Probabilistic inference and syllogisms

Traditionally, syllogisms are arguments with two premises and one conclusion which are constructed by propositions of the form “All S are P ” and “At least one S is P ” and their respective negated versions. We will discuss probabilistic notions of the existential import and the basic sentences type. We will develop an intuitively plausible version of the syllogisms that is able to deal with uncertainty, exceptions and nonmonotonicity. We will develop a new semantics for categorical syllogisms that is based on subjective probability. Specifically, we propose de Finetti’s principle of coherence and its generalization to lower and upper conditional probabilities as the fundamental corner stones for the new semantics. Coherence allows for dealing with partial and imprecise assessments. Moreover, it is especially suitable for handling zero antecedent probabilities (i.e., here conditioning events may have probability zero): This is relevant for studying the probabilistic interpretation of the existential import. Then, we will generalize our probabilistic interpretation of the basic syllogistic concepts to construct probabilistic versions of selected syllogisms. Finally, we will relate them to inference rules in nonmonotonic reasoning.