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. 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. ∗This paper is published in the Journal of Logic and Computation, 21 (2011): 351– 374. At the end of the 3rd paragraph of Section 2.2, this preprint adds a left and right parenthesis, in red, to the logical type assigned to the Π function, thereby correcting an error. The authors would like to thank Uri Nodelman for his observations on the first draft of this paper. We’d also like to thank Bernard Linsky for observations on the second draft, which led us to reconceptualize the significance of our results within a more historical context. We’d also like to acknowledge one of the referees of this journal, whose comments led us to clarify and better document the claims in the paper. Paul Oppenheimer and Edward N. Zalta 2