Majority logic and majority spaces in contrast with ultrafilters

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 dn 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.