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/lp/cambridge-university-press/how-much-propositional-logic-suffices-for-rosser-s-essential-SkprGFSGpP
- Publisher
- Cambridge University Press
- Copyright
- © The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
- ISSN
- 1755-0211
- eISSN
- 1755-0203
- DOI
- 10.1017/S175502032000012X
- Publisher site
- See Article on Publisher Site
Abstract
Abstract In this paper we explore the following question: how weak can a logic be for Rosser’s essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson’s Q is essentially undecidable in intuitionistic logic, and P. Hájek proved it in the fuzzy logic BL for Grzegorczyk’s variant of Q which interprets the arithmetic operations as nontotal nonfunctional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a much weaker arithmetic theory, a version of Robinson’s R (with arithmetic operations also interpreted as mere relations). Our result is based on a structural version of the undecidability argument introduced by Kleene and we show that it goes well beyond the scope of the Boolean, intuitionistic, or fuzzy logic.
Journal
The Review of Symbolic Logic – Cambridge University Press
Published: Jun 1, 2022
Keywords: 03B25; 03B47; 03B52; undecidability; substructural logic; Robinson arithmetic