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

### Authors

Paul E. Oppenheimer and Edward N. Zalta
### Reference

*Journal of Logic and Computation*, 21 (2011): 351–374.
### Abstract

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. By contrast,
Whitehead & 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 & 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
but not in functional type theory. This shows that relational type
theory is more general than functional type theory. The
simplification offered by Church in his functional type theory is an
over-simplification: one can't assimilate predication to functional
application.

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