A Supraclassical Probabilistic Entailment Relation

Abstract

The paper presents an original supraclassical nontrivial plausible entailment relation $\vapprox$ that employs Kolmogorov's probability theory. Its crucial feature is the primitiveness of a conditional probability, which one calculates with the help of the method of truth tables for classical propositional logic. I study the properties of the entailment relation in question. In particular, I show that while being supraclassical, i.e., all classical entailments and valid formulas are $\vapprox$-valid, but not vice versa, it is not trivial and enjoys the same form of inconsistency as classical entailment ⊧ does. I specify the place of the proposed probability entailment relation in certain classifications of nonclassical entailment relations. In particular, I use Douven's analysis of some probabilistic entailment relations that contains dozens of properties that are crucial for any probabilistic entailment relation, as well as Hlobil's choosing your nonmonotonic logic: shopper’s guide, due to the fact that $\vapprox$ is not monotonic, and Cobreros, Egré, Ripley, van Rooij's entailment relations for tolerant reasoning. At last, I perform a comparative analysis of classical, the proposed, and some other entailment relations closely related to the latter: those introduced by Bocharov, Markin, Voishvillo, Degtyarev, Ivlev, where the last two entailment relations are based on the so-called principle of reverse deduction, which is an intuitively acceptable way to connect classical and probabilistic entailment relations.