A DEFENSE OF LEIBNIZ’S MAIN DEFINITION OF COMPLETE CONCEPTS

Abstract

In the present paper, I intend to show the role in Leibniz’s overall philosophy of his definition of a complete notion

as a concept that contains every predicate of a certain subject. I will try to argue that this definition consists in an exoteric and more intuitive or accessible device directed to those unfamiliar with the intricacies of his views on logic and also that it plays a pedagogical role in his writings. In addition, I intend to explain away the seeming mismatch between this definition and that of individual substance, as well as Leibniz’s definitions of truth and proposition based on the relation of concept containment. In doing so, I will try to show that the definition above of a complete notion provides an important argument for the existence of such notions based on our ordinary talk about individuals and also how my proposal deals with Donald Rutherford’s problem of this definition’s extensional correctness vis-a-vis accidents.

Keywords:
Leibniz; Complete Notions; Proposition; Truth; Exoteric

1 - Proposition, Truth, and Conceptual Containment

In Generales Inquisitiones (hereafter GI) §195, Leibniz presents us with a definition of proposition according to which “A proposition is that which states what term is or is not contained in another” (A, VI, 4, 786/LEIBNIZ, 2021, 131).¹ This concise definition entails one of Leibniz’s most controversial views, namely, his famous Concept Containment Theory of Truth (hereafter CCTT), which is usually formulated as stating that, in every true proposition, the predicate is contained in the subject. Leibniz does in fact in some passages use either this exact formulation or something very akin to it, even in some of his major writings. In Principium Scientiae Humanae, for instance, he says that “True is the proposition whose predicate is contained in the subject” (A, VI, 4, 671). In his Discourse on Metaphysics (hereafter DM) §8, he also says:

Now, it is evident that all true predication has some ground in the nature of things, and when a proposition is not identical, that is to say, when the predicate is not explicitly contained in the subject, it must be contained in it virtually, and this is what the philosophers call inesse [emphasis in the original]. Thus the term of the subject must always include that of the predicate, so that he who understood perfectly the notion of the subject would also judge that the predicate belongs to it. (AA, VI, 4, 1540/LEIBNIZ, 2020, 13, emphasis added, except where noted.)

However, in other writings, he is far more restricted and offers only a qualified formulation of the theory. In his correspondence with Arnauld, in discussing precisely the summary to his DM, he says: “[...] 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” (AA, II, 2, 80/LEIBNIZ, 1967, 63, emphasis added, translation slightly modified). Likewise, in the Principium Scientiae Humanae, he later adds that “This takes place in every true affirmative proposition, universal or singular, necessary or contingent” (A, VI, 4, 671). Thus Leibniz limits his CCTT to affirmative propositions only, and he also usually does not mention particular propositions in his full description of the theory.² This is certainly for the best, for, if the predicate was contained in the subject even in the case of, say, negative propositions, a clear problem would arise. As Leibniz states in GI §42: “‘A contains B’ and ‘A does not contain B’: of these propositions, one is true and the other false, i.e. they are opposites” (A, VI, 4, 55/LEIBNIZ, 2021, 71). But, if we accept that, in every true proposition, the predicate is contained in the subject, then it seems that assuming that the proposition ‘A does not contain B’ is true would entail that the predicate, i.e., B, is contained in the subject, i.e., A. This would make the opposite proposition, i.e., ‘A contains B’, true and, consequently, make ‘A does not contain B’ false, contradicting our hypothesis.

Moreover, this unrestricted version of CCTT would render negation as a proper propositional connective, i.e., one that negates the whole proposition as opposed to one that negates only the predicate, entirely unavailable to Leibniz, as Castañeda seems to think it was:

One of the fundamental principles in Leibniz's philosophy is this: "In every proposition the (concept of the) predicate is included in the (concept of the) subject." [...] Thus, there cannot be true negative propositions, i.e., true propositions in which negation has the largest possible scope. Hence, for profound philosophical reasons Leibniz is committed to the view that negation cannot be, in the end, a genuine propositional connective, but only a term operator. (CASTAÑEDA, 1976, 483)

It is true that Leibniz sometimes identified the negation of the predicate with the negation of the whole proposition, in Lenzen’s notation: ¬(AεB) ↔ Aε¬B.³ However, he also seemed to have corrected himself on and again on different occasions.⁴ Furthermore, this problem does not arise if we take into account Leibniz’s qualified formulation of the CCTT. It is not the case that a negative proposition is true if and only if its predicate is contained in its subject because what this kind of proposition states is that its predicate is not contained in its subject. Rather, a negative proposition should precisely be true if and only if its predicate is not contained in its subject.

On the other hand, as far as particular propositions go, Leibniz explicitly says that “If I say ‘some man is a denier’, man does not contain ‘denial’” (AA, VI, 4, 762/LEIBNIZ, 2021, 85). However, he still occasionally thinks of particular propositions in terms of the containment relation. A particular affirmative proposition ‘Some A is B’ can be said to be true, according to Leibniz, if and only if A contains B by means of what Adams terms a consistent addition (ADAMS, 1994, 60). And a particular negative proposition ‘Some A is not B’ can be said to be true if and only if A contains not-B by means of a consistent addition. Given the relations in the square of opposition, particular propositions can also be seen as (propositional) negations of universal propositions. Universal propositions are in turn simply of the form ‘A is B’-affirmative-and ‘A is not-B’- negative-, which mean the same as ‘A contains B’ and ‘A contains not-B’ respectively. Thus particular propositions can be seen as having the following form: ‘A does not contain not-B’/‘A is not not-B’ (particular affirmative) and ‘A does not contain B’/‘A is not B’ (particular negative).

Finally, for Leibniz, singular propositions also say of concepts that one is either contained or not contained in the other. He in fact argues for the existence of such concepts from his definition of true proposition: “And that there is such a concept is evident from the definition of a true proposition explained a little earlier” (AA, VI, 4, 553). Singular propositions are just universal propositions whose subject is a certain type of concept called a complete notion. If I say that ‘Barlach is a sculptor’, what I am really saying is that the concept ‘sculptor’ is contained in the concept of Barlach or his complete notion: “[...] when we say that Alexander is robust, we do not wish to say anything but that robust is contained in Alexander’s notion, and the same goes for every other one of his predicates” (A, VI, 4, 553). Also, in the case of singular propositions, Leibniz seems to say that the propositional negation ‘A does not contain B’ and the predicative negation ‘A contains not-B’ are equivalent (G VII, 211). We will also return to this point in the next section, where we will discuss what complete notions are.

2 - Leibniz’s Definitions of Complete Concepts

In explaining what he means by a ‘complete notion,’ Leibniz gives us a couple of different answers. Wolfgang Lenzen, in his article, Logical Criteria for Individual(concept)s (2004b), provides us with a survey of different kinds of such answers, which correspond to different strategies of definition-or criteria, as Lenzen calls them-of such concepts on Leibniz’s part. He also offers what he calls contemporary reconstructions of such definitions.

Not all of them were discussed by Leibniz himself though. One of them specifically, i.e., the criterion based on Sommers’ and Englebretsen’s Wild Quantity Thesis (hereafter WQT)⁵, was explicitly put forward by Lenzen. The WQT is the claim according to which any singular proposition ‘A is B’ is equivalent to both the particular ‘Some A is B’ and the universal ‘Every A is B.’ Lenzen’s definition can roughly be read as saying that a complete notion is a concept A that has the property of containing a concept B if and only if there is a concept C such that AC is consistent and contains B (LENZEN, 2004b, 106):

$I\left(B\right)\leftrightarrow P\left(B\right)\wedge \forall Y\left(P\left(Y\right)\wedge Y\epsilon B\to B\epsilon Y\right)$

Although it is arguable that Leibniz does in fact subscribe to WQT-and both Englebretsen and Lenzen do make compelling cases for this reading⁶ -, Leibniz never presented this criterion for complete notions as his own. Lenzen is fully aware of that. The other definitions are either explicitly given by Leibniz in his body of work or partial reconstructions of such definitions. In this paper, we will focus only on these and will not further discuss Lenzen’s definition here.

Before discussing these criteria, however, Lenzen calls attention to a requirement of such concepts that is usually not explicitly mentioned in Leibniz’s accounts but is of the utmost importance. This we can call the Kauppian requirement, as it seems to have been first noted by Raili Kauppi in her important study Über die Leibnizsche Logik (1960, 168). This condition establishes that every complete notion is conceptually consistent or, in Leibniz’s own terminology, possible. To say that a concept A is possible is, according to Leibniz, to say that A does not contain, for any concept B, both B and its negation not-B (AA, VI, 4, 749). Another way to look at the Kauppian requirement is to think that it establishes that (conceptual) consistency is a mark of the second-order concept of (conceptual) completeness. Now that we have established this important condition, let us take a look at some of Leibniz’s definitions of complete notions.

2.1 - The Superfluity Definition

The first definition we will explore is based on the idea of superfluity of an added concept. I will call this definition ‘superfluity definition.’ Leibniz here attempts to define a complete notion A by means of its property of, for any concept B, the conjunction AB being either redundant or inconsistent. This means that any concept B that would be added to A as a new mark would either (a) be superfluous, repeating some information already contained in A-and ultimately be eliminable by the idempotency law AA=A (AA, VI, 4, 782)-or (b) result in an inconsistent concept, contradicting some information contained in A, for instance, not-B. One can grasp the idea behind this definition thinking of a complete notion as being ‘full to capacity,’ that is, anything you can add is either already in there or makes the whole thing fall apart. Leibniz presents this definition in GI

§72:

Therefore, suppose that we have BY with Y any indefinite term, which is superfluous, i.e. so that ‘some Alexander the Great’ and ‘Alexander the Great’ are the same, then B is an individual. <If there is a term BA [si sit terminus BA] and B is an individual, A will be superfluous, i.e., if BA=C, then B=C>. (AA, VI, 4, 762/LEIBNIZ, 2021, 85)

However, according to the Kauppian requirement, not every concept that has that property is a complete notion, but only the consistent ones. This is important because inconsistent concepts satisfy this property trivially since they contain every concept.⁷ Thus, every added concept is both superfluous and maintains the inconsistency of the original concept, thus also trivially resulting in an inconsistent concept. (The fact that inconsistent concepts contain every concept also entails that, in reality, there is only one inconsistent concept, since they all contain the exact same concepts, i.e., all of them, and so have precisely the same marks.) We would then get the following reconstruction (LENZEN, 2004b, 103):

$I\left(B\right)\leftrightarrow P\left(B\right)\wedge \forall Y\left(P\left(Y\right)\wedge Y\epsilon B\to B\epsilon Y\right)$

Lenzen’s exegetical argument for the introduction of the Kauppian requirement here is based on reading “si sit terminus BA” not as “If there is a term BA” or “Let there be a term BA” but as “Let the term BA be”, that is, “Let there be a consistent term BA.” This is rooted in the fact that Leibniz frequently expresses the consistency of a concept A not only by saying ‘A is possible’ but also simply by saying ‘A is.’⁸

2.2 - The Disjunctive Definition

We can also define a complete notion as a concept A such that, for every concept B, A contains either B or not-B, but not both. That is, a complete notion is such that it contains, for every pair of contradictory concepts, either one or the other. I will call this the ‘disjunctive definition.’ This formulation is largely due to Mates (1986, 63). However, Leibniz does say something very close to it in a passage from De Perfecta Notione Substantiarum (1677), where he characterizes a complete notion as a concept that is “perfect, or such that in that concept there is contained an answer to everything that can be asked about the/a thing (de re)” (AA, VI, 4, 1350-1). Also, the Kauppian requirement is satisfied here simply by reading the disjunction in the definition as exclusive.

2.3 - The Subject Containment Definition

The next definition is based on the property of a complete notion of containing every concept in which it is contained. Let us call this definition the ‘subject containment definition.’ According to Leibniz, if a concept B contains a complete notion A, then A also contains B, so that, by the antisymmetry of the containment relation [(PεQ∧QεP)→P=Q] (AA, VI, 4, 753), A and B are actually identical. Leibniz presents this definition, for instance, in the following passage of Notationes Generales:

Furthermore, a singular substance is one that cannot be said of anything else. Or if a singular substance is said of something else, they will be the same. Specifically, if, from the fact that A is B, it can also be inferred that B is A, or that B and A are the same, then A or B will be called a singular substance, or a thing subsisting by itself, so that, if it is said that Peter is the Apostle who denied Christ since it is established that there is only one such Apostle, or because it can be inferred conversely that the Apostle who denied Christ is Peter, therefore the Apostle who denied Christ will be a person or singular substance. (AA, VI, 4, 554-5)

Again, this property is also trivially satisfied by the inconsistent concept. Since the inconsistent concept contains every concept, it will also contain any concept that contains it and will thus be identical to it. Hence, the Kauppian requirement comes into play again. A complete notion can thus be defined as a consistent concept A such that, for every concept B, if B contains A then A contains B. In Lenzen’s notation (LENZEN, 2004b, 105):

$I\left(B\right)\leftrightarrow P\left(B\right)\wedge \forall Y\left(P\left(Y\right)\wedge Y\epsilon B\to B\epsilon Y\right).$

Finally, let us move on to Leibniz’s remaining approach to defining complete notions. I shall call it the “substance expression definition.” This definition will be the main focus of this paper.

2.4 - The Substance Expression Definition

Leibniz’s substance expression definition is probably his most well-known and acclaimed definition of complete notion. This is also definitely his go-to definition. Most of Leibniz’s attempts at defining complete notions are some kind of variation of it. A pretty intuitive reading of this definition is to see it as defining a complete notion as a concept that contains every property of a given (either actual or merely possible) substance or individual. We encounter a definition of this kind in the following passage of the Notationes Generales: “[...] if the term A involves all the terms B and C and D, etc., which can be said about the same thing, the term A will express the singular substance itself, or the concept of the singular substance is a complete term containing everything that can be said about it [de illa dici possunt]” (AA, VI, 4, 553). The Latin text of the passage is pretty clear about how the complete notion or term contains all the predicates of the substance, ‘singular substance [in the accusative, substantiam singularem]’ being the nearest feminine expression preceding the pronoun ‘illa.’ This will come in handy later. Leibniz follows with an illuminating example. He says:

Thus, if someone is robust, and fierce, and learned, and king, and leader of the army, and victor of the battle of Arbela, and other such things that are said about Alexander the Great; surely God, seeing the singular essence of Alexander the Great, will see some complete concept in which all these things are virtually [virtute] contained, or from which all these things follow. (AA, VI, 4, 553)

The example highlights the point I am trying to make here, namely, that the complete notion of Alexander contains everything that is said of him, the substance, the individual, or the person. Leibniz is clearly distinguishing Alexander the Great himself from his complete concept, and he clearly states it is Alexander’s predicates (being robust, fierce, learned, king, etc.) that are contained in his complete notion. This reading of the definition in hand is in accordance with Mates' (1986, 62-3) and Lenzen’s (2004b, 88). The latter reconstructs the definition in the following manner: “C is the complete concept of the individual c iff for every concept (or attribute) B: C contains B iff B(c).” However, he then goes on to reject this definition. I will present his argument for doing so in the next section and then try to defend Leibniz’s definition.

3 - Lenzen’s Argument

The substance expression definition, read in this way, surely stands out from the bunch. For one, it is the only definition of complete notion that does not define it through a relation that holds between it and other concepts. Rather, it defines such (complete) concepts by means of a relation that holds between them and another kind of entity, namely, ‘things’ or ‘singular substances.’ This is in fact why Lenzen, in discussing this kind of definition, argues that it does “not adequately reflect Leibniz’s logical conception of an individual’s complete concept” (LENZEN, 2004b, 89-90). To back up his claim, he puts forward the following argument:

[...] the logical relation of predication (as it is somewhat hidden behind the brackets of the formula B(c)), is not, as such, available for Leibniz. Leibniz never makes any syntactic distinctions between a singular term and a general term, nor does he see any structural differences between a universal predication like ‘(Every) Man is mortal’ and a singular predication like ‘Caesar is mortal’. Both sentences, for Leibniz, have one and the same logical form: ‘C is B’, or ‘C contains B’. In what follows this fundamental relation of conceptual containment shall be abbreviated by CεB. Hence in Leibniz’s logic the predication B(c) must equally be represented by CεB, so that the decisive criterion of I ND 1, “C contains B iff B(c)”, becomes the undiscriminating tautology: CεB iff CεB! (LENZEN, 2004b, 89-90)

As we have seen earlier, Leibniz sees propositions as asserting either that a concept is or is not contained in another and that he treats singular propositions as a special case of a universal one: A singular proposition is just a universal proposition whose subject is a complete notion, which, in turn, is just a special kind of concept. So it seems like Leibniz’s propositions are, in contemporary terminology, second-order propositions. They talk about concepts in terms of the (second-order) relations of containment and non-containment. They only talk about particulars or individuals indirectly, that is, by means of concepts-especially their complete notions. However, if this is the case, as Lenzen stresses, it seems that a certain conception of predication is unavailable to Leibniz.

Leibniz can only talk about predication as basically another name for the containment relation, the subject and the predicate being the relata of such relation, namely, the container and the containee respectively. There seems to be no room in Leibniz’s logic for the notion of predication understood as the attribution of a first-order concept to a singular term in the contemporary sense, that is, something like a proper name or a demonstrative. As we have said earlier, when we say that Alexander is robust, we only mean that the complete notion of Alexander contains the concept of being robust. But then it seems that our substance expression definition of complete notion is lost.

If we are to define a complete notion of a given (possible) individual a as a concept A such that, for every concept B, a is B if and only if A contains B, the copula ‘is’ in ‘a is A’ must express a first-order predication to the individual a, not containment in its complete notion. Otherwise, as Lenzen has shown, if we take ‘a is B’, to mean ‘A contains B’, the definition is trivial and is satisfied by every concept, for it only says that a complete notion A of an individual a is a concept that contains every concept it contains. But if this is so, then it seems that Leibniz’s most employed and famous-perhaps also the most intuitive and approachable-definition of complete notion is compromised. In the next section, I will try to defend it.

3.1 - The Role of the Substance Expression Definition in Leibniz’s Corpus

To fully and properly appreciate the importance of Leibniz’s substance expression definition of complete concepts, one must look into its place and role in his overall philosophy. To do so, I think one must take into consideration the distinction between the exoteric and esoteric or acroamatic modes of writing or philosophizing. This is, of course, a distinction that Leibniz himself makes use of. The terms ‘esoteric’ and ‘exoteric’ concern, in their root, something relating to or belonging to an inner or outer circle respectively. In this sense, esoteric modes of writing or philosophizing have roughly to do with a more rigorous and precise way of dealing with a given subject, which is proper-so to speak-to the ‘initiated’, whereas their exoteric counterpart consists of more ‘popular’ or ‘accessible’ ways of handling things.

I think Leibniz’s substance expression definition of complete notion can be seen as a largely exoteric device. As I see it, this kind of definition basically addresses someone standing outside of Leibniz’s philosophy and who is not familiar with the details and intricacies of his logic but who is still well-versed in traditional Aristotelian logic. I believe this kind of definition is meant as a clear-cut, quick, and intuitive way of defining complete notions by way of traditional logical resources, such as the more ordinary outlook of a first-order predication, as opposed to Leibniz’s technical understanding of predication as conceptual containment, which is particular to his logic.

In trying to make this point, I shall draw on John Whipple’s work on the distinction between exoteric and esoteric modes of presentation, especially as laid out in his 2015 article Leibniz and the Art of Exoteric Writing. In this paper, Whipple claims that the role of Leibniz’s exoteric writings is mainly pedagogical:

the primary aim of Leibniz’s exoteric writings is not to conceal heretical or politically dangerous views in order to avoid persecution. The primary aim is pedagogical: exoteric texts are designed to serve as intellectual stepping- stones that can help his readers traverse the gulf between received opinions and esoteric truth. (WHIPPLE, 2015, 4)

He also later draws a distinction between exoteric/esoteric modes of presentation and exoteric/esoteric content, based on Leibniz’s use of the conceptual pair. Concerning the idea of exoteric/esoteric content, he says:

The second sense of “esoteric” concerns the content rather than the form of one’s philosophy. Leibniz took his metaphysics to be esoteric in the sense that it involved a range of purely intelligible concepts and principles, many of which were far removed from received opinions or “common sense.” (WHIPPLE, 2015, 3)

I think it is a well-established view in Leibniz scholarship that the doctrine of complete concepts turned out to be a highly esoteric aspect of his philosophy, given that no mention of it appears in his published writings.⁹ One might even say- in view of its unintuitive character-that it is a good example of esoteric content in his philosophy. However, one might argue that this was not always the case-at least in Leibniz’s eyes. As a matter of fact, Leibniz, for instance, did submit in 1686 a summary composed of the section headings of what came to be known as the Discourse on Metaphysics to Antoine Arnauld, a prominent figure at the time, arguably in order to ‘test the waters’ and see how his philosophy would behave in reaching wider audiences.

These section headings, of course, featured a compact presentation of Leibniz’s doctrine of complete concepts, and this is exactly one of the key components of the summary that startled Arnauld and got the long correspondence between the two philosophers going. Just before mentioning Leibniz’s thesis that ‘the individual concept of each person contains once for all everything that will ever happen to him, Arnauld famously says that he found “in these thoughts so many things that frighten me and that almost all men, if I am not mistaken, will find so shocking, that I do not see what use such a work can be, which will clearly be rejected by everybody” (AA, II, 2, 9/LEIBNIZ, 1967, 9).

As Whipple seems to suggest, this reaction is probably what caused Leibniz to omit his doctrine of complete concepts in his later published writings (WHIPPLE, 2015, 19, footnote 61). However, I do think Leibniz’s submission of the summary of the Discourse on Metaphysics to Arnauld suggests that the text was meant for a somewhat wider audience, or at least-so to speak-to the ‘uninitiated’ or ‘outsiders’, who were unfamiliar with Leibniz’s developments in logic, for example. This, I believe, is in conformity with the view of his substance expression definition of complete concepts being an exoteric device, for I read the definition of complete notions provided in the Discourse on Metaphysics as a substance expression definition. (I intend to argue in favor of this claim in the next section of the paper.) One can identify in this move some aspects of the strategies Leibniz usually employs in his exoteric writings.

Whipple presents a careful survey of these strategies in his paper. The list includes: (a) selective omission of controversial aspects of Leibniz’s philosophy, which can be both inferred from the text or require supplementation (b) the existence of both a surface and a deep reading of the text (c) the use of language that is tailored so as to sound more familiar to the audience (d) appeal to eclectic well-regarded references (e) presenting theses as hypothesis even though he believes he has a demonstration he could provide to it (f) sensible analogies, metaphors, imaginative thought experiments, stories, and anecdotes that help illustrate his points and make them easier to grasp (WHIPPLE, 2015, 19). I believe that Leibniz’s substance expression definition of complete notions, as-I will argue-it appears in the Discourse, is also closely related with-yet not exactly an instance of-strategies (b) and (c). To be clear, I do not regard Leibniz’s strategy in presenting his substance expression definition of complete notion as merely a matter of tailoring one’s language. I believe it is substantially different from the other definitions we have presented here. However, I still think it qualifies as an exoteric device and it resembles strategies (b) and (c) in that Leibniz chooses a definition that is more accessible to his reader in virtue of her background over another one which is in a sense more precise. He is defining complete notions by means of familiar conceptual resources available to his readers, even though he can provide a more rigorous account that is more in accordance with his views on logic-such as his thesis that every proposition only affirms or denies a relation of containment between concepts.

Here the pedagogical role of Leibniz’s exoteric writing emphasized by Whipple stands out: he is using this definition provisionally, as a ‘stepping stone’ to spell out his views. This strategy is especially useful when the subject he intends to get at is not logic itself, but in which matters of logic are relevant to a certain degree, as it is the case with the Discourse on Metaphysics. It saves time and effort when the more rigorous esoteric definition would require too much insight into Leibniz’s own views on logic and strenuous explanation, especially if the reader or correspondent in question is not particularly interested in the subject. Moreover, the other definitions do have more of a kind of ‘mathematical air’ to them, which, Leibniz says in a letter to Burnett, ‘puts people off’ (rebute les gens, G 3, 320) and is a major obstacle to esoteric writing. And that is less of a problem with the substance expression definition, which is very straightforward.

4 - The Substance Expression Definition in the Discourse on Metaphysics

In advancing the reading that Leibniz’s substance expression definition of complete notions is exoteric in nature, I have put forward the claim that the definition of complete notions presented in the Discourse on Metaphysics is a substance expression definition. I now intend to argue in favor of this claim. Of course, it is important to acknowledge and address the fact that, as I have pointed out earlier in the paper, Leibniz introduces something that looks a lot like CCTT in Discourse §8, where the definition in question appears. It is what takes us from the merely nominal definition (or explication) of individual substance first presented to the supposedly ‘real’ one, by means of a consideration of what it is “to be attributed truly to a certain subject.”

Leibniz then claims that “all true predication has some ground in the nature of things” from what he seems to conclude that “when a proposition is not identical, that is to say, when the predicate is not explicitly contained in the subject, it must be contained in it virtually, and this is what the philosophers call inesse” (AA, VI, 4, 1540/LEIBNIZ, 2020, 13), from which he takes to follow that “the term of the subject must always include that of the predicate, so that he who understood perfectly the notion of the subject would also judge that the predicate belongs to it” (AA, VI, 4, 1540/LEIBNIZ, 2020, 13). I believe this is in fact a formulation of CCTT, although this version of the definition, as we have seen earlier in Section 1, is not as detailed as others are, including the one appearing in the correspondence with Arnauld, in terms of the appropriate qualifications to his theory of truth.

Leibniz also does not explicitly say here that every proposition only relates concepts in terms of containment and non-containment, although he seems to be gradually building towards it, first stating the theory more vaguely in terms of ‘subjects’ and ‘predicates’, then speaking of the ‘term of the subject’, and then of the ‘notion of the subject.’ Taken as a whole, the passage does seem to imply, albeit very subtly, that propositions are about concepts or notions. I believe this is important, not only because it suggests a ‘stepping stone’ approach but also because it hints toward what I take to be the main function of the introduction of CCTT here, mirroring what happens in another one of Leibniz’s texts, the Notationes Generales, i.e., arguing for the existence of complete concepts.

Sure enough, right after presenting his views on truth, Leibniz presents his ‘real’ definition of individual substance, according to which “the nature of an individual substance or of a complete being is to have a notion so complete that it should be sufficient to contain and to allow deduction from it of all the predicates of the subject to which this notion is attributed” (AA, VI, 4, 1540/LEIBNIZ, 2020, 13). Now, it seems pretty clear that what Leibniz is trying to do here is to define individual substance as something that has a complete concept. He says that explicitly in another one of his writings of the time: “For the nature of a singular substance is such that it has a complete notion, in which all the predicates of the same subject are involved” (AA, VI, 4, 1625).

However, it should also be clear that, in the passage from the Discourse of Metaphysics, ‘attributing to a subject’ must mean some kind of first-order predication. Otherwise, if ‘attributing’ means asserting a relation of containment between concepts, then we get the problem pointed out by Lenzen, that is, the definition will state that an individual substance a is something that has a concept A such that, for every B, A contains B if and only if A contains B, which is a trivial condition satisfied by any concept whatsoever and, consequently, does not single out complete notions. Instead, the definition should be taken to be stating that an individual substance a is something that has a concept A such that, for every B, a is B if and only if A contains B.

We saw earlier in the paper how, according to Lenzen, “Leibniz never makes any syntactic distinctions between a singular term and a general term, nor does he see any structural differences between a universal predication like ‘(Every) Man is mortal’ and a singular predication like ‘Caesar is mortal’” (LENZEN, 2004b, 89-90). However, I believe there are passages where Leibniz seems to notate a placeholder for the individual substance or the subject of the attribution and the placeholder for the complete notion differently, thus indicating a difference between the two relations, of ‘first-order predication’ and ‘concept- containment.’ This happens precisely in the context of a substance expression characterization of complete notions, as is the case with the Discourse on Metaphysics.

For instance, in the following passage, Leibniz uses an S to notate the subject and a P to notate the predicates of S, while using A to notate the complete notion: “Let P be any predicate of subject S, demonstrable from A itself. Then A will be a substantive term” (AA, VI, 4, 940). Here he also seems to notate complete notions and the subjects of the attribution differently:

Let there be a proposition: Y is B, and let Y be any subject of which B can be said, and let Z be any attribute of Y. I say that if it can be demonstrated from Y being B that Y is Z, then B will be a complete term expressing the substance. (AA, VI, 4, 389).

One might object that he is using the placeholders for the subject on a par with the placeholders for the predicates by using either S for the subject and P for the predicates or letters from the end of the alphabet in both cases, contrasting both of these with the complete notion, notated with a letter

from the beginning of the alphabet. This is true. I believe Leibniz is here trying to highlight the complete notion in contrast to the ‘ordinary’ and familiar (first-order) relation between the subject and the predicate. However, there are also passages where Leibniz notates both the incomplete predicates of the subject and the complete notion as letters from the beginning of the alphabet and does not use a placeholder for the subject at all, as in this passage we have quoted earlier from the Notationes Generales:

if the term A involves all the terms B and C and D, etc., which can be said about the same thing, the term A will express the singular substance itself, or the concept of the singular substance is a complete term containing everything that can be said about it [de illa dici possunt]. (AA, VI, 4, 553)

I take Leibniz to be highlighting here the difference in the status of the subject of attribution or the thing of which all these predicates can be said from the predicates/concepts themselves, both complete and incomplete.

Having argued that this relation of attribution to a subject in Discourse §8 must be some kind of first-order predication and not the relation of conceptual containment, now it is finally time to address the main issue at hand: After all, did CCTT not suggest the picture that a proposition relates concepts or notions in terms of conceptual containment (or non- containment in the more detailed version)? Is the predicate not contained in the notion of the subject in a true proposition? Well, I believe the CCTT in Discourse §8 serves one main purpose, i.e., that of introducing or providing a reason for the existence of this odd kind of concept, i.e., complete notions. This is also the same role CCTT plays in the just quoted Notationes Generales. In Leibniz’s words:

And it is clear that such a concept exists based on the definition of a true proposition explained a little earlier. For when we say "Alexander is robust," we mean nothing other than that "robust" is contained within the notion of Alexander, and the same applies to the other predicates of Alexander. (AA, VI, 4, 553)

I believe the idea here is mainly that if every proposition consists in either an assertion or a denial of a containment relation between concepts, apparently the proposition expressed by a singular ordinary sentence like “Alexander is robust” should involve a concept A which, being the subject- concept in the proposition in question, would contain the predicate ‘robust’ in case “Alexander is robust” is true and not contain it if its false. Likewise, this concept should contain each and every concept that is true of the very same individual, i.e., Alexander. This would qualify it as a complete notion according to our substance expression definition thereof.

It is important to point out that Leibniz does not always use this particular characterization of complete notions to use them in the definition of an individual substance. For example, in a passage we have already quoted, he seems to do exactly that with a disjunctive characterization of a complete notion: “[...] substances alone should be called the lowest, or individual species - namely, those whose concept is perfect, or such that in that concept there is contained an answer to everything that can be asked about the thing” (AA, VI, 4, 1350-1/LEIBNIZ, 2006, 40). But the substance expression definition provides Leibniz with a neat argument in favor of the existence of such concepts which seems not to be available for the remaining definitions.

I think a more compact version of the argument just described is implicit in Leibniz’s consideration of what being truly attributed to a subject consists in. The idea here is that if “the term of the subject must always include that of the predicate, so that he who understood perfectly the notion of the subject would also judge that the predicate belongs to it”, there must also be some concept that plays the role of subject in singular (true) propositions. Leibniz seems also to think that this concept should be the same in every ordinary singular (true) proposition about the same individual and so contain every one of that individual's predicates.

Leibniz thus introduces a version of CCTT in order to argue in favor of the existence of a kind of concept, i.e., complete notions, so as to define what an individual substance is. However, his subscription to CCTT also seems to commit him to a kind of reduction of predication to containment-at least in the case of true propositions, but, as we have seen earlier, CCTT rather seems to stem from the general view that every proposition either asserts or denies a relation of containment between concepts. But when it is time to characterize complete notions in the Discourse, it seems like Leibniz appeals to an ordinary relation of first- order predication. This is no accident for this particular characterization seems to provide him with just the kind of argument he is after, relating concept containment to ordinary first-order predication.

As I am trying to suggest, this ought to be seen as an exoteric device used for pedagogical reasons. Ultimately, predications are in general reduced to concept containment. So the substance expression definition of complete notions is, in a sense, not official or fully rigorous. This kind of definition does not constitute full-blooded Leibnizianism. However, it does provide a way into his metaphysics and logic, both in the sense that it is straightforward and easier to grasp and in the sense that it allows for an analytic argument for the existence of such notions based on our ordinary first-order predications. Substance expression definitions of complete notions thus play an essential role in Leibniz’s overall philosophy.

For the exact same reasons, I believe Leibniz’s characterization of concept containment in terms of first- order predication in his definition of proposition in Praecognita ad Encyclopaediam sive Scientiam universalem (1678/79) should also be read as an exoteric device.¹⁰ The definition is as follows:

A proposition is that which expresses that one of two attributes or terms of a thing, called the predicate, is contained in the other, which we call the subject, so that to whatever the subject is attributed, the predicate must also be attributed (ita ut cui subjectum tribuitur, eidem et praedicatum sit tribuendum). (AA, VI, 4, 135)

If every proposition only relates concepts or attributes of a thing by means of the concept containment relation we cannot further attribute these concepts to things in first- order predications, for these attributions are supposed to be themselves propositions and the latter are only supposed to relate concepts in terms of containment and non- containment relations. I believe we can only cash out containment in terms of the more familiar ordinary first- order predication from outside Leibniz’s system, in a kind of ‘metalanguage’ of philosophical exchange, which is by its own nature more resourceful and permissive so as to make room for the sake of the argument for conceptual distinctions that one does not accept.

5 - Rutherford’s Problem

Finally, I would like to point out how my reading seems to prevent a problem first introduced by Donald Rutherford in his article Truth, Predication and the Complete Concept of an Individual Substance (1988) from taking place. The problem is the following. Leibniz seems to define what an individual substance is in terms of possession of a complete concept. This means that possession of a complete concept is a necessary and sufficient condition for it to be an individual substance. This is also supposed to be based on CCTT. One might take this to mean that, given any individual A and any concept B, if ‘A is B’ is true then, given CCTT, B must be contained in the concept of A, so A’s concept must be complete in the sense that it contains all of A’s predicates. However, in Rutherford’s words:

The problem with this argument is that it seems to deliver too much. If it is based solely on Leibniz’s general account of propositional truth, then there is no reason why it should not be applied to a proposition whose subject term appears to designate a species. Man is rational, sensate, two-legged, etc. Thus we may reason that the concept man must include the concepts of all and only those predicates which are attributed truly to man, and infer that man is in this sense complete. But if we accept this conclusion then (2) [If the concept of X is complete, X is an individual substance] must be false; possession of a complete concept will not be a sufficient condition for something’s being an individual substance. (RUTHERFORD, 1988, 131)

Sometimes this problem is also put in terms of distinguishing a complete notion (notio completa), which only individual substances are supposed to have, from what Leibniz calls, in a remark in his copy of a letter to Arnauld, a full notion (notio plena), which accidents also have. According to Leibniz’s definitions: “A full concept contains all the predicates of the thing, e.g., heat; a complete concept all the predicates of the subject, e.g. this hot thing (hujus calidi)” (AA, II, 2, 71/LEIBNIZ, 1967, 54, translation slightly modified). Complete notions are also, according to Leibniz, the same as a substance’s full notion: “They coincide in individual substances” (AA, II, 2, 71/LEIBNIZ, 1967, 54). So, how are we then to prevent accidents from having complete concepts so that Leibniz’s definition of individual substance by means of complete concepts is extensionally correct, since ‘accidents’ and ‘individual substances’ are supposed to be mutually exclusive terms?

Well, we have said earlier how, in Leibniz’s definition of individual substance ‘the nature of an individual substance or of a complete being is to have a notion so complete that it should be sufficient to contain and to allow deduction from it of all the predicates of the subject to which this notion is attributed’, attribution to a subject is supposed to mean first-order predication and not concept containment. Similarly, in Leibniz’s definition of accident later in the text ‘the accident is a being whose notion does not include everything that can be attributed to the subject to which this notion is attributed’ (AA, VI, 4, 1540/LEIBNIZ, 2020, 13, emphasis added) attribution to a subject is also supposed to mean first-order predication and not concept containment.

So, an individual substance a is something that has a concept A such that, for every concept B, A contains B if and only if a is B, ‘a is B’ being a first-order predication. An accident, on the other hand, is something that has a concept A such that, given any subject of first-order predications a such that a is A, it is not the case that, for every concept B, A contains B if and only if a is B, ‘a is B’ being, in this case, also a first-order predication. I believe the example Leibniz gives us immediately after stating his definition of accident makes it clear that ‘being attributed to a subject’ as it appears in the definition is first-order: ‘Thus, the quality of King that belongs to Alexander the Great, taken in abstraction from the subject, is not sufficiently determinate for one individual, and does not include the other qualities of the same subject, nor does it include everything that the notion of this prince contains’ (AA, VI, 4, 1540/LEIBNIZ, 2020, 13, emphasis added).

As we can see, the quality of King is seen here as belonging to the individual Alexander the Great and it is an accident because it does not include the other qualities of the same subject (Alexander himself). Complete concepts thus are concepts that contain all of the predicates truly attributed to a (single) given subject of first-order predication and incomplete or general concepts are concepts that do not. An individual substance is something that has a complete concept and an accident is something that has an incomplete concept. Finally, we may say that full notions are concepts that contain every predicate of a certain subject, either in the sense of first-order predications or in the sense of subordination or containment between concepts or predications about accidents.

Also, one must keep in mind that both the substance expression definition of individual substance and its correspondent definition of accidents are exoteric, and first- order predication is an exoteric device par excellence. Strictly speaking, according to Leibniz’s theory of propositions (of which CCTT is a logical consequence), the latter only assert or deny containment relations between concepts. So, Leibniz is not strictly committed to accidents being subjects of predication just as much as he is not committed to substances themselves being subjects of predications (but only their conceptual counterparts, i.e., complete notions). Rather, talk of substances and accidents as subjects of predication should be understood-in full-blooded Leibnizian terms-as talk of complete notions and general concepts containing other concepts (or themselves). So the problem does not seem to arise at all.

One might of course object that appealing to first-order predication here is a circular endeavor, for this is supposed to be defined in terms of predication about individuals and so employing this resource in our definition of complete notion would make the definition of individual substance promptly circular. Rutherford seems to remark on this particular problem:

By a ‘subject of predication’ Leibniz might just mean the subject term of a proposition. But if that is so, then without some modification (which Leibniz himself could easily have made) (3) [X is a complete concept (notion, term) if and only if it contains all the predicates of the subject of which it is predicable] comes uncomfortably close to defining the notion of ‘full concept’ considered above. Alternatively, by ‘subject’ Leibniz might mean ‘substance’ or ‘notion of substance’. Many of his discussions of the complete concept of a human being indeed give the impression that this is all that is at stake (e.g. GP II 43; VE 1.191. 2.380). If this were so, however, (3) would obviously be unhelpful as an elucidation of the idea of individual substance. (RUTHERFORD, 1988, 133)

There are a number of strategies one can use to answer this objection. First, I do not think that first-order predication must necessarily be cashed out in terms of predication about individuals. Granted that Leibniz does not use this contemporary terminology, I believe he makes it pretty clear from context that this is the kind of predication that is at stake, especially from examples. Second, the expression ‘subject’ might be characterizing something weaker than ‘individual substance’, despite the fact the notion of predication at hand is first-order. Rutherford makes a compelling case for this alternative by means of the idea of subjectum cum praedicato (RUTHERFORD, 1988, 138-9).

Finally, the definition at hand might in fact be circular after all. As Rutherford himself makes it clear, ‘This is a possibility that cannot be ruled out’ (RUTHERFORD, 1988, 133). There is textual evidence in favor of this interpretation. It may have been just a poor choice for Leibniz to pick this definition of complete notions to define what an individual substance is. As we have seen earlier, he does eventually use other kinds of definitions of complete notion to do the job, and, in this case, the problem does not seem to come up. However, even then, a case can be made that circularity might not be as much of an issue as it might seem at first. I think the following passage by Mates can shed some light on the point:

Leibniz's philosophy has no "beginning," that is, no unique, logically primitive set of axioms. Contrary to what many commentators seem to have supposed, he does not treat his philosophical principles as a deductive system in which certain propositions are to be accepted without proof and the rest are to be deduced from these. Instead, it is clear that he regards his doctrine simply as a network of important truths that have many interesting logical interrelationships. He deduces the various principles from one another in different orders and combinations. Often he gives alternative definitions of the same concept, sometimes even showing how to derive these from one another. It is obvious that he has no particular order of theorems and definitions in mind. (MATES, 1989, 4)

In the passage just quoted, Mates seems to imply that, as far as Leibniz’s approach to philosophy is concerned, there is no unique order of definitions to be presented. Leibniz seems to allow for occasionally defining two different concepts A and B one in terms of the other. I think Leibniz’s choice of order of which concept he defines in terms of which usually relies heavily on which one of these his audience is most familiar with. But just as we might, in accordance with contemporary logical practice, define necessity in terms of possibility, e.g., ‘necessary is that whose opposite is not possible’, and define possibility in terms of necessity, e.g., ‘possible is that whose opposite is not necessary’, we may also define an individual substance as something which possesses a complete concept and a complete concept in turn as a concept that contains all the predicates of an individual substance.

One might say that what matters in both these cases is actually the ‘logical interrelationship’ between the concepts that makes them interdefinable. So even if Leibniz’s definition of individual substance as possession of a complete concept combined with his substance expression definition of the latter, not everything is lost. In that case, the concepts can be said to hang onto each other like in a web. If I ask who Esau is and you tell me that he is Jacob’s brother and I ask you in turn who Jacob is and you only answer that he is Esau’s brother, our conversation was not completely uninstructive. Sure, it would be nice to relate them to someone I actually know, but I can still take from it that there are these two brothers, one named Jacob and the other named Esau.

References

• ADAMS, R. M. Leibniz: Determinist, Theist, Idealist. New York, Oxford University Press, 1994.
• CASTAÑEDA, H-N. Leibniz’s Syllogistico-Propositional Calculus. Notre Dame Journal of Formal Logic, vol. XVII, n° 4, 1976, pp. 481-500.
• DI BELLA, S. The Complete Concept of an Individual Substance. In: ANTOGNAZZA, M.R. (ed.). The Oxford Handbook of Leibniz. Oxford, Oxford University Press, 2018.
• ENGLEBRETSEN, G. A Note on Leibniz’s Wild Quantity Thesis. Studia Leibnitiana, Bd. 20, H. 1, 1988, pp. 87-89.
• GRACIA, J.J.E. Individuals as Instances. The Review of Metaphysics, Vol. 37, No. 1, Sep., 1983, pp. 37-59.
• KAUPPI, R. Über die Leibnizsche Logik mit besonderer Berücksichtigung des Problems der Intension und der Extension. Helsinki: Acta Philosophica Fennica, 1960.
• LEIBNIZ, G.W. Opuscules et Fragments Inédits de Leibniz: Extraits des Manuscrits de la Bibliotèque Royale de Hanovre. Paris, Felix Alcan 1903. Edited by Louis Couturat.
• LEIBNIZ, G.W. The Leibniz-Arnauld Correspondence. Manchester: Manchester University Press, 1967. Translated by H.T. Mason.
• LEIBNIZ, G.W. Sämtliche Schriften und Briefe. Akademie Verlag, 1999.
• LEIBNIZ, G.W. Die philosophischen Schriften von Gottfried Willheim Leibniz. Hildelsheim. Georg Olms Verlag, 2002.
• LEIBNIZ, G.W. The Shorter Leibniz Texts: A Collection of New Translations. London: Continuum, 2006.
• LEIBNIZ, G.W. Discourse on Metaphysics. Oxford: OUP, 2020. Translated by Gonzalo Rodriguez-Pereyra.
• LEIBNIZ, G.W. General Inquiries on the Analysis of Notions and Truths. Oxford: OUP , 2021. Translated by Massimo Mugnai.
• LENZEN, W. “‘Non est’ non est ‘est non’ - Zu Leibnizens Theorie der Negation”. Studia Leibnitiana 18, 1986, p. 1-37.
• LENZEN, W. Leibniz’s Logic. In: GABBAY, D. & WOODS, J. (eds.). The Rise of Modern Logic: From Leibniz to Frege (Handbook of the History of Logic, vol. 3). Amsterdam, Elsevier. 2004a, p. 1-83.
• LENZEN, W. Logical Criteria for Individual(concept)s. In: CARRARA, M., NUNZIANTE, A.M. & TOMASI, G. (eds.). Individuals, Minds, and Bodies: Themes from Leibniz. Stuttgart, Steiner, 2004b, p. 87-107.
• RODRIGUEZ-PEREYRA, G. Leibniz on Substance in the Discourse of Metaphysics. In: LODGE, P. & STONEHAM, T. Locke and Leibniz on Substance. New York, Routledge, 2015.
• RUSSELL, B. A Critical Exposition of the Philosophy of Leibniz. London. Routledge, 1992.
• RUTHERFORD, D. Truth, Predication, and the Complete Concept of an Individual, Studia Leibnitiana , Sonderheft 15, Stuttgart: Franz Steiner Verlag, 1988, pp. 130-44.
• RUTHERFORD, D. Metaphysics: The late period. In: N. Jolley (ed.), The Cambridge Companion to Leibniz. Cambridge: Cambridge University Press, pp. 124-174.
• SOMMERS, F.; ENGLEBRETSEN, G. An Invitation to Formal Reasoning: The Logic of Terms. Altershot: Ashgate Publishing Company, 2000.

• 1
  This particular formulation corresponds exclusively to a proposition in its ‘tertii adjecti’ form, i.e., ‘A is (not) B’ (AA, VI, 4, 779). However, every proposition in ‘tertii adjecti’ form can be converted into one in ‘secundi adjecti’ form, ‘AB is (not)’ and vice- versa (AA, VI, 4, 780; AA, VI, 4, 749). In either case, every proposition is regarded as establishing properties of or relations between concepts.
• 2
  See also AA, VI, 4, 1515.
• 3
  See, for example, AA, VI, 4, 813, §21. See LENZEN (1986).
• 4
  See, for example, AA, VI, 4, p. 300; C 237.
• 5
  See, for instance, SOMMERS & ENGLEBRETSEN, 2000.
• 6
  See ENGLEBRETSEN (1988) and LENZEN, 2004a, 76.
• 7
  See LENZEN, 2004a, 16.
• 8
  See, for example, AA, VI, 4, 786.
• 9
  Some might say instead that Leibniz abandoned the thesis altogether. Rutherford seems to suggest a reading of that nature, according to which the later ‘law of the series theory’’ substitutes the ‘complete concept theory’ (RUTHERFORD, 1995, 128). See DI BELLA (2018) for a different reading.
• 10
  I also read the following passage, where Leibniz defines an individual substance as something of which a complete concept is predicated as exoteric in nature: “If A is B and B is a complete term, A will be a singular substance, or a certain subject, which is vulgarly called an individual” (AA, VI, 4, 626).

• Article info
  CDD: 149.7
• Funding:
  Funded by CAPES; 88882.425706/2019-01/88887.466978/2019-00.

Publication Dates

• Publication in this collection
  16 Dec 2024
• Date of issue
  2024

History

• Received
  19 Oct 2024
• Accepted
  30 Oct 2024