Zeely Bridge to Teribithia Consider the context of these brief sentences. How do they illustrate a thematic concern in the book. Trace the polarities inherent in each of the following as they occur throughout the story. Are the statements in any way ironic?
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.
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. 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.
InLotfi A. Zadeh introduced a distinction between "Type 1 fuzzy sets" without uncertainty and " Type 2 fuzzy sets " with uncertainty, which has been widely accepted. Philosophy[ edit ] 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.
Sciences[ edit ] In mathematics and statisticsa 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 parametersand which can be usefully described with imprecise categories such as "high", "medium" or "low" using some kind of scale or conceptual hierarchy.
Fuzzy logic In mathematics and computer sciencethe 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.
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.
In principle, this allows us to give a definite, precise answer to the question, "To what extent is something the case?A writer uses a bridge statement, or bridge sentence, to link one idea to another and create a smooth transition between ideas. John Trimble explains in "Writing with Style: Conversations on the Art of Writing" that essays should maintain a steady flow by "bridging" ideas for the reader.
JC & E* I have to tell you that I’m with Milan & JC on this f comment. It’s completely depressing. Thanks for the S&G Bridge over troubled water song – I think. Before you read/watch/listen to “If You Can Read This I Can Prove God Exists,” read THIS first. ( words – 2 minutes) – then come back and continue reading.
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Lord of . The variable raw contains a string with 1,, characters. (We can see that it is a string, using type(raw).)This is the raw content of the book, including many details we are not interested in such as whitespace, line breaks and blank lines.
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