Higher‐order Logic Reconsidered

In Stewart Shapiro, Oxford Handbook of Philosophy of Mathematics and Logic. Oxford and New York: Oxford University Press (2005)
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Abstract

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 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.

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The Logic of Finite Order.Simon Hewitt - 2012 - Notre Dame Journal of Formal Logic 53 (3):297-318.

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