Introduction
1 Genesis and Structure of the ‘General Inquiries’
Leibniz composed the General Inquiries on the Analysis of Notions and Truths [Generales inquisitiones de analysi notionum et veritatum] (‘GI’ henceforth) during the year 1686, the same year in which he began to correspond with Arnauld and wrote the Discourse on Metaphysics. 1 The correspondence with the philosopher and theologian Antoine Arnauld (1612–94) constitutes one of the main sources for the study of Leibniz’s philosophy, and the same holds for the Discourse, which offers a first systematic account of notions like those of complete concept of an individual, pre-established harmony between soul and body, and substantial form (something very similar to the ‘monad’ of Leibniz’s mature philosophy).2
The GI is a necessary supplement to the correspondence with Arnauld and the Discourse in so far as it develops a central topic of Leibniz’s metaphysics and shows the intimate connection that links Leibniz’s philosophy with the attempt to create a new kind of logic. It is in the GI, indeed, that Leibniz articulates for the first time his favourite solution to the problem of contingency, and it is in the GI that he displays the main features of his logical calculus.
At first glance, the GI gives the impression of a ‘compact’ and coherent work: it begins with a fairly long introduction where several topics are discussed (philosophy of logic, metaphysics, and grammar), and then a list of paragraphs of various lengths follows, marked with numbers from 1 to 200. To the sequence of paragraphs, however, there is no corresponding systematic and coherent development of a logical calculus. It is only towards the end of the essay that Leibniz proposes a set of principles from which the theorems previously proved can be derived; and he
1 Cf. Antognazza (2009: 239–41). 2 Cf. Discourse and A VI, 4B: 1529–88.
attains this result without explicitly discussing the relationship of his final outcome with the other principles previously proposed: these simply survive in the body of the text as evidence of the steps that have evolved to produce the final outcome.3
On at least two occasions he revives his old project of employing numbers to express propositions but then, after a while, he abandons this issue and abruptly begins to develop a different topic.4 As George Parkinson remarked, the GI is a difficult work ‘in which Leibniz often seems to be groping his way’.5 This, however, does not undermine the extraordinary value of the GI, which is very rewarding for everyone interested in logic, philosophy of logic, and metaphysics (besides Leibniz’s thought). As Marko Malink and Anubav Vasudevan point out, the GI: does not take the form of a methodical presentation of an antecedently worked-out system of logic, but rather comprises a meandering series of investigations covering a wide range of topics. As a result, it can be difficult to discern the underlying currents of thought that shape the treatise amidst the varying terminology and conceptual frameworks adopted by Leibniz at different stages of its development.6
Given this composite structure of the GI, in what follows I devote two sections to introduce each of the two main topics of this work: logic and metaphysics.
Section 2 (‘Logic’) begins with a preparatory account of Leibniz’s project for a universal characteristic and focuses on the relationships between rational grammar and logic. Then, I will discuss the general structure and the main ingredients of Leibniz’s logical calculus as presented in the GI.
Section 3 (‘Metaphysics’) is centred on the problem of contingency, which caused a lot of trouble for Leibniz from the beginning of his correspondence with Arnauld until the end of his life. I attempt to explain,
3 Malink andVasudevan (2016: 685–6).
4 Leibniz’s idea of employing numbers to designate concepts (and propositions) traces back to the DAC (1666): 4–5; 161. An extensive discussion of Leibniz’s use of numbers in his logical essays can be found in chapter 3 of Schriften zur Syllogistik
5 LP: xxvi–xxvii. 6 Malink and Vasudevan (2016: 686).
first, the nature of this problem and then to show how Leibniz reckoned he had solved it: in the GI, indeed, we find, even though it is expressed in a tentative way, the core of his solution based on infinite analysis.
2 Logic
2.1 The Characteristic Art and the Rational Grammar
The GI is an essential part of the project for the constitution of what Leibniz calls characteristic art (ars characteristica). In Latin the word character means ‘sign’ or ‘mark’, and the characteristic art was conceived as a system of signs provided with rules for performing three different tasks: encoding concepts, forming propositions, and inferring propositions from propositions.
The first embryonic idea of the characteristic art can be traced back to the Dissertation on Combinatorial Art, which Leibniz wrote when he was 19 years old (1665–6).7 The Dissertation contains many seeds from which Leibniz’s philosophy will take its mature form. In particular, in the Dissertation Leibniz elaborates a project that can be summarized as follows:
(1) By means of analysis, each concept should be decomposed into its component parts until the first concepts are reached.
(2) Once the first concepts are reached, combine them and produce all kinds of complex concepts.
(3) At the same time choose a system of simple signs to designate the first concepts, so that any complex of signs can be univocally associated with each complex concept.8
Leibniz believes that the best signs to employ are numbers. If the chosen signs are letters or marks different from numbers, we will have a kind of universal language: a language, that is, of pure concepts, accessible to everyone. If numbers (in particular prime numbers) are employed to
7 See DAC: 1–4. 8 Cf. DAC: 4.
designate the first concepts, then we will have the possibility of transforming each logical argument into a calculus.
After his stay in Paris (1676–9), Leibniz enriches his project for the constitution of the characteristic art by the following tasks:
(1) Define a method for developing an analysis of each concept.
(2) Define a method for recombining the first concepts and thus producing all complex concepts.
(3) Define a set of rules for developing a very general calculus based on either the relations of coincidence or of containment holding for terms and propositions.
Leibniz assigns the task of realizing points 1 and 2 to a discipline that he calls general science (scientia generalis) and that he divides into two parts: analysis and synthesis. 9
Since the concepts that Leibniz had in mind were those employed in everyday life and in the sciences of his time, if they were to be analysed as required by the constitution of the characteristic art, a kind of general repository was needed to store them (in some order). Leibniz believed that an encyclopedia of all knowledge acquired by mankind during the centuries could play the role of such a ‘repository’. Thus, the task of constructing an encyclopedia is integrated into the project for the characteristic art.10
Leibniz, however, was uncertain about the structure of the encyclopedia, whether it should be systematic, beginning with principles and axioms and then including all truths that can be derived from the principles, or whether it should contain all items in alphabetical order. Several manuscripts with sketches and projects of the two possible structures clearly show Leibniz’s irresolution on this point.11
9 On scientia generalis, see A VI, 4A: LII–LXXXVII, 352–74, 544; L: 233. On analysis and synthesis, see L: 173–6, 184–8, 229–34 and Schneider (1970). On the relationships between scientia generalis and Leibniz’s projects for an encyclopedia of the sciences, see Pelletier (2018).
10 Cf. Philosophical Essays: 8; A VI, 4A: 84, 138, 257, 338–60.
11 Cf. A VI, 4A: 257, 338–49, 430.
A series of essays written around the same time as the GI shows that in this period Leibniz intended to build the universal language on the basis of a very austere grammar that he called rational grammar ( grammatica rationalis).12 In these essays, Leibniz investigated the grammar of a fragment of Latin (the Latin written and spoken mainly by scientists and philosophers of his time) aiming to reduce it to a limited number of elements. To realize this task, in a long essay entitled Analysis of Particles (Analysis particularum) Leibniz investigates the behaviour and meaning of several Latin particles.13 In a text explicitly devoted to philosophical language he splits the terms (vocabula) of the natural language into words (voces) and particles (particulae). Words are nouns, verbs, and adverbs; particles are prepositions, conjunctions, pronouns, and even inflexions and cases. As Leibniz remarks, ‘Words constitute the matter, particles the form of the discourse (oratio).’14
Leibniz’s distinction is analogous to that of medieval logicians between categorematic and syncategorematic terms and roughly corresponds to a more general distinction between fundamental (radicalis) and auxiliary (servilis) expressions that Leibniz wants to introduce into characteristic. 15 ‘Fundamental’ or basic expressions are substantives and adjectives; ‘auxiliary’ expressions are particles. Leibniz characterizes his project as follows:
Everything in discourse can be analysed into the noun substantive ‘being’ or ‘thing’, the copula, i.e. the substantive verb ‘is’, adjectives, and formal particles.16
An important task that Leibniz assigns to rational grammar is that of finding a treatment of relational arguments that would permit them to be handled by the methods of what he regards as logic.17 Statements, and consequently arguments containing relations, indeed, were quite troublesome to people who accepted traditional logic based on Aristotle’s syllogistic.
12 Cf. AVI, 4A: 102–5, 112–17, 267, 338–9, 344–5, 528.
14 A VI, 4A: 882. 15 A VI, 4A: 643.
16 LP: 16 (translation slightly modified); A VI, 4A: 886.
13 A VI, 4A: 646–67.
17 LP: xx.
A well-known argument in syllogistic form is the following:
(A)
(1) All men are mortals.
(2) All Greeks are men.
(3) Therefore, all Greeks are mortals.
A typical argument involving relations, instead, is this:
(B)
(1) Socrates is Sophroniscus’ son.
(2) Therefore, Sophroniscus is Socrates’ father.
According to the mnemonic verses employed by the schoolmen, (A) is a syllogism traditionally classified as an instance of the mode Barbara. During the seventeenth century (B) became known as a case of inversion of relation. 18 In each sentence of syllogism (A) a predicate is attributed to a subject, whereas the premise and conclusion of the argument based on the inversion of relation state that a relation (son, father) holds between two individuals (Socrates, Sophroniscus). The traditional syllogism employs three terms (in the example above: ‘man’, ‘Greek’, and ‘mortal’) and concludes thanks to the role played by the so-called middle term (in the example above: ‘man’); the inversion of relation does not have a middle term and was considered, at least by some seventeenth-century logicians, as a direct (or immediate) inference. The two arguments are quite different, and it is impossible to express the inversion of relation as a syllogism, maintaining at the same time all the constraints that characterize a syllogism in its traditional form.
Aristotle believed that arithmetic, geometry, optics, and ‘in general those sciences which make enquiry about the cause’ carry out their demonstrations through the first syllogistic figure.19 As Jonathan Barnes remarks:
In his Elements Euclid first sets down certain primary truths or axioms and then deduces from them a number of secondary truths or theorems. Before ever Euclid wrote, Aristotle had described and commended that
18 Cf. Jungius (1957): 89–93. 19 Aristotle, An. post. 79a17–24.
rigorous conception of science for which the Elements was to provide a perennial paradigm. All sciences, in Aristotle’s view, ought to be presented as axiomatic deductive systems—that is a main message of the Posterior Analytics. And the deductions which derive the theorems of any science from its axioms must be syllogisms—that is the main message of the Prior Analytics. 20
Aristotle’s view was accepted by the great majority of his followers and by those who shared the logical theory developed in Prior Analytics. The ancient philosopher and physician Galen of Pergamon (third century ad ) was probably the first to claim that the Aristotelian syllogism was unsuitable for handling relations and relational arguments. Galen, indeed, introduced a new class of inferences that he called relational syllogisms to handle relations.21 These syllogisms differed, according to him, from categorical and hypothetical syllogisms:
There is also another, third, species of syllogism useful for proofs, which I say come about in virtue of something relational, while the Aristotelians are obliged to number them among the predicative syllogisms.22
Since Galen, only a very small number of logicians and philosophers have dared to oppose the received view inspired by Aristotle.23 Among these dissenting voices, that of Joachim Jungius (1587–1657) was one of the most interesting. A professor of natural sciences in Hamburg, the author of a logic handbook, the Logica Hamburgensis (1638), and highly regarded by Leibniz, Jungius believed that traditional logic needed to be expanded with additional inferences involving relations that he assumed to be primitive and not reducible to syllogisms.24
20 Barnes (2007: 360).
22 Galen (1974: xvi, 1).
21 Barnes (2007: 419–24, 431–3).
23 During the period from around 900 to 1200, some authors belonging to the Arabic tradition were well aware of the difficulties implied by attempting to express relational inferences in the form of an Aristotelian syllogism. From the thirteenth century onwards, Arabic thinkers continued to discuss the problem of relational inferences, still maintaining a logical framework largely inspired by traditional syllogistic doctrines (cf. Khaled El-Rouayheb (2010)), but they seem to have exerted no significant influence on authors belonging to the cultural milieu originating in the Latin tradition.
24 One of these inferences was the inversion of relation just mentioned above; another was the so-called inference from the right to the oblique (a recto ad obliquum), of which the following is an example:
2.2
Arguments Containing Relations: Rational Grammar and Logic
Relations and relational arguments caused Leibniz two different (but related) kinds of problems: an ontological problem on the one hand, and a logical-syntactic problem on the other. The ontological problem was determined by the basic ontological view prevailing at the time among Western philosophers. According to this view, the entire world was composed of individual beings only, called ‘substances’, and their inhering accidents. These accidents were thought of as strictly ‘monadic’, i.e. they could not simultaneously inhere in more than one substance. Yet, since relations link together two or more substances, the problem arises about the nature of ‘polyadic’ properties corresponding to relations and relational accidents.
The logical-syntactic problem was determined by the structure of the proposition that Leibniz and the great majority of his contemporaries considered basic. For Leibniz the elementary form of any proposition (this too inspired by Aristotle) was subject-copula-predicate. As we have seen, however, relational sentences do not have such a form and therefore do not easily conform to the structure of a traditional syllogism.
To solve the first, the ontological problem, Leibniz adopted a strategy analogous to that of Peter Auriol, a medieval thinker who lived about three centuries before him.25 Leibniz explicitly recognizes the existence of polyadic predicates, but only in so far as they denote ‘merely mental
Every circle is a figure [Omnis circulus est figura]; Therefore, who describes a circle, describes a figure [Ergo Quicumque circulum describit, figuram describit].
This inference was called ‘from the right to the oblique’ since, whereas in the first premise all terms (circulus [‘circle’], figura [‘figure’]) are in nominative (‘right’) case, they are in a case different from nominative (‘oblique’) in the conclusion (circulum, figuram [accusative]). See Jungius (1957: 123).
25 Peter Auriol (1280–1322) was a French Franciscan who taught at Bologna, Toulouse, and Paris. Concerning the nature of a relation, he wrote in his commentary on the Sentences: a relation is only in apprehension, having no being in things, because that which exists as one and simple and connects [attingit] two really distinct things seems to be only the work of the intellect, otherwise the same simple and individual thing will be in several things separated from each other. But it is clear that a relation connects two distinct things, one as a foundation and the other as term. Then, being something indivisible and simple, it cannot be in the extra-mental reality, but only in the consideration of the intellect.
(Auriol, In I Sent., d. 30)
things’.26 Thus, if Paris loves Helen, the relation of loving, according to Leibniz, has a merely mental nature. In the ‘real world’ there is the couple of individual substances called respectively ‘Paris’ and ‘Helen’, each with its internal properties, but there is not a ‘real property’ connecting them as a ‘bridge’ and corresponding to ‘loving’.
As regards the second problem, which was logical-syntactic in nature, Leibniz proposes a solution strictly connected with the previous one. Given a sentence of the general form ‘R(a, b)’ stating that a certain relation ‘R’ holds between two subjects ‘a’ and ‘b’, he reduces it to another sentence in which an auxiliary expression (‘in so far as’, ‘by that very fact’, etc.) connects two sentences in form, such as ‘a is P’ and ‘b is Q’, with ‘P’ and ‘Q’ each corresponding to one of a pair of correlated terms, such as lover–beloved, murdering–murdered, and so forth. Thus, in a text entitled Grammatical Thoughts, probably written in the autumn of 1678, Leibniz proposes the following analysis of the sentence ‘Paris loves Helen’:
(a) Paris is a lover, and by that very fact [et eo ipso] Helen is a loved one.27
Analogously, in the case of ‘Caius is killed by Titius’:
(b) In so far as Titius is murdering, therefore Caius is murdered.28
The two sentences (a) and (b), however, are not logically equivalent to the sentences that they are supposed to analyse: (a), for instance, is true in any case in which Paris loves a woman different from Helen and Helen is loved by someone distinct from Paris.29 And (b) could be true even though Titius does not murder Caius, but someone else, and Caius is murdered by a fourth person.
Leibniz was well aware that there are different types of relations and that the above-mentioned analyses apply mainly to asymmetrical relations.30 He therefore proposes a slightly different treatment for symmetrical
26 Kauppi (1960: 58–60); Mugnai (2012).
27 A VI, 4A: 114–15. Cf. Mates (1986: 213–18).
28 A VI 4A: 651.
29 Leibniz seems to interpret the connectives ‘and by this very fact’ (et eo ipso) and ‘in so far as’ (quatenus) as a kind of ‘very strong’ conditional, but he does not elaborate this point.
30 Relations, that is, for which, if it holds that a is R to b, from this it does not follow that b is R to a (i.e. R(a, b) does not imply R(b, a)).
relations. Given a sentence of the general form ‘a is similar to b’, with ‘a’ and ‘b’ names of individuals, he suggests that it should be analysed as ‘a is P’ and ‘b is P’, with ‘P’ designating a property common to both a and b. 31 The rationale behind this analysis (well known to Ockham and other medieval logicians) is that two or more subjects are similar if they share at least a property.32 In this case too, the truth of ‘a is P’ and ‘b is P’ is a necessary condition for the truth of ‘a is similar to b’, but the latter assertion is not logically equivalent to the conjunction of the first two sentences.
What Leibniz attempts to convey with these grammatical transformations is the idea that relations ‘supervene’ on the internal properties of two or more subjects belonging to the real world—a result of which we may be aware only by ‘thinking together’ the related items. Leibniz coins the word ‘concogitabilitas’ [‘co-thinkability’] to express the act of thinking with which we usually grasp relations.33
In the GI, relations are strongly connected with so-called ‘oblique terms’ (termini obliqui), i.e. terms in a case other than the nominative. Thus, Leibniz associates the treatment of oblique terms with that of partial terms, such as ‘same’ or ‘similar’. He calls ‘integral’ a term that may play the role of subject or predicate in a proposition. Homo [‘man’] and Caesar, for instance, are integral, whereas idem [‘same’] and similis [‘similar’] are partial. 34 Leibniz considers as partial both proper and common nouns, when they are in a case other than the nominative. The core of Leibniz’s distinction may be represented as follows:
Terms
Integral: Ens [‘Being’]; doctus [‘wise’]; Caesar; similis Alexandro [‘similar to Alexander’]; ensis Evandri [‘Evander’s sword’] . . . etc.
Partial: similis [‘similar’]; idem [‘identical’]; Evandri [‘of Evander’]; Alexandri [‘of Alexander’] . . . etc.
31 Cf. A VI, 4A: 11. 32 Cf. William of Ockham (1974: 281).
33 Cf. Kauppi (1960: 49); Mugnai (2012).
34 Obviously, in the proposition ‘similar expresses a relation’, the term similar plays the role of a subject, even though it is a partial term. Here, however, this term is not considered according to its ordinary meaning: it is mentioned but not used. A medieval logician would say that in this proposition it has ‘material supposition’.
Employing a Fregean expression, we may say that Leibniz distinguishes two kinds of terms: the saturated and the unsaturated. The latter are what he calls ‘partial terms’, those that are saturated are ‘integral terms’. Partial terms cannot enter the logical calculus unless they have been ‘saturated’. To saturate a partial term, Leibniz uses an ad hoc linguistic device. The proposition ‘Caesar is similar to Alexander’ becomes ‘Caesar is similar to a thing, which is Alexander’. This is the same logico-linguistic analysis of oblique terms proposed in a letter to Johannes Vagetius (1633–91) in which Leibniz suggests transforming the expression ‘qui discit graphicen’ [‘he who learns painting’] into the equivalent proposition ‘qui discit rem, quae est graphice’ [‘he who learns a thing which is painting’].35 Thus, in the GI, Leibniz transforms ‘the sword of Evander’ into ‘the sword, which is a thing of Evander’. In his essays on rational grammar Leibniz moves even a step further, reducing ‘the sword, which is a thing of Evander’ to ‘the sword, which is a thing Evandrian’ [ensis, qui est res Evandria], in perfect agreement with the fundamental tenet of his logical-grammatical research: not to differentiate between nouns and adjectives.
Since rational grammar should make explicit the intimate structure of relational sentences, arranging them in such a way that they can be processed in the logical calculus, it is quite natural that Leibniz should think of rational grammar as a prerequisite for logic. As Parkinson remarks:
Leibniz says that logic needs supplementation rather than expansion, and that this supplementation must come from ‘rational grammar’, which will transform relational arguments into forms which traditional logic can handle.36
This explains, the long series of remarks on language and grammar at the beginning of the GI, introducing the logical calculus properly speaking.37
35 Vagetius was a pupil of Jungius, and when the latter passed away, he inherited a remarkable quantity of unpublished texts on logic written by Jungius. Vagetius supervised a second edition of Jungius’ Logica Hamburgensis (1681) and added a long ‘Afterword’ to it (Vagetius 1977). Leibniz not only read the ‘Afterword’ and made some remarks on it (A VI, 4B: 1117–21), but even had the opportunity of having a look at Jungius’ unpublished papers, the majority of which were later lost in a fire.
36 LP: xx.
37 Schupp (1982: 144): ‘From the point of view of a systematic construction of the sciences, logic comes after rational grammar. This ordering is obeyed even in the GI, whose first part (without numbered paragraphs) discusses some basic grammatical distinctions.’
2.3 The Structure of the Proposition: Extension and Intension
As I observed above, Leibniz assumes that the elementary form of any statement is that of ‘subject-copula-predicate’, the subject being either a proper noun, like ‘Caesar’, ‘John’, or ‘Eva’, or a common noun, like ‘man’, ‘dog’, or ‘table’. The predicate is a name of a property, like ‘being red’, ‘being a man’, or ‘being a philosopher’. Subject and predicate are terms, and a term is the conceptual content associated with a Latin noun.38 The copula expresses the relation of containment or inherence, which may be considered from two different points of view, in so far as it concerns concepts or individuals (real or merely possible).39
In the Elements of a Calculus, composed before the GI, in 1679, the relation of containment is introduced as follows:
every true universal affirmative categorical proposition simply signifies some connection between predicate and subject (a connection based on the nominative case, which is what is always meant here). This connection is that the predicate is said to be in the subject, or to be contained in the subject.40
And again in the Elements Leibniz explains the difference between the two points of view according to which we can interpret a proposition:
(11) Two terms which contain each other but do not coincide are commonly called ‘genus’ and ‘species’. These, in so far as they compose concepts or terms (which is how I regard them here) differ as part and whole, in such a way that the concept of the genus is a part and that of the species is a whole, since it is composed of genus and differentia. For example, the concept of gold and the concept of metal differ as part and whole; for in the concept of gold there is contained the concept of metal and something else, e.g. the concept of the heaviest
38 Cf. A VI, 4A: 288 (LP: 39): ‘By “term” I understand, not a name, but a concept, i.e. that which is signified by a name; you could also call it a notion, an idea.’
39 Cf. A VI, 6: 486. 40 LP: 17–18 (translation slightly modified); AVI, 4A: 197.
among metals. Consequently, the concept of gold is greater than the concept of metal.
(12) The Scholastics speak differently; for they consider, not concepts, but instances which are brought under universal concepts. So they say that metal is wider than gold, since it contains more species than gold, and if we wish to enumerate the individuals made of gold on the one hand and those made of metal on the other, the latter will be more than the former, which will therefore be contained in the latter as a part in the whole.41
Thus, given, for example, the proposition ‘Every man is mortal’, these are the two different ways of interpreting it:
1. Every individual ‘falling under’ the concept of man belongs to the collection (aggregate, set, or class) of individuals falling under the concept being mortal.
2. The concept associated with the word ‘man’ has among its component parts the concept associated with the word ‘animal’.
In the New Essays, more than twenty years after the Elements, Leibniz comes back to the difference between the two points of view and gives a name to each of them:
This manner of statement deserves respect; for indeed the predicate is in the subject, or rather the idea of the predicate is included in the idea of the subject. . . . The common manner of statement concerns individuals, whereas Aristotle’s refers rather to ideas or universals. For when I say Every man is an animal, I mean that all the men are included amongst all the animals; but at the same time I mean that the idea of animal is included in the idea of man. ‘Animal’ comprises more individuals than ‘man’ does, but ‘man’ comprises more ideas or more attributes: one has more instances, the other more degrees of reality; one has the greater extension, the other the greater intension.42
41 LP: 20 (A VI, 4A: 199–200).
42 NE: 486. On the distinction between intension and extension, see Kauppi (1960: 7–12); Lenzen (1983); Swoyer (1995).
