Majority logic and majority spaces in contrast with ultrafilters.
Item
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Title
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Majority logic and majority spaces in contrast with ultrafilters.
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Identifier
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AAI3204974
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identifier
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3204974
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Creator
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Salame, Samer.
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Contributor
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Adviser: Rohit Parikh
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Date
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2006
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Language
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English
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Publisher
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City University of New York.
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Subject
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Computer Science
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Abstract
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In this thesis I will extend graded modal logic (GML ) first discussed in [9, 10] to a logic that can capture the concept of majority. I will present the modal system MJL that will capture our intuition about majority and prove soundness and completeness for this system. I will also discuss May's theorem with infinite population. Graded modal logic, as presented in [7], extends propositional modal systems with a set of modal operators ⋄n n∈N that express "there are more than n accessible worlds such that...". I extend GML with a modal operator W that can express "there are at least half of the accessible worlds such that...". The semantics of W is straightforward provided that there are only finitely many accessible worlds; however if there are infinitely many accessible worlds the situation becomes much more complex. In order to deal with such situations, we introduce the notion of majority space. A majority space is a set W together with a collection of subsets of W intended to be the weak majority (at least half) subsets of W. We then extend standard Kripke structure with a function that assigns a majority space over the set of accessible states to each state. Given this extended Kripke semantics, majority logic is proved sound and complete.;Part of this thesis is devoted to talk about May's theorem with infinite population. We will talk about three different kinds of anonymity: finite, bounded and infinite. We will compare all three of them together. Given an infinite subset A that we call majority, if we remove a finite set of elements of A then are we going to get a set that we still call majority? This answer will be a generalized property of majority sets over finite spaces.;I will also talk about majority spaces in contrast with ultrafilters. I will present another way to construct ultrafilters and majority spaces by using the limits of sequences over that family of sets. I will also argue that ultrafilters have a dictatorship flavor while the majority spaces have a democracy flavor.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
