The Hamiltonian Syllogistic

TL;DR

This paper re-examines Hamilton's doctrine of predicate quantification, demonstrating that a deductive system for his extended syllogistic language can be constructed with modern techniques, revealing new logical forms.

Contribution

It introduces a formal deductive system for Hamilton's extended syllogistic language, including dualized quantifiers, with modern proof techniques.

Findings

A deductive system for Hamilton's language exists with certain qualifications.

The dualization of implicit existential quantifiers yields new valid sentence forms.

The system differs from classical syllogistic in meaningful ways.

Abstract

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 are q" is to be understood as "All p are some q". Second, these implicit quantifiers can be meaningfully dualized to yield novel sentence-forms, such as, for example, "All p are 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.