MoL-2007-05:
Baskent, Can
(2007)
*Topics in Subset Space Logic.*
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## 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.

Item Type: | Report |
---|---|

Report Nr: | MoL-2007-05 |

Series Name: | Master of Logic Thesis (MoL) Series |

Year: | 2007 |

Date Deposited: | 12 Oct 2016 14:38 |

Last Modified: | 12 Oct 2016 14:38 |

URI: | https://eprints.illc.uva.nl/id/eprint/777 |

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