Notre Dame Journal of Formal Logic

Stable Formulas in Intuitionistic Logic

Nick Bezhanishvili and Dick de Jongh

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In 1995 Visser, van Benthem, de Jongh, and Renardel de Lavalette introduced NNIL-formulas, showing that these are (up to provable equivalence) exactly the formulas preserved under taking submodels of Kripke models. In this article we show that NNIL-formulas are up to frame equivalence the formulas preserved under taking subframes of (descriptive and Kripke) frames, that NNIL-formulas are subframe formulas, and that subframe logics can be axiomatized by NNIL-formulas. We also define a new syntactic class of ONNILLI-formulas. We show that these are (up to frame equivalence) the formulas preserved in monotonic images of (descriptive and Kripke) frames and that ONNILLI-formulas are stable formulas as introduced by Bezhanishvili and Bezhanishvili in 2013. Thus, ONNILLI is a syntactically defined set of formulas axiomatizing all stable logics. This resolves a problem left open in 2013.

Article information

Notre Dame J. Formal Logic, Volume 59, Number 3 (2018), 307-324.

Received: 18 August 2014
Accepted: 2 December 2015
First available in Project Euclid: 4 May 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B55: Intermediate logics 03B20: Subsystems of classical logic (including intuitionistic logic)

intuitionistic logic intermediate logics subframe logics monotonic maps stable logics axiomatization


Bezhanishvili, Nick; de Jongh, Dick. Stable Formulas in Intuitionistic Logic. Notre Dame J. Formal Logic 59 (2018), no. 3, 307--324. doi:10.1215/00294527-2017-0030.

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