The novelty of fuzzy logic is, that it "breaks with the traditional principle that formalisation should correct and avoid, but not compromise with, vagueness". The basic idea of fuzzy logic is that a real number is assigned to each statement written in a language, within a range from 0 to 1, where 1 means that the statement is completely true, and 0 means that the statement is completely false, while values less than 1 but greater than 0 represent that the statements are "partly true", to a given, quantifiable extent. Susan Haack comments: "Whereas in classical set theory an object either is or is not a member of a given set, in fuzzy set theory membership is a matter of degree; the degree of membership of an object in a fuzzy set is represented by some real number between 0 and 1, with 0 denoting no membership and 1 full membership." "Truth" in this mathematical context usually means simply that "something is the case", or that "something is applicable". This makes it possible to analyze a distribution of statements for their truth-content, identify data patterns, make inferences and predictions, and model how processes operate. Petr Hájek claimed that "fuzzy logic is not just some "applied logic", but may bring "new light to classical logical problems", and therefore might be well classified as a distinct branch of "philosophical logic" similar to e.g. modal logics.
Fuzzy logic offers computationally-oriented systems of concepts and methods, to formalize types of reasoning which are ordinarily approximate only, and not exact. In principle, this allows us to give a definite, precise answer to the question, "To what extent is something the case?", or, "To what extent is something applicable?". Via a series of switches, this kind of reasoning can be built into electronic devices. That was already happening before fuzzy logic was invented, but using fuzzy logic in modelling has become an important aid in design, which creates many new technical possibilities. Fuzzy reasoning (i.e., reasoning with graded concepts) turns out to have many practical uses. It is nowadays widely used in: The programming of vehicle and transport electronics, household appliances, video games, language filters, robotics, and driverless vehicles. Fuzzy logic washing machines are gaining popularity. All kinds of control systems that regulate access, traffic, movement, balance, conditions, temperature, pressure, routers etc. Electronic equipment used for pattern recognition, surveying and monitoring (including radars, satellites, alarm systems and surveillance systems). Cybernetics research, artificial intelligence, virtual intelligence, machine learning, database design and soft computing research. "Fuzzy risk scores" are used by project managers and portfolio managers to express financial risk assessments. Fuzzy logic has been applied to the problem of predicting cement strength. It looks like fuzzy logic will eventually be applied in almost every aspect of life, even if people are not aware of it, and in that sense fuzzy logic is an astonishingly successful invention. The scientific and engineering literature on the subject is constantly increasing.
In mathematics and statistics, a fuzzy variable (such as "the temperature", "hot" or "cold") is a value which could lie in a probable range defined by some quantitative limits or parameters, and which can be usefully described with imprecise categories (such as "high", "medium" or "low") using some kind of scale or conceptual hierarchy.
|Food item||Contains bread||Bread is separately baked||Bread contains the other ingredients during eating||Two separate bread layers||"Sandwich" is in the name (U.S.)||Made with slices from English sandwich bread loaf||Unweighted score||Classified as|
|Peanut butter and jelly sandwich||Yes||Yes||Yes||Yes||Yes||Yes||Yes||7||Sandwich|
|Bacon, lettuce, and tomato sandwich||Yes||Yes||Yes||Yes||Yes||Yes||Yes||7||Sandwich|
|Toast sandwich||Yes||Yes||Yes||Yes||Yes (despite inner 3rd bread slice)||Yes||Yes||7||Sandwich|
|Croque-monsieur||Yes||Yes||Yes (but re-cooked)||No (due to cheese on outside)||Yes||No||Yes||5||Sandwich|
|Banh mi||Yes||Yes||Yes||Yes||Maybe||Maybe (sometimes called "banh mi sandwich")||No (baguette)||5||Roll (UK/Australia) or sandwich (US)|
|Panini||Yes||Yes||Yes (but re-toasted)||Yes||Yes||No (only in Italian)||No||5||Pressed sandwich (e.g. with the Cuban sandwich)|
|with bun||Yes||Yes||Yes||Yes||Yes||No||No (hamburger bun or bread roll)||5||Burger (UK/Australia), sometimes disputed as a sandwich vs. hamburger (US) due to tradition and the use of bun instead of bread.|
|Hamburger without bun||Yes||No||No||No||No||No||No||1||Burger (patty) with toppings|
|Hot dog with bun||Yes||Yes||Yes||Yes||No||No||No (hot dog bun)||4||Disputed. Some classify as a sausage sandwich. Others classify as a hot dog (a type of non-sandwich sausage dish due to tradition or the vertical orientation of the bread sides.|
|Submarine sandwich||Yes||Yes||Yes||Yes||Maybe||Yes||No (hoagie roll)||5.5||Roll (UK/Australia) or sandwich (US)|
|Pita pocket||Yes||Yes||Yes||Yes||No||No||No||4||Pocket sandwich|
|Wraps and burritos||Yes||Yes||Yes||Yes||No||No||No||4||Disputed. Legal classification varies by jurisdiction.|
|Tacos and quesadillas||Yes||Yes||Yes||Yes||No||No||No||4||Disputed, with some classifying as non-sandwich tortilla-based dishes, either due to separate culinary tradition (Spain vs. UK) or the vertical nature of bread sides in tacos.|
|Calzone||Yes||Yes||No||Yes||No||No||No||3||Dumpling or folded pizza|
|Cha siu bao||Yes||Yes||No||Yes||No||No||No||3||Dumpling|
|Open-faced sandwich||Yes||Yes||Yes||No||No||Yes||Yes||5||Open-faced sandwich|
|Sandwich cake||Yes||Maybe (cake is bread-like)||No||No||Yes||Maybe ("layer cake" in US, "sandwich" in UK)||No||3||Cake (mostly named by analogy due to repeated layering)|
|Salad with croutons||Yes||Yes||Yes||No||No||No||No||2||Salad|
|Ice cream cone with ice cream||Yes||No||No||No||No||No||No||1||Pastry|
|Ice cream sandwich||Yes||No||No||No||No||Yes||No||2||Sandwich cookie (named by analogy to bread sandwiches)|
|Aluminium foam sandwich||No||No||No||No||No||Yes||No||1||(named by analogy to bread sandwiches)|