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It's well known that some special classes of formulas have got specific model-theoretic properties in classical logic. For instance, universal, existential, positive, and Horn formulas are stable on subsystems, extensions, homomorphic images, and products, respectively. Moreover, each formula stable on subsystems is equivalent to a universal formula, and so on.

My question is about analogues of such statements for intuitionistic logic. Could you please tell me where to find an overview of these results?

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You can see "Stable Formulas in Intuitionistic Logic" by N.Bezhanishvili and D.De Jongh. In another D. de Jongh's article were introduced "NNIL formulas" and was shown that these formulas are the ones that are preserved under submodels of Kripke Models. Basically this article developes properties of these formulas and it is shown that NNIL-formulas are subframe formulas and that all subframe logics can be axiomatized by NNIL-formulas. If you want to know what is a "Subframe formula", read Zakharyaschev 1989,1996.

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