Foundations of reasoning with uncertainty via real-valued logics.

Interest in logics with some notion of real-valued truths has existed since at least Boole and has been increasing in AI due to the emergence of neuro-symbolic approaches, though often their logical inference capabilities are characterized only qualitatively. We provide foundations for establishing the correctness and power of such systems. We introduce a rich class of multidimensional sentences, with a sound and complete axiomatization that can be parameterized to cover many real-valued logics, including all the common fuzzy logics, and extend these to weighted versions, and to the case where the truth values are probabilities.