Page:Mind (New Series) Volume 12.djvu/196

182 B. EUSSELL : probability of all the events that are possible in a certain con- tingency, the one which is most probable would certainly happen (p. 239). This view seems to rest upon a false theory of prob- ability, but I cannot discover precisely what theory, or whether any definite theory at all. It seems certain, however, that the most probable of a number of events is never certain unless all the others are impossible. The whole theory of probability appears to belong to a world apart, having nowhere any contact with the world of certainty ; and this is fortunate, for the logical analysis of probability, so far as I have been able to discover, is as yet wholly unaccomplished. Leaving this twofold division, let us examine the rest of Leibniz's general science. The analysis either of ideas or of truths, he says, may be infinite ; but the foundation of all truths is the same, namely that the predicate is contained in the subject (pp. 184, 208 ff.). Consequently there are no indemonstrable axioms except the law of identity or contradiction, though for the present it is necessary to accept some axioms without proof. Axioms are proved by means of definitions, but their truth rests on the law of identity, not on definitions. Definitions are not arbitrary, as Hobbes maintained, for their objects must be shown to be possible, i.e., not contradictory. The best way of proving this is to analyse a notion completely, for all simple notions are compatible inter se. Here Leibniz was faced by an insuperable difficulty, which was one great source of error in his philosophy. We saw that he believed all synthesis of simple concepts into complex ones to be of a single type, the type which is now called logical multiplica- tion. Hence he was unable to explain how simple ideas, all compatible inter se, could generate incompatible complexes (p. 432). He remarks himself (Gcrh. vii., 195) : "It is yet unknown to men what is the reason of the incompossibility of different things, or how it is that different essences can be opposed to each other, seeing that all purely positive terms seem to be compatible ". The fact is, that the notion "not-a" is formed by a synthesis of quite a different kind from logical multiplication : there is not a class of nots and a class of a's whose common part is "not-a ". Thus incompatibility is only explicable by admitting a synthesis which is not that of two predicates, such as the analytic theory of judgment requires ; and yet, until we have such negative predi- cates as "not-<z," there is no possibility of contradiction, and therefore no field for the application of the analytic criterion of truth. And when this one new form of synthesis has been ad- mitted, it becomes easy to see that there are others, of which the chief are logical addition and relative multiplication. 1 Thus a more careful consideration of negative terms and of the conditions of incompatibility would have sufficed to show Leibniz the falsity 1 Relative multiplication is the kind of synthesis which, from two rela- tions of father to son, obtains a relation of grandfather to grandson.