Towards fuzzy type theory

In the paper, the formal type theory is generalized to fuzzy one. The structure of truth values is assumed to be the IMTL-algebra (on [0, 1], the algebra of left continuous t-norms with involutive negation) since the formulation of FTT based on it preserves the elegancy of classical formulation. Some properties of theories of fuzzy type theory are demonstrated and the completeness saying that each consistent theory has a model is proved. We will follow the way of the development of the classical type theory as elaborated especially by A. Church and L. Henkin.