Classical mathematical logic, built on the conceptually simple core of propositional Boolean calculus, plays a crucial role in modern computer science. A critical limit to its applicability is the underlying bivalent principle that forces all propositions to be either true or false. Propositional logics of graded notions (such as tall, rich, etc.) have been deeply studied for over two decades but their predicate extensions (accommodating, among others, modalities and quantifiers) are still only very partially developed and scarcely applied to particular computer science problems. The overall goal of the proposed project is to develop predicate graded logics in two complementary directions: (1) studying logical systems in full generality in order to provide a solid mathematical framework and (2) applying achieved results to three particular problems in computer science which heavily involve graded notions: representation of vague and uncertain knowledge, valued constraint satisfaction problems, and modelling of coalition games.