Axiomatizing hybrid products: How can we reason many-dimensionally in hybrid logic?

The aim of this paper is to propose a two-dimensional hybrid logic in order to formalize inferences containing both spatial and temporal propositions. The semantic idea behind the proposal is to name both horizontal and vertical lines in a 2D-plane by two kinds of nominals. This is a generalization of the idea of naming a point in one-dimensional hybrid logic. I give an axiomatization of the proposed two-dimensional hybrid logic and show that it enjoys a general completeness result (called pure completeness) with respect to product Kripke frames. Moreover, in order to capture T×WT×W-frames studied by R.H. Thomason (1984), I introduce the notion of a dependent product frame, which enables us to represent the dependence of space over time. I also give a complete axiomatization of this dependent two-dimensional hybrid logic, and, as a corollary, reveal that a hybridization of T×WT×W-logic enjoys strong completeness.