The aim of this work is to revisit the proposal made by Dag Westerståhl a decade ago when he provided a modern reading of the traditional square of opposition and of related structures. We propose a formalization of this modern view and contrast it with the classical one. We discuss what may be a modern hexagon of opposition and a modern cube, and show their interest in particular for relating quantitative expressions.

The aim of this paper is to make sense of the Keynes–Johnson octagon of oppositions. We will discuss Keynes' logical theory, and examine how his view is reflected on this octagon. Then we will show how this structure is to be handled by means of a semantics of partition, thus computing logical relations between matching formulas with a semantic method that combines model theory and Boolean algebra.

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. Blanché’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. (shrink)

This special issue is related to the 6th World Congress on the Square of Opposition which took place at the Orthodox Academy of Crete in November 2018. In this introductory paper we explain the context of the event and the topics discussed.