The Kuznetsov-Gerciu and Rieger-Nishimura Logics: The Boundaries of the Finite Model Property
Guram Bezhanishvili, Nick Bezhanishvili, Dick de Jongh
Abstract:
We give a systematic method of constructing extensions of the
Kuznetsov- Gerciu logic KG without the finite model property (fmp for
short), and show that there are continuum many such. We also introduce
a new technique of gluing of cyclic intuitionistic descriptive frames
and give a new simple proof of Gerciu's result that all extensions of
the Rieger-Nishimura logic RN have the fmp. Moreover, we show that
each extension of RN has the poly-size model property, thus improving
on [Gerciu]. Furthermore, for each function f:\omega->\omega, we
construct an extension Lf of KG such that Lf has the fmp, but does not
have the f-size model property. We also give a new simple proof of
another result of Gerciu characterizing the only extension of KG that
bounds the fmp for extensions of KG. We conclude the paper by proving
that RN.KC = RN + (¬p v ¬¬p) is the only pre-locally tabular extension
of KG, introduce the internal depth of an extension L of RN, and show
that L is locally tabular if and only if the internal depth of L is
finite.