This paper explores the space of (propositional) probabilistic logical languages, ranging from a purely `qualitative' comparative language to a highly `quantitative' language involving arbitrary polynomials over probability terms. While talk of qualitative vs. quantitative may be suggestive, we identify a robust and meaningful boundary in the space by distinguishing systems that encode (at most) additive reasoning from those that encode additive and multiplicative reasoning. The latter includes not only languages with explicit multiplication but also languages expressing notions of dependence and conditionality. We show that the distinction tracks a divide in computational complexity: additive systems remain complete for $\mathsf{NP}$, while multiplicative systems are robustly complete for $\exists\mathbb{R}$. We also address axiomatic questions, offering several new completeness results as well as a proof of non-finite-axiomatizability for comparative probability. Repercussions of our results for conceptual and empirical questions are addressed, and open problems are discussed.

翻译：本文探索（命题）概率逻辑语言的空间，范围涵盖从纯粹的‘定性’比较语言到涉及概率项上任意多项式的高度‘定量’语言。尽管定性与定量的讨论可能具有启发性，但我们通过区分（至多）编码加法推理的系统与编码加法和乘法推理的系统，在此空间中识别出稳健且有意义的分界线。后者不仅包括显式包含乘法的语言，还包括表达依赖与条件性概念的语言。我们表明这一区分对应计算复杂性上的分界：加法系统仍保持对$\mathsf{NP}$的完全性，而乘法系统稳健地保持对$\exists\mathbb{R}$的完全性。我们还探讨了公理化问题，提出了若干新的完全性结果，并证明了比较概率的非有限公理化性。本文讨论了我们结果对概念与经验问题的影响，并提出了未解决的开放性难题。