PP-2014-19: Stable formulas in intuitionistic logic

PP-2014-19: Bezhanishvili, Nick and de Jongh, Dick (2014) Stable formulas in intuitionistic logic. [Report]

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Abstract

NNIL-formulas are propositional formulas that do not allow nesting of implication to the left. These formulas were introduced in [16], where it was shown that NNIL-formulas are (up to provable equivalence) exactly the formulas that are preserved under taking submodels of Kripke models. In this paper we show that NNIL-formulas are up to frame equivalence the formulas that are preserved under taking subframes of (descriptive and Kripke) frames. As a result we obtain that NNIL-formulas are subframe formulas and that all subframe logics can be axiomatized by NNIL-formulas. We also introduce ONNILLI-formulas, only NNIL to the left of implications, and show that ONNILLI-formulas are (up to frame equivalence) the formulas that are preserved in monotonic images of (descriptive and Kripke) frames. As a result, we obtain that ONNILLI-formulas are stable formulas as introduced in [1] and that ONNILLI is a syntactically defined set of formulas that axiomatize all stable logics. This resolves an open problem of [1].

Item Type: Report
Report Nr: PP-2014-19
Series Name: Prepublication (PP) Series
Year: 2014
Uncontrolled Keywords: Intuitionistic logic, intermediate logics, monotonic maps, truth-preservation, axiomatization
Subjects: Logic
Date Deposited: 12 Oct 2016 14:37
Last Modified: 12 Oct 2016 14:37
URI: https://eprints.illc.uva.nl/id/eprint/507

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