Abstract
In 1666, five years after Dalgarno published the Ars Signorum, and in the year that the first version of Wilkins’s Essay was destroyed by the fire, a short treatise entitled Dissertatio de Arte Combinatoria was published in Leipzig. It was written by a nineteen-year-old student of philosophy and law, whose name was Gottfried Wilhelm Leibniz. At first glance, this treatise on combinations seems to have little in common with the linguistic projects described in the previous chapters. But in fact it contained, in seminal form, all the ingredients as well as a first sketch of the most ambitious project for a philosophical language to be conceived in the seventeenth century, and probably ever. Unlike Dalgarno and Wilkins, Leibniz never reached definitive results in this regard, but he worked on the project throughout his life
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Notes
Couturat 1901: 169; Erdmann’s mistake was first pointed out by Trendelenburg.
Couturat 1901: 323-388; Rescher 1954; Kauppi 1960: 129-243; Burkhardt 1980; Lenzen 1990 provide admirable treatments of this subject.
The Latin text has been printed in Gerhardt 1880, and in the Academy edition (A 6 1, 165ff.). English translations of portions of the text are L 73-84 and Ñ 1-11. A German translation of parts of the text is S 29-59.
For instance, in a letter to Conring (1671) he refers to the ‘absurdities of the Lullists’ (A 2 1 81), and in a letter to Hesenthaler of the same year he says that the Lullists have corrupted and discredited the art of combinations (A 2 1 200).
Cf. e.g. Jungius 1957[1638/1681]: 211[298]. Leibniz often expressed his great admiration for Jungius.
Jungius 1957[1638/1681]: 215[306] also uses the example of a threefold division of ‘republic’, but in a different context from the one in which Leibniz treats it.
E.g. L 229 (=G VII 292), L 470 (=G VII 526), NE III 3 §10.
Cf. e.g. Bacon, Novum Organon, Aphorism XIV: “The syllogism consists of propositions, propositions consist of words, words are symbols of notions. Therefore if the notions themselves (which is the root of the matter) are confused and over-hastily abstracted from the facts, there can be no firmness in the superstructure” (Works I: 158, IV: 49).
Couturat 1901: 46 has criticised Leibniz’s calculation for overlooking PA propositions that are not derived from UA ones by subalternation or conversion. Dürr 1949: 24ff. points out against Couturat that Leibniz explicitly limits his discussion to ‘eternal truths’.
Cf. Kauppi 1960: 143-144.
Leibniz supposes that PN propositions are deducable from UN ones just as PA are deduced from UA, that is, either by subalternation (no A is Â-> some A is not B) or by partial conversion (no A is Â-> some  is not A). Since simple conversion is valid for UN propositions (no A is Â-> no  is A), inference by partial conversion leads to exactly the same result as simple conversion plus sub-alternation. This is in contrast with what is the case with affirmative propositions, because simple conversion is not valid for UA propositions (not: all A are Â-> all  are A). This is why the truth conditions for UA and PA propositions, which likewise depend on subalternation and partial conversion, are nevertheless different from each other. Since Leibniz assumes that partial conversion and subalternation exhaust the possible cases in which particular propositions are true, he cannot account for cases in which some PN is true and the corresponding UN is false.
The ‘A’ in this expression is probably inserted because what is here rendered as ‘of the’ is expressed in the original by the Greek article της, which Leibniz probably regarded as requiring a syntactic antecedent.
Cf. Mates 1986: 62 for a lucid account of this general point.
Parkinson translates: “for the variation of angles or of extremities in lines, kinds of relations”. This is a correct rendering of the original, which reads: “pro variatione angulorum aut Terminorum in lineis genera relationum” (G IV 73). The problem is that this does not make any sense: that ‘kinds of relations’ are supposed to represent ‘variation of angles’ is incomprehensible by itself, and from the context it can be inferred that something like the reverse is intended. It seems plausibe, therefore, to assume that the original text contains an error in that the words ‘relationum’ and ‘angulorum’ have changed places. Parkinson’s rendering ‘Terminorum’ by ‘extremities’ is clearly occasioned by the context, but is at variance with Leibniz’s use of ‘Terminus’ in the sense of ‘term’ in the same passage. Schmidt 1960: 59 renders ‘Terminorum’ by ‘Ausdrücken’ as is very probably correct, but provides an even less understandable translation: “fur die Variation von Winkeln oder von Ausdrücken in Linien die Arten der Relationen”.
Acta Eruditorum Lipsiae 1691, 63; G IV 103. Cf. C 175: “Il y a plus de 20 ans (...) que je m’avisa d’une methode qui nous méne infalliblement à l’analyse generale des connoissances humaines (...) comme on peut juger par un petit traité que je fis imprimer à lors, où il y a quelques choses qui sentent le jeune homme et l’apprentif, mais le fonds est bon, et j’y basti depuis la dessus”.
G VII 126, 185, 292, C 346, Guhrauer 1846 II, Anhang 55.
Loemker’s translation contains an error here, mistranslating ‘combinatio’ as ‘comparison’. This is unfortunate, as it obscures the fact that Leibniz is here alluding to combination or synthesis, as opposed to analysis. The philosophical language supports both methodical procedures, and hence “includes both the arts of discovery and judgment”. Cf. 5.3.5.
“I read much later what I had written when I was fourteen, and it pleased me very much”. One of the things he had written at that age, was his idea concerning the categories of propositions. Guhrauer 1846 II, Anhang 55.
It has been seen however (2.2.4) that Ward 1654: 21 sketched a similar programme.
Couturat assumes on internal grounds that this patron was Boineburg. Couturat states: “Cette lettre doit dater de la premiere jeunesse de Leibniz (1666-1672)” (C 30), which seems quite plausible. However, the terminology used in the opening sentence of the letter suggests that Leibniz must have at least heard of Wilkins’s Essay, as it begins thus: “Since your Excellency takes delight in thoughts about a philosophical language, which others also call a universal characteristic”. Leibniz first heard of the Essay in 1668 (cf. 5.3.1).
26 September / 6 October 1668, G VII 9, A 2 1 10-11.
Letter from William Curtius to Leibniz, 13/23 October 1670, A 2 1 67.
To Martin Fogel, a professor of logic and metaphysics in Hamburg (14/24 January 1671.
On universal synthesis and analysis, or the art of discovery and judgment, L 229-234, G VII 292-298 (presumably about 1679). About Kircher’s Ars magna sciendi sive combinatoria Leibniz says: “I hoped to find similar matters discussed in it, but when it did appear later, I found that it had merely revived the Lullian art or something similar to it but that the other had not even dreamed of the true analysis of human thoughts” (L 230).
Hofmann 1974: 24-25. On January 22nd 1673, Hooke noted in his diary: “Libnitius shewd Arithmetical engine” (Robinson and Adams (eds.) 1935: 23). 26 When exactly the letter was written is also uncertain, but this was presumably some time in 1673 (A 2 1 xxix).
On terminology, see 5.3.7 below.
It does not matter, says Leibniz, whether we want to establish a universal writing or a language, for it is easy to produce either one of them by the same efforts (A 2 1 240). Cf. 5.3.8 for discussion.
E.g. A 2 1 271: “L’esprit humain ne s9auroit aller fort avant en raisonnant, sans se servir des caracteres: et les caracteres bien choisis ont cela de merveilleux, qu’ils laissent pour ainsi dire les marques des pensées sur le papier; et nous donnent le moyen d’estre infallibles”.
G VII 57, G VII 64, G VII 125, G VII 168, C 285 = A 6 4 800, A 6 4 643.
It is unclear to me why Leibniz chose this particular instance, as I cannot see why this should be an improvement. Perhaps the replacement of numerical symbols by letters was considered to be so valuable that it was seen as an improvement wherever it was done. Struik 1967: 103 notes that Descartes wrote aa for a2, “though he has a3 for aaa”.
E.g. L 166, G VII 184, L 261, L 670, A 2 1 384.
Cf. A 2 1 384-385; D 168; C 152; L 249-250.
Dascal 1987: 44 gives the following, incomplete, list of places where Leibniz uses this expression: A 6 1 327, A 6 2 477-478, A 6 2 94, A 1 1 79, G VII 78.
Couturat 1903: 435 prints one sentence. The close link between the alphabet of human thoughts and the formal language is also clearly stated in G VII 204-207 (D 181-184).
In the Nova Methodus (1667) he likewise equates’ simple terms’ with’ sensible qualities’ (L 89), but elsewhere he characterizes the primitives in other ways; cf. 5.3.4.
Kneale quotes the following passage: “The art of symbolism (ars characteristica) is the art of so forming and ordering symbols that they reproduce (referant) thoughts, or have the same relations among themselves that thoughts have among themselves” (1966: 207).
That Leibniz refers to relations between ideas rather than relations between things is possibly relevant in this context. This point has been disregarded by Kneale, who treats Leibniz’s argument as though it unambiguously concerned correspondences between language and the world.
For the distinction between ‘contain’ and ‘involve’ cf. D 183: “Let a character be said to be contained in a formula if it is clearly a part of it. (...) Let a character be said to be involved (in a formula)if at least through the substitution of equipollents it may finally be shown to be contained (therein).”
For a clear exposition of this point, see Mates 1986: 130-132. Some relevant texts: A 6 4 552 (S 475); C 402-403 (PM 94-95); C 261; C 367 (P 56-57); A6 6 287.
Grua 537. However, this text was written in April 1679, whereas the letter quoted in the text (A 2 1 497), in which the problems concerning the primitives are summarized, dates from December of that year.
Nouveaux essais sur l’entendement humain (NE), written 1703-1704, contains a detailed discussion of Locke’s An essay concerning human understanding (1689). Cf. 5.4.3.
L 646. Cf. Couturat 1901: 214ff; Parkinson 1966: xxxiii; Mates 1986: 152ff.
C 514, PM 8. I have added ‘enough’ so as to render’ satis’ which Parkinson leaves untranslated. The original reads: “Non videtur satis in potestate humana esse Analysis conceptuum, ut scilicet possimus pervenire ad notiones primitivas, seu ad ea quae per se concipiuntur”.
Burkhardt’s claim to the contrary (1980: 171, 263) is based, among other things, on what Leibniz says in a letter to Foucher (1687, G I 392). But he says there that we must not assume that analysis down to the primitives is always possible, which is obviously not equivalent to saying that we must assume that such an analysis is never possible.
Meditationes de Cognitione, Veritate et Ideis (MCVI), L 291-295. Cf. 5.4.4.
Cf. C 187: we only have a negative, and provisional criterion for deciding whether a given notion is primitive or not, for we know when a notion is not primitive, namely when it occurs in a proposition which is necessarily true but not proved. In such cases we are certain that the notion can be defined, i.e. resolved into simpler ones. If the process of definition is continued, we arrive at the most simple ideas. Leibniz adds that he does not believe that this process might go on infinitely. Cf. also G II 227 (L 525), written 1701: “It will be difficult to give examples of a concept from which no representative mark can be removed. Primitive concepts lie concealed in derivative ones but are hard to distinguish in them”.
Discourse on Metaphysics § 24, PM 33, L 319. Parkinson (1966: xxxv) says it is surprising that “as late as the early months of 1686” Leibniz should think that this is possible, since “he had decided that human beings cannot reduce derivative concepts to those that are absolutely primary”. Perhaps this decision was not as final as Parkinson apparently assumes, if it was unambiguously made at all.
Cf. Mates 1986: 153 for a list of different expressions of the principle.
Cf. A 6 4 161: “La Caracteristique que je me propose ne demande qu’une espece d’Encyclopedic nouvelle. L’Encyclopedic est un corps, où les connaissances humaines les plus importantes sont rangées par ordre. Cette Encyclopedie estant faite selon l’ordre que je me propose, la caracteristique seroit quasi toute faite, cependant ceux qui y travailleront n’en scauroient pas le dessein: croyant de travailler seulement à une Encyclopedie” (April 1679(?)).
Some of the relevant texts are G VII 43-45, 49-53, 54-56; 57-63; 64-65; 66-77; C 30-41; C 217; C 218-222; C222-224. Couturat 1901: 119-175 provides an excellent and detailed account of this subject.
A 2 1 242. Couturat 1901: 142 remarks that this constitutes a hardly implicit invitation to start a collaborative project.
G VII 157-173; G VII 174-183.
G VII178; similar remarks are at C 30 and C 228.
E.g. Heinekamp 1972: 462; Kauppi 1960: 25.
G IV 46, 52, 64, 78. Couturat (1901: 457-472) has convincingly argued that these references should not be taken to show that Leibniz was in any way indebted to Hobbes for his logical insights.
A 2 1 56-59, L 105-107. Hobbes, who was in his eighties, never bothered to answer. Some four years later, Leibniz wrote a second, more critical letter on political philosophy, which likewise remained unanswered (A 2 1 244-245).
Arnauld & Nicole 1970[1662]: 170 (II, 7). As Loemker notes (L 185), and as is confirmed by the passage mentioning geometry (see below), the Port Royal logic probably influenced Leibniz’s argumentation in this text. It should be noted, however, that Leibniz reaches opposite conclusions: both truth and falsity, he concludes, are properties of thoughts. Burkhardt (1980: 252), ignoring the dialogue form of the text, misinterprets B’s hesitation to accept this conclusion as expressing Leibniz’s own view.
Cf. Ishiguro 1990: 28 for a similar interpretation. By contrast, Kauppi (1960: 39) interprets the ‘phosphorus’ example as being aimed at showing that similar means of expression may express different things, overlooking the fact that ‘lucifer’ and ‘phosphorus’ are names of the same thing, though both words also have other meanings. Possibly Kauppi was led astray by the word ‘but’ introducing B’s remark: “But the Greek ‘phosphorus’ has the same relation to ‘phos’ and ‘phero’”. This ‘but’ (Latin ‘at’) is puzzling, as it suggests an opposition to the preceding sentence about the Latin ‘lux’ and ‘fero’. However, Leibniz’s using ‘but’ here might be explained by taking into account that  at this point is trying to cast doubt on the view that there is no similarity between characters and things.
nr]‘Le_phosphore de M. Krafft, ou liqueur de terre seiche de sa composition qui jettent continuellement de grands eclats de lumiere’, Journal des savants, August 2, 1677 (L 176).
Dascal 1987: 73. Although I disagree with some of Dascal’s conclusions, the present discussion owes a great deal to his treatment of the subject.
This is not to say that Leibniz’s thesis on the importance of signs for thinking has escaped Burkhardt’s thoroughness. On the contrary, Burkhardt gives due recognition to this thesis (1980: 178), but unfortunately not in the context where it is essential to do so.
In his zeal to force the Dialogue into the three-place pattern, Burkhardt goes so far as to say that Leibniz in this text also makes mention of a level of concepts between signs and things: this is because he refers to general or rational grammar, which is the same for all languages. But in fact the only thing Leibniz says about this in the text is that all languages somehow correspond in that they have the same relation to things. Again, no mention of concepts, which is quite understandable given the structure of the argument.
The New Essays consists of a very detailed commentary on Locke’s Essay (1689). The NE was written in 1703-1704, but not published until 1765.
In the Introduction to the New Essays, Leibniz remarks: “It is true that the necessity of having to follow the thread of another person’s argument in making my remarks has meant that I have been unable to think of achieving the graces of which the dialogue form is capable”. Apparently, Leibniz has occasionally tried to achieve some of these graces after all.
Locke 1689 (further: Essay), Book IV, ch. V, § 4.
This is not to say that Leibniz was a ‘conceptual realist’, i.e. that he held that concepts have an independent ontological status. On the contrary, in his view only individual substances exist.
Dascal, 1975, 1976, 1977, 1980. I have used the English versions of these articles in Dascal 1987.
Pombo (1987: 110) takes this very sentence to express a reaffirmation of what was said in the first, but I think that Pombo is wrong here, and that she apparently overlooks the word ‘always’, without which the sentence would indeed mean what she says it does. Adding emphasis on ‘It is neither possible nor desirable’, as she does, cannot change the fact that the sentence as it stands merely denies the possibility of always distinctly observing things or ideas, and hence carries the implication I draw attention to, and which is clearly confirmed by the sequel, as Pombo is likewise concerned to show. Moreover, it is remarkable that Pombo suppresses the phrase immediately following the one she emphasizes, namely: “itaque compendii causa signa pro ipsis adhibentur”. If signs constituted thinking, what could they be an abbreviation of?
Leibniz uses the same example elsewhere, e.g. C 154.
An even shorter way of removing this difficulty is to translate ‘perficitur’ as ‘is perfected’ instead of ‘is performed’, as the following translation does: “All human reasoning is perfected through the use of signs or characters.” (Rutherford 1995: 260). But this is almost certainly a mistranslation, not only because it involves deviating from the standard meaning of ‘perficere’, but also as it renders the use of the universal quantifier ‘omnis’ stylistically awkward.
This remark is very similar to one occurring in DAC, G IV 35: “we often grasp a number, however large, all at once in a kind of blind thought, namely, when we read figures on paper which not even the age of Methusalem would suffice to count explicitly” (L 76).
It must be noted however that e.g. the discovery of counterfeit gold may induce adaptation of nomenclature, for instance coining a new term for counterfeit gold while retaining ‘gold’ for real gold, or coining new terms for both materials, while retaining ‘gold’ for the species to which they both belong. NE IV VI 4.
As has been noted above (5.4.2), Burkhardt surprisingly includes the Dialogue among the texts allegedly supporting this point.
LH IV 7  4, 13-14 (printed in S 479, German translation S 465). In English, this reads approximately: “If A is  and  is a complete term then A will be a singular substance, or some subject which is usually called an individual. For only a singular substance has a complete concept”.
It should be noted that Arnauld and Nicole explicitly state that the word ‘term’ applies to both words and concepts: “tant les idées universelles que les noms communs se peuvent appeller termes généraux”. Nicole & Arnauld 1970[1662]:86(I,6).
Cf. Mates 1988: 124ff., Ishiguro 1990: 17ff. Burkhardt 1980: 94-96 gives a useful survey of the different meanings of ‘terminus’ as used by Leibniz.
Cf. e.g. C 159: “The most perfect scientific method does not start from things that are posterior by nature and composite and specific, which are those that strike the senses, but from notions and truths that are most simple and general, which are first observed by the intellect, from where it descends gradually to more specific and composite notions”.
This interdependence has later been termed ‘the law of reciprocity’ (Kauppi 1960: 8).
A distinction similar to the one Leibniz draws here was proposed in the logic of Port Royal, but in this work it is made with regard to ideas: each idea includes (enferme) attributes making up its ‘compréhension’, and applies to (convient à) subjects constituting its ‘etendue’: “J’appelle compréhension de l’idée, les attributs qu’elle enferme en soi, & qu’on ne lui peut ôter sans la détruire (...); J’appelle étendue de l’idée, les sujets à qui cette idée convient” (Arnauld & Nicole 1970[1662]: 87 (I, 6)). Although Leibniz does not explicitly say so, the context makes it fairly obvious that his distinction between extension and intension applies to words.
It is unclear whether or not Leibniz deliberately used 8 in this example, since 8 can only be factorized into primes as 2x2x2, whereas Leibniz probably supposed that only products of different prime numbers (concepts composed of different primitives) would be used.
C 42-92. Cf. Couturat 1901: 326-335, Lukasiewicz 1951: 126ff, Kauppi 1960: 145-153, Parkinson 1966: xxi-xxii for discussion.
This table was printed earlier in Maat 1995: 176. I suggested that Leibniz may have borrowed not just the entries, but some parts of the definitions as well, adding however that in most cases the definitions differ completely (1995: 175). Nevertheless, Dutz 1996: 51-55 has argued at length for the difference between Wilkins’s and Leibniz’s definitions, on the erroneous assumption that he was refuting my position.
The editors of the Academy edition suggest, contrary to fact, that the only thing Wilkins provides in his tables is a ‘mere division of concepts’ (A 6 2 487).
Couturat 1901: 63-75, Dascal 1990, Mugnai 1992: 59-83, Schneider 1994: 226-236 provide more elaborate, but equally limited discussion.
“Petrus scribit pulchre. Id est Petrus scribit aliquid pulchrum seu Petrus scribit, et quod Petrus scribit est pulchrum. Petrus stat pulchre. Id est Petrus est pulcher quatenus est stans” (A 6 4 106).
Cf. A 3 3 235-236: in a letter of July 1680, Hooke inquires whether Leibniz “hath yet perfected his arithmeticall Instrument”.
Hooke wrote apiece called Mathematical Language around 1686, which was never published (Royal Society Classified Papers XX, no. 72; Slaughter 1982: 182). In my opinion, this text contains nothing that could indicate Hooke was in fact working on a scheme that was even remotely similar to Leibniz’s.
Broadsheet carrying page number 442; Leibniz manuscript LH IV 7 D, bl. 3-4.
Maat 1995: 173-175 provides illustration and discussion.
VE 382-393 = A 6 4 42-53. The editors have established that this was probably written in 1685 or 1686. Leibniz omits Wilkins’s genus ‘military relation’, which may have been a slip that was caused by Leibniz’s aversion to warfare.
This was stated in a personal communication by prof. Heinrich Schepers, at the time the director of the Leibniz-Forschungsstelle in Münster.
The high editorial standards of the Academy edition are exemplified by the meticulous records of these notes in A 6 3 169-188.
The editors of the Academy edition note: “Aus welchem Grunde und in welcher Absicht Hodann am unteren Rand des Lexicon Grammatico-Philosophicum das Vorkommen der Vokale und Konsonanten am Silbenanfang in der Reihenfolge der Druckspalten 1-8 registriert hat, ist nicht zu erkennen” (A 6 3 169). It is very probable however, that Hodann, while compiling the list of definitions using Dalgarno’s alphabetical Lexicon Latino-Philosophicum (cf. 5.5.2), had to consult the lexicon of radical words frequently in order to make sense of Dalgarno’s compound words. An index-like survey of the type entered by Hodann facilitates locating radical words on the broadsheet, which otherwise may be cumbersome.
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Maat, J. (2004). Leibniz: The Art of Thinking. In: Philosophical Languages in the Seventeenth Century: Dalgarno, Wilkins, Leibniz. The New Synthese Historical Library, vol 54. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1036-8_5
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