This paper investigates logical consequence defined in terms of probability distributions, for a classical propositional language using a standard notion of probability. We examine three distinct probabilistic consequence notions, which we call material consequence, preservation consequence, and symmetric consequence. While material consequence is fully classical for any threshold, preservation consequence and symmetric consequence are subclassical, with only symmetric consequence gradually approaching classical logic at the limit threshold equal to 1. Our results extend earlier results obtained by J. Paris in a SET-FMLA setting to the SET-SET setting, and consider open thresholds beside closed ones. In the SET-SET setting, in particular, they reveal that probability 1 preservation does not yield classical logic, but supervaluationism, and conversely positive probability preservation yields subvaluationism.

翻译：本文针对经典命题语言，基于标准概率概念，研究以概率分布定义的逻辑后承关系。我们考察了三种不同的概率化后承概念，分别称为实质后承、保真后承与对称后承。实质后承在任何阈值下均完全保持经典性，而保真后承与对称后承则属于次经典后承；仅当阈值极限趋近于1时，对称后承才逐渐逼近经典逻辑。我们的研究将J. Paris在SET-FMLA框架下取得的早期结果推广至SET-SET框架，并同时考察开区间阈值与闭区间阈值。特别地，在SET-SET框架中，研究结果表明：概率1保真并不产生经典逻辑，而是产生超赋值主义；反之，正概率保真则产生次赋值主义。