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"... 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 ..."

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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 forthcoming in the Journal of Logic and Computation. 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 1.

### Edward N. Zalta 2 Reflections on Mathematics ∗

"... Though the philosophy of mathematics encompasses many kinds of questions, my response to the five questions primarily focuses on the prospects of developing a unified approach to the metaphysical and epistemological issues concerning mathematics. My answers will be framed from within a single concep ..."

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Though the philosophy of mathematics encompasses many kinds of questions, my response to the five questions primarily focuses on the prospects of developing a unified approach to the metaphysical and epistemological issues concerning mathematics. My answers will be framed from within a single conceptual framework. By ‘conceptual framework’, I mean an explicit and formal listing of primitive notions and first principles, set within a well-understood background logic. In what follows, I shall assume the primitive notions and first principles of the (formalized and) axiomatized theory of abstract objects, which I shall sometimes refer to as ‘object theory’. 1 These notions and principles are mathematics-free, consisting only of metaphysical and logical primitives. The first principles assert the existence, and comprehend a domain, of abstract objects, and in this domain we can identify (either by definition or by other means) logical objects, natural mathematical objects, and theoretical mathematical objects. These formal principles and identifications will help us to articulate answers not only to the five questions explicitly before us, but also to some of the other fundamental questions in the philosophy of mathematics raised below. 1. Why were you initially drawn to the foundations of mathematics and/or the philosophy of mathematics? As a metaphysician, I’ve always been interested in data that consists of (apparently) true sentences and valid inferences that appear to be about

### Edward N. Zalta 2 Essence and Modality ∗

"... In the course of research on modal logic over the past 60 years, it has become traditional to define an essential property in modal terms as follows: (E) F is essential to x =df ✷(E!x → Fx), where ‘E!x ’ asserts existence and abbreviates ‘∃y(y = x)’. Kit Fine has ..."

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In the course of research on modal logic over the past 60 years, it has become traditional to define an essential property in modal terms as follows: (E) F is essential to x =df ✷(E!x → Fx), where ‘E!x ’ asserts existence and abbreviates ‘∃y(y = x)’. Kit Fine has