Hans Mittelmann's Benchmarks for Optimization Software.
Origins[ edit ] Problems of vagueness and fuzziness have probably always existed in human experience. Sometimes people have to think, when they are not in the best frame of mind to do it, or, they have to talk about something out there, which just isn't sharply defined.
Across time, however, philosophers and scientists began to reflect about those kinds of problems, in much more systematic ways. Sorites paradox[ edit ] The ancient Sorites paradox first raised the logical problem of how we could exactly define the threshold at which a change in quantitative gradation turns into a qualitative or categorical difference.
With 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: According to the modern idea of the continuum fallacythe fact that a statement is to an extent vague, does not automatically mean that it is invalid.
The problem then becomes one of how we could ascertain the kind of validity that the statement does have. Loki's wager[ edit ] The Nordic myth of Loki's wager suggested that concepts that lack precise meanings or precise boundaries of application cannot be usefully discussed at all.
This idea sounds simple enough, but it had large implications. The intellectual origins of the species of fuzzy concepts as a logical category have been traced back to a diversity of famous and less well-known thinkers,  including among many others EubulidesPlatoCiceroGeorg Wilhelm Friedrich Hegel Karl Marx and Friedrich Engels Friedrich NietzscheHugh MacColl Charles S.
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 logicians and philosophers have also tried to analyze the characteristics of fuzzy concepts as a recognized species, sometimes with the aid of some kind of many-valued logic or substructural logic. An early attempt in the post-WW2 era to create a theory of sets where set membership is a matter of degree was made by Abraham Kaplan and Hermann Schott in They intended to apply the idea to empirical research.
Kaplan and Schott measured the degree of membership of empirical classes using real numbers between 0 and 1, and they defined corresponding notions of intersection, union, complementation and subset.
Zinov'ev and others  Inthe American philosopher William Alston introduced the term "degree vagueness" to describe vagueness in an idea that results from the absence of a definite cut-off point along an implied scale in contrast to "combinatory vagueness" caused by a term that has a number of logically independent conditions of application.
Ackermann and Nicholas Rescher respectively. Zadeh is credited with inventing the specific idea of a "fuzzy concept" in his seminal paper on fuzzy sets, because he gave a formal mathematical presentation of the phenomenon that was widely accepted by scholars.
This finding is intuitively quite appealing, because people say "this product is more or less good" or "to a certain degree, he is a good athlete", implying the graded structure of concepts.
In his classic paper, Zadeh called the concepts with a graded structure fuzzy concepts and argued that these concepts are a rule rather than an exception when it comes to how people communicate knowledge.
Moreover, he argued that to model such concepts mathematically is important for the tasks of control, decision making, pattern recognition, and the like. Zadeh proposed the notion of a fuzzy set that gave birth to the field of fuzzy logic The fact that a concept is fuzzy does not prevent its use in logical reasoning; it merely affects the type of reasoning which can be applied see fuzzy logic.
If the concept has gradations of meaningful significance, it is necessary to specify and formalize what those gradations are, if they can make an important difference.
Not all fuzzy concepts have the same logical structure, but they can often be formally described or reconstructed using fuzzy logic or other substructural logics.Sondheim in Two Contexts. by Gerald Moshell. Let me bring up the matter of hearing Stephen Sondheim's music in two different contexts.
Sweeney Todd Context Essay Introduction “ Sweeney Todd – The Demon Barber of Fleet Street” is a well known play and musical capturing the audience’s attention as to what can happen behind closed doors. “Sweeney Todd: Demon Barber of Fleet Street” Review By Laura Harding I got the pleasure of spending an evening at the Devonport Choral Society’s production “Sweeney Todd: Demon Barber of Fleet Street” on the 29th of May at the Devonport Entertainment and Convention Centre.
His work has won numerous Tony Awards, the most recent being for Sweeney Todd (). Sondheim's mentor was the renowned lyricist Oscar Hammerstein II, who was a neighbor and family friend. Representation of representation’s representation Different levels of representation in Tim Burton’s ‘Sweeney Todd’ Written Assignment in the seminar „Media and Representation“ Södertörns Högskola School of Culture and Education Media, Communication and Cultural Analysis Handed in to Fredrik Stiernstedt Submitted by .
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