Symmetric and contrapositional quantifiers

In his pioneering work Mostowski (1957) made the very insightful suggestion that quantifiers are special higher order relations, relations between relations (where sets are unary relations). This move led eventually to the generalised quantifier theory (GQT). In particular it made it possible to study ”classical” quantifiers using familiar tools of the elementary theory of relations. This simple observation gave rise to various new results basically, but not exclusively, in the domain of natural language semantics (Barwise and Cooper 1981, van Benthem 1986, Keenan and Stavi 1986, Keenan 1993, Keenan and Westerstahl 1997, Zuber 2005). Furthermore, since arguments of relations are in this case Boolean objects, the combinatorics of possibly related arguments is much richer. Thus we can see what happens not only when we commute arguments of a relation but also when in addition we replace these arguments by their various Boolean compounds. The purpose of this paper is to explore one such possibility and study in particular a special class of quantifiers, called contrapositional that is quantifiers Q corresponding to a binary relation between sets such that Q(X)(Y ) = Q(Y ′)(X ′), where X ′ is the Boolean complement of X. We will see that contrapositional quantifiers are closely related to symmetric quantifiers, that is quantifiers corresponding to symmetric relations. The framework which will be used to study symmetric and contrapositional quantifiers is that of Boolean semantics (Keenan and Faltz 1986). One of the advantages of this approach is that it allows us to use a general definitional format which makes it easier to see Boolean structure of analysed objects and permits simple comparisons and generalisations in many cases. I will study symmetry and contraposition of quantifiers corresponding to binary relations between sets, formally type 〈1, 1〉 quantifiers, generically called here simple quantifiers. Other quantifiers, called here higher type quantifiers, which are quantifiers corresponding to binary relations over n-ary relations, n > 1, will also