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A linguistic truth set in which each element is a linguistic truth value is discussed. Ranking linguistic truth values based on Graded Mean Integration Representation method is discussed also. We then give a decreasing linguistic truth set and an increasing linguistic truth set by using the above ranking method, and present a Not function of linguistic truth value combined by the above decreasing linguistic truth set and the increasing linguistic truth set. A minimum function and a maximum function based on representations of linguistic truth values are introduced. In addition, some natural operations of linguistic logic combined by minimum function and maximum function, and Not function are presented. Some properties of our presented natural operations are presented, and are proved. Furthermore, some application examples of linguistic logical statements are discussed finally.

This paper reports some novel approach on linguistic logic with our intention to realize CWW, Computing With Words, via a simple example which consists of only five words. As a by product, this simple example of the linguistic logical system may serve as a mathematical model, modeling the degree of truthfulness in daily usage. The five words set of a linguistic variable modeling the degree of truthfulness are; true, nearly true, undecided, nearly false and false. We subjectively choose trapezoidal fuzzy numbers as our linguistic truth values in order to model our linguistic logic system. Firstly, some natural operations and linguistic logic operators are defined to suit our objective of developing a closed linguistic variable set. Then the computation of linguistic truth values for this linguistic logical system is developed in order to facilitate us to perform the linguistic inferences. Properties of these natural operations can be derived accordingly. It is perhaps quite rewarding to see numerous linguistic truth relations defined on a single linguistic truth set and linguistic implications ended up with numerous linguistic truth tables. In addition, the linguistic inferences of generalized modus ponens and generalized tollens determined by linguistic compositional rules based on the linguistic truth relation and some natural operations are introduced. The simple examples of the linguistic inferences of the various generalized tautologies are illustrated. Finally, we have proved via a simple dictionary that a closed and self consistent linguistic logical system indeed can be constructed and it is possible to move a chunk of information as modeled by a fuzzy set to a higher level according to the theory of semiotics. These results have shown some promise in realizing the appealing theory of CWW.

There are much more publications in open literature on the topic of linguistic variables as compared with Computing With Words (CWW). This paper intends to change that situation by demonstrating the feasibility of a working system. In order to accomplish this goal, a very focused example of an insurance evaluation system, no matter how rudimentary, is presented. Lotfi A. Zadeh has persistently stressed the importance of CWW for a contemporary society and he is absolutely right on target. We have proceeded our project, taking the advantage of the previously established works of Chen, Hsieh and Wang, such as linguistic logic, uncertainty numbers modeling, fuzzifications and defuzzifications schemes, etc. Basically, the Linguistic Evaluating System (LES), consists of Linguistic Descriptions Evaluating Algorithm (LDEA), Linguistic Descriptions Base (LDB), and Fuzzy Knowledge Base (FKB). The previously published works on linguistic methodology and linguistic logic make it possible to develop our LDEA. This main algorithm intends to appraise the linguistic descriptions based upon the newly proposed preferred evaluation rules. In addition, the measuring evaluations of the linguistic descriptions, which have been based on the innovative combination of different IF-THEN fuzzy rules, which in term has been based upon the use of the linguistic distance measurements, reflected the preference of a humane expert. If then, only then, the desired user friendly applications of the essential thrust of risk assessment required for insurance industry has been finally realized.