Аннотация:In the seventies it was discovered that sometimes modal propositional formulas have a rather big expressive power: there are modal formulas without first-order equivalents on Kripke frames. The main typical means for obtaining such results are the Löwenheim-Skolem theorem and the compactness theorem. However, by the Lindström theorem, these effects are very strong: both theorems together characterize first-order logic completely. It is natural to raise the question: what specific properties of first-order formulas are true for modal formulas (on interesting classes of frames).
Here we will consider the following well-known first-order effect: if a theory has arbitrary large finite models, then it has an infinite model. Of course, a modal variant of this property must be relativized to rooted frames: say that a formula has cat-property if it has an infinite rooted frame whenever it has arbitrary large finite rooted frames. In our considerations we will not differentiate formulas and logics which are axiomatized by these formulas.
