Hamilton’s cumular conception of quantifying particles: an exercise in third-order logic

Abstract

Sir William Hamilton is remembered for his proposal to extend the four traditional categoricals to eight by quantifying predicate as well as subject terms. He intended the quantifying particles to be understood in a ‘collective’ or ‘cumular’ manner rather than in a ‘distributive’ or ‘exemplar’ one, but commentators from De Morgan onwards have worked primarily from the latter perspective, comforted in the 20th century by the fact that it translates readily into the language of first-order logic with identity. Formal representation of the cumular approach needs more sophisticated resources, and the paper shows how it may be carried out using selection functions in the language of third-order logic. It also reviews a number of variants, some equivalent and others not so, as well as their reductions to second-order logic, and situates historical sources, both before and after Hamilton, with respect to the web of formal constructions.