Two-Dimensional Modal Logics

This chapter contains a technical introduction to the world of multi-dimensional modal logics. We will treat some relatively simple logics with a two-dimensional semantics. In section 2.1, we introduce the family of modal operators we are going to study, with their two-dimensional semantics. In sections 2.2 and 2.3, we study two-dimensional modal logic with unary operators. These sections can be seen as an appetizer for the α-dimensional case which is treated in chapter 5. Section 2.4 deals with the modal logic of composition. This section is an introduction to chapter 3, which is completely devoted to logics with composition as their main connective. Section 2.5, finally, is about two-dimensional tense logic, a subject which is taken up again in chapter 4. We conclude this chapter with some historical notes on the logics described here.