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Frege"s Begriffsschrift did not come into being bereft of any connection with the philosophical tradition. It can be understood as a revival of the Leibnizian ideal of a lingua characteristica, which is, at the same time, a calculus ratiocinator (Chapter I). Frege uses the Begriffsschrift as a tool in the carrying out of the logistic program of establishing the concept of number and the laws of arithmetic on a purely logical basis. This program was first developed in Frege"s Foundations of Arithmetic, in conjunction with a basic critique of Husserlian psychologism, Mill"s empiricism and Kant"s view on number. At the same time, in opposition to the proponents of formal arithmetic, he developed profound insights not only into arithmetic as a game but also into the game and calculus in general. Unfortunately, he used this knowledge only in a critique of formalism, since he was himself interested in the construction of a content-ful arithmetic, in which the numbers are introduced as the extensions of certain concepts and thereby as purely logical objects. Characteristic of the construction in the Foundations is a process of abstraction or transposition which reduces all identities between abstract objects to the equality of extensions of concepts, and the latter are presupposed as a sort of basic phenomenon (Chapter II). The extensions of concepts themselves became the problem in the final carrying through of logicism in the Basic Laws of Arithmetic, which makes use of an expansion and interpretation of the Begriffsschrift. In addition, a new theory of judgement and a new doctrine on function and object enable us to speak of a second period in Frege"s thought, beginning in the area of 1890. But it can be shown that the first of these is already basically applied in the Foundations of 1884, and the earlier interpretation by Scholz has to be corrected. According to the new doctrine, concepts and relations are only special cases of functions conceived as references of "unsaturated" expressions. These can be of different levels, according to the type of argument: but Frege did not want to carry the levels beyond three. Regardless of this, it can be shown that the majority of possible functions are not included even in a continuation of Frege"s structure of levels, since he explained "levels" only for homogeneous functions (Chapter III).

DOI: 10.1007/978-94-017-2981-9_10

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