304 J. N. KEYNES'S STUDIES IN FORMAL LOGIC. Given be is DE or D/ or hi, C is aB or DEFG or BFH, ~Bcd is eK or hi, Kef is d, i is BC or Cd or C/ or H, ABCDEFG is H or I, DEFGHisB ABA; is / or h, ADFI& is H, ADEFH is B or C or G or K ; show that A is K. I have no space to do more than briefly indicate some of the criticisms suggested by these methods. In the first place are they actually more compendious and effective than those which we already possess? It is hardly possible to answer this inquiry summarily, because so much depends upon the extent to which, in any given example, we make appeal to such intermediate formulas as we may have already established and can thus take for granted. But, speaking generally, it seems to me that Mr. Keynes's plan is at least as effective as any other when we are dealing with propositions couched originally in the predicative form P is Q, but I rather prefer methods of the Boolian kind where our data fall into the equational type P = Q. I think also, but may be liable to prejudice here, that the speculative questions which give these advanced logical processes their main value e.g., the nature of limiting cases, the interpretation of symbolic forms are more instructively set before us when we make appeal to some extent to mathematical or other symbols. But the further question arises whether these methods can fairly claim to belong to the Common Logic at all. It must be frankly admitted that no symbols are introduced beyond those universally employed in the older treatises, with the slight exception of the abbreviation x for not-X. On the other hand the spirit of the methods is throughout of the mathematical type in regard to the generalisations employed; and some of the processes as, for instance, the combination of propositions with disjunctive predi- cates are to all intents and purposes translations into common language of the ' multiplication ' of Boole and others. I feel tolerably sure that Mr. Keynes himself would not have worked out his scheme unless he had been a thorough adept in the more symbolic methods, and I rather doubt if there are many of the adherents of the old scheme who will be able, without his train- ing, adequately to appreciate these very ingenious modifications of it. J. VENN.
