The paper presents an infinite hierarchy PR m [ m = 1, 2, . . . ] of sound and complete axiomatic systems for modal logic with graded probabilistic modalities , which are to reflect what I have elsewhere called the Bolding-Ekelöf degrees of evidential strength as applied to the establishment of matters of fact in law-courts. Our present approach is seen to differ from earlier work by the author in that it treats the logic of these graded modalities not (...) only from a semantical or model-theoretic viewpoint but from a prooftheoretical and axiomatic stance as well. A paramount feature of the approach is its use of so-called systematic frame constants as labels of diverse grades of probability. Apart from this novel feature our approach can be seen to go back to pioneering work by Lou Goble in 1970. (shrink)