David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jonathan Jenkins Ichikawa
Jack Alan Reynolds
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Noûs 44 (2):329-339 (2010)
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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References found in this work BETA
Hartry Field (2001). Truth and the Absence of Fact. Oxford University Press.
George Boolos (1998). Logic, Logic, and Logic. Harvard University Press.
Stewart Shapiro (1991). Foundations Without Foundationalism: A Case for Second-Order Logic. Oxford University Press.
Herbert B. Enderton (1972). A Mathematical Introduction to Logic. New York,Academic Press.
Vann McGee (1997). How We Learn Mathematical Language. Philosophical Review 106 (1):35-68.
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