Frege's Recipe

Abstract

In this paper, we present a formal recipe that Frege followed in his magnum opus “Grundgesetze der Arithmetik” when formulating his definitions. This recipe is not explicitly mentioned as such by Frege, but we will offer strong reasons to believe that Frege applied it in developing the formal material of Grundgesetze. We then show that a version of Basic Law V plays a fundamental role in Frege’s recipe and, in what follows, we will explicate what exactly this role is and explain how it differs from the role played by extensions in his earlier book “Die Grundlagen der Arithmetik”. Lastly, we will demonstrate that this hitherto neglected yet foundational aspect of Frege’s use of Basic Law V helps to resolve a number of important interpretative challenges in recent Frege scholarship, while also shedding light on some important differences between Frege’s logicism and recent neo-logicist approaches to the foundations of mathematics

The Semantics of Value-Range Names and Frege’s Proof of Referentiality

“…Frege (1893), §11) for the sole purpose of enabling him to define the application operator in such a way that it guarantees logical flexibility, austerity and conciseness in pursuit of the logicist project and, in particular, could be most effective in the projected proofs of the basic laws of cardinal arithmetic and real analysis. In the course of analyzing Frege's Grundgesetze-definitions, Cook and Ebert (2016) also comment on Frege's definition of the application operator. According to Cook and Ebert, this definition is the sole exception to what they call Frege's generalized recipe for identifying the objects falling under some mathematical concept C (for details see pp.…”

“…337f.). I recommend the reader of my essay to carefully study Cook & Ebert (2016). It is a subtle, well-argued and stimulating investigation mainly of Frege's definitional methodology in Grundgesetze and related issues.…”