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496 HUGH MACCOLL : statement a) ; then, if a is a factor of u, IB must also be a- factor of u". In spite however of the mathematical anal- ogies obtained by the above interpretation of indices, I found it generally more convenient to employ the subscript form u a as a synonym for u : a, and to reserve indices for other uses. What chiefly led me to this decision was the discovery that in dealing with implications of the higher degrees (i,e. implications of implications) a calculus of two dimensions (unity and zero) is too limited, and that for such cases we must adopt a three-divisional classification of our statements. We have often to consider not merely whether a statement is true or false, but whether it is a certainty, like '2 + 3 = 5; an impossibility, like 2 + 3 = 8; or a variable (neither always true nor always false), like x = 4. To illustrate the meaning of a variable statement, we may suppose x in the last statement to be taken at random out of three possible and equally probable values 2, 4, 0. If this experiment be repeated often enough, the statement (x = 4) will be sometimes true and sometimes false ; its chance of being true will, in fact, be one-third* The formula (a : 0) : (u a : u p ) is another example of a certainty ; for it holds good whether its elementary constituents (u, a, 0) be certainties, impossi- bilities, or variables separately or conjointly. But this- does not apply to the converse implication (u a : Up) : (a : /3), which fails for some values of its constituents, as, for ex- ample, when u is a certainty, a a variable, and /8 an impossibility. This necessity for a 1 three-divisional classification of statements naturally suggested the adoption of some corre- sponding modification in notation ; so I chose the symbol e (as in my fourth paper in the Proceedings of the Mathematical Society) to replace unity as the symbol of certainty; rj (instead of zero) as the symbol for an impossibility ; and as a suitable symbol to denote a statement which is neither a certainty nor an impossibility, whose chance of being true is neither unity nor zero, and which, therefore, may fitly be called a variable. For distinguishing these three classes of statements the notation of indices is most con- venient. The three equational symbols (a = e), (j3 = 77), and 1 For other divisions and a logic of 3" dimensions, see note at the end of this paper.