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.
"Stable Formulas in Intuitionistic Logic." Notre Dame J. Formal Logic 59 (3) 307 - 324, 2018. https://doi.org/10.1215/00294527-2017-0030