xnH PREFACE TO THE SECOND EDITION. 



Even the mathematical axiom that ' equals added to 

 equals make equals,' is deduced from the principle of 

 substitution. At p. 95 of Erdmann's edition, we find : " Si 

 eidem addantur coincidentia fiunt coincidentia. Si A oc B, 

 erit A + oc B + 0. Nam si in propositione A + oc A 

 -f C (quae est vera per se) pro A semel substituas P (quod 

 facere licet per Def. I. quia A cc B) fiet A + C a B + C 

 Q. E. Dem." This is unquestionably the mode of deducing 

 the several axioms of mathematical reasoning from the 

 higher axiom of substitution, which is explained in the 

 section on mathematical inference (p. 162) in this work, 

 and which had been previously stated in my Substitution 

 of Similars, p. 16. 



There are one or two other brief tracts in which Leibnitz 

 anticipates the modern views of logic. Thus in the 

 eighteenth tract in Erdmann's edition (p. 92), called 

 " Fundamenta Calculi Ratiocinatoris, he says : " Inter ea 

 quorum unum alteri substitui potest, sal vis calculi legibus, 

 dicetur esse sequipollentiam." There is evidence, also, that 

 he had arrived at the quantification of the predicate, and 

 that he fully understood the reduction of the universal 

 affirmative proposition to the form of an equation, which is 

 the key to an improved view of logic. Thus, in the tract 

 entitled "Difficultates Qusedam Logicse," 1 he says : "Omne^l 

 est B ; id est eequivalent AB et A, seu A non B est non-ens." 



It is curious to find, too, that Leibnitz was fully ac- 

 quainted with the Laws of Commutativeness and " Simpli- 

 city" (as I have called the second law) attaching to logical 

 symbols. In the " Addenda ad Specimen Calculi Univer- 

 salis" we read as follows. 2 " Transpositio literarum in 

 eodem termiuo nihil mutat, ut db coincidet cum ba, seu 

 animal rationale et rationale animal." 



"Repetitio ejusdem liter in eodem termino est inutilis, 

 ut b est aa; vel bb est a; homo est animal animal, vel 

 homo homo est animal Sufficit enira dici a est b, seu 

 homo est animal." 



1 Erdmann, p. 102. a Ibid p 9 g 



