ABSTRACT
This chapter begins by reprising in more rigorous terms the role of intentional content in defining the various ordering relations. The discussion addresses a technical issue that has divided commentators who have interpreted the orderings algebraically: whether ideas are “dual” to extensions in the algebraic sense of duality. Formally, the operation of privative negation is well-defined in the formal semantics of gradable adjectives. The general picture that emerges is that the structure of ideas in the Logic is that it shares features with the structure of concepts and species in medieval logic: ideas are formed by abstraction and restriction, species are organized in a tree hierarchy, and species are sometimes defined by privative negation. The structural properties of species and their relation to privative negation are also discussed in the chapter. Key formal definitions and reconstructions, however, are provided in the Appendix for those who are interested in or who would prefer a more rigorous account.
