MoL-2007-05: Topics in Subset Space Logic

MoL-2007-05: Baskent, Can (2007) Topics in Subset Space Logic. [Report]

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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

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