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.