Probability propagation rules for Aristotelian syllogisms

TL;DR

This paper develops a probabilistic semantics for Aristotelian syllogisms using coherence-based probability assessments, extending classical logic to include defaults, negations, and generalized quantifiers, and analyzing validity through probability propagation rules.

Contribution

It introduces a coherence-based probabilistic framework for syllogisms, generalizes de Finetti's theorem for conditional probabilities, and connects classical syllogistic logic with nonmonotonic reasoning.

Findings

All traditionally valid syllogisms are valid in the probabilistic semantics.

Probability propagation rules can analyze syllogisms with generalized quantifiers.

Reductio by conversion does not work, but reductio ad impossibile can be applied.

Abstract

We present a coherence-based probability semantics and probability propagation rules for (categorical) Aristotelian syllogisms. For framing the Aristotelian syllogisms as probabilistic inferences, we interpret basic syllogistic sentence types A, E, I, O by suitable precise and imprecise conditional probability assessments. Then, we define validity of probabilistic inferences and probabilistic notions of the existential import which is required, for the validity of the syllogisms. Based on a generalization of de Finetti's fundamental theorem to conditional probability, we investigate the coherent probability propagation rules of argument forms of the syllogistic Figures I, II, and III, respectively. These results allow to show, for all three figures, that each traditionally valid syllogism is also valid in our coherence-based probability semantics. Moreover, we interpret the basic syllogistic sentence types by suitable defaults and negated defaults. Thereby, we build a bridge from our probability semantics of Aristotelian syllogisms to nonmonotonic reasoning. Then we show that reductio by conversion does not work while reductio ad impossibile can be applied in our approach. Finally, we show how the proposed probability propagation rules can be used to analyze syllogisms involving generalized quantifiers (like Most).