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 ⋄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.