A light at the end of the tunnel

Graded properties, expressed by graded predicates, are ubiquitous in human discourse and reasoning. They are characterized by the fact that utterances of (simple) sentences involving them are graded, that is, admit mutual comparison. Typical examples are vague properties (such as red, old, tall, rich), i.e. those that do not establish a clear distinction between objects that satisfy them and those that do not, and hence have borderline cases.1 Mathematically, we may say that graded properties are those that can be modelled by a function from the universe of discourse to a set (or a set of tuples) of ordered grades. In the example, it could naturally be the mapping that sends each angle to its number of degrees. Arguably, graded properties are epistemologically necessary, as they allow putting together many similar notions that would otherwise collapse the conceptual system (and the language) with too many properties (and predicates). It is an economic necessity to reason with one predicate red, instead of having infinitely many for each possible level in the color spectrum. Reasoning with graded properties is successfully and correctly carried out in many contexts (not withstanding the fact that natural language has enough devices to provide higher levels of precision whenever necessary). Logic is the science of correct reasoning and, as such, has to deal with all forms of valid consequence potentially carried out by rational beings. During most of its history as a formal science, Logic has tried to explain correct reasoning by means of a classical paradigm, that of logic based on bivalence principle. Despite of its many merits and achievements, such approach does violence to many properties, forcing crisp definitions (splitting graded properties into many crisp ones) in contexts that did not require them. Indeed, natural language allows satisfactory communication and correct reasoning in terms of graded predicates. The classical logical analysis seems, therefore, too artificial and detached from actual reasoning. However, to the best of our knowledge, it appears that a serious research program aimed at understanding reasoning with graded predicates in all its complexity from the point of view of Logic has never been carried out. Moreover, it has been argued by Stenning and van Lambalgen [2] that Logic, despite its origins, has become increasingly divorced from the