Stable formulas in intuitionistic logic
Nick Bezhanishvili, Dick de Jongh
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].