Nothingness is said in many ways: Leibniz’s concepts of nothingness¹

O nada é dito de muitas maneiras: os conceitos de nada de Leibniz

Abstract:

This paper aims to survey and discuss Leibniz’s definitions and uses of nothingness (nihil, le rien, le néant), especially in his middle writings. There are arguably three senses of nothingness in Leibniz’s essays from the late 1670s to the mid-1680s: (i) the concept that does not contain any other concept, (ii) the complete notion which contains the negation of every perfection, (iii) names without associated concepts. Leibniz also often speaks of nothingness as a kind of (iv) general inexistence. I will argue that both concepts (i) and (ii) are consistent and that (ii) is a negative counterpart to God’s complete notion. I, then, present some arguments in favor of the view that (ii) does exist in God’s understanding, yet cannot be instantiated, which entails that there are two non-coextensive senses of possibility, i.e., consistency and instantiability: There would be a complete notion which is not the complete notion of a (possible) individual substance.

Keywords:
Nothingness; Leibniz; Privation; Complete Notions; Possibility

Resumo:

O presente artigo pretende levantar e discutir as definições e usos leibnizianos do termo ‘nada’ (nihil, le rien, le néant), especialmente em seus escritos intermediários. Parece haver três sentidos do termo, nos ensaios redigidos por Leibniz, do final da década de 1670 a meados da década de 1680: (i) o conceito que não contém nenhum outro conceito, (ii) a noção completa que contém a negação de cada perfeição, (iii) nomes sem conceitos associados a eles. Leibniz também frequentemente menciona o nada como uma espécie de (iv) inexistência geral. Defende-se que ambos os conceitos (i) e (ii) são consistentes e que (ii) é uma contraparte negativa da noção completa de Deus. Apresentam-se, então, alguns argumentos a favor da leitura de que (ii) existe, no intelecto divino, embora não possa ser instanciado. Disso se segue que há dois sentidos não coextensivos de possibilidade, isto é, consistência e instanciabilidade: Haveria uma noção completa que não é a noção completa de uma substância individual (possível).

Palavras-chave:
Nada; Leibniz; Privação; Noções Completas; Possibilidade

INTRODUCTION

As someone who subscribes to a version of the celebrated thesis that “nothing comes from nothing” (Leibniz, 1990, p. 435-436/A VI, 6, 435-436; 1999, p. 1635/A VI, 4, 1635), one might assume that Leibniz had little to no interest in the concept of nothingness. However, the tentative essays, in which he attempted to define the fundamental concepts of metaphysics from the late 1670s to the mid-1680s, suggest otherwise. In these writings, Leibniz seems to be almost as concerned with the concepts of nothing (nihil) and non-being (non-ens) as he was with their positive counterparts, i.e., something (aliquid) and being (ens), respectively. As a result, there seems to be, as I shall argue, many senses of ‘nothingness’ scattered throughout Leibniz’s body of work. This paper aims to survey these senses and discuss their relation to some of the most important concepts and theses of Leibniz’s logic and metaphysics. Such a discussion, however, will require a preliminary outline of his views on propositions, truth and negation.

According to Leibniz, a proposition is “[…] that which states what term is or is not contained in another” (Leibniz, 1999, p. 786/A VI, 4, 786).³ Universal affirmative propositions, such as ‘Every A is B’ or simply ‘A is B,’ state that a concept B is contained in the concept A. Universal negative propositions, such as ‘Every A is not B’ or simply ‘A is not-B,’ state that the negative concept not-B is contained in the concept A. Singular propositions are just universal ones whose subject-concepts are complete notions. Following the relations expressed in the square of opposition, particular affirmative propositions, ‘Some A is B,’ can be seen as negations of universal negative propositions, while particular negative ones, ‘Some A is not B,’ can be seen as negations of universal affirmative propositions. Thus, they respectively state that not-B is not contained in A (A is not not-B) and that B is not contained in A (A is not B).⁴

From this view on propositions follows Leibniz’s famous concept-containment account of truth, according to which, “[...] always, in every true affirmative proposition, necessary or contingent, universal or singular, the notion of the predicate is always in some way included in that of the subject - praedicatum inest subjecto-or else I do not know what truth is” (2009, p. 80/A II, 2, 80). Very important here is the distinction between universal negative ‘A is not-B’ and particular affirmative propositions ‘A is not B.’ Sometimes, Leibniz distinguished sharply between these two kinds of propositions. Sometimes, however, he did not distinguish them at all, but rather saw the negation of the whole proposition and the negation of its predicate as equivalent.⁵ For instance, Leibniz crossed out the following passage from his Primaria Calculi Logici Fundamenta:

Given any term, either A inheres in it (inest) or not-A inheres in it. If A does not inhere, then not-A will inhere, and conversely. Therefore, ‘A not inhering’ is equivalent to ‘not-A inhering.’ Or, A ∞ Ynot-B, and A not-∞ ZB are equivalent; or A ∞ A not-B, and A not-∞ AB are equivalent. Therefore, this is wrong (Ergo male, 1903, p. 237/C 237).

A more resolute statement of the distinction between propositional and predicative negation can be found in the following passage from De Negatione (1679):

It is one thing, therefore, to negate a proposition, and another to negate a predicate. Thus, we shall say: ‘non’ prefixed to the sign negates the proposition; ‘non’ prefixed to the copula negates the predicate, so that we may have a definite rule. [...] ‘non’ prefixed to the proposition signifies its contradictory; whereas ‘non’ prefixed to the copula negates the predicate (Leibniz, 1999, p. 300/A VI, 4, 300).

The inference ‘A is not-B → A is not B’ does hold if A is a consistent concept, but not the other way around (Leibniz, 1999, p. 213/A VI, 4, 213). Furthermore, Leibniz also accepted that the equivalence ‘A is not-B ↔ A is not B’ holds when A is a complete notion. This, in fact, is the source of an important definition of a complete notion, which we will discuss later in the paper:

Contradictory terms are those of which one is positive and the other is the negation of this positive, as ‘man’ and ‘not-man.’ Concerning these, the rule must be observed: if two propositions are presented about precisely the same singular subject, in which one of the contradictory terms is the predicate of one proposition, and the other of the contradictory terms is the predicate of the other proposition, then necessarily one proposition will be true and the other false. I say, however, of precisely the same subject; for example, this gold is a metal, this gold is not a metal (Leibniz, 1999, p. 217-218/A VI, 4, 217-218).

Leibniz’s fluctuation between strictly distinguishing and equating the negation of a whole proposition and the negation of its predicate is crucial for our goal, since, as we shall see in the next section, most of his definitions of the concept of nothingness are given by means of negative predications of one kind or the other. Let us, then, take a look at some of Leibniz’s definitions of his fundamental metaphysical concepts in his definitional essays from the late 1670s to the mid-1680s.

1 ‘ALIQUID,’ ‘NIHIL,’ ‘ENS,’ ‘NON-ENS,’ ‘RES’

As we saw earlier, Leibniz usually considers the concept of ‘nothing’ as opposed to that of ‘something,’ while that of ‘non-being’ is opposed to that of ‘being.’ Most of these concepts are typically defined by some kind of predication. Consequently, since Leibniz generally views predication in terms of containment, they can also be seen as defined in terms of which concepts they do or do not contain. Sometimes Leibniz elucidates the concept of ‘something’ by means of examples: “Something: A, likewise B, likewise C” (Leibniz, 1999, p. 306/A VI, 4, 306), “Something is A or B, etc.” (Leibniz, 1999, p. 875/A VI, 4, 875), and “Something: whatever can be thought, like A, or B, or C, or any other term whatsoever” (Leibniz, 1999, p. 938/A VI, 4, 938). The absence of a negation preceding the term is supposed to indicate a certain positive aspect of these terms. This is explicitly stated elsewhere, on occasions when Leibniz purports to define the concept by appealing to predications.

This happens, for instance, in De Ente, Existente, Aliquo, Nihilo et Similibus (1683-6 [?]), where we find the following definition: “Something is A, if it is established that A is either B or C or D, and so on. I understand B, C, D to be positive” (Leibniz, 1999, p. 570/A VI, 4, 570). This definition is closely related to one presented in Definitiones Notionum Metaphysicarum atque Logicarum (1685[?]): “A being or something is that to which a positive term applies [competit], as A, B, C, provided, namely, that in its explanation it is not to be resolved into a merely privative one” (Leibniz, 1999, p. 625/A VI, 4, 625). This suggests the view that whatever we would call ‘something’ should contain, at least, one positive concept, that is, a concept which “[…] can be distinctly thought without negation. That is, if A∞B, C, etc., and B∞D, E, etc., and C∞F, G, etc., as indicated above, and a negative term like non-L never arises, except if it is a negation of a negative” (Leibniz, 1999, p. 938/A VI, 4, 938).

‘Something’ is also eventually defined as applying to everything which is thinkable: “Something: that which can be thought” (Leibniz, 1999, p. 937/A VI, 4, 937). It is important to note here that ‘thinkable’ is conceived as generally not implying possibility or consistency. A contradictory concept is also conceived as thinkable: “Something: any term that can be thought, whether possible or impossible, e.g., A, B, C; and generally X” (Leibniz, 1999, p. 930/A VI, 4, 930). The property of encompassing even contradictory concepts is a recurring feature of the concept of something: “Term or Something A: anything that can be thought (as if a Being), even if perhaps it is not possible, e.g., a perpetual mechanical motion” (Leibniz, 1999, p. 934/A VI, 4, 934).⁶ Leibniz, thus, seems to allow not only for talk about contradictory things, but even thought about them.⁷

Only consistent concepts, however, can be distinctly thought: “Possible: whatever can be distinctly thought, or whose notion does not involve a contradiction” (Leibniz, 1999, p. 938/A VI, 4, 938). This also makes clear that a concept can be called ‘something’ even if it contains negative or privative concepts, for the definition of impossibility or inconsistency of a concept A requires contradictory concepts B and not-B, one being the negation of the other, to be both contained in A: “Non-Being or Impossible: that whose definition involves A and not-A, i.e., that which implies a contradiction” (Leibniz, 1999, p. 930/A VI, 4, 930). Finally, the connection with the notion of possibility also brings us to the relationship between the concepts of something and being. Leibniz generally regards these as distinct concepts, the former being wider in scope than the latter. This is, however, not always the case.

Leibniz usually associates the concept of being with the idea of possibility and the concept of non-being with that of impossibility. Possibility is, as customary, characterized in terms of conceptual consistency: “Being or possible is A, if, when substituting A for an equivalent, there never arises L, not L (i.e., a contradiction). If, however, it does arise, A will be impossible or a non-Being” (Leibniz, 1999, p. 875/A VI, 4, 875). He also characterizes being as that which can be understood or distinctly thought (Leibniz, 1999, p. 937-938/A VI, 4, 937-938). A notable exception is the Generales Inquisitiones (1686), where being is conflated with positive and non-being with merely and wholly privative (Leibniz, 1999, p. 740/A VI, 4, 740). As we have seen, the Definitiones Notionum Metaphysicae atque Logicarum (1685[?]) also identifies being and something. ‘Thing’ is also generally regarded as synonymous with being and possible,⁸ which justifies Leibniz’s use of ‘A is possible,’ ‘A is a being,’ and ‘A is a thing’ as different ways to assert the consistency of a concept.

The most recurrent definitions of nothingness, on the other hand, appeal to some kind of negative predication involving the concept: “If N is not A, and N is not B, and N is not C, and so on, N will be said to be Nothing” (Leibniz, 1999, p. 551/A VI, 4, 551). Leibniz usually views these repeated negative predications, which define the concept of nothingness, as being justified by, or even as elucidating, what he terms a common saying or axiom,⁹ according to which nihili nulla sunt attributa (‘of nothing, nothing is an attribute’ or ‘nothing has no attributes’): “This is what people commonly say, that nothing has no attributes” (Leibniz, 1999, p. 551/A VI, 4, 551). Some variation of this definition can be found in several of Leibniz’s definitional essays. However, in order to correctly interpret his definition of the concept of nothingness, one should first ask (a) what kind of negative predication is involved in the definition, and (b) what kind of concept is being (negatively) predicated of nothingness.

If we remember our discussion in the previous section, the two senses of negative predication (the negation of the whole proposition and the negation of its predicate), which Leibniz sometimes conflates and sometimes sharply distinguishes, are crucial for answering the question (a). Depending on which kind of negative predication we have in mind, the definition will pick out entirely different concepts. This concern is validated by a passage from De Calculo Analytico Generale (1678/1679[?]). There, after defining something in the following manner: “A, B, etc., is something. Let it be that A is B or A is not B; A and likewise B we shall call something” (Leibniz, 1999, p. 146/A VI, 4, 146), Leibniz adds: “Hence, I always want positive terms, and if there are any that are negative, I prefer to free them from negation by transferring them into the copula” (Leibniz, 1999, p. 146/A VI, 4, 146).

For one thing, in this passage, Leibniz is making general use of the negation externalization rule ‘A is not-B → A is not B.’ Now, this rule does, in fact, hold for consistent concepts, even if we distinguish propositional negations from predicative negations. It corresponds to the inference from a negative universal proposition E to a negative particular proposition O (the negation of the affirmative universal A) by subalternation. However, Leibniz’s general use of the rule, reducing every instance of ‘A is not-B’ to ‘A is not B,’ suggests he has something like the equivalence ‘A is not-B ↔ A is not B’ in mind, in the sense that he takes them to mean the same thing, the transference of the negation from the predicate to the copula entailing no loss in meaning. This means that all the concepts (negatively) predicated of nothingness are positive, but this might be only because every negation, applied to the predicate, was transferred to the copula.

This highlights the importance of considering whether Leibniz conflates propositional and predicative negation when interpreting his definition of nothingness in a given text. Since he went back and forth in distinguishing between propositional and predicative negations, it is not clear whether he had only one of these in mind or whether he tried out both of them on different occasions. Either way, both definitions, taken at face value, yield different and interesting results. One thing is clear: Leibniz did attempt to define nothingness by means of negative predicates. This happens in a few of his writings. In Definitiones: Aliquid, Nihil, Non-Ens, Ens (1688-9[?]), he states: “Nothing: not-A and not-B and not-C, etc., or generally not-X; hence nothing has no attributes” (Leibniz, 1999, p. 930/A VI, 4, 930). In Definitiones: Terminus vel Aliquid, Nihil (1688-9[?]), a similar formulation is available: “Nothing: not-A and not-B and not-C, etc.; or not-Y (hence nothing has no attributes or predicates)” (Leibniz, 1999, p. 934/A VI, 4, 934).

In Generales Inquisitiones (1686), Leibniz similarly defines non-being by means of attributions of negative predicates: “Non-being is that which is merely privative, or the privative of all, that is, non-Y; that is, not-A, not-B, not-C, etc.” (Leibniz, 1999, p. 740/A VI, 4, 740). Here, Leibniz is taking ‘non-being’ as a synonym for what he called ‘nothingness’ in both Definitiones: Aliquid, Nihil, Non-Ens, Ens (1688-9[?]) and Definitiones: Terminus vel Aliquid, Nihil (1688-9[?]), instead of regarding it as the direct opposite of being, understood as a possible or consistent concept. This synonymy, between nothingness and this alternative meaning of non-being, is clear from other writings where Leibniz mentions both of them in the definition. In Definitiones Notionum Metaphysicarum atque Logicarum, for instance, we find the following version of the definition of nothingness: “Nothing is that to which only a merely negative term applies; namely, if N is not A, nor B, nor C, nor D, and so on, so that no positive term can be found which is its predicate, then N is said to be nothing” (Leibniz, 1999, p. 625/A VI, 4, 625). He goes on to say that: “Thus, the common axiom, ‘non-being has no attributes,’ contains the definition of nothingness, or of non-Being, itself” (Leibniz, 1999, p. 625/A VI, 4, 625).

In De Mundo Praesenti (1684-6[?]), on the other hand, Leibniz defines non-being in the following way: “Non-being is that whose attributes are only negative. That is, if A is neither B nor C nor D, and so on to infinity, it will be Nothing” (Leibniz, 1999, p. 1506/A VI, 4, 1506). As we can see, both works seem to use ‘nothing’ and ‘non-being’ (in this alternative sense) intersubstitutively. Another crucial feature of these two works is that they verbally describe what Leibniz is intending with his formalization. He says in Definitiones Notionum Metaphysicarum atque Logicarum that “Nothing is that to which only a merely negative term applies” so that “[…] no positive term can be found which is its predicate” (Leibniz, 1999, p. 625/A VI, 4, 625). And, in De Mundo Praesenti, he says that nothingness is “[…] that whose attributes are only negative” (Leibniz, 1999, p. 1506/A VI, 4, 1506) This is important because we can see that Leibniz is formalizing what, in his verbal descriptions, he considers predications of negative concepts as negations of the copula.

This goes to show that Leibniz is here probably taking for granted the conflation between propositional negation and negation of the predicate ‘A is not-B ↔ A is not B,’ as suggested by the aforementioned passage from De Calculo Analytico Generale (1678/1679[?]). Now, since Leibniz describes a proposition with a negated copula, ‘N is not B,’ as N having a negative or privative predicate, one could consider every one of Leibniz’s definitions of nothingness by means of negative predications in light of his subscription to the equivalence ‘A is not-B ↔ A is not B.’ One could, then, regard the definitions by means of negative predicates as standard, and the ones employing negated copulas as a deviation due to Leibniz’s occasional subscription to the equivalence just mentioned. We would, then, ultimately end up with a single definition of nothingness by containment of negative concepts, instead of one by the failure to contain certain concepts.

Yet, since Leibniz’s subscription to the equivalence ‘A is not-B ↔ A is not B’ was, in fact, intermittent, it could also be argued that, if we distinguish propositional from predicative negations, we end up with two kinds of definitions, instead of reducing the ones with negated copulas to the ones with negative predicates. The two approaches to the definitions would, then, correspond to alternative attempts at analyzing the concept: one according to which the concept of nothingness fails to contain a multitude of concepts, and one according to which it contains a multitude of negative concepts. Let us now turn to the second question we raised: which concepts are these that are contained or not contained in the concept of nothingness? We know that, if the concept of nothingness is to be defined by the containment of certain concepts, these concepts ought to be negative, but one may still ask which negative concepts are contained in the concept of nothingness.

Different answers to the question may affect whether ‘nothingness’ consists in a single concept, with the properties described in its definition, or whether it is, in fact, a general (second-order) property which may apply to several (first-order) concepts, as it is the case with ‘something’ or ‘being’ (many concepts can be said to be something or a being insofar as they are respectively positive or consistent). I will argue in favor of the first alternative. Leibniz’s illustrative examples, “If N is not A, and N is not B, and N is not C, and so on, N will be said to be Nothing” (Leibniz, 1999, p. 551/A VI, 4, 551) or “Non-being is that which is merely privative, or the privative of all, that is, non-Y; that is, not-A, not-B, not-C, etc.” (Leibniz, 1999, p. 740/A VI, 4, 740) are obviously supposed to be not only infinite (“That is, if A is neither B nor C nor D, and so on to infinity, it will be Nothing,” 1999, p. 1506/A VI, 4, 1506), but also somehow exhaustive. Hence, the expression employed in Generales Inquisitiones to characterize the concept as omnium privativum: the concept either contains every negative concept or fails to contain each and every concept (at least of a given kind).

This would ensure that ‘nothingness’ is, according to both readings, a single concept satisfying the properties described in the definitions given in the corpus leibinitianum. It is also my understanding that the definition of the concept of nothingness, by means of containment of negative concepts, includes an implicit closure clause; that is, the concept of nothingness is that concept which contains every negative concept and no other concept. This is a key step in establishing the possibility or the consistency of the concept of nothingness in both these readings, as we will discuss later. If we take Leibniz’s definition of nothingness by means of negative predications, ‘N is not B,’ at face value, that is, as stating that a concept N is called Nothing if and only if every concept X is not contained in N, then the concept of nothingness would be the same as what Lenzen (2004, p. 28) calls the ‘empty concept,’ the concept which does not contain any other concept (but itself by the reflexivity of the containment relation).

Alternatively, we may consider Leibniz’s definition by means of negative predications as establishing that a concept N is called Nothing if and only if, for every positive concept X, X is not contained in N. This would make ‘nothing’ a general second-order property which applies to many concepts; that is, any one which contains only negative concepts, but not necessarily all of them. For instance, let us imagine that concept A is comprised only of negative concepts not-C and not-D, which, in turn, are both comprised only of negative concepts, and concept B is comprised of both not-C and not-D, but also of a third negative concept not-E, itself comprised only of negative concepts, neither not-E nor its components being contained in not-C or not-D. The concept of ‘nothingness’ applies equally to both A and B. It seems that this is Angelelli’s view (1977, p. 12).

In order to proceed in our analysis and explore Leibniz’s definition of ‘nothingness’ by means of predications of only negative concepts, we should examine what it means to contain only negative concepts. This is related to how the negation and conjunction or composition of concepts operate in Leibniz’s logic. If a concept A contains a negative concept not-B and also a different negative concept not-C, does it not also contain the complex concept notC⊕notB? Isn’t notC⊕notB a positive conjunctive concept of two negative concepts whose negation is not(notC⊕notB)? So, how exactly is a concept supposed to contain only negative or privative concepts? For one thing, it seems that, by a negative or privative concept, Leibniz does not merely mean a concept expressed by a term, whose main logical operator is a negation, and, by a positive concept, a term which does not satisfy this description. As we have already seen, a positive concept is one that can be expressed without resorting to any negative terms, and a negative concept is supposedly one that cannot.

This is crucial because, by means of Leibniz’s own algebra of concepts, one can always rewrite a complex term involving negations and conjunctions, like notA⊕notB, whose main logical operator is not negation, as a term which does have negation as its main logical operator, like not(not(notA⊕notB)) or even not(not(not(A⊕B)⊕not(A⊕notB))⊕not(not(A⊕B)⊕not(notA⊕B))), and vice-versa. If we can, by this process, rewrite the concept so as to get rid of negations, the concept is positive. Otherwise, it is supposedly (at least partially) negative. But Leibniz also speaks of purely negative or privative concepts, as we have seen. It seems that he uses this terminology in, at least, two senses. Purely negative concepts may be (a) immediate negations of positive concepts or (b) conjunctions of such negations (I will argue that this is the case of the concept of nothingness defined by means of predications of negative concepts).

It is in the first sense that, in the New Essays (1704), Leibniz seems to speak of “[…] the privative nature of rest” because “[…] all it involves is a denial of motion in the body” (Leibniz, 1999, p. 130/A VI, 6, 130). Regarding the second sense, on the other hand, I believe Leibniz regards conjunctions of only negative concepts as also merely negative or privative in a sense, even though their main operator is a conjunction. It is even possible that, before his calculi of real addition, he did not consider it an operator at all, but only the immediate combinations of concepts, as his notation by juxtaposition suggests. Just like a concept is called positive depending on which kind of concepts it contains, a concept can also be called purely negative if it is made up only of negative concepts, even if its main operator is not negation. The analogy here is probably the idea of sums of zeros being equal to zero: 0+0+...+0 = 0.

Furthermore, one should note, as Antognazza argues (2014, p. 132), that, in contrast with much of the tradition dating back to Aristotle (1831, p. 1004/1004a10-1004a31), Leibniz does not seem to distinguish between privation and negation (Leibniz, 1999, p. 405/A VI, 4, 405; 1999, p. 740/A VI, 4, 740). In what follows, I shall argue that there is a sense in which we can speak of nothingness as containing only negative or privative concepts. I think Leibniz’s commitment to his conceptual atomism can provide an adequate solution to this problem. He makes his subscription to the thesis mentioned above clear in the following passage from Calculus Ratiocinator (1679):

A primitive term or simple requisite is that which is conceived by itself, or which lacks another requisite. For example, suppose d is ae, and a is bc, and b is fg; but f itself cannot be further resolved, then f will be a primitive term. There are some primitive terms, for if nothing were conceived by itself, then nothing at all would be conceived. There are many (plures) primitive terms: otherwise, there would be no variety in composites (Leibniz, 1999, p. 277/A VI, 4, 277).

There is no doubt that Leibniz accepted the existence of simple or primitive concepts. Their existence is required by the existence of complex concepts, which are nothing but combinations of them. These primitive concepts are sometimes called perfections: “I call perfection every simple quality, which is positive and absolute” (1980, p. 578/A VI, 3, 578; 1980, p. 575/A VI, 3, 575). The privative concepts that are all contained in the concept of nothingness are the immediate negations of these simple concepts or perfections. The concept of nothingness can, thus, be understood as that concept which contains every privation, in the sense of the immediate negation of each perfection or simple concept (and consequently every conjunction thereof). In this sense, Leibniz’s concept of nothingness, defined by predications of negative concepts, is merely privative in the second distinguished sense.

Let us now handle Leibniz’s third sense of nothingness, which is, as we will see, very different from the other two. According to this third kind of definition, the expression ‘nothing’ applies to signs or names that are not associated with any thoughts. In Definitiones: Aliquid, Nihil, Opposita, Possibile (1688-9[?]), Leibniz introduces his definition: “Nothing: that which can be named but cannot be thought, as ‘Blitiri’” (Leibniz, 1999, p. 937/A VI, 4, 937). The example, common in Scholastic philosophy, is meant to be pure gibberish. Unlike something like ‘square circle,’ whose parts ‘square’ and ‘circle’ do convey meaning, the term ‘nothing,’ according to the definition above, is supposed to apply to a mere string of graphic signs or sounds totally devoid of meaning. Also, as we have discussed beforehand, Leibniz does allow not only for talk, but even obscure thought about contradictory things (non-beings). In this third sense, however, ‘nothing’ does not admit thought, obscure or distinct.

Finally, Leibniz also makes a fourth, more traditional, use of the term nothingness (le rien), according to which nothingness consists in a kind of general inexistence. In this sense, as with the preceding one, nothingness is not a concept. If, in the preceding sense, nothingness indicates a certain absence of meaning or of concepts associated with a name (Nomen sine Notione, 1999, p. 528/A VI, 4, 528), in this sense, nothingness is an absence of being, understood as absence of existence: “[...] if there had ever been nothing, there would always have been nothing, since a being cannot be produced by nothingness (le rien)” (Leibniz, 1999, p. 436/A VI, 6, 436). It is in this sense, especially, as in the passage just quoted, that Leibniz seems to subscribe to the maxim ex nihilo nihil fit (Leibniz, 1999, p. 1635/A VI, 4, 1635),¹⁰ in the sense of the causal inertness of this general inexistence, while elsewhere, as we will shortly discuss, he speaks of things emerging, participating, or being composed of God and nothingness.

I do not think these four senses of nothingness can be united into a single one. As we have seen, once one distinguishes negative predications from predications of negative concepts, one must also distinguish between (a) nothingness as the empty concept and (b) nothingness as the concept which contains the negation of every primitive concept. Both of these senses of nothingness, in turn, must be distinguished from the third sense according to which nothingness is a mere string of signs totally devoid of meaning, which does not allow for thinking,¹¹ while the first two are precisely defined by true predications. All of these are, on the other hand, distinct from the fourth sense, which is neither a concept nor indicates a mere absence of a concept associated with a sign, but rather a general inexistence. Moreover, as stated in the New Essays, Leibniz does allow that words, like ‘ignorance,’ do signify privative ideas, not merely the absence of ideas, because the “act of denial is positive” (Leibniz, 1999, p. 130/A VI, 6, 130; 1990, p. 276, A VI, 6, 276).¹²

2 NOTHINGNESS AS A COMPLETE CONCEPT, POSSIBILITY AND EXISTENCE

There are a couple of different kinds of definitions of complete notions available in Leibniz’s corpus. According to one of the most well-known, a complete notion is a concept A such that, for every concept B, A contains either B or not-B, but not both. A complete notion is thus, by definition, a consistent concept, because, given any pair of contradictory concepts, it contains either one or the other, but never both at once.¹³ An example of this kind of definition can be seen in De Perfecta Notione Substantiarum (1677), where Leibniz defines a complete notion as a concept so “[…] perfect, or such that in that concept there is contained an answer to everything that can be asked about the/a thing [de re]” (Leibniz, 1999, p. 1350-1/A VI, 4, 1350-1). Leibniz also highlights this definitional feature in a passage quoted above (Leibniz, 1999, p. 217-218/A VI, 4, 217-218).

Mates formulates a version of this definition in the following manner: “[…] a complete individual concept is a concept that contains, for every simple attribute, either that attribute or its negation, but not both” (1986, p. 63). Mates’ formulation appeals to Leibniz’s conceptual atomism; it defines a complete notion in terms of its containing simple concepts or their negations. One might think the definition we gave above is more general. It defines a complete notion by means of it containing concepts, simple or not, or their negations. However, if a concept satisfies Mates’ formulation, it also satisfies the unqualified version; that is, if it contains, for every simple concept, either that concept or its negation, then, it will also contain, for every concept, either that concept or its negation. This is because the containment of a complex concept is defined purely in terms of the containment of simpler concepts by means of negation and conjunction.

This means that nothingness, defined as the concept that contains the negation of every simple concept, is a complete notion and is, thus, trivially consistent. Nothingness, understood in this sense, contains only the negations of simple concepts, the negation of concepts that contain simple concepts, and all of their conjunctions. Thus, it contains only one of each contradictory pair of concepts. Hence, it is possible in the sense of being consistent. This would also make nothingness a sort of dual counterpart of the complete notion of God, that is, while the concept of God contains every and only perfections, the concept of nothingness contains every and only privations. In fact, this makes this sense of nothingness prone to the same argument Leibniz lays out in Quod Ens Perfectissimum Existit (1676?) for establishing the internal consistency of the concept of God,¹⁴ the missing link in Anselm’s and Descartes’ versions of the ontological argument.

If the concept of God is consistent because it contains only perfections, and perfections are all mutually compatible, the same can be said of nothingness and privations. Inconsistency arises rather when we combine concepts with their negations. This dual opposition between the concept of God and that of nothingness establishes a picture in which the complete notion of every possible creature, containing both perfections and privations, due to its creaturely limitation, stands between these two extremes of pure perfection and pure privation in proportion to its degree of reality: the more perfect being closer to God, and the less perfect to nothingness. Leibniz, however, goes beyond that picture. In De Organo sive Arte Magna Cogitandi (1679[?]), he actually presents a proposal according to which everything “emerges” (prodeant) from God himself, i.e., pure Being (Ente puro), and nothingness or privation (Leibniz, 1999, p. 158/A VI, 4, 158).

Leibniz seems to treat God and nothingness here almost as simple concepts (things conceived through themselves) by the combination of which everything else can be composed (componi) in analogy with his binary system, by means of which “[…] every number is expressed by unity and nothingness” (Leibniz, 1999, p. 158/A VI, 4, 158), that is, the binary digits, 1 and 0. Since Leibniz seems to be treating God and nothingness as simple concepts, one might think that nothingness, understood as the empty concept, fits the bill a little better. However, it is important to point out that, elsewhere, he also deals with the problem of reconciling God’s simplicity with the variety of ideas (1885, p. 576/G VI, 576).¹⁵ In a later work, called Dialogue Effectif sur la Liberté de L’Homme et L’Origine du Mal (1695), these views are somewhat resumed. There, however, nothingness (le Néant) is described as being infinite, eternal, and as having “[…] many attributes in common with God” (1948, p. 364/Gr 364).

Leibniz clarifies that nothingness is infinite in the sense that “[…] it includes an infinity of things” (1948, p. 364/Gr 364), which supports its interpretation as a complete notion containing infinitely many other concepts. The eternity of nothingness also corroborates the reading of nothingness as a full-fledged inhabitant of the understanding of God alongside every other complete notion and the eternal truths. In fact, the divine understanding is nothing, but the “[…] region of possibles” (2009, p. 49/A II, 2, 49), and nothingness is, as we have shown, consistent on the same grounds by which the concept of God is. Finally, one must read the statement that nothingness has many attributes in common with God as stating the dual opposition between them. Since the concept of God contains only perfections and nothingness contains only privations, they do not contain any concepts in common.

They are, however, structurally similar, as one is the negative or positive counterpart of the other: they mirror each other and, thus, share second-order attributes (such as consistency itself). In this dialogue, Leibniz speaks of God and nothingness “[…] entering the composition of things” by a kind of participation, that is, through the perfections or privations they have in common with God and nothingness respectively (1948, p. 365/Gr 365). However, the analogy with the binary system remains: “[…] in arithmetic, zeros joined to ones make up different numbers, such as 10, 100, 1000” (1948, p. 364/Gr 364). Furthermore, this parallel between a proof of the consistency of the concept of God and one of nothingness naturally raises the question of the connection between nothingness, thus understood, and existence.

Leibniz’s argument of consistency is supposed to bridge the gap in the ontological argument and, thus, provide a sound a priori proof of the existence of God: the concept of God contains all and only perfections. Existence is a perfection. Thus, it contains existence. It is also consistent. Hence, God exists. It seems like the duality between the concept of God and nothingness would provide us with a correlative proof of inexistence: nothingness contains all and only privations. Non-existence is a privation. Thus, it contains non-existence. Hence, nothingness only exists as a concept; it is not instantiated. Therefore, there is a kind of mismatch between the possibility of nothingness, understood as internal consistency, and the one understood as possible existence or instantiation. It is not merely that God would not create nothingness because, being wholly privative, it would not add anything to the perfection of the actual world. One cannot even picture what it would be like, for an existent entity, to satisfy the concept of nothingness.

This shows not only that internal conceptual consistency and possible instantiation are not coextensive, but also that there is no strict one-to-one correspondence between complete notions and possible individual substances. Consequently, one can also say that the definition of a complete notion, as a concept which contains, for each pair of contradictory concepts, either one or the other but not both, is not coextensive with the definition of a complete notion as a concept which contains every property of a possible individual. Perhaps one should distinguish between complete notions in general and complete notions of an individual substance, as Leibniz sometimes specifies. Every possible substance has a complete notion, but not necessarily the other way around. This is somewhat problematic because Leibniz does seem to suggest that there is such a one-to-one correspondence: “There can be as many singular substances as there are distinct combinations of all compatible attributes” (Leibniz, 1999, p. 306/A VI, 4, 306).

One might respond that the term ‘attribute’ has a somewhat positive connotation, so that it would be an abuse to count a purely privative complete notion as a combination of compatible attributes despite its completeness and consistency. The inexistence of nothing as an actual being, alongside its purely conceptual existence, is actually important for Leibniz in order for him to answer the charges of Manicheanism against his thesis that there is an “infinite cause capable of counterbalancing the influence of divine goodness” (1885, p. 604/G VI, 604). Leibniz writes in the Dialogue: “B: You yourself will acquit me of this charge of Manicheanism when I name this other principle. A: Then please name it now, sir. B: It is nothingness” (1948, p. 364/Gr 364). One can say that things arise from God and nothingness only at a conceptual level; as far as actual things go, the phrase ‘ex nihilo nihil fit’ still stands.

CONCLUDING REMARKS

There is, perhaps, a way in Leibniz’s mature metaphysics to reestablish a one-to-one correspondence between complete notions and possible substances by means of the concept of a bare monad (monade toute nue), that is, monads endowed with perception alone. The perceptions of bare monads are confused or indistinct. They are not distinct enough to configure sensation and memory, as the souls of animals have. Rather, they live in a state of stupor (étourdissement). In works like the Monadology (1714) and the Principles of Nature and Grace (1714), Leibniz frequently associates degrees of reality or perfection with the degrees of distinction in perceptions; a substance is “[…] perfect to the extent that it possesses distinct perceptions” (1885, p. 604/G VI, 604). According to Leibniz, every substance perceives everything, albeit confusedly. We do not distinguish ourselves from God in terms of what we perceive. We perceive all the same things as God.

However, God, being absolutely perfect, does perceive everything with a maximum degree of distinctness, while we do so only somewhat confusedly. The duality between God and nothingness might suggest that the latter is actually the complete concept of a bare monad whose perceptions have the least degree possible of distinction. This would reinstate the one-to-one correspondence between complete notions and substances. It is questionable, however, whether there is indeed a unique maximum possible degree of confusion, that is, whether perception admits a least degree of distinction while remaining a state that represents “multiplicity (multitude) in a unity” (1885, p. 608/G VI, 608). Be that as it may, Leibniz does not have such elements as the concept of bare monad available to him in his middle writings, so that he would have been committed to an uninstantiable complete notion. Also, as we have seen, this inexistence has an important role, that of providing him with a defense against the charge of Manicheanism.

REFERENCES

• ANGELELLI, I. En Torno Al Principio ‘Nihili Nulla Sunt Attributa’. Anuario Filosófico, v. 10, n. 2, p. 9-17, 1977.
• ANTOGNAZZA, M. R. Metaphysical Evil Revisited. In: NEWLANDS, S.; JORGENSEN, L. (orgs.). New Essays on Leibniz’s Theodicy. Oxford: Oxford University Press, 2014. p. 112-134.
• ARISTOTLE. Aristotelis Opera. Edited by Immanuel Bekker. Berlin: Georg Reimer, 1831 (V. 8).
• KAUPPI, R. Über die Leibnizsche Logik mit besonderer Berücksichtigung des Problems der Intension und der Extension. Helsinki: Acta Philosophica Fennica, 1960.
• LACERDA, T. A Filosofia Expressiva de Leibniz. São Paulo: EDUSP, 2025.
• LEIBNIZ, G. W. Die Philosophischen Schriften von Gottfried Wilhelm Leibniz. Berlin: Weidman, 1885. [G] (V. VI).
• LEIBNIZ, G. W. Opuscules et fragments inédits de Leibniz. Edited by Louis Couturat. Paris: Presses Universitaires de France, 1903.
• LEIBNIZ, G. W. Textes Inédits. Edited by Gaston Grua. Paris: Presses Universitaires de France, 1948. [Gr]
• LEIBNIZ, G. W. Sämtliche Schriften und Briefe. Akademie, 1980. [A] (Series VI, V. 3).
• LEIBNIZ, G. W. Sämtliche Schriften und Briefe. Akademie, 1990. [A] (Series VI, V. 6).
• LEIBNIZ, G. W. Sämtliche Schriften und Briefe. Akademie, 1999. [A] (Series VI, V. 4).
• LEIBNIZ, G. W. Sämtliche Schriften und Briefe. Akademie, 2009. [A] (Series II, V. 2).
• LENZEN, W. ‘Non est’ non est ‘est non’: Zu Leibnizens Theorie der Negation. Studia Leibnitiana, v. 18, n. 1, p. 1-37, 1986.
• LENZEN, W. Leibniz’s Logic. In: GABBAY, D.; WOODS, J. (org.). The Rise of Modern Logic: From Leibniz to Frege (Handbook of the History of Logic. Amsterdam: Elsevier, 2004. p. 1-84 (Vol. 3).
• MATES, B. The Philosophy of Leibniz: Metaphysics and Language. Oxford. Oxford University Press, 1986.

• ¹
  I would like to thank Professors Guido Imaguire, Ulysses Pinheiro, Vivianne de Castilho, Edgar Marques, Tessa Lacerda, and Abel Casanave for their suggestions and comments on earlier stages of my research on the topic, as well as CAPES for funding my doctoral studies. I am also grateful to reviewers A and B, and to Editor Marcos Antônio Alves for their feedback on an earlier draft of this paper.
• ³
  This formulation applies exclusively to propositions in tertii adjecti form. However, every proposition in tertii adjecti form can be converted into a proposition in secundi adjecti form and vice versa (Leibniz, 1999, p. 780/A VI, 4, 780; 1999, p. 749/A VI, 4, 749).
• ⁴
  In their secundi adjecti form, however, the roles are switched: universal propositions are expressed as negations of particular propositions.
• ⁵
  See Lenzen (1986).
• ⁶
  See also Leibniz, 1999, p. 938/A VI, 4, 938.
• ⁷
  This should be understood in connection with Leibniz’s account of nominal definitions as not establishing the consistency of a given definiendum, and with the distinction between notion and idea in paragraph XXV of the Discourse on Metaphysics (Leibniz, 1999, p. 1569/A VI, 4, 1569), not to be confused with the distinction made in paragraph XXVII of the same work (Leibniz, 1999, p. 1572/A VI, 4, 1572).
• ⁸
  Eventually, however, Leibniz identifies ‘thing’ and ‘something’ (Leibniz, 1999, p. 1499/A VI, 4, 1499). He also alternatively defines ‘thing’ as a congruent phenomenon (Leibniz, 1999, p. 570/A VI, 4, 570).
• ⁹
  See Angelelli (1977).
• ¹⁰
  For an in-depth analysis of this sense of nothingness, see Lacerda (2025, p. 107-117).
• ¹¹
  On occasion, Leibniz seems to link this definition of nothingness according to which ‘nothing is whatever cannot be thought’ but only named with his characterization of nothingness by means of negative predications: “Nothing is that from which is removed whatever can be thought, so that it is not A, nor B, nor C, nor D, nor etc.” (Leibniz, 1999, p. 938/A VI, 4, 938). It seems like we are taking away every mark of a concept so that, in the end, we are not left with anything, that is, with no concept at all, in accordance with the interpretation of nothingness as a name with no corresponding concept. This is, in a sense, very similar to what we get in the interpretation of nothingness as the empty concept. In the latter, however, we would have true negative predications holding of a concept that does not contain anything but itself.
• ¹²
  As we will see in the next section, (b) is actually not only a consistent concept, which should give it a place in the divine understanding, but also a complete notion.
• ¹³
  On the consistency requirement for complete notions, see Kauppi (1960, p. 168).
• ¹⁴
  See also Leibniz, 1999, p. 626/A VI, 4, 626.
• ¹⁵
  On this issue, see Lacerda (2025, p. 102-107).
• Data availability statement:
  The dataset for this article is available in the SciELO Dataverse of the Trans/Form/Ação Journal, at the link: https://doi.org/10.1590/0101-3173.2026.v49.n1.e026002

• Responsible editor:
  Marcos Antonio Alves

Data availability

The dataset for this article is available in the SciELO Dataverse of the Trans/Form/Ação Journal, at the link: https://doi.org/10.1590/0101-3173.2026.v49.n1.e026002

Publication Dates

• Publication in this collection
  06 Feb 2026
• Date of issue
  2026

History

• Received
  02 Oct 2025
• Accepted
  07 Nov 2025
• Reviewed
  10 Nov 2025
• Published
  12 Jan 2026