NNIL-formulas revisited: universal models and finite model property

Abstract

NNIL-formulas, introduced by Visser in 1983-1984 in a study of Σ 1 -subsitutions in Heyting Arithmetic, are intuitionistic propositional formulas that do not allow nesting of implication to the left. The main results about these formulas were obtained in a paper of 1995 by Visser and others. It was shown that NNIL-formulas are exactly the formulas preserved under taking submodels of Kripke models. In the present paper an observation by Bezhanishvili and de Jongh of NNILformulas as the formulas backwards preserved by monotonic maps of Kripke models is applied to construct a universal model for NNIL. This universal model shows that NNIL-formulas are also exactly the ones that are backwards preserved by monotonic maps. The methods developed in constructing the universal model are used in this paper in a new direct proof that these logics have the finite model property.