We present a logical relationship between a small number of intuitive properties for measures of belief and the axioms of probability theory. The relationship was first demonstrated several decades ago but has remained obscure. We introduce the proof and discuss its relevance to research on reasoning under uncertainty in artificial intelligence. In particular, we demonstrate that the logical relationship can facilitate the identification of differences among alternative plausible reasoning methodologies. Finally, we make use of the relationship to examine popular non-probabilistic strategies.