Chapter

The Competition

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/10.1093/0198250290.003.0009
 The Competition

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Compares second‐order logic, with standard semantics, to other proposed foundations. It is in the spirit of the book thesis that there is no unique foundation for mathematics, nor a unique underlying logic of a given branch of mathematics. Different purposes suggest different logics. The examples considered here include languages with free second‐order variables, Boolos's attempt to assimilate (monadic) second‐order quantifiers to the plural construction of natural language, infinitary logic, ω‐logic, weak second‐order logic, logics with cardinality quantifiers, ancestral logic, substitutional quantification, and first‐order set theory.

Keywords: ancestral; Boolos; cardinality; foundationalism; infinitary; omega logic; plural; set theory; substitutional

Chapter.  23194 words. 

Subjects: Philosophy of Mathematics and Logic

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