Higher-order fuzzy logic, partial fuzzy type theory, fuzzy generalized quantifiers, intermediate quantifiers, Bochvar connectives, Sobocinski connectives, partial fuzzy sets

Background: In computer science, one often meets the requirement to deal with partial functions. They naturally raise, for example, when a mistake such as the square root of a negative number or division by zero occurs, or when we want to express the semantics of the expression “Czech president in 18th century” because there was no such president before 1918. Method: In this paper, we will extend the theory of intermediate quantifiers (i.e., expressions such as “most, almost all, many, a few”, etc.) to deal with partially defined fuzzy sets. First, we extend algebraic operations that are used in fuzzy logic by additional value “undefined”. Then we will introduce intermediate quantifiers using the former. The theory of intermediate quantifiers has been usually developed as a special theory of higher-order fuzzy logic. Results: In this paper, we introduce the quantifiers semantically and show how they can be computed. The latter is also demonstrated in three illustrative examples. Conclusion: The paper contributes to the development of fuzzy quantifier theory by its extension by undefined values and suggests methods for computation of their truth values.