Definability and Interpolation: Model-theoretic investigations
Eva Hoogland

Abstract:
In this thesis we study definability and interpolation.  These are
properties of logics such as compactness or decidability that have
been established as yardsticks by which to measure the behavior of
logics.  What do they look like?  In a slogan, the _Beth
(definability) property_ states that implicit definability equals
explicit definability.  These notions will be explained in full
detail in the thesis. The gist is that implicit definability is a
semantic concept whereas explicit definability is a syntactic
phenomenon.  To say that the two forms of definability coincide (as
the Beth property does) may therefore be regarded as an indication
that there is a good balance between syntax and semantics of a logic.

Proving that a given logic S has the Beth property usually proceeds
by way of proving the _interpolation property_ for S.  This
property requires that any validity \phi -> \psi has an _interpolant_.
That is, there exists a formula \theta in the common language of
\phi, \psi, such that \phi -> \theta and \theta -> \psi are again 
validities.  Apart from its connection with definability, interpolation 
is also an interesting notion in itself which points to a well-behaved 
deductive system.
   

The objectives of this dissertation are fourfold. We successively

 * Provide ``everything you always wanted to know about definability
   and interpolation but were afraid to ask.'' 

 * Relate definability to the algebraic property of surjectiveness
   of epimorphisms.

 * Offer tools for proving and disproving definability theorems and 
   interpolation theorems. 

 * Present plenty of examples that show that the interpolation
   property is much stronger than the definability property.  To this
   end, we do two detailed case studies, viz., of guarded fragments of
   first order logic and of interpretability logics.


Chapter 2: 
The aim of this introductory chapter is to make the reader familiar
with the main themes of this dissertation: definability and
interpolation. The chapter is written in a rather informal manner with
an emphasis on giving simple examples.  We discuss the precise
relationship between the relevant properties, summarize the state of
the art, and provide ample references to the literature.  This chapter
also takes a look at the matter from an algebraic perspective.

Chapter 3: 
This chapter is of an abstract algebraic nature in which algebraic
equivalents of several Beth definability properties are given.  We
also supply many applications of these characterizations.  The chapter
contains an introduction to the abstract algebraic framework(s) we are
working in.

Chapter 4 and Chapter 5:
These chapters can be seen as case studies in which we extend known
methods for proving interpolation and definability.  The fourth
chapter concerns interpretability logics (these are non-standard modal
logics), the final chapter deals with guarded fragments of first order
logic.