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- W79002562 abstract "A mathematical theory T is categorical if, and only if, any two models of T are isomorphic. If T is categorical, it can be shown to be semantically complete: for every sentence φ in the language of T, either φ follows semantically from T or ¬φ does. For this reason some authors maintain that categoricity theorems are philosophically significant: they support the realist thesis that mathematical statements have determinate truth-values. Second-order arithmetic (PA) is a case in hand: it can be shown to be categorical and semantically complete. The status of second-order logic is a controversial issue, however. Worries about the purported set-theoretic nature and ontological commitments of second-order logic have been influential in the debate. Recently, a number of authors – most notably Vann McGee and Shaughan Lavine – have argued that one can get some of the advantages of secondorder axiomatisations – categoricity, in particular – while walking free of the standard objections against second-order logic. In so arguing they appeal to the notion of an open-ended schema. In “open-ended arithmetic”, induction holds not only for formulas in the current language, but for formulas in any extension of that language. Both McGee and Lavine stress that this is not merely a variant of second-order logic. In this paper, we discuss the defensibility of this claim on McGee’s part. McGee’s account involves a metatheoretic rule that provides information about extensions of the language are legitimate. We argue that in the presence of this rule open-ended schemas are as ontologically suspect as standard second-order quantification." @default.
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- W79002562 date "2007-01-01" @default.
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- W79002562 title "McGee on open-ended schemas" @default.
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