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. Our multidimensional sentences form a very rich class. Each of our multidimensional sentences describes a set of possible truth values for a collection of formulas of the real-valued logic, including which combinations of truth values are possible. Our completeness result is strong, in the sense that it allows us to derive exactly what information can be inferred about the combinations of truth values of a collection of formulas given information about the combinations of truth values of a finite number of other collections of formulas. 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. The generality of this work, compared to many previous works on special cases, may provide insights for both existing and new real-valued logics whose inference properties have never been characterized. This work may also provide insights into the reasoning capabilities of deep learning models.
Keywords: fuzzy logic; real-valued logic; strongly complete axiomatization.