Extending Modal Logic
Maarten de Rijke
Abstract:
This thesis is concerned with extensions of the standard modal language.
After the introduction in chapter 1, chapter 2 develops a general perspective
on modal logic according to which modal languages are primarily many-sorted
descreption languages for relational structures, mainly concerned with the
fine-structure of model theory. Moreover, the chapter presents a number of
central themes such as `expressivity', `combinations of modal logics',
`transfer of properties of modal logics to richer languages' and `connections
between modal logics'.
Chapters 3, 4 and 5 look into {\em specific} extended modal systems, e.g.
modal logics with a difference operator, a dynamic modal logic, and modal
systems that correspond to Peirce algebras. Some applications are sketched,
and the above themes are discussed for these systems. Furthermore, a method
for axiomatic completeness in systems with difference operators is presented
(chapter 3), and applied (chapters 4 and 5).
Chapters 6 and 7 are concerned with more {\em general themes} in extended
modal logic.
Chapter 6 develops the model theory of classes of basic modal logics with
the help of bisimulations. This results in general theorems about
definability and preservation. In addition, we give a characterization of
basic modal logic analogous to the well-known Lindstr\"om theorem for
first-order logic. Chapter 7 looks at extended modal formulas as classical
higher-order conditions on the underlying semantic structures. This chapter
formulates abstract and general algorithms that reduce higher-order
conditions corresponding to certain extended modal formulas to simpler
formulas.