In attempting to formulate the logical structure of quantum physics it is necessary to refer frequently to probabilities; even the basic axioms of this system of thought which progressively pervades the various branches of physical science involve an explicit appeal to probability concepts. Now it has been customary in mathematical and physical investigations to examine very critically and with utmost care all consequences of a set of postulates when such a set is given, but the postulates themselves, in particular the questions with regard to their consistency and their relation to other fundamental suppositions, often receive unduly little attention. The mystifying power of the term axiom, applied to any set of assumptions, whether basic or not, appears to insure its immunity against inquisitive attacks from the quarters of undesirable sceptics. It is only too natural that such a situation should exist, for one science always starts where another leaves off, and axioms mostly have their place in the boundary region between two, in a domain, hence, over which neither can claim complete jurisdiction and in which neither is perfectly competent to operate. Probability, upon which quantum mechanics is axiomatically founded, lies precisely at the triple junction of mathematics, physics, and philosophy, and partakes of the uncertainty and diffuseness that is likely to affect the concepts arising in such neglected, yet much disputed, boundary fields.