Relations Versus Functions at the Foundations of Logic: Type-Theoretic Considerations

Though Frege was interested primarily in reducing mathematics to logic, he succeeded in reducing an important part of logic to mathematics by defining relations in terms of functions. In contrast, Whitehead and Russell reduced an important part of mathematics to logic by defining functions in terms of relations (using the definite description operator).We argue that there is a reason to prefer Whitehead and Russell’s reduction of functions to relations over Frege’s reduction of relations to functions. There is an interesting system having a logic that can be properly characterized in relational type theory (RTT) but not in functional type theory (FTT). This shows that RTT is more general than FTT. The simplification offered by Church in his FTT is an over-simplification: one cannot assimilate predication to functional application.