TY - JOUR
T1 - Open-endedness, schemas and ontological commitment
AU - Pedersen, Nikolaj Jang Lee Linding
AU - Rossberg, Marcus
N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.
PY - 2010/6
Y1 - 2010/6
N2 - 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.
AB - 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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U2 - 10.1111/j.1468-0068.2010.00742.x
DO - 10.1111/j.1468-0068.2010.00742.x
M3 - Article
AN - SCOPUS:77952932181
VL - 44
SP - 329
EP - 339
JO - Nous
JF - Nous
SN - 0029-4624
IS - 2
ER -