The traditional square of opposition and generalized quantifiers ∗

The traditional square of opposition dates back to Aristotle’s logic and has been intensely discussed ever since, both in medieval and modern times. It presents certain logical relations, or oppositions, that hold between the four quantifiers all, no, not all, and some. Aristotle and traditional logicians, as well as most linguists today, took all to have existential import, so that “All As are B” entails that there are As, whereas modern logic drops this assumption. Replacing Aristotle’s account of all with the modern one (and similarly for not all) results in the modern version of the square, and there has been much recent debate about which of these two squares is the ‘right’ one. My main point in the present paper is that this question is not, or should not primarily be, about existential import, but rather about patterns of negation. I argue that the modern square, but not the traditional one, presents a general pattern of negation one finds in natural language. To see this clearly, one needs to apply the square not just to the four Aristotelian quantifiers, but to other generalized quantifiers of that type. Any such quantifier spans a modern square, which exhibits that pattern of negation but, very often, not the oppositions found in the traditional square. I provide some technical results and tools, and illustrate with several examples of squares based on quantifiers interpreting various English determiners. The final example introduces a second pattern of negation, which occurs with certain complex quantifiers, and which also is representable in a square.