Historically, logic has been the foundation and in the same time the pinnacle of (human) reasoning. However during the 20th century reasoning was claimed by cognitive science and got `divorced' from logic (to quote Stenning and van Lambalgen). Nevertheless, there is a clear trend to amend the rift and employ various (!) logical formalisms to analyse and model various (!) reasoning tasks. This trend is stimulated by numerous artificial reasoning scenarios stemming from computer science which are, nowadays, the most visible ‘market’ for logic.
One of the issues which is not yet sufficiently thematized in this process, is the problem of graded predicates, i.e., predicates whose natural semantical treatment is hard to press into the black-and-white Boolean paradigm (e.g. predicates like “red”, “tall”, or “intelligent”). While there is an extensive logical literature about this issue, works aiming at understanding the role of graded predicates in actual reasoning are few and a convincing overall theory is still lacking.
After a general introduction I design (following natural and clearly articulated design choices) an abstract semantical framework for logics where predicates can take more than two values. For such logics I first describe their propositional part and then, under certain additional assumptions, give their Hilbert-style axiomatizations. Finally I showcase an example of how these logics can be applied to study reasoning in the formalized contexts involving graded predicates.
At the end I would like to initialise a very general discussion between mathematicians, philosophers and computer scientists about the role which (not only these) logics can play in analysing and performing reasoning in the formalized contexts (with or without graded predicates) and ultimately in the study of natural reasoning scenarios and in the process of their transformation into the formalized ones.