Stable formulas in intuitionistic logic
نویسندگان
چکیده
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].
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