Frege's two notions of "extension"
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2023-06-28
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Universidade Federal de Goiás
Resumo
The goal of this paper is to answer a question proposed by Richard
Heck in the paper “Formal Arithmetic Before Grundgesetze”. In that paper,
Heck inquires as to the reasons why it took almost eight years for Frege to
honor his promises of concluding his grandiose project of grounding mathe matics on logic. Although Heck tried to answer his own question, we think
that a more adequate philosophical discussion regarding Frege’s delay can still
be offered. This paper will try to fill in that gap by presenting what we under stand was the central problem faced by Frege on Die Grundlagen der Arithmetik:
the lack of a standard criterion to fix the meaning of identity propositions of
mathematics. We believe that Frege’s initial proposal of a dual character for
“identity propositions” was the cause of all his problems in fixing his defini tion of numbers in Die Grundlagen. In the aforementioned eight-year period,
Frege’s challenge became that of finding a criterion capable of unifying his
treatment of identities. In our account, the German philosopher finally de cided to fill in this gap by providing a new construal of “extension”, one which
included some important refinements on his previous account of that notion.
The new concept thus construed allowed Frege to unify his treatment of iden tity propositions by including in his system a universal and flexible criterion
for deciding the truth of any identity proposition. Frege’s new construal of
“extension” was supported by his famous basic law V. So, our claim will be
that Frege’s resistance and doubts about the inclusion of axiom V as a logical
law in his system were the primary cause of that delay.
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Frege, Extension, Identity
Citação
VELLOSO, Araceli. Frege's two notions of "extension". Philósophos, Goiânia, v. 28, n. 1, p. 1-34, jan./jun. 2023. DOI: 10.5216/phi.v28i1.75351. Disponível em: https://revistas.ufg.br/philosophos/article/view/75351/39970. Acesso em: 26 dez. 2023.