The Historical ‘Triumph’ Of First‐Order Languages

Stewart Shapiro

in Foundations without Foundationalism

Published in print March 2000 | ISBN: 9780198250296
Published online November 2003 | e-ISBN: 9780191598388 | DOI:
 The Historical ‘Triumph’ Of First‐Order Languages

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Traces the history of the dispute between those logicians who hold that only first‐order languages are legitimate and those who advocate second‐order languages. In the early decades of the twentieth century, first‐order languages were sorted out for special study. Starting around then, the main proponents of first‐order logic include Skolem, von Neumann, Weyl, and Gödel. Hilbert, Bernays, and Zermelo favoured—and used—higher‐order languages. On the contemporary (or near contemporary) scene, Quine remains the main opponent of higher‐order logic. It is often suggested that the early historical disputes are full of confusion over the different syntax, or proof theory, and model‐theoretic semantics, over the notion of set and property, over the range of logic and the range of categoricity, and over the applicability of the Löwenheim–Skolem theorems. However, many of the remarks by historical figures have counterparts in the current debate. Once presuppositions are noted, some of the pronouncements are at least relatively clear and, in some cases, remarkably clear, even by modern lights.

Keywords: Bernays; first‐order language; Gödel; Hilbert; history of logic; von Neumann; Quine; second‐order language; set; Skolem; Löwenheim–Skolem; Weyl; Zermelo

Chapter.  14874 words. 

Subjects: Philosophy of Mathematics and Logic

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