Abstract

This paper expands upon the work by Wang (Proceedings of TARK, pp. 493–512, 2017) who proposes a new framework based on quantifier-free predicate language extended by a new bundled modality $\exists x\Box$ and axiomatizes the logic over S5 frames. This paper first gives complete axiomatizations of the logics over K, D, T, 4, S4 frames with increasing domains and constant domains, respectively. The systems w.r.t. constant domains feature infinitely many additional rules defined inductively than systems w.r.t. increasing domains. In the second part of the paper, we show that these rules are admissible in the systems without them by providing a general strategy for proving completeness theorems for the logics without these rules over constant domain models (also increasing domain models).