Probing the quantitative–qualitative divide in probabilistic reasoning

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 $\mathrm{NP}$, while multiplicative systems are robustly complete for $\exists R$. 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.