Graded square of opposition, graded hexagon of opposition, graded cube of opposition, fuzzy natural logic, generalized intermediate quantifiers, generalized syllogisms

The main objective of this paper is devoted to two main parts. First, the paper introduces logical interpretations of classical structures of opposition that are constructed as extensions of the square of opposition. Blanch\'{e}'s hexagon as well as two cubes of opposition proposed by Morreti and pairs Keynes-Johnson will be introduced. The second part of this paper is dedicated to a graded extension of the Aristotle's square and Peterson's square of opposition with intermediate quantifiers. These quantifiers are linguistic expressions such as ``most'', ``many'', ``a few'', and ``almost all'', and they correspond to what are often called ``fuzzy quantifiers'' in the literature. The graded Peterson's cube of opposition, which describes properties between two graded squares, will be discussed at the end of this paper.