This paper undertakes a re-examination of Sir William Hamilton’s doctrine of the quantification of the predicate. Hamilton’s doctrine comprises two theses. First, the predicates of traditional syllogistic sentence-forms contain implicit existential quantifiers, so that, for example, All p is q is to be understood as All p is some q. Second, these implicit quantifiers can be meaningfully dualized to yield novel sentence-forms, such as, for example, All p is all q. Hamilton attempted to provide a deductive system for his language, along the lines of the classical syllogisms. We show, using techniques unavailable to Hamilton, that such a system does exist, though with qualifications that distinguish it from its classical counterpart.
Syllogisms Natural language and logic Complexity Proof-theory