Fuzzy Logic Theory of Evaluating Expressions and Comparative Quantifiers ∗ Vil´ em Nov´ ak University of Ostrava Institute for Research and Applications of Fuzzy Modeling 30. dubna 22, 701 03 Ostrava 1, Czech Republic e-mail: Vilem.Novak@osu.cz Abstract The paper presents a formal the- ory of the meaning of evaluating lin- guistic expressions developed within  Lukasiewicz fuzzy type theory. Fur- thermore, a class of comparative generalized quantifiers is also intro- duced as special formulas based on the above theory. Keywords: Evaluating linguistic expressions, intension, extension, fuzzy type theory, generalized quan- tifiers. 1 Introduction Majority of applications of fuzzy logic uses ex- pressions such as “small, very small, medium, more or less big, deep, very intelligent, rather narrow, medium important, very tall, ex- tremely nice, about 100, not too expensive” and many others. These expressions fall into the class of the, so called, evaluating linguistic expressions †) . Though narrow from linguistic point of view, this class plays an extremely important role in human life. In general, eval- uating expressions are specific expressions of natural language, which characterize position on a bounded ordered scale. In this paper, we provide a logical theory of evaluating linguistic expressions that is based * The research was partially supported by the grant 201/04/1033 of the GA ˇ CR and partially by the project 1M0572 of the M ˇ SMT ˇ CR. †) In the sequel, we will usually omit the adjective “linguistic”. on the analysis of their structure. We will demonstrate that their inherent vagueness is a manifestation of the, more or less hidden, phe- nomenon described as sorites paradox. As the basic means, we took a formal fuzzy type the- ory (FTT) that is, a higher order fuzzy logic. The reason is that any model of the meaning of words must be able to distinguish between intension and extension (cf. [5]). Fuzzy type theory can fulfil this task in a very elegant way. Besides their general use in everyday language and various kinds of special evaluations, they can serve as a basis for a theory of the, so called, comparative quantifiers that form a special class of generalized quantifiers. We will outline their theory in Section 4. Results of this paper can be classified as a contribu- tion to the theory of precisiated natural lan- guage as proposed by L. A. Zadeh in [15] since each model of our theory provides a concrete “implementation” of these ideas that can be applied in practice. 2 Preliminaries Due to lack of space, we refer the reader to the papers [9, 10] for the formalism, axioms and the basic properties of FTT. In this pa- per, we will use the  Lukasiewicz fuzzy type theory ( L-FTT) whose structure of truth val- ues is  Lukasiewicz Δ algebra. We will only recall that interpretation of a formula A αβ of type αβ is a function M β −→ M α where M α ,M β are sets assigned to the types α, β, respectively. Important is the notion of a fuzzy equality = α considered on each set