 The fact that the notions of derivability and model-theoretic validity are definable in standard mathematics seems to have been a very attractive feature of them among practicing logicians. But this attractive feature of course does not justify by itself taking either notion as an adequate characterization of logical truth. On most views, with a mathematical characterization of logical truth we attempt to delineate a set of formulae possessing a number of non-mathematical properties. Which properties these are varies depending on our pretheoretic conception of, for example, the features of modality and formality. (By “pretheoretic” it's not meant “previous to any theoretical activity”; there could hardly be a “pretheoretic” conception of logical truth in this sense. In this context what's meant is “previous to the theoretical activity of mathematical characterization”.) But on any such conception there will be external, non-mathematical criteria that can be applied to evaluate the question whether a mathematical characterization is adequate. In this last section we will outline some of the basic issues and results on the question whether derivability and model-theoretic validity are adequate in this sense.