This book examines three connected aspects of Freges logicism: the differences between Dedekinds and Freges interpretation of the term logic and related terms and reflects on Freges notion of function, comparing its understanding and the role it played in Freges and Lagranges foundational programs. It concludes with an examination of the notion of arbitrary function, taking into account Freges, Ramseys and Russells view on the subject. Composed of three chapters, this book sheds light on important aspects of Dedekinds and Freges logicisms. The first chapter explains how, although he shares Freges aim at substituting logical standards of rigor to intuitive imports from spatio-temporal experience into the deductive presentation of arithmetic, Dedekind had a different goal and used or invented different tools. The chapter highlights basic dissimilarities between Dedekinds and Freges actual ways of doing and thinking. The second chapter reflects on Freges notion of a function, in comparison with the notions endorsed by Lagrange and the followers of the program of arithmetization of analysis. It remarks that the foundational programs pursued by Lagrange and Frege are crucially different and based on a different idea of what the foundations of mathematics should be like. However, despite this contrast, the notion of function plays similar roles in the two programs, and this chapter emphasizes the similarities. The third chapter traces the development of thinking about Freges program in the foundations of mathematics, and includes comparisons of Freges, Russells and Ramseys views. The chapter discusses earlier papers written by Hintikka, Sandu, Demopoulos and Trueman. Although the chapters main focus is on the notion of arbitrary correlation, it starts out by discussing some aspects of the connection between this notion and Dedekind Theorem.