Mathematical logic plays a crucial role in the 21st century society, indeed, in an era where information has become a key aspect of our lives, it provides a powerful mathematical framework for the study of the structure and the dynamics of information. A particularly interesting aspect of mathematical logic is that it allows to take in account uncertainty, vagueness or subjectivity, yet in the rigorous setting proper to mathematics.Drawn from the applications, disciplines like many-valued logic, modal logic, reasoning under uncertainty, etc. have quickly developed a weaponry of sophisticated investigation tools that enabled the disclosure of a number of substantial facts about their expressiveness, functionality and scope. Nonetheless the investigations in these topics are far from being concluded, rather they are ready to be shifted to another level which could take into account inter-relations between those topics and offer an exchange of knowledge and methodology among their practitioners.This project fits into this tendency shift, proposing an investigation of new semantics for many-valued logic with tools traditionally developed in modal and intuitionistic logic. Indeed, while the algebraic approach has enabled a fine comprehension of many-valued logic, leading in few years to a very reasonable framework of study, the time is mature for seeking new interpretations and concrete representations of these logical systems. As recent results put in evidence, the instruments developed in fifty years of studies on modal and intuitionistic logic, ranging from dualities to correspondence theory, are certainly the most qualified to help addressing this issues.The applicant has a strong background in many-valued logic and has already shown independency and vocation to research. The host institution, widely considered a pivotal research centre in modal and intuitionistic logic, is probably the best place in Europe to carry out such a research.