Abstract

We prove a completeness theorem for Kf, an extension of K by the operator ⋄f that means “there exists a finite number of accessible worlds such that … is true, plus suitable axioms to rule it. This is done by an application of the method of consistency properties for modal systems as in [4] with suitable adaptations. Despite no graded modality is invoked here, we consider this work as pertaining to that area both because ⋄f is a definable operator in the graded infinitary system Kω10, and because this idea was the original source for the development of graded modalities.