Topics in Subset Space Logic
Can Baskent
Abstract:
In this thesis, we will first provide a comprehensive outlook of
subset space logic in detail in order to set the basis for our future
discussions of the subject.
Then, we will import some simple truth preserving operations which are
familiar from (basic) modal logic and provide their definitions in our
new language. Furthermore, we will observe that these operations are
valid in subset space logic as well. As expected, validity preserving
operations will enable us to point out definable and non-definable
properties in the language of subset space logic.
Furthermore, we will discuss several important extensions of subset
space logic. These extensions will be indispensably significant in our
future discussion. In addition to that, we will follow the tradition
and present a game theoretical semantics for subset space logic. We
will therefore introduce evaluation games and bisimulation
games. Moreover, we will even present bisimulation games and
evaluation games for the extended languages. This can be seen as a
continuation of topological games.
Equipped with all these tools, we will observe that the subset space
logic is strong enough to axiomatize the dynamic aspects of knowledge
change, in particular, the public announcement logic. We will then
provide the full axiomatization of subset space public announcement
logic and its then straightforward completeness proof. As long as the
research area of "geometry of knowledge" is considered, we believe, it
is significant to see that public announcement logic works well in the
subset space language.
All these discussions will lead us to take a closer look at the notion
of shrinking - which can be considered as the temporal and perhaps the
dynamic operator of the subset space logic. We will observe that, in
fact, the shrinking operator is not a remote concept in formal
sciences. We will motivate our point with several examples chosen from
the broader research areas in various branches of logic such as
methodology and philosophy of mathematics, belief revision etc. These
considerations yet will not able us to formalize the improved concept
of shrinking. However, we will suggest one approach to analyze the
conceptual framework for the shrinking operator - which is
unfortunately far from being complete and precise. However, we
believe, this initiation of discussions on the shrinking operator will
emphasize the significance of the aforementioned operator.
After that as a third point, we will consider the multi-agent version
of subset space logic. However, it will turn out that it is not as
nice as it is expected to be. We will suggest several methods to
formalize the concept. Thereon we will import some basic results from
modal logic and observe that they are valid in the subset space logic
as well.
Last, but not least, we will recall the concept of common knowledge,
and point out the definition of common knowledge in the language of
basic and extended subset space logic. As another contribution, we
will consider the extensions of public announcement logic with an
additional operator together with a general notion of common knowledge
(called relativized common knowledge). We will then easily prove the
completeness of public announcement logic extended with these
aforementioned operators in the extended language of subset space
logic by reducing it to already known completeness results.
Finally, we will conclude with some open problems and future work
ideas that might bring some light to the shaded areas in the subset
space logic - the logic which we believe has the necessary tools per
se to analyze many conceptual frameworks in logic.