How did Russell’s theory of types attempt to avoid self-referential contradictions?

Bertrand Russell’s theory of types was developed to resolve logical paradoxes, such as Russell’s paradox, which arose from the unrestricted use of predicate functions and the assumption that one could form a set of all sets that do not contain themselves.123

To avoid these self-referential contradictions, Russell introduced a hierarchical system of logical types. The fundamental mechanism is as follows:

• Hierarchical Structure: Entities are organized into a hierarchy where a set or predicate of a certain type can only contain or apply to entities of a lower type.24
• Restriction on Membership: By enforcing this structure, the system ensures that an expression like $x\in X$ is only meaningful if $X$ is of a higher type than $x$. This prevents the formation of sets that contain themselves, as a set cannot be a member of its own type.24
• Vicious Circle Principle: Russell sought to eliminate "impredicative" definitions—those that define an object by referencing a totality to which the object itself belongs. This principle mandates that definitions must proceed from more basic elements to more inclusive ones, thereby excluding circular reasoning.243