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The Oxford Handbook of Philosophy of Mathematics and Logic$
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Stewart Shapiro

Print publication date: 2005

Print ISBN-13: 9780195148770

Published to Oxford Scholarship Online: July 2005

DOI: 10.1093/0195148770.001.0001

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Higher‐order Logic Reconsidered

Higher‐order Logic Reconsidered

(p.781) Higher‐order Logic Reconsidered
The Oxford Handbook of Philosophy of Mathematics and Logic

Ignacio Jané (Contributor Webpage)

Oxford University Press

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

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