The Hamiltonian Syllogistic
Ian Pratt-Hartmann
TL;DR
The paper analyzes Hamiltonian syllogistic forms, which dualize predicate quantification to extend the classical syllogistic. By formalizing languages $\mathcal{H}$, $\mathcal{H}^\dagger$, and $\mathcal{H}^{*\dagger}$, it distinguishes their proof-theoretic and computational properties from the classical case: there is no finite sound-and-complete rule system for $\mathcal{H}$, but a finite sound-and-refutation-complete system exists; satisfiability for the dagger languages is NP-time complete, while complete indirect systems exist for $\mathcal{H}^\dagger$ and $\mathcal{H}^{*\dagger}$. The results clarify the trade-offs between grammatical robustness of predicate-quantified forms and the feasibility of complete syllogistic reasoning, linking to related relational syllogistic results. Overall, the work maps the landscape of Hamiltonian predicate quantification in syllogistic form and provides precise, finite, and indirect-rule systems where appropriate. The findings have implications for the logical analysis of quantified predicates and the computational complexity of their syllogistic reasoning.
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.