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.
| Original language | English |
|---|---|
| Pages (from-to) | 329-339 |
| Number of pages | 11 |
| Journal | Nous |
| Volume | 44 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - 2010 Jun 1 |
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All Science Journal Classification (ASJC) codes
- Philosophy
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Open-endedness, schemas and ontological commitment. / Pedersen, Nikolaj J.l. Linding; Rossberg, Marcus.
In: Nous, Vol. 44, No. 2, 01.06.2010, p. 329-339.Research output: Contribution to journal › Review article
TY - JOUR
T1 - Open-endedness, schemas and ontological commitment
AU - Pedersen, Nikolaj J.l. Linding
AU - Rossberg, Marcus
PY - 2010/6/1
Y1 - 2010/6/1
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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UR - http://www.scopus.com/inward/citedby.url?scp=77952932181&partnerID=8YFLogxK
U2 - 10.1111/j.1468-0068.2010.00742.x
DO - 10.1111/j.1468-0068.2010.00742.x
M3 - Review article
VL - 44
SP - 329
EP - 339
JO - Nous
JF - Nous
SN - 0029-4624
IS - 2
ER -