Reasoning about belief requires specific tools, so as to take into account their particular properties, among which their subjective nature and their gradualness. The first, essential, that is due to the non-factual nature of beliefs, can be represented in the modal logic formalism. Gradualness allows to modulate beliefs, e. g. distinguishing between a "firm" belief and a "weak" belief. Manipulating such graded beliefs needs to extend modal logics in order to increase their expressiveness.
In a general modal framework, we propose to establish a proportional semantics for weighted modalities, based on classical Kripke models. We examine weighted modal axioms extending their classical counterparts and propose a typology based on four categories, depending on the enrichment of classical axioms and on the equivalence with the classical property of accessibility relation. We also propose a logical system for representing and manipulating graded beliefs, based on a representationalist conception of belief and using fuzzy set theory. We study the induced axiomatics, in particular regarding our proposed axioms for the general weighted modal logic. Finally, we propose two applications of these theoretical models : a model checking tool for weighted modal formulae and an artificial player for a cooperative game in which decision making is based on graded belief reasoning.