On the Distinction between Relational and Functional Type Theory
By Paul E. Oppenheimer and Edward N. Zalta
It is commonly believed that it makes no difference whether one starts with relational types or functional types in formulating type theory, since one can either start with relations as primitive and represent functions as relations or start with functions as primitive and represent relations as functions. It is also commonly believed that the formula-based logic of relational type theory is equivalent to the term-based logic of functional type theory. However, in this paper, the authors argue that there are systems with logics that can be properly characterized in relational type theory, but not in functional type theory.
Source: Online Papers in Philosophy