Notre Dame Journal of Formal Logic

Propositional Logics of Closed and Open Substitutions over Heyting's Arithmetic

Albert Visser

Abstract

In this note we compare propositional logics for closed substitutions and propositional logics for open substitutions in constructive arithmetical theories. We provide a strong example where these logics diverge in an essential way. We prove that for Markov's Arithmetic, that is, Heyting's Arithmetic plus Markov's principle plus Extended Church's Thesis, the logic of closed and the logic of open substitutions are the same.

Article information

Source
Notre Dame J. Formal Logic Volume 47, Number 3 (2006), 299-309.

Dates
First available in Project Euclid: 17 November 2006

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1163775437

Digital Object Identifier
doi:10.1305/ndjfl/1163775437

Mathematical Reviews number (MathSciNet)
MR2264699

Zentralblatt MATH identifier
1113.03053

Subjects
Primary: 03F50: Metamathematics of constructive systems
Secondary: 03F30: First-order arithmetic and fragments 03B20: Subsystems of classical logic (including intuitionistic logic)

Keywords
propositional logic constructive arithmetical theories realizability

Citation

Visser, Albert. Propositional Logics of Closed and Open Substitutions over Heyting's Arithmetic. Notre Dame J. Formal Logic 47 (2006), no. 3, 299--309. doi:10.1305/ndjfl/1163775437. https://projecteuclid.org/euclid.ndjfl/1163775437


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References

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