Frame based formulas for intermediate logics (Q1005954)

In the study of propositional logics, particularly of intermediate logics and normal modal logics, the so-called Jankov-de Jongh formulas defined by finite frames have been applied as a powerful tool to many important results in several topics such as axiomatizability, finite model property, splitting, construction of continuum many logics, and so on. Based on a similar idea Fine, Zakharyaschev, and others introduced certain variations, e.g., subframe and cofinal subframe formulas, and showed they also work for other interesting results. In this paper, the author generalizes these formulas to the notion of frame-based formulas by extracting principal features, and shows that many central issues on the above-mentioned applications can be discussed in a uniform framework thus generalized. One of main results, for example, is a general criterion for an intermediate logic to be axiomatized by the frame-based formulas, which gives rise to a simple proof of (finite) axiomatizability of locally tabular (tabular) logic by Jankov-de Jongh formulas. In addition to some observations on the above-mentioned applications, it is also shown that there exists some intermediate logic that is not axiomatizable by frame-based formulas. The paper is devoted to the argument on intermediate logics, but the author also refers to the case of normal modal logics in paralell and remarks that a similar approach is available for transitive normal modal logics.