¹ A critical exposition of the philosophy of Leibniz, Cambridge: at the University Press, 1900. Page references are to this book, which is based on lectures delivered in the spring of 1899. In the introduction to the second edition (1938) of Principles of Mathematics (1903) Russell states that most of it was written in 1900. The striking differences between Leibniz and the Principles make it possible to date certain developments of Russell's thought rather precisely.

² The monadology, etc., with introduction and notes by Robert Latta, Oxford, Clarendon Press, 1898.

³ Lectures on the philosophy of Leibniz, Oxford, Clarendon Press, 1949.

⁴ See also “Intentionality,” in Semantica, p. 177–216 (Archivio di Filosofia, 1955).

⁵ Something is in this sense a part of something else if and only if the two exemplify a descriptive (not logical) relation of a certain logical structure. If that is not seen, one may well wonder how anything can exemplify a character (e.g., extension) that is not exemplified by any of its parts (e.g. points). This, it would seem, is the real monster that lurks, for Leibniz, in the labyrinth of the continuum. Accordingly, he finds no difficulty in a continuum of attributes (e.g., hues); for he does not think of them as parts of a compound (e.g., color). Everybody agrees that biographically the continuity issue was the seed from which Leibniz's system grew. No doubt this is so; yet I shall, because I think I safely can, ignore the issue completely. If I am right in that then something is gained for the understanding of the ideas themselves, both those considered and those neglected, as well as for the structural understanding of Leibniz's thought.

⁶ A broader meaning of ‘ontology’ may be obtained by omitting ‘descriptive’ from this sentence. Since this meaning is neither interesting nor very important historically, I shall ignore it and, for brevity's sake, use ‘undefined’ instead of ‘undefined descriptive’. These notions are really syntactical, as is another distinction I shall presently introduce. But the ideal language is an interpreted syntactical schema; that is why I prefer to call them linguistic rather than syntactical. For the notion of an ideal language, its commonsensicality, and the distinction, within it, between descriptive and logical signs and expressions, see The Metaphysics of Logical Positivism, Longmans, Green & Co., 1954 (hereafter referred to as MLP).

⁷ It is also an ingredient of the notion of concreteness. Let me say, then, that ‘concrete’ and ‘abstract’ are banned from this essay. For I have found not only that they are a pair of troublemakers but also that in philosophy I can do nicely without them. In the psychology of thought they are of some use.

⁸ To say the same thing in the language of Principia Mathematica, which I shall employ in the next section, terms (and variables) are of different types; particulars are the undefined descriptive constants of type O. The possibility of an ideal language without this (syntactical) distinction among terms has been considered only recently. See “Particularity and the new nominalism,” Methodos, 6, 1954.

⁹ This is why it is worth while to distinguish two meanings of the pair ‘dependent-independent’. By collapsing them one commits one's self, perhaps unnecessarily and certainly prematurely, to calling a substance the cause of its attributes. As will be seen in the next section, this use may, without one's noticing it, prejudge the explication of ‘cause’. Notice also that wherever change is mentioned, so is, implicitly, time.

¹⁰ See also “On nonperceptual intuition,” Phil. Phenom. Res., 10, 1949 (reprinted in MLP).

¹¹ The definition is: ‘R ₁(f,g)’ for ‘(x)[f(x) ⊃ g(x)]’. ‘R ₁’ is logical because the definiens contains no descriptive constants. As is customary. I distinguish constants from variables, which are logical signs, by subscripts.

¹² More accurately, if (I) is considered as a definition of ‘=’, this (defined logical) sign can be shown to be an adequate transcription of the English ‘identical’. See “The identity of indiscernibles and the formalist definition of identity,” Mind, 62, 1953 (reprinted in MLP).

¹³ More accurately, it is sufficient that (E) hold for ‘φ’ and ‘Ψ’ of type 2.

¹⁴ The qualification excludes such predicates as φ₁, where ‘φ ₁(x)’ stands by definition for ‘(x = α) V (x = b)’. ‘φ₁(α)’ is of course analytic.

¹⁵ As with ‘A’ and ‘B’, I use capitals for type 2 and add superscripts to them to mark the higher types.

¹⁶ I refrain deliberately from mentioning at this point “perceptions” and ideal things (entia rationis). See fn. 29 and Section Five.

¹⁷ That the existent itself produces or creates, in a lesser sense of ‘create’, its own attributes is here beside the point.

¹⁸ It seems to me that haecceitates are not the only Scotist feature in Leibniz's thought. He also holds, for instance, that every created monad, including spirits, has primary matter. The attributes corresponding to primary matter are, in bodies, inertia and impenetrability. Do not then the eternal things which correspond to these attributes jointly constitute a forma corporeitatis? And is there not some structural similarity between the evolution and involution of immortal monads and the doctrine of rationes seminales, which, if not Scotist, is at least Franciscan? Ignorant as I am, I say these things with great hesitation; but it might be worth a scholar's while to trace the cues. One wonders how many Scotist works the young Leibniz devoured in his father's library.

¹⁹ I am aware of the ambiguity of the unqualified term in Leibniz and shall therefore soon replace it by ‘analytic’.

²⁰ One may well wonder, even if 3. is granted for human choices, whether the via affirmativa carries that far. The medievals distinguished more nicely in such matters.

²¹ I dodge, as expendable for my purposes, the essentially theological issue of the nature of the principle of sufficient reason as such. Yet my argument sides with Latta and Joseph by rejecting implicitly Russell's contention that any part of it, except possibly that embedded in 3, is analytic. If the principle itself were held to be analytic, as it well might by one who combines Leibniz's arguments with Spinoza's theology, the distinction between the two kinds of truth would disappear. At one point (p. 24) Russell, because of another misunderstanding, says that it might disappear. See fn. 25.

²² I neglect here and subsequently statements mentioning God.

²³ Russell does, in fact, glimpse the distinction (p. 59) or, rather, he comes close to the notion of an individual which, as such, cannot be a nature. But he concludes, rashly and wrongly, on this score alone, not only that there are no substances, but that the substance notion is irremediably confused.

²⁴ Letter of September 28th, 1686.

²⁵ And, therefore, all (noncontradictory) sentences? See fn. 21.

²⁶ This judgment is based on the report of Couturat (La logique de Leibniz, 1901) and the logical fragments he edited (Opuscules et fragments inédits de Leibniz, 1903). Nowhere in these two volumes have I found the slightest hint that Leibniz (or, for that matter, Couturat; see also his own L'algèbre de la logique, 1905) realized the need for a schema containing both subject (x, y, ⃛; a, b, ⃛) and predicate (f, g, ⃛; f ₁, g ₁, ⃛) variables and constants. Nor, I judge, in spite of some most intriguing anticipations (pp. 17, 60), did Russell himself realize it at the time he delivered the Leibniz lectures. Notice, incidentally, that la langue universelle, not being designed for philosophical purposes, is not in the current sense an ideal language.

²⁷ Leibniz usually employs Latin capitals; I use Greek letters in order to avoid confusions with the notation of Section Two.

²⁸ Quite consistently within his pattern, Leibniz brushes aside the distinction between accidents and attributes.

²⁹ I say official because his inventory is upon his own account incomplete. See fn. 16 and Section Five.

³⁰ Let ‘a’, ‘b’, ‘c’ be the names of particulars which, as it happens, are green. A “nominalist” who purports to define ‘green(x)’ by ‘(x = a) V (x = b) V (x = c) …'makes, at a lower type level, the same mistake.

³¹ See also “Particularity and the new nominalism,” Methodos, 6, 1954.

³² Unaware as he was of the type distinction, Leibniz did of course not realize the role of (E) which is explained in Section Two.

³³ The passage on p. 28 reads “amounts to little more than the law of identity”; but the context leaves no doubt about its meaning.

³⁴ The error is facilitated by Leibniz's writing ‘αβγδ ⃛ is α’ for the first and having no transcription for the second.

³⁵ One contributory cause of this error was Russell's preoccupation with the relations issue. For he ascribes to Leibniz at this point the desire to get rid of the causal relation which connects two successive states of a substance by replacing it, in the earlier state, with an attribute of activity. The realism issue also interferes. The attributes other than activity are referred to as “phenomenal.” See Section Five.

³⁶ This is, within the system, the major function of natures.

³⁷ See fns. 16 and 29.

³⁸ Ideal things are not ideas. The latter are the eternal things; for the former Leibniz has no official place. But he was of course aware of them and even has a name for them. The point is that he didn't do anything decisive about them in a “systematic” way, just as in the case of the “tie” between a character (eternal thing, idea) and “its” attributes. (Perceived relations, too, are sometimes called ideal. Like all “rationalists” Leibniz struggles with the concept-percept dichotomy.)

³⁹ Limiting one's self, as I do, to relations among individuals and thus sidestepping the issue of extension, just as I sidestepped that of continuity (see fn. 5), does not affect what I wish to discuss.

⁴⁰ I disregard the Wiener-Kuratowski method of replacing relational predicates by nonrelational ones of a higher type. Within logic this possibility is important and not merely, as Russell once called it, a trick. But to take it into account in a discussion like this is merely to burden one's self with a lot of verbiage without adding anything to the substance.

⁴¹ This is clearly a philosophical question, not one about Leibniz.

⁴² See, for instance, his famous analysis of ‘being at the same place’ in the fifth letter to Clarke.

⁴³ This is not to deny that Leibniz, who anticipated so much, also anticipated one part or version of the early positivists' so-called meaning criterium when he argued against Clarke that there can be no such thing as absolute space since, if the world were displaced in it, we could not know it. His own subsumption of this argument under the identity of indiscernibles and the principle of sufficient reason is mistaken.

⁴⁴ One who reads this essay as if I were a realist will not be misled. This is not the place to present my analysis of the realism issue. It may be found in MLP.

⁴⁵ The context makes it clear that this something more is not extension (see fn. 39).

⁴⁶ This appeal to Kant's authority fits well with the Kantian view Russell then held on geometry (see Section Two). The other German whom he then knew best and who influenced him most was probably Lotze, who was notoriously a great student of Leibniz.

⁴⁷ Some of these questions are explored in C. D. Broad, The mind and its place in nature (chapter 4).

⁴⁸ Twenty five years later Broad (l.c.) examined it, very interestingly, on its own merit. See fn. 53.

⁴⁹ E.g., in order to say that a nontemporal character f ₁ is exemplified at a certain time, a relational second-type predicate of coincidence (‘Co’) such that ‘Co(f ₁, f ₁)’ transcribes ‘f ₁ coincides with t ₁’. A little reflection shows that this relation does not at all have the properties of simultaneity. Notice also that the indices I used (t ₁, t ₂, …) are merely parts of the names of the temporal characters. Only after a linear order has been introduced among these characters by means of an undefined relation may the indices be used to indicate position in this order.

⁵⁰ See fns. 1 and 26.

⁵¹ I assume thus that the ‘t ₁’ in ‘f ₁'(a, t ₁)’ is of type zero. Formally one could assign it to type one. However, the victory would be hollow, not necessarily because ‘f ₁‘ would then be of the rather complex type (0, 1) but, rather, because it would still be true that the ‘t ₁’ occur always as subjects and never as predicates. A predicate being always exemplified by an individual would be transcribed by ‘(x)[ti(x) ∪ f ₁(α, x)]. The character ‘ti’ (being a moment) could be defined in terms of the order relation that obtains between any two of these peculiar “individuals” and no others.

⁵² A little reflection shows that it would not do to make the ‘a’, ‘b’, ⃛, into attributes. I say attributes rather than characters because the word may, perhaps quite properly, remind some of Spinoza. Again, the “world” I consider in this paragraph may remind some others of Whitehead. But, again, not being a thorough student of either Spinoza or Whitehead, I say these things only with hesitation.

⁵³ This is the alternative C. D. Broad explored. See fn. 48.

⁵⁴ Thoughtless ones say that the particulars of their ideal language refer to physical objects. Thus they inherit, without knowing it, the greatest difficulty of all substance philosophies.

⁵⁵ See “Two cornerstones of empiricism,” Synthese, 8, 1953 (reprinted in MLP).

⁵⁶ E.g., sense data, but not their characters; hence “some” and not “all.”

⁵⁷ Notice that I do not use the argument according to which contemporary physics has disposed of absolute space and time. See “Professor Quine on analyticity,” Mind, 64, 1955.