Probabilistic interpretations of the square of opposition

Abstract

We investigate the square of opposition from a probabilistic point of view. Probability allows for dealing with exceptions and uncertainty. We will interpret the corners of the square by means of (precise or imprecise) conditional probability assessments. They will be defined within the framework of coherence, which originally goes back to de Finetti. In this framework probabilities are conceived as degrees of belief, where conditional probability is defined as a primitive concept. Coherence allows for dealing with partial and imprecise assessments. Moreover, the coherence approach is especially suitable for dealing with zero antecedent probabilities (i.e., here conditioning events may have probability zero): This is relevant for studying different probabilistic interpretations of the existential import. In this talk, we will discuss probabilistic notions of the existential import and present probabilistic interpretations of universally affirmative and negative as well as particular affirmative and negative propositions. After choosing appropriate probabilistic constraints for defining the four basic types of propositions and the existential import, we will present a probabilistic version of the traditional square of opposition. We will discuss in what sense the traditional relations—contradictories, contraries, sub-contraries, and sub-alternations— are also contained in the probabilistic square of opposition. Moreover, we will generalize our probabilistic interpretation of the basic syllogistic concepts to construct probabilistic versions of selected syllogisms. We will also relate them to inference rules in nonmonotonic reasoning. Finally, we will discuss how probabilistic syllogisms could serve as a rationality framework for human reasoning about quantifiers within the so-called “new psychology of reasoning”.