PREFACE TO THE SECOND EDITION. xvii 



" Eadem sunt quorum unum potest substitui alter! salva 

 veritate. Si sint A et B, et A ingrediatur aliquam pro- 

 positionem veram, et ibi in aliquo loco ipsius A pro ipso 

 substituendo B fiat nova propositio eque itidem vera, idque 

 semper succedat in quacunque tali propositione, A et B 

 dicuntur esse eadem ; et contra, si eadein sint A et B, 

 procedet substit'utio quarn dixi." 



Leibnitz, then, explicitly adopts the principle of sub- 

 stitution, but he puts it in the form of a definition, saying 

 that those things are the same which can be substituted 

 one for the other, without affecting the truth of the 

 proposition. It is only after having thus tested the same- 

 ness of things that we can turn round and say that A and 

 P>, being the same, may be substituted one for the other. 

 It would seem as if we were here in a vicious circle ; for 

 we are not allowed to substitute A for B, unless we have 

 ascertained by trial that the result is a true proposition. 

 The difficulty does not seem to be removed by Leibnitz' 

 proviso, "idque semper succedat in quacunque tali pro- 

 positione." How can we learn that because A and B may 

 be mutually substituted in some propositions, they may 

 therefore be substituted in others ; and what is the criterion 

 of likeness of propositions expressed in the word "tali" ? 

 Whether the principle of substitution is to be regarded as a 

 postulate, an axiom, or a definition, is just one of those fun- 

 damental questions which it seems impossible to settle in the 

 present position of philosophy, but this uncertainty will not 

 prevent our making a considerable step in logical science. 



Leibnitz proceeds to establish in the form of a theorem 

 what is usually taken as an axiom, thus (Opera, p. 95) : 

 '' Theorema I. Quae sunt eadem uni tertio, eadem sunt 

 inter se. Si A a B et B a C, erit A a C. Nam si in 

 propositione A cc B (vera ea hypothesi) substituitur G in 

 locum B (quod facere licet per Def. I. quia B oc C ex 

 hypothesi) fiet A oc (7. Q. E. Dem." Thus Leibnite 

 precisely anticipates the mode of treating inference with 

 two simple identities described at p. 51 of this work. 



I 



