# Probabilistic squares and hexagons of opposition under coherence

## TL;DR

This paper develops a probabilistic framework for the square and hexagon of opposition using coherence-based probability, introducing new constructions and applications with generalized quantifiers.

## Contribution

It introduces a novel probabilistic interpretation of the square and hexagon of opposition based on coherence theory, including new constructions and applications.

## Key findings

- Probabilistic versions of the square and hexagon of opposition are constructed.
- The framework incorporates imprecise probability assessments and generalized quantifiers.
- New relations of acceptability are analyzed within coherence-based probability theory.

## Abstract

Various semantics for studying the square of opposition and the hexagon of opposition have been proposed recently. We interpret sentences by imprecise (set-valued) probability assessments on a finite sequence of conditional events. We introduce the acceptability of a sentence within coherence-based probability theory. We analyze the relations of the square and of the hexagon in terms of acceptability. Then, we show how to construct probabilistic versions of the square and of the hexagon of opposition by forming suitable tripartitions of the set of all coherent assessments. Finally, as an application, we present new versions of the square and of the hexagon involving generalized quantifiers.