Chapter

Theory

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/10.1093/0198250290.003.0003
 Theory

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Show how to extend an ordinary, first‐order language to one with free second‐order variables, ramified and unramified second‐order variables, and then bound third‐ and higher‐order variables, together with constants for relations on relations. Deductive systems for each language are presented, along with brief discussions of the axioms and rules of inference in light of the intended interpretation of higher‐order variables as ranging over sets or properties. Three different model‐theoretic semantics are developed. In standard semantics, relation variables range over the entire class of relations on (or subsets of) the domain. With Henkin semantics, each model contains a specified class of relations and functions that the higher‐order variables range over. With first‐order semantics, the language is regarded as a many‐sorted first‐order language, with the predication or membership relation non‐logical. It is shown that Henkin semantics is equivalent to first‐order semantics.

Keywords: deduction; Henkin; language; model theory; semantics

Chapter.  9573 words. 

Subjects: Philosophy of Mathematics and Logic

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