Bertrand Russell, A Critical Exposition of the Philosophy of Leibniz (1900).


      Shortly after the publication of the first edition of this book, its principal thesis -- namely, that Leibniz's philosophy was almost entirely derived from his logic -- received overwhelming confirmation from the work of Louis Couturat. His "La Logique de Leibniz" (1901), supported by his collection of MSS. overlooked by previous editors, entitled "Opuscules et Fragments inedits de Leibniz'' (1903), showed that the "Discours de Metaphysique" and the letters to Arnauld, upon which I had to rely almost exelusively for my interpretation, were mere samples of innumerable writings expressing the same point of view, which had remained buried among the mass of documents at Hanover for over two centuries. No candid reader of the "Opuscules" can doubt that Leibniz's metaphysics was derived by him from the subject-predicate logic. This appears, for example, from the paper "Primae Veritates" (Opuscules, pp. 518-523), where all the main doctrines of the "Monadology'' are deduced, with terse logical rigor, from the premiss:

"Semper igitur praedicatum seu consequens inest subjecto seu antecedenti, et in hoc ipso consistit natura veritatis in universum . . . . . Hoc autem est in omni veritate affirmativa universali aut singulari, necessaria aut contingente" (1b. p. 518). "Always therefore the predicate or consequent inheres in the subject or antecedent, and in this fact consists the nature of truth in general . . . But this is true in every affirmative truth, universal or singular, necessary or contingent.''

      Wherever my interpretation of Leibniz differed from that of previous commentators, Couturat's work afforded conclusive confirmation, and showed that the few previously published texts upon which I had relied had all the importance that I had attributed to them. But Couturat carried inorthodoxy further than I had done, and where his interpretation differed from mine, he was able to cite passages which seemed conclusive. The Principle of Sufficient Reason, he maintains, asserts simply that every true proposition is analytic, and is the exact converse of the Law of Contradiction, which asserts that every analytic proposition is true. The Identity of Indiscernibles, also, is expressly deduced by Leibniz from the analytic character of all true propositions; for after asserting this he proceeds:

"Sequitur etiam hinc non dari posse duas res singulares solo numero differentes: utique enim oportet rationem reddi posse cur sint diversae, quae ex aliqua in ipsis differentia petenda est" (Ib. p. 519). "It even follows from this that there cannot be two singular things which differ oonly numerically; for it must be possible to give a reason why they are diverse, which is to be sought in some difference between them."

      Leibniz's logic was, therefore, at least in his most lucid moments, simpler than that with which I have credited him. In particular, the Law of Sufficient Reason is interpreted in #14 of the present work in a manner which is quite different from Couturat's, not compatible with the texts upon which he relies, and less consistent with Leibniz's logic. At the same time, there are abundant texts to support the view which I took. This is an instance of Leibniz's general duality: he had a good philosophy which (after Arnauld's criticisms) he kept to himself and a bad philosophy which he published with a view to fame and money. In this he showed his usual acumen: his bad philosophy was admired for its bad qualities, and his good philosophy, which was known only to the editors of his MSS., was regarded by them as worthless, and left unpublished. For example, he composed, in 1686, a work on mathematical logic, and wrote on the margin "hic egregie progressus sum"; but no editor before Couturat accepted his estimate of his own work. In another MS., he sent out Euler's diagrams for all the moods of the syllogism; in yet another, he gave De Morgan's formula: A or B = not (not A and not B). 'These are merely samples of results or methods, known by the names of subsequent discoverers, which should have been known as Leibniz's, but for the bad taste of his editors and his own preference for cheap popularity. I think it probable that as he grew older he forgot the good philosophy which he had kept to himself, and remembered only the vulgarized version by which he won the admiration of Princes and (even more) of Princesses. If Couturat's work could have been published in his lifetime, he would, I feel sure, have hated it, not as being inaccurate, but as being indiscreetly accurate.

      Buried among his fragments on logic, there is a curious definition of existence.

"Definiri potest Existens, quod cum pluribus compatibile est quam quodlibet aliud incompatibile cum ipso" (Opuscules, p. 360). "The existent may be defined as that which is compatible with more things than is anything incompatible with itself."

Again, after saying "The existent is what has being or possibility, and something more," he proceeds:

"Ajo igitur Existens esse Ens quod cum plurimis compatibile est, seu Ens maxime possibile, itaque omnia coexistentia aeque possibilia sunt" (Ib. p. 376). "I say therefore that the existent is the being which is compatible with most things, or the most possible being, so that all coexistent things are equally possible."

Strange consequences follow if Leibniz intended this to be, in the strict sense, a definition of "existence." For, if it was so intended, there was no act of Creation: the relations of essences are among eternal truths, and it is a problem in pure logic to construct that world which contains the greatest number of coexisting essences. This world, it would follow, exists by definition, without the need of any Divine Decree; moreover, it is a part of God, since essences exist in God's mind. Here, as elsewhere, Leibniz fell into Spinozism whenever he allowed himself to be logical; in his published works, accordingly, he took care to be illogical.

      Mathematics, and especially the infinitesimal calculus, greatly influenced Leibniz's philosophy. The truths which we call contingent are, according to him, those in which the subject is infinitely complex, and only an infinitely prolonged analysis can show that the predicate is contained in the subject. Every substance is infinitely complex, for it has relations to every other, and there are no purely extrinsic denominations, so that every relation involves a predicate of each of the related terms. It follows that every singular substance involves the whole universe in its perfect notion" (Opuscules, p. 521). For us, accordingly, propositions about particular substances are only empirically discoverable; but to God, who can grasp the infinite, they are as analytic as the proposition "equilateral triangles are triangles." We can, however, approximate indefinitely to the perfect knowledge of individual substances. Thus, speaking of St. Peter's denial of Christ, Leibniz says: "The matter can be demonstrated from the notion of Peter, but the notion of Peter is complete, and so involves infinites, and so the matter can never be brought to perfect demonstration, but this can be approached more and more nearly, so that the difference shall be less than any given difference." Couturat comments on "this quite mathematical locution, borrowed from the infinitesimal method" (La Logique de Leibniz, p. 213n). Leibniz is fond of the analogy of irrational numbers. A very similar question has arisen in the most modern philosophy of mathematics, that of the finitists. For example, does pi at any point have three successive 7's in its decimal expression? So far as people have gone in the calculation, it has not. It may be proved hereafter that there are three successive 7's at a later point, but it cannot be proved that there are not, since this would require the completion of an infinite calculation. Leibniz's God could complete the sum, and would therefore know the answer, but we can never know it if it is negative. Propositions about what exists in Leibniz's philosophy, could be knoun a priori if we could complete an infinite analysis, but, since we cannot, we can only know them empirically, though God can deduce them from logic.

      At the time when I wrote "The Philosophy of Leibniz," I knew little of mathematical logic, or of George Cantor's theory of infinite numbers. I should not now say as is said in the following pages, that the propositions of pure mathematics are "synthetic." The important distinction is between proposition deducible from logic and propositions not so deducible; the former may advantageously be defined as "analytic," the latter as "synthetic." Leibniz held that, for God, all propositions are analytic; modern logicians, for the most part regard pure mathematics as analytic, but consider all knowledge of matters of faet to be synthetic.

      Again, I should not now say: "It is evident that not every monad can have an organic body, if this consists of other subordinate monads" (p. 150). This assumes that the number of monads must be finite, whereas Leibniz supposed the number to be infinite. "In every particle of the universe," he says, "a world of infinite creatures is contained" (Opuscules, p. 522). Thus it is possible for every monad to have a body composed of subordinate monads, just as every fraction is greater than an infinite number of other fractions.

      It is easy to construct an arithmetical scheme representing Leibniz's view of the world. Let us suppose that to each monad is assigned some rational proper fraction m, and that the state of each monad at time t is represented by m f(t), where f(t) is the same for all the monads. There is then a correspondence, at any given time, between any two monads and also between any one monad and the universe; we may thus say that every monad mirrors the world and also mirrors every other monad. We might suppose the body of the monad whose number is m to be those monads whose numbers are powers of m. The number m may be taken as measuring the intelligence of the monad; since m is a proper fraction, its powers are less than m, and therefore a monad's body consists of inferior monads. Such a scheme is of course merely illustrative, but serves to show that Leibniz's universe is logically possible. His reasons for supposing it actual, however, since they depend upon the subject-predicate logic, are not such as a modern logician can accept. Moreover, as is argued in the following pages, the subject-predicate logic, taken strictly, as Leibniz took it, is incompatible with the plurality of substances.

      Except in regard to the points mentioned above, rny views as to the philosophy of Leibniz are still those which I held in 1900. His importance as a philosopher has become more evident than it was at that date, owing to the growth of mathematical logic and the simultaneous discovery of his MSS. on that and kindred subjects. His philosophy of the empirical world is now only a historical curiosity, but in the realm of logic and the principles of mathematics many of his dreams have been realized, and have been shown at last to be more than the fantastic imaginings that they seemed to all his successors until the present time.

September, 1937