Completeness Theorems for $$\exists \Box $$-Fragment of First-Order Modal Logic

The paper expands upon the work by Wang [4], who proposes a new framework based on quantifier-free predicate language extended by a new modality \(\exists x\Box \) and axiomatizes the logic over S5 frames. This paper gives the logics over K, D, T, 4, S4 frames with increasing and constant domains. And we provide a general strategy for proving completeness theorems for logics w.r.t. the increasing domain and logics w.r.t. the constant domain respectively.