Curry's paradox in contractionless constructive logic
type: Item
instance of: scholarly article
title: Curry's paradox in contractionless constructive logic
zbMATH Open document ID: 0873.03032
DOI: 10.1007/BF00247001
author: Seiki Akama
published in: Journal of Philosophical Logic
publication date: 1996-06-09T00:00:00Z
review text: The paper shows the consistency of a naive abstraction principle in a logic which is \textit{D. Nelson's} logic of constructible falsity [J. Symb. Logic 14, 16-26 (1949; Zbl 0033.24304)] minus the contraction principle \(((\alpha\to(\alpha\to\beta))\to (\alpha\to \beta))\). The result is established proof-theoretically, via a suitable normal form theorem. Comparisons are made with results for the similar logics of \textit{F. B. Fitch} [Symbolic logic. An introduction (1952; Zbl 0049.00504)], \textit{D. Prawitz} [Natural deduction (1965; Zbl 0173.00205)] and the BCK logic of \textit{H. Ono} and \textit{Y. Komori} [J. Symb. Logic 50, 169-201 (1985; Zbl 0583.03018)]. Although the paper calls this system `naive set theory', there is no extensionality axiom; nor does it discuss the consistency of adding one. -- Adding one to the naive abstraction schema in the slightly stronger Łukasiewicz' infinite-valued logic is provably inconsistent [see \textit{R. White}, J. Philos. Logic 8, 509-534 (1979; Zbl 0418.03037)].
Mathematics Subject Classification ID: 03B60
Mathematics Subject Classification ID: 03E70
Mathematics Subject Classification ID: 03F05
zbMATH DE Number: 885307
zbMATH Keywords: logic of constructibe falsity minus the contraction principle
zbMATH Keywords: consistency
zbMATH Keywords: naive abstraction
zbMATH Keywords: normal form theorem
reviewed by: Graham Priest
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 885307