Open-endedness, schemas and ontological commitment

Noûs 44 (2):329-339 (2010)
Abstract
Second-order axiomatizations of certain important mathematical theories—such as arithmetic and real analysis—can be shown to be categorical. Categoricity implies semantic completeness, and semantic completeness in turn implies determinacy of truth-value. Second-order axiomatizations are thus appealing to realists as they sometimes seem to offer support for the realist thesis that mathematical statements have determinate truth-values. The status of second-order logic is a controversial issue, however. Worries about ontological commitment have been influential in the debate. Recently, Vann McGee has argued that one can get some of the technical advantages of second-order axiomatizations—categoricity, in particular—while walking free of worries about ontological commitment. In so arguing he appeals to the notion of an open-ended schema—a schema that holds no matter how the language of the relevant theory is extended. Contra McGee, we argue that second-order quantification and open-ended schemas are on a par when it comes to ontological commitment
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DOI 10.1111/j.1468-0068.2010.00742.x
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Truth and the Absence of Fact.Hartry Field - 2001 - Oxford University Press.
Logic, Logic, and Logic.George Boolos - 1998 - Harvard University Press.
How We Learn Mathematical Language.Vann McGee - 1997 - Philosophical Review 106 (1):35-68.

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