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HOW MUCH PROPOSITIONAL LOGIC SUFFICES FOR ROSSER’S ESSENTIAL UNDECIDABILITY THEOREM?

Part of: General logic

Published online by Cambridge University Press:  29 June 2020

GUILLERMO BADIA ,
PETR CINTULA ,
PETR HÁJEK  and
ANDREW TEDDER
GUILLERMO BADIA
Affiliation:
SCHOOL OF HISTORICAL AND PHILOSOPHICAL INQUIRY UNIVERSITY OF QUEENSLANDST LUCIAQLD4072, AUSTRALIAE-mail: guillebadia89@gmail.com
PETR CINTULA
Affiliation:
INSTITUTE OF COMPUTER SCIENCE OF THE CZECH ACADEMY OF SCIENCESPRAGUE 8, 182 00, CZECH REPUBLICE-mail: cintula@cs.cas.czE-mail: ajtedder.at@gmail.com
PETR HÁJEK
Affiliation:
INSTITUTE OF COMPUTER SCIENCE OF THE CZECH ACADEMY OF SCIENCESPRAGUE 8, 182 00, CZECH REPUBLICE-mail: cintula@cs.cas.czE-mail: ajtedder.at@gmail.com
ANDREW TEDDER
Affiliation:
INSTITUTE OF COMPUTER SCIENCE OF THE CZECH ACADEMY OF SCIENCESPRAGUE 8, 182 00, CZECH REPUBLICE-mail: cintula@cs.cas.czE-mail: ajtedder.at@gmail.com
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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.

Keywords

undecidabilitysubstructural logicRobinson arithmetic

MSC classification

Primary: 03B25: Decidability of theories and sets of sentences
Secondary: 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) 03B52: Fuzzy logic; logic of vagueness
Type
Research Article
Information
The Review of Symbolic Logic , Volume 15 , Issue 2 , June 2022 , pp. 487 - 504
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Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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Footnotes

This paper is an extension and generalisation of work started by Cintula and Hájek before the unfortunate death of the latter in 2016. Hájek is included among the authors in recognition of his work on this topic, and with the blessing of his family, but it should be noted that he was not able to contribute directly to the final version of this paper.

References

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