Calculi for Constructive Communication, a Study of the Dynamics of Partial States
Jan Jaspars
Abstract:
This thesis presents a mathematical logical analysis of the
infrastructure of partial worlds, and demonstrates how its
model-theoretical treatment can be used for a constructive
formalization of the dynamics of a group of reasoning and
communicating agents.
Following G\"ardenfors' influential general view on epistemic
dynamics, Jaspars first specifies the means for static representation
of information, and then presents the dynamics of such epistemic
registrations. The static side consists of a straightforward partial
variant of the {\em possible worlds semantics} of modal logic. The
alternative aspect of this semantics in this thesis is its dynamics;
whereas classical possible world semantics is purely eliminative --
information growth equals elimination of possibilities -- partial
semantics permits to add a constructive component as well. The key
issue of this thesis is to point out how such different ways of
information flow can peacefully cohabit in the theory of partial
possible worlds. On the basis of this construction-elimination
dynamics Jaspars defines relatively simple sequential calculi for
reasoning about interacting agents.
Essential linguistic ingredients which are formalized in these calculi
are {\em intentional modalities} and the representation of {\em mutual
epistemic information} of a group of agents. The general aim is to
show how a variety of dynamic interpretations of communicative
actions, and principles of pragmatic rationalism of communicating
agents, can be stipulated in terms of partial modal formalisms and
their dynamic extensions.
The first part introduces the basic logical equipment for this
enterprise in partial logic. Motivations, formal model-theoretic
interpretations and sequential derivation systems are presented.
Part two presents the technical streamlining of completeness and
decidability proof procedures for the systems of part one. By means of
a generalization of the well-known Henkin proof procedure, Jaspars
shows that partial logics do not have to be more troublesome than
their regular two-valued counterparts.
As a contribution to general modal logic, the last chapter of part two
presents a bit of the correspondence theory of partial modal logic.
Definability and completeness for so-called Geach extensions of the
minimal partial modal logic are established. These results combine the
notion of accessibility in modal logic and the information orders that
naturally arise from partial semantics.