Aristotelian Diagrams for the Proportional Quantifier ‘Most’

Abstract

In this paper, we study the interaction between the square of opposition for the Aristotelian quantifiers (‘all’, ‘some’, ‘no’, and ‘not all’) and the square of opposition generated by the proportional quantifier ‘most’ (in its standard generalized quantifier theory reading of ‘more than half’). In a first step, we provide an analysis in terms of bitstring semantics for the two squares independently. The classical square for ‘most’ involves a tripartition of logical space, whereas the degenerate square for ‘all’ in first-order logic (FOL) involves a quadripartition, due to FOL’s lack of existential import. In a second move, we combine these two squares into an octagon of opposition, which was hitherto unattested in logical geometry, while the meet of the original tri- and quadripartitions yields a hexapartition for this octagon. In a final step, we switch from FOL to a logical system, which does assume existential import. This yields an octagon of the well known Lenzen type, and its bitstring semantics is reduced to a pentapartition.