Chapter

Higher‐order Logic Reconsidered

Ignacio Jané

in The Oxford Handbook of Philosophy of Mathematics and Logic

Published in print March 2005 | ISBN: 9780195148770
Published online July 2005 | e-ISBN: 9780199835560 | DOI: http://dx.doi.org/10.1093/0195148770.003.0026

Series: Oxford Handbooks in Philosophy

Higher‐order Logic Reconsidered

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Second-order languages, canonically understood, allow quantification over all sets of objects in the range of the first-order variables. In this chapter two arguments are given against the suitability of using second-order consequence (defined in the Tarskian way) as the consequence relation of axiomatic theories. According to the first argument, second-order languages are inadequate for axiomatizing set theory because of the strong set-theoretic content coded by second-order consequence. The second more general argument is directed against the determinacy of second-order consequence, that is, against the assumption that this is a definite relation. Only taking a strong realist view of set theory can one maintain that it is.

Keywords: second-order logic; second-order language; quantification; consequence; Tarski; set theory; axiomatic theory; determinacy

Chapter.  14226 words. 

Subjects: Philosophy of Mathematics and Logic

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