In this paper we first recall some results for conditional events, compound conditionals, conditional random quantities, p-consistency, and p-entailment. We discuss the equivalence between conditional bets and bets on conditionals, and review de Finetti's trivalent analysis of conditionals. But we go beyond de Finetti's early trivalent logical analysis and his later ideas, aiming to take his proposals to a higher level. We examine two recent articles that explore trivalent logics for conditionals and their definitions of logical validity and compare them with the approach to compound conditionals introduced by Gilio and Sanfilippo within the framework of conditional random quantities. As we use the notion of p-entailment, the full deduction theorem does not hold. We prove a Probabilistic Weak Deduction Theorem for conditional events. After that we study some variants of it, with further results, and we present several examples. Moreover, we illustrate how to derive new inference rules related to selected Aristotelian syllogisms. We focus on iterated conditionals and the invalidity of the Import-Export principle in the light of our Probabilistic Weak Deduction Theorem. We use the inference from a disjunction, A or B, to the conditional, if not-A then B, as an example to show the invalidity of this principle. We introduce a General Import-Export principle by examining examples and counterexamples. In particular, when considering the inference rules of System P, we find that a General Import-Export principle is satisfied, even if the assumptions of the Probabilistic Weak Deduction Theorem do not hold. We also deepen further aspects related to p-entailment and p-consistency. Finally, we briefly discuss some related work relevant to AI.

中文翻译：

---

关于三价逻辑、概率弱推导定理和一般进出口原则


在本文中，我们首先回顾了条件事件、复合条件、条件随机量、p 一致性和 p 蕴涵的一些结果。我们讨论了条件投注和条件投注之间的等价性，并回顾了 de Finetti 对条件的三价分析。但是，我们超越了 de Finetti 早期的三价逻辑分析和他后来的想法，旨在将他的建议提升到一个更高的水平。我们研究了最近的两篇文章，这些文章探讨了条件的三价逻辑及其对逻辑有效性的定义，并将它们与 Gilio 和 Sanfilippo 在条件随机量框架内引入的复合条件方法进行了比较。当我们使用 p 蕴涵的概念时，完全演绎定理不成立。我们证明了条件事件的概率弱演绎定理。之后，我们研究了它的一些变体，并获得了进一步的结果，并提供了几个例子。此外，我们说明了如何推导出与选定的亚里士多德三段论相关的新推理规则。我们根据我们的概率弱演绎定理，专注于迭代条件和进出口原则的无效性。我们用从析取 A 或 B 到条件（如果不是 A 则 B）的推论来说明这个原则的无效性。我们通过研究示例和反例来介绍一般导入-导出原则。特别是，在考虑系统 P 的推理规则时，我们发现满足一般进出口原则，即使概率弱演绎定理的假设不成立。我们还进一步深化了与 p 蕴涵和 p 一致性相关的方面。最后，我们简要讨论了一些与 AI 相关的工作。