Chapter

Foundationalism and Foundations of Mathematics

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/10.1093/0198250290.003.0002
 Foundationalism and Foundations of Mathematics

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According to the foundational conception, logic is the ideal of justification, showing which propositions can be justified on the basis of which propositions. According to another, semantic conception of logic, an argument is valid if its conclusion is true under every interpretation of the language in which the premises are true. If one maintains both conceptions of logic, it is desirable, but not essential, for them to converge on a single system. This raises informal analogues of soundness and completeness, which are then compared to their more formal counterparts. The conclusion is that a system that conforms to both conceptions is too weak for important purposes. Sacrificing the analogue of completeness suggests a rejection of the foundational conception and motivates the consideration of second‐order logic.

Keywords: completeness; deduction; foundationalism; foundations; logic; mathematics; semantics; soundness; validity

Chapter.  16036 words. 

Subjects: Philosophy of Mathematics and Logic

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