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Open‐endedness, Schemas and Ontological Commitment
Author(s) -
Pedersen Nikolaj Jang Lee Linding,
Rossberg Marcus
Publication year - 2010
Publication title -
noûs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.574
H-Index - 66
eISSN - 1468-0068
pISSN - 0029-4624
DOI - 10.1111/j.1468-0068.2010.00742.x
Subject(s) - computer science , sociology , epistemology , philosophy
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.