In the first edition I inserted a section (vol. i. p. 25), on “Anticipations of the Principle of Substitution,” and I have reprinted that section unchanged in this edition (p. [21]). I remark therein that, “In such a subject as logic it is hardly possible to put forth any opinions which have not been in some degree previously entertained. The germ at least of every doctrine will be found in earlier writings, and novelty must arise chiefly in the mode of harmonising and developing ideas.” I point out, as Professor T. M. Lindsay had previously done, that Beneke had employed the name and principle of substitution, and that doctrines closely approximating to substitution were stated by the Port Royal Logicians more than 200 years ago.
I have not been at all surprised to learn, however, that other logicians have more or less distinctly stated this principle of substitution during the last two centuries. As my friend and successor at Owens College, Professor Adamson, has discovered, this principle can be traced back to no less a philosopher than Leibnitz.
The remarkable tract of Leibnitz,[2] entitled “Non inelegans Specimen Demonstrandi in Abstractis,” commences at once with a definition corresponding to the principle:—
“Eadem sunt quorum unum potest substitui alteri salva veritate. Si sint A et B, et A ingrediatur aliquam propositionem veram, et ibi in aliquo loco ipsius A pro ipso substituendo B fiat nova propositio æque itidem vera, idque semper succedat in quacunque tali propositione, A et B dicuntur esse eadem; et contra, si eadem sint A et B, procedet substitutio quam dixi.”
Leibnitz, then, explicitly adopts the principle of substitution, 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 sameness of things that we can turn round and say that A and B, 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 propositione.” 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 fundamental 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. Quæ sunt eadem uni tertio, eadem sunt inter se. Si A ∝ B et B ∝ C, erit A ∝ C. Nam si in propositione A ∝ B (vera ea hypothesi) substituitur C in locum B (quod facere licet per Def. I. quia B ∝ C ex hypothesi) fiet A ∝ C. Q. E. Dem.” Thus Leibnitz precisely anticipates the mode of treating inference with two simple identities described at p. 51 of this work.
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 ∝ B, erit A + C ∝ B + C. Nam si in propositione A + C ∝ A + C (quæ est vera per se) pro A semel substituas B (quod facere licet per Def. I. quia A ∝ B) fiet A + C ∝ 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, salvis calculi legibus, dicetur esse æquipollentiam.” 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 Quædam Logicæ,”[3] he says: “Omne A est B; id est æquivalent AB et A, seu A non B est non-ens.”
It is curious to find, too, that Leibnitz was fully acquainted with the Laws of Commutativeness and “Simplicity” (as I have called the second law) attaching to logical symbols. In the “Addenda ad Specimen Calculi Universalis” we read as follows.[4] “Transpositio literarum in eodem termino nihil mutat, ut ab coincidet cum ba, seu animal rationale et rationale animal.”
“Repetitio ejusdem literæ in eodem termino est inutilis, ut b est aa; vel bb est a; homo est animal animal, vel homo homo est animal. Sufficit enim dici a est b, seu homo est animal.”