# Foundations of Reasoning with Uncertainty via Real-valued Logics

arXiv: https://arxiv.org/abs/2008.02429 · HTML: https://mlsanity.com/paper/2008.02429/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Aug 2020.

## Built on by (most cited first, 2 of 2)

- [Indoor and Outdoor 3D Scene Graph Generation via Language-Enabled Spatial Ontologies](https://mlsanity.com/paper/2312.11713/) (2023)
- [A First Polynomial Non-Clausal Class in Many-Valued Logic](https://mlsanity.com/paper/2110.12901/) (2021)

## Abstract (arXiv)

Real-valued logics underlie an increasing number of neuro-symbolic approaches, though typically their logical inference capabilities are characterized only qualitatively. We provide foundations for establishing the correctness and power of such systems. We give a sound and strongly complete axiomatization that can be parametrized to cover essentially every real-valued logic, including all the common fuzzy logics. Our class of sentences are very rich, and each describes a set of possible real values for a collection of formulas of the real-valued logic, including which combinations of real values are possible. Strong completeness allows us to derive exactly what information can be inferred about the combinations of real values of a collection of formulas given information about the combinations of real values of several other collections of formulas. We then extend the axiomatization to deal with weighted subformulas. Finally, we give a decision procedure based on linear programming for deciding, for certain real-valued logics and under certain natural assumptions, whether a set of our sentences logically implies another of our sentences.