On Examination of Medical Data with Approximate Reasoning - Fuzziness and Medicine - page 255

18.2 Measuring and Reasoning with Imprecise Concepts

in Statistics

18.2.1 Linguistic Variables and Approximate Reasoning

In traditional measuring we may draw a distinction between qualitative, compar-
ative and quantitative concepts or variables, but in statistical analysis we usually
regard them as being precise entities. For example, when Likert or Osgood scales
are used, their values are assumed to be precise values although they are actually
imprecise by nature. Thanks for the fuzzy systems, we can also operate with both
precise and imprecise values, as well as with the linguistic values. Hence, in ad-
dition to the precise (or crisp) numerical values or intervals, we may also use such
values as approximately 5 , approximately from 4 to 6 or very young . In statistics this
means that we may also use fuzzy linguistic variables. One usable method for gen-
erating linguistic values for linguistic variables within Likert or Osgood scales is as
follows [15]:

1. Specify such two primitive terms for each variable which are antonyms (if pos-
sible). Young and old seem appropriate to age on Osgood scale. Fully agree
and fully disagree on Likert scale.
2. Specify such linguistic modifiers (adverbs) as very , fairly or more or less ,and
construct usable symmetrical scales with the modifiers and primitive terms.
Five or seven values are widely used in this context. For example, young -
fairly young - middle-aged ( neither young nor old )- fairly old - old .
3. We can also use negation not. For example, not fairly young .
4. Use such connectives for compound expressions as and ,or and if - then .
5. If necessary, specify such quantifiers as all , most , some or none .
6. We may also use crisp numerical values or such fuzzy numbers as approxi-
mately 5 or approximately between 4 and 6 .

Given a linguistic variable and its reference set (universe of discourse), each lin-
guistic value refers to a certain fuzzy set, and in practice we operate with the corre-
sponding membership functions of these sets in a computer environment (Fig. 18.1).
Various shapes for these functions, such as triangular, bell-shaped and trapezoidal,
are available, and we may specify them according to our expertise or empirical data
[1],[5], [21].
The fuzzy systems have already proved their applicability in various disciplines
even though several models still use only more or less numerical methods. In a
sense, they have mainly applied fuzzy sets and not fuzzy linguistic logic. However,
recently Zadeh has established the principles of his novel fuzzy extended logic,
FLe , which is a combination of “traditional” provable and “precisiated” fuzzy logic
as well as a novel meta-level “unprecisiated” fuzzy logic [25]. He states that in the
precisiated case the objects of discourse and analysis can be imprecise, uncertain,
unreliable, incomplete or partially true, whereas the results of reasoning, deduc-
tion and computation are expected to be provably valid in the traditional sense. In
his meta-level unprecisiated logic, in turn, membership functions and generalized