Provability Logic and Admissible Rules
Rosalie Iemhoff

Abstract:
%Nr: DS-2001-04
%Author: Rosalie Iemhoff
%Title: Provability Logic and Admissible Rules

This thesis consists of two parts. In the first part  the provability and 
preservativity logic of Heyting Arithmetic are studied, and the second part 
contains results in intuitionistic propositional logic. The two parts are 
connected via admissible rules; they play a central role in the provability 
logic of  Heyting Arithmetic and are the main topic of the second part of the 
thesis.

Up till now there are  no axiomatizations known for provability logics of 
constructive theories.  However, in the first part of the thesis it is shown 
that for many well-known properties of Heyting Arithmetic that are 
expressible in provability logic, it is known whether they belong to the 
logic or not. Therefore, it is argued that the system studied in the thesis  
forms at least a very natural fragment, if not all,  of the provability logic 
of Heyting Arithmetic. The principles of this system are studied from the 
modal point of view.  Therefore, this part of the thesis can also be viewed 
as a study in intuitionistic modal logic, in which surprising  frame 
properties become visible.  It is shown that the given system is complete 
with respect to a certain class of frames. The principles are also studied 
separately  and proved to be independent.

The second part of the thesis is about intuitionistic propositional logic. 
First, a basis for the admissible rules of this logic   is established. Then 
it is shown that intuitionistic propositional logic is characterized by these 
rules plus the  Disjunction Property. In a similar way it is shown that every 
finite part of the basis plus the  Disjunction Property characterizes one of 
the Gabbay-de Jongh logics. This shows that the characterization of 
intuitionistic  propositional logic is optimal; no finite part of the basis 
characterizes it.