Modal Categorical Inferences in Quarc

We investigate basic forms of inference involving modal notions and quantifiers, called modal categorical inferences. We do so by extending Quarc, a novel logic that assigns a primary role to quantified phrases, with modalities from the hexagon of opposition. We show that there are two possible readings of de dicto modalities (called symmetric and asymmetric, respectively), as opposed to the unique reading of de re modalities. We focus on the asymmetric reading of de dicto modalities and explore the logical relations that obtain between them. These are proven in a natural deduction system, accompanied by an appropriate syntax and semantics, and graphically represented via a dodecagon of opposition. Moreover, we show that the asymmetric reading, in contrast to the symmetric one, preserves all properties of the hexagon for basic modal notions. Thus, it provides a particularly attractive basis on which to further investigate quantified modal reasoning.