Two principles in many-valued logic

Abstract: Classically, two propositions are logically equivalent precisely when they are true under the same logical valuations. Also, two logical valuations are distinct if, and only if, there is a formula that is true according to one valuation, and false according to the other. By a real-valued logic we mean a many-valued logic in the sense of Petr H\'ajek that is complete with respect to a subalgebra of truth values of a BL-algebra given by a continuous triangular norm on [0, 1]. Abstracting the two foregoing properties from classical logic leads us to two principles that a real-valued logic may or may not satisfy. We prove that the two principles are sufficient to characterise {\L}ukasiewicz and G\"odel logic, to within extensions. We also prove that, under the additional assumption that the set of truth values be closed in the Euclidean topology of [0,1], the two principles also afford a characterisation of Product logic.