# Probing the Quantitative-Qualitative Divide in Probabilistic Reasoning

arXiv: https://arxiv.org/abs/2307.05659 · HTML: https://mlsanity.com/paper/2307.05659/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jul 2023.

## Built on by (most cited first, 3 of 3)

- [Standards for Belief Representations in LLMs](https://mlsanity.com/paper/2405.21030/) (2024)
- [On Probabilistic and Causal Reasoning with Summation Operators](https://mlsanity.com/paper/2405.03069/) (2024)
- [Probabilistic and Causal Satisfiability: Constraining the Model](https://mlsanity.com/paper/2504.19944/) (2025)

## Abstract (arXiv)

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 mathsfNP, while multiplicative systems are robustly complete for existsℝ. 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.