Chapter

Second‐Order Logic and Rule‐Following

Stewart Shapiro

in Foundations without Foundationalism

Published in print March 2000 | ISBN: 9780198250296
Published online November 2003 | e-ISBN: 9780191598388 | DOI: http://dx.doi.org/10.1093/0198250290.003.0008
 Second‐Order Logic and Rule‐Following

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To speak roughly, there are three positions toward mathematical theories and their logic. One option, unpopular among mathematicians, is to follow Skolem and deny that there is an intended range, unique up to isomorphism, for the variables of mathematical theories like arithmetic and real analysis. The second option is to claim that the ranges of the variables of such mathematical theories are determined up to isomorphism somehow, but that this range is understood only informally. Any model‐theoretic semantics must fail to capture our intuitive grasp of mathematical theories. The third perspective follows Church and recognizes the intuitive presuppositions of classical mathematical theories and includes these in the formal logic. The presuppositions of second‐order logic concerning the range of the variables may be more perspicuous than those of mathematics, but they are not more troublesome. The issues of second‐order logic are related to the Wittgensteinian issue of rule‐following. The Skolemite sceptic is analogous to a sceptic concerning the correct application of rules.

Keywords: church; informal; isomorphism; model theory; rule‐following; scepticism; Skolem; Wittgenstein

Chapter.  8462 words. 

Subjects: Philosophy of Mathematics and Logic

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