EQ-algebra-based fuzzy type theory and its extensions

In this paper, we introduce a new algebra called ‘EQ-algebra’, which is an alternative algebra of truth values for formal fuzzy logics. It is specified by replacing implication as the main operation with a fuzzy equality. Namely, EQ-algebra is a semilattice endowed with a binary operation of fuzzy equality and a binary operation of multiplication. Implication is derived from the fuzzy equality and it is not a residuation with respect to multiplication. Consequently, EQ-algebras overlap with residuated lattices but are not identical with them. We choose one class of suitable EQ-algebras (good EQ-algebras) and develop a formal theory of higher-order fuzzy logic called ‘basic fuzzy type theory’ (FTT). We develop in detail its syntax and semantics, and we prove some basic properties, including the completeness theorem with respect to generalized models. The paper also provides an overview of the present state of the art of FTT.