Page:Mind (New Series) Volume 12.djvu/202

 188 B. RUSSELL : culus presented itself to him as that species of the Calculus of Combination which is subject to the law of tautology (aa = a) as- well as to the commutative law. This and the geometrical calculus- were the two that he endeavoured to develop out of the infinity of algorithms that appeared to him possible (pp. 320-321). M. Couturat's researches into Leibniz's work on Symbolic Logic are exceedingly interesting : they show the great progress he had made, and the precise causes of his failure. He occupied himself with this subject principally at three periods, 1679, 1686, and 1690. The second of tuese dates is interesting, for M. Couturat has found a long MS. which completes the Discours de Mcta physique and shows its connection with Leibniz's logical studies. The editors, as our author remarks, are the more unpardonable in having omitted this MS., as Leibniz has written on it : " Hie egregie progressus sum " (pp. 344-345). The system of 1679 represents simple concepts by primes, and conceives their combination on the analogy of arithmetical multi- plication. At first, Leibniz thought one number would do for each concept ; but he soon found that negative terms were required, and for these he employed negative numbers. Here, however, the rules of composition could no longer be made analogous to- those of arithmetic. In order that a complex notion should be possible, it w r as necessary and sufficient that the positive and negative numbers representing it should have no common factor. He proves many theorems, notably one which he calls " prae- clarum theorema " : If a is & and c is d, then ac is bd. He also- arrives at the logical definition of cardinal numbers, recently re- vived by Frege and Schroder : thus* he says that m is one when,, if a is m and b is m, it follows that a and b are identical (p. 342). Once only he represented by multiplication what we call logical addition, and obtained the law of tautology for this case also ; but he was unable to develop this idea, because he preferred the point of view of intension (p. 343). In the system of 1686, Leibniz discovered the double interpre- tation of formulas, according as single letters stand for concepts. or propositions (p. 354). But he involved himself in hopeless difficulties owing to his determination to rescue scholastic logic at all costs. His calculus rightly refused to justify faulty conversions, or to give existential import to universal terms. He remarks : " All laughers are men, therefore some man laughs ; but the first is true even if no man langhs, while the second is not true unless- some man actually langhs " (p. 359). To avoid this difficulty, he says that all terms are to be tacitly assumed to exist (p. 360) ; nevertheless he has to admit the impossible, i.e., that there are- general terms which do not exist (p. 349). If he had had less, respect for scholastic logic, M. Couturat concludes (p. 354), the- Algebra of Logic would have been constituted some 200 years sooner. The system of 1690 adds little to its predecessors. Leibniz.