Another important feature of fuzzy systems is the ability to define "hedges," or
modifier of fuzzy values. These operations are provided in an effort to maintain
close ties to natural language, and to allow for the generation of fuzzy statements
through mathematical calculations. As such, the initial definition of hedges and
operations upon them will be quite a subjective process and may vary from one
project to another. Nonetheless, the system ultimately derived operates with the
same formality as classic logic. The simplest example is in which one transforms
the statement "Jane is old" to "Jane is very old." The hedge "very" is usually
defined as follows:
m"very"A(x) = mA(x)^2 Thus, if mOLD(Jane) = 0.8, then mVERYOLD(Jane) = 0.64.
Other common hedges are "more or less" [typically SQRT(mA(x))], "somewhat,"
"rather," "sort of," and so on. Again, their definition is entirely subjective, but
their operation is consistent: they serve to transform membership/truth values in a
systematic manner according to standard mathematical functions.
A more involved approach to hedges is best shown through the work of Wenstop
in his attempt to model organizational behavior. For his study, he constructed
arrays of values for various terms, either as vectors or matrices. Each term and
hedge was represented as a 7-element vector or 7x7 matrix. He ten intuitively
assigned each element of every vector and matrix a value between 0.0 and 1.0,
inclusive, in what he hoped was intuitively a consistent manner. For example, the
term "high" was assigned the vector 0.0 0.0 0.1 0.3 0.7 1.0 1.0
and "low" was set equal to the reverse of "high," or 1.0 1.0 0.7 0.3 0.1 0.0 0.0
Wenstop was then able to combine groupings of fuzzy statements to create new
fuzzy statements, using the APL function of Max-Min matrix multiplication.
These values were then translated back into natural language statements, so as to
allow fuzzy statements as both input to and output from his simulator. For
example, when the program was asked to generate a label "lower than sortof
low," it returned "very low;" "(slightly higher) than low" yielded "rather low,"
etc. The point of this example is to note that algorithmic procedures can be
devised which translate "fuzzy" terminology into numeric values, perform
reliable operations upon those values, and then return natural language statements
in a reliable manner.