Imprecise probability assessments and the Square of Opposition

There is a long history of investigations on the square of opposition spanning over two millenia. A square of opposition represents logical relations among basic sentence types in a diagrammatic way. The basic sentence types, traditionally denoted by A (universal affirmative: ''Every S is P''), E (universal negative: ''No S is P''), I (particular affirmative: ''Some S are P''), and O (particular negative: ''Some S are not P''), constitute the corners of the square, and the logical relations--contradiction, contrarity, subalternation, and sub-contrarity--form the diagonals and the sides of the square. We investigate the square of opposition from a probabilistic point of view. To manage imprecise assessments which generally are non-closed or non-convex sets, we generalize the notions of coherence for interval-valued probability assessments to the case of imprecise (in the sense of set-valued) probability assessments. We interpret a basic sentence type as a pair (F,I), where F is a sequence of conditional events and I is an imprecise probability assessment on F. Moreover, by means of the notion of g-coherence, we introduce the above mentioned logical relations among our probabilistic interpretation of the sentences. Then we show how to construct probabilistic versions of the square of opposition by forming suitable tri-partitions. Finally we present applications of the probabilistic square of oppositions to study defaults and the semantics of quantified statements.