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A Logical Reconstruction of Leibniz’s Argument for His Complete Concept Conception of the Nature of Substance in Discours §8

In: History of Philosophy & Logical Analysis
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Ralf Busse Department of Philosophy, Faculty of Philosophy and Philology, Johannes Gutenberg University Mainz Mainz

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Abstract

This paper develops a valid reconstruction in first-order predicate logic of Leibniz’s argument for his complete concept definition of substance in §8 of the Discours de Métaphysique. Following G. Rodriguez-Pereyra, it construes the argument as resting on two substantial premises, the “merely verbal” Aristotelian definition and Leibniz’s concept containment theory of truth, and it understands the resulting “real” definition as saying not that an entity is a substance iff its complete concept contains every predicate of that entity, but iff its complete concept contains every predicate of any subject to which that concept is truly attributable. An account is suggested of why Leibniz criticises the Aristotelian definition as merely nominal and how he takes his own definition to overcome this shortcoming: while on the Aristotelian basis the predication relation could generate endless chains, so that substances as endpoints of predication would be impossible, Leibniz’s definition reveals lowest species as such endpoints, which he therefore identifies with individual substances. Since duplicate lowest species make no sense, the Identity of Indiscernibles for substances follows. The reading suggests a Platonist interpretation according to which substances do not so much have but are individual essences, natures or forms.

Abstract

This paper develops a valid reconstruction in first-order predicate logic of Leibniz’s argument for his complete concept definition of substance in §8 of the Discours de Métaphysique. Following G. Rodriguez-Pereyra, it construes the argument as resting on two substantial premises, the “merely verbal” Aristotelian definition and Leibniz’s concept containment theory of truth, and it understands the resulting “real” definition as saying not that an entity is a substance iff its complete concept contains every predicate of that entity, but iff its complete concept contains every predicate of any subject to which that concept is truly attributable. An account is suggested of why Leibniz criticises the Aristotelian definition as merely nominal and how he takes his own definition to overcome this shortcoming: while on the Aristotelian basis the predication relation could generate endless chains, so that substances as endpoints of predication would be impossible, Leibniz’s definition reveals lowest species as such endpoints, which he therefore identifies with individual substances. Since duplicate lowest species make no sense, the Identity of Indiscernibles for substances follows. The reading suggests a Platonist interpretation according to which substances do not so much have but are individual essences, natures or forms.

1 Introduction

In §8 of his Discours de Métaphysique (= DM, 1539–15411), Leibniz argues for his Complete Concept Conception of Individual Substances (CCS) on the basis of his Concept Containment Theory of Truth (CCT). G. Rodriguez-Pereyra (2015; see also 2020, 69–80) has defended a novel interpretation of DM §8, which overcomes two points of misinterpretation. The main aim of this paper is to vindicate and elaborate on his line of interpretation by offering a precise logically valid reconstruction of Leibniz’s argument in first-order predicate logic. Towards the end of the paper, I will venture some interpretative conjectures concerning Leibniz’s rather Platonist conception of substance in DM. I will almost completely be focussed on DM §8, which represents an important and fascinating moment in Leibniz’s thinking on substance and deserves an adequate interpretation in its own right.2

In section (2) I will start with an overview of §8. In section (3) I will highlight Rodriguez-Pereyra’s two interpretative achievements, first, the observation that Leibniz infers CCS not from CCT alone but from CCT plus the Aristotelian ‘nominal’ definition of substance ADS, and, secondly, an adequate reading of CCS. In section (4) I will develop a formalisation of Leibniz’s argument in first-order predicate logic. As I explain in section (5), in order to prove the validity of that formalised argument it must be shown that from CCT plus certain auxiliary premises the co-extensionality of the Aristotelian definiens for substance-hood and the Leibnizian definiens logically follows. Since the premises can be claimed to hold necessarily (if at all), this would prove the necessary equivalence of the Aristotelian proto-definiens and the definiens of Leibniz’s proposed real definition. Leibniz would have shown that the entities adequately identified as substances by his real definition are exactly the entities singled out as substances by the initial Aristotelian definition. However, assuming a straightforward formalisation of ADS only one direction of the required equivalence holds. (6) The full equivalence can be proved when the reading of ADS is slightly weakened. Since the weakening formally allows for duplicate substances, I will address the Identity of Indiscernibility. In section (7) I will suggest an answer to the question of why Leibniz criticises ADS as ‘merely verbal’ and considers CCS a real definition, highlight his identification of individuals with lowest species, which also explains why he embraces the Identity of Indiscernibles, and indicate a corresponding Platonist reading of DM. Section (8) concludes.

2 The Argument of DM §8 in Context

At the textual surface, Leibniz’s proceeding in §8 is clear enough:3

(i) On the basis of his preceding discussion of miracles, he motivates the guiding question of “what […] an individual substance is”. (ii) He takes up (at least one half of) Aristotle’s definition of (primary) substance (ADS) as a subject to which “several predicates are attributed” but which “is attributed to no other”. (iii) He criticises ADS as “not sufficient” and “merely verbal”, indicating that what is missing is an account of “what it is to be attributed truly to a certain subject”. (iv) In order to fill this gap, he introduces CCT, according to which in a true subject-predicate proposition “the subject term must always contain the predicate term”. (v) He infers his real definition of substance CCS, according to which the nature of an individual substance is to have a complete notion, in a sense to be made precise in the following. (vi) He illustrates CCS by the example of the substance Alexander the Great and his accident of being king. (vii) He draws some first and already far-reaching metaphysical conclusions, namely “that from all time in Alexander’s soul there are vestiges of everything that has happened to him and marks of everything that will happen to him and even traces of everything that happens in the universe, even though God alone could recognize them all.”

To this Vestiges-and-Marks thesis and this first formulation of his expression thesis he in §9 adds further “paradoxa” that he claims to follow from CCS, most importantly the No-Duplicate-Substances thesis, or Identity of Indiscernibles for substances, “that it is not true that two substances can perfectly resemble and differ only in number”, the Infimae Species thesis that “every individual is a lowest species”, and a more explicit version of the Expression thesis that each substance “expresses, however confusedly, everything that happens in the universe, whether past, present, or future”. It is of considerable interpretative importance to inquire into what argument Leibniz takes himself to be able to provide for a definition of substance that he expects to have all those astonishing consequences.

3 Rodriguez-Pereyra’s Two Major Interpretative Achievements

Rodriguez-Pereyra’s (2015, 152) first major interpretative achievement is that he resists previous accounts according to which Leibniz infers CCS from CCT alone. For example, J. W. Nason (1981, 11) writes that the “most important single argument in support of Leibniz’s doctrine of monads [= individual substances] is that drawn from the logical nature of true affirmative propositions.” D. Rutherford (1988, 130) interprets Leibniz as implying that CCS “is a consequence of his view that the truth of a proposition is determined by the containment of its predicate-term in its subject-term”. N. Jolley (2005, 48) comes close to such a reading when he writes that “[f]rom the concept-containment theory of truth Leibniz’s thesis that individual substances have complete concepts follows as a special case” and that “his analysis of the nature of individual substances is derived from [his] theory of truth”.4

First, however, one-premise interpretations are in danger of mispresenting Leibniz as thrusting aside the Aristotelian definition ADS and replacing it by CCS. Leibniz does not discard the definition of substance as a subject of predication that cannot be predicated of any other subject, but only states that it does not suffice for metaphysical purposes, because it is merely verbal. He is perfectly clear on what part of the traditional definition it is that prevents it from providing a real definition: ADS relies on a notion of something being truly predicated of a subject but does not explain what true predication consists in. It is here that Leibniz steps in with CCT. While it has been acknowledged in the literature that Leibniz aims at an enhancement rather than a replacement of the Aristotelian definition,5 we so far lack a precise reconstruction of Leibniz’s argument for CCS as resting on ADS as one of its premises.

Secondly, Leibniz could hardly hope to validly infer CCS from CCT alone, simply because CCS contains the expression “individual substance” not featuring in CCT at all. Some initial understanding of individual substance must be added to CCT in order to be able to infer a real definition of substance from it, and this initial notion is provided by ADS. Leibniz’s argument for CCS has two substantial premises, ADS and CCT.

Rodriguez-Pereyra’s (2015, 152–153) second attainment is an adequate interpretation of the argument’s conclusion that, in Leibniz’s own words,

(CCS) the nature of an individual substance or of a complete being is to have a notion so complete that it is sufficient to contain and to allow us to deduce from it all the predicates of the subject to which this notion is attributed.

My Emphasis

On the usual reading, the apparent definite description “the subject to which this notion is attributed” refers back to “an individual substance”, so that CCS contains nothing more than universal quantifications over candidate individual substances and over predicates:

(CCSstandard) For every s: s is an individual substance just in case the concept of s is such that, for every predicate f, if f is a predicate of s, then f is contained in the concept of s.

B. Look interprets CCS in this manner:6

[…] x is a substance if and only if x has a complete individual concept [emphasis in the original] (CIC), that is, a concept that contains within it all predicates of x [my emphasis] past, present, and future.

Look 2013, Section 4.1

G. H. R. Parkinson recognises Leibniz’s distinction between subjects and predicates, on the one hand, and their concepts, on the other, but essentially offers the same interpretation:

[…] the concept of whatever can be truly predicated of a substance must be contained in the concept of the substance.

Parkinson 1965, 125; my emphasis

Similarly Chr. Mercer, following R. Sleigh:

[…] a being is an individual substance if and only if its concept contains all and only the concepts of those entities that may be attributed to it.

Mercer 2001, 474; my emphasis

Rodriguez-Pereyra appears to have adhered to this traditional (mis)construal himself when he wrote:

That substances have complete individual concepts means that a substance’s individual concept includes everything that is true of it.

Rodriguez-Pereyra 2014, 57; my emphasis

Even Robert Adams wrote, in a famous early paper:

Leibniz expresses this notion of completeness [of individuals] by saying that the concept of an individual implies every predicate of that individual.

Adams 1979, 9; my emphasis

First, however, this reading does not fit in with Leibniz’s example of Alexander and his accident of being king:

[…] taken in abstraction from the subject, the quality of being a king which belongs to Alexander the Great is not determinate enough to constitute an individual and does not include the other qualities of the same subject, nor does it include everything that the notion of this prince includes.

My emphasis

Here the candidate entity s to be ‘tested’ for being an individual substance is Alexander’s quality of being a king. But the phrase “the same subject”, which corresponds to the phrase “the subject to which this notion is attributed” in Leibniz’s formulation of CCS, does not refer to candidate entity s, the quality of being king, but to Alexander, the entity that has this quality. CCS must thus be construed as containing a further reference, one to subjects, or instances, of the concept of the candidate entity s being ‘tested’ for qualifying as an individual substance. Secondly, only this more complex construal of CCS matches a parallel account of individuals in Leibniz’s logical papers closely related to DM, the Generales Inquisitiones (1686): “[…] nec homo continet omnia quae de eo dici possunt de quo ipse” (§71, 762), “Nor does ‘man’ contain everything that can be attributed to that to which it itself [= ‘man’] can be attributed” (my translation). Thirdly, as Rodriguez-Pereyra (2015, 150; but see 153–155) argues, CCSstandard should better not be what Leibniz has in mind, because an accident too has a complete concept in the inadequate sense of CCSstandard.

Given the apparent singular terms “the subject to which this notion is attributed”, “this same subject”, and “that to which it itself can be attributed”, it might be thought that one ought to construe this additional reference to subjects either as singular reference or, if this is analysed by a Russellian definite description, as existential quantification. However, the singular terms appear at the grammatical surface only. They may be interpreted as means of reference to arbitrary entities: ‘Consider an arbitrary subject of the notion of candidate substance s. Then every predicate of this subject must be contained in the notion of s.’ The further reference to subjects is neither singular nor existential, but is universal quantification as well. This choice may be implicit in Rodriguez-Pereyra’s plural reference to (all of) “the subjects to which [a candidate entity’s] concept is attributed” (2015, 153). So I suggest the following reading of CCS:

(CCScorrect) For every s: s is an individual substance just in case the concept of s is such that, for every entity x, if x is a subject of the concept of s, then for every predicate f, if f is a predicate of x, then f is contained in the concept of s.

Thus, a substance is an entity the concept of which contains every predicate of every instance of that concept; no instance of the concept of a substance exceeds in its properties what is already contained in that concept. B. Mates hit the nail on its head when he summarised:7

A concept is complete (and thus the concept of an individual substance) if and only if it contains every property of whatever [!] falls under it.

Mates 1986, 193

4 Formalising Leibniz’s Argument in Predicate Logic

Let me prefix a methodological preliminary before setting out to formalising Leibniz’s argument. It might be objected that reconstructing Leibniz’s metaphysical argument by first-order predicate logic is anachronistic and that the argument would have to be reconstructed in the light of Leibniz’s own logic, such as that of the Generales Inquisitiones. However, this objection would rest on a misunderstanding of the relation between a reconstructed theory and the logical and other means by which it is reconstructed. Leibniz’s philosophy forms our object of interpretative reconstruction. When we reason about his philosophy, be it his metaphysics or his logic, we naturally use the best logical means available to us, and arguably standard predicate-logic is among those means. Also, if with the help of first-order logic Leibniz’s argument in §8 can be reconstructed as logically valid, this establishes that his conclusion logically follows from his premises, assuming that standard logical consequence is truth-preserving. Such a result is an interpretative achievement independently of whether Leibniz’s own logic is rich enough to capture the structure of his metaphysical argumentation. After all, one would have a hard time to find a treatment of multiply quantified formulas containing relational predicates in Leibniz’s logical writings.

So let us rephrase CCScorrect in first-order predicate logic and then look for corresponding formalisations of the argument’s two main premises ADS and CCT. In CCS, Leibniz refers to the (complete) concept of the candidate entity by existential quantification: the nature of a substance is “to have a notion so complete […]” In other places, however, he uses singular terms, such as “the notion of this prince” in the example of Alexander. I suggest that we construe this as a functional term with a functional expression ‘n(…)’ standing for a function that maps entities on concepts (or notions) of those entities. We also need a one-place predicate ‘S …’ for being an individual substance. In addition, two two-place predicates are required, ‘C …—’ for conceptual containment (in …—is conceptually contained) and ‘A …—’ for true attribution (to …—can truly be attributed). CCS then becomes:

In words: s is a substance just in case, for every x to which the concept of s n(s) can truly be attributed, every predicate f truly attributable to x is contained in n(s).

The first premise ADS defines an individual substance as an entity that cannot truly be attributed to any other entity (while it is a subject of predication itself; we can leave this part implicit):

The second substantial premise is CCT, the real definition of true predication in terms of conceptual containment. In his motivation of CCT, Leibniz is dealing with two different levels that may be dubbed ontic and conceptual, respectively:

[…] when the predicate is not explicitly contained in the subject, it must be contained in it virtually. That is what the philosophers call in-esse, when they say that the predicate is in the subject. Thus the subject term [le terme du sujet] must always contain the predicate term [celuy du predicat], so that one who understands perfectly the notion of the subject would also know that the predicate belongs to it.

Leibniz first mobilises the ontological intuition that in all cases of true predication the predicate, i.e. the predicated item—what we may call the predicable—must be ‘in’ the subject, i.e. the entity of which the predicable is predicated. Then he ascends to the level of concepts (terms, notions) and states that this ontic structure of in-esse must be mirrored, as it were, by a relation of conceptual containment between the concept (or “term”) of the predicable and the concept of the subject entity. CCT expresses this mirroring relation between the ontic and the conceptual level: a predicable can be truly attributed to a subject just in case the concept of the subject contains the concept of the predicable:

To the two premises ADSPL and CCTPL, we must add three auxiliary premises concerning concepts and conceptual containment. First, while this idea of a conceptual mirroring of an ontic in-esse motivates CCT, Leibniz does not keep the ontic and the conceptual level strictly apart.8 In DM §8, he is fairly careful on the side of the subject of predication, presumably because he wants to uphold a clear distinction between a substance and its complete concept. However, he is rather liberal on the side of what is predicated. For example, in CCS he writes that all predicates (not: the concepts of those predicates) of instances of the concept of the candidate substance must be contained in that latter’s concept. We should therefore embrace a trans-categorial auxiliary premise to the effect that a predicable is contained in a concept just in case its concept is:

In addition, we assume that iterating the concept forming function n(…) is trivial: the concept of the concept of an entity is simply the entity’s concept again:

This principle would hardly be plausible if applying the n-operation to a first-order concept generated a second-order concept containing predicates such as being an abstract entity or containing other concepts. But if we think of the operation as delivering a concept that captures the qualitative content of the argument to which it is applied, (C-2) is reasonable, because plausibly a concept capturing the qualitative content of a given concept simply is that same concept again—at least if concepts are purely qualitative, as Leibniz appears to assume.

Leibniz uses the attribution relation A in a categorially liberal way, too. Thus, by using the embedded antecedent ‘Axn(s)’ in CCSPL, it is implied that not only ontic predicables but also concepts can be attributed to subjects. Together with CCTPL, C-1 and C-2 allow us to deal with this categorial promiscuity of true attribution. For example, ‘Axy’ is coextensional (coext, for short)9 with ‘Cn(x)n(y)’ (by CCTPL), which is coext with ‘Cn(x)n(n(y))’ (by C-2), which is coext with ‘Axn(y)’ (by CCTPL). Hence every occurrence of ‘Axy’ can be replaced by ‘Axn(y)’ and vice versa.

A third auxiliary premise is to the effect that conceptual containment is transitive in the following specific sense: n(x) contains n(y) only if all predicates contained in n(y) are also contained in n(x):

C-3 will be employed first in section 6, where I will suggest a repair to Leibniz’s argument.

A possible objection to this stage-setting is that it treats accidents such as being king as universals or ‘repeatables’, while Mates (1986, 196–197) in particular has argued that accidents, the non-substantial predicative entities at the ontic (as contrasted with the conceptual) level are particular, so that they are trope-like rather than universal; only concepts are universal on this nominalist interpretation. Tropist interpretations have forcefully been contested, however (Rodriguez-Pereyra 2014, ch. 14). Also, even assuming that a trope view of accidents is Leibniz’s position at the time of the Discours, it is not particularly highlighted in DM §§8–9 nor, as far as I can see, in DM as a whole. On the contrary, we saw that Leibniz is liberal if not careless as regards the distinction between the ontic and the conceptual level as far as predicates are concerned. Thus, for the purpose of a reconstruction of §8, auxiliary premise (C-1) seems justified. We could do justice to a trope interpretation by introducing equivalence classes of accidents under a duplication relation (cf. the duplication predicate for substances in section 7) and revising the principles accordingly, to the effect that a substance falls under the concept of a given accident (which may be the individual accident of another substance) just in case a duplicate of that accident can be attributed to it. Yet the manoeuvre would complicate the reconstruction in a way hardly illuminating for Leibniz’s central argumentative intention.

The above considerations reveal that in discussing §8 we should not think of a substance’s complete concept as a construction out of the many predicates that can be attributed to that substance. Thus, we should not assume that one can ideally arrive at Alexander’s complete concept by first listing all his predicates,—being king, being educated by Aristotle, having defeated Darius, etc.,—and then conjoining them into a single super-concept (cf. Mates 1986, 87–88 for such a model). Whatever Leibniz may suggest in other contexts, this can hardly be his view in DM §8. For the envisaged construction presupposes a notion of predicates truly being attributable to the subject in question, Alexander, which, on Leibniz’s CCT, can only be defined on the basis of a prior conception of the concept or notion of that subject. The concept-forming operation n(…) must be accepted as primitive in a sense, as it cannot be defined in terms of conjoining all the predicates attributable to the entity to which that operation is applied—not if Leibniz is serious about CCT accounting for true attribution by conceptual containment. In §8 Leibniz uses the picture of God “seeing Alexander’s individual notion or haecceity” rather than constructing it. A plausible interpretation is that an entity’s concept captures what can be conceptually grasped by an ideal being completely investigating that entity all by itself, i.e. its intrinsic qualitative nature. The complete concept of a substance, in particular, captures that substances intrinsic qualitative nature.10

5 Is Leibniz’s Argument Logically Valid?

The complete formalised argument with its four premises runs as follows:

What would we need to show in order to prove the argument’s logical validity? A universally quantified biconditional CCSPL ‘"s: Ss ↔ λ’ (‘λ’ for Leibniz) is inferred from a quantified biconditional ADSPL ‘"s: Ss ↔ α’ (‘α’ for Aristotle) with the help of three additional premises CCTPL and C-1+2. The argument aims to show that the very same range of entities that are singled out as substances by the nominal Aristotelian definition are, in light of CCT, more adequately featured as substances by the inferred Leibnizian definition. Thus, CCT plus the auxiliary premises must prove the Aristotelian definiens α and the Leibnizian definiens λ of substance-hood to be co-extensional.

More technically, let set Γ comprise CCTPL and C-1+2. What must be shown is that all Γ-models that are models of ‘"s: Ss ↔ α’ are also models of ‘"s: Ss ↔ λ’. Since α and λ contain the same single free variable ‘s’ as formula ‘Ss’, these two quantified biconditionals are true just in case ‘Ss’ and α, in the first case, and ‘Ss’ and λ, in the second, are co-extensional in the sense of being satisfied by the same range of entities. Thus, all Γ-models M such that M(‘Ss’) = M(α) must also be such that M(‘Ss’) = M(λ), where M(…) is the evaluation function generated by M that assigns to a monadic formula the set of entities satisfying the formula. By the Euclidicity (a = b Ù a = cb = c) and transitivity (a = b Ù b = ca = c) of identity, this is so just in case all Γ-models M such that M(‘Ss’) = M(α) are also such that M(α) = M(λ). So in order to prove the argument’s logical validity, we must show that from all four premises of the argument (Γ plus ADSPL, which amounts to M(‘Ss’) = M(α)) the extensional equivalence of the Aristotelian and the Leibnizian definiens of substance α = ‘¬$x: sx Ù Axs’ and λ = ‘"x: Axn(s) → "f(Axf → Cn(s)f)’ logically follows, i.e. ‘"s: α ↔ λ’.

In a standard calculus, we can reduce this task to two core problems. We must prove that from the premise set in question, Γ plus ADSPL, both ‘"s: α → λ’ and ‘"s: λ → α’ logically follow. In the first case, we would use the premise set to establish the conditional ‘α(a) → λ(a)’ of arbitrary instances of α and λ, α(a) = ‘¬$x: ax Ù Axa’ and λ(a) = ‘"x: Axn(a) → "f(Axf → Cn(a)f)’, and afterwards generalise to obtain ‘"s: α → λ’. Similarly in the second case, where we would seek to establish the converse ‘λ(a) → α(a)’ and then generalise. We will see that only the first of these two desired core entailments holds: the premises plus α(a) entail λ(a), but λ(a) and the premises do not entail α(a). In order to obtain full logical validity, we will need to revise the argument’s first premise ADSPL (see section 6).

5.1 The Premises Entail ‘α(a) → λ(a)’

Note that ‘S’ features in ADSPL but neither in α(a) nor in λ(a). Nor is ADSPL a logical falsehood, which would render the inference trivially valid. The conditional ‘α(a) → λ(a)’ must therefore be inferred with the sole help of CCTPL and C-1+2; ADSPL as a premise runs at idle; it plays a role only by an instance of its definiens clause, α(a), forming the antecedent of the conditional to be inferred. We assume the contradictory of λ(a), ¬λ(a), and infer the contradictory of α(a), ¬α(a). Informally speaking, the inference is straightforward. λ(a) is to the effect that the concept of substance a has no instances richer in properties than what is already involved in that concept. ¬λ(a) says that there is such an instance, which, by CCTPL and C-1+2, entails that a certain property can be attributed to this entity but not to a; which entails that a is attributed to an entity numerically different from a; which is ¬α(a).

Assume the contradictory of λ(a), ¬λ(a),

which is equivalent to

Thus, if λ(a) is false, there must be some entity x, call it ‘b’, that is an instance of n(a) and is such that some predicate f, call it ‘p’, can be attributed to b without being contained in n(a); so we have the following arbitrary instances of existential generalisations,

From (3) we infer

From (4) we infer, operating on its second conjunct,

By the non-identity of discernibles, (9) entails

From this and (7) we infer

which by existential generalisation yields

the contradictory of α(a) = ‘¬$x: ax Ù Axa’.

In order to satisfy the Aristotelian definiens for being an individual substance, then, an entity must also satisfy Leibniz’s complete concept definiens, given CCTPL and the two auxiliary premises concerning the categorially liberal use of conceptual containment. In this sense, Leibniz’s definiens follows from the Aristotelian definiens plus the additional premises. This, however, does not establish that the Leibniz’s definition CCSPL follows from the Aristotelian definition ADSPL plus the additional premises. Leibniz’s definiens may still be wider than Aristotle’s. And so it is.

5.2 The Premises Do Not Entail ‘λ(a) → α(a)’

Here is a model that verifies λ(a) but not α(a): Let the domain comprise just two objects, D = {a, b}; we use ‘a’ and ‘b’ as names both in the object- and in the metalanguage. Let n be the identity mapping, let ‘a’ denote a, and let ‘A’ and ‘C’ be interpreted by the set of all combinations of objects from the domain: I(‘A’) = I(‘C’) = {<a,a>, <a,b>, <b,a>, <b,b>}. Then λ(a) = ‘"x: Axn(a) → "f(Axf → Cn(a)f)’ comes out true, but α(a) = ‘¬$x: ax Ù Axa’ is false, because there is an object distinct from I(‘a’) = a, namely b, to which a can be attributed. Artificial as it is, this model indicates the crucial difference between α(a) and λ(a). The Leibnizian λ(a) says that to no instance of n(a) a predicable can be attributed that is not already contained in n(a). This, however, is compatible with n(a) having an instance, say b, numerically different from a, to which the same range of predicable entities f can be attributed, among them a itself. The Aristotelian α(a), by contrast, states that a cannot be attributed to any numerically different entity such as b.

6 Leibniz’s Argument Repaired

What kind of case is excluded by α(a) but not by λ(a)? λ(a) says that all predicables of all instances of n(a) are already contained in n(a). This is compatible with there existing a numerically different entity b also satisfying the definiens in CCSPL and having the very same concept as a, so that n(a) = n(b). It is very plausible to construe this as a case of duplicate substances. Neither CCTPL nor C-1+2 (nor C-3) exclude such a case. They merely require that the very same range of predicables can be attributed to a and to b.

It might be thought that the Leibnizian definiens does exclude duplicate substances, because among the predicables over which variable ‘f ’ ranges are ‘haecceitistic’ properties such as being identical to a and being identical to b; and since being identical to b cannot, as a matter of fact, be truly attributed to a, b and a could not have the same concepts. Such a reasoning would be in accord with Rodriguez-Pereyra interpretation of Leibniz’s commitment to the Identity of Indiscernibles in DM on the basis of CCS:

The argument is that individual substances have complete concepts that permit to deduce everything that is true of them. Since they permit to deduce everything about them, they permit to deduce facts about the identity of substances. But those complete concepts are purely qualitative. Therefore, there cannot be two substances that resemble each other perfectly. This is, I think, a valid argument, but with very controversial premises. Indeed, the premise that complete concepts are purely qualitative would not be granted by a denier of the Identity of Indiscernibles.

Rodriguez-Pereyra 2014, 60

However, in reconstructing Leibniz’s argument for CCS, it seems wise not to rely on metaphysically loaded background assumptions such as the possibility of haecceitistic properties but to remain neutral on the possibility (ultimately to be rejected) of duplicate substances.

Neither would it be correct to understand the difference between α(a) and λ(a) as being that while the latter does not rule out duplicates, the former does. As it stands, α(a) merely demands that, as a brute fact, a could not be attributed to such a duplicate. So the ground of the failure of λ(a) to entail α(a) is not that the latter rules out a metaphysical scenario, duplicate substances, not ruled out by λ(a). Instead, α(a) merely involves a per fiat construal of true attribution to the effect that a substance can only be attributed to itself but not even to other duplicates of itself, if such there be.

The wise thing to do, therefore, is to liberalise the construal of true attribution in ADSPL and to see whether Leibniz’s argument proves valid with a revised version of ADSPL (it does!). Such a revised version of the nominal definition serving as Leibniz’s starting point no longer says that a substance is never attributed to any numerically distinct entity, but merely that a substance can only be attributed to its duplicates, the substance itself included:

As two background premises, we add a Principle of Duplication to the effect that entities x and y are duplicates just in case their concepts are identical (which entails reflexivity of duplication),

and a Principle of Concept Identity, which says that concepts are identical just in case they contain the same predicates,

Together with CCTPL and C-1+2, these two premises entail the lemma

i.e., entities are duplicates just in case the same predicates can be attributed to them. (For ‘Cn(x)f’ coext by C-1 ‘Cn(x)n(f)’ coext by CCTPL ‘Axf’; similarly ‘Cn(y)f’ coext by C-1 and CCTPL ‘Ayf’.)

We do not directly assume LEM-D, because for DUP and CID we can claim intuitive support, which is less clear for LEM-D. Given a background understanding of duplicates as entities that share their intrinsic qualitative natures, DUP is a meaning postulate to the effect that the concept of an entity captures that entity’s intrinsic nature. As suggested in section 4 above, we can hypothesise that this comes close to Leibniz’s understanding in §8 of what it is to be the concept of an entity and the (complete) concept of a substance in particular. I take CID to be a plausible principle in the Leibnizian context independently of how exactly concept of and containment are construed. The essential job of Leibnizian concepts is to contain or be contained in other concepts, so if a criterion of concept identity is sought after, it must arguably be stated in terms of what concepts contain or are contained in. The new premises have the somewhat anomalous result that if there were duplicate substances, they would be attributable to each other—anomalous certainly from Leibniz’s perspective, because he would hardly accept that at the ontic level one substance can be in another. Yet since the resulting metaphysics is expected to exclude duplicate substances anyway, this technical possibility does no damage to the reconstruction.

6.2 The Premises Entail ‘α*(a) → λ(a)’

We will now need the transitivity principle for the C-relation introduced but not employed in the first attempted reconstruction,

By the now familiar mode of reasoning with coextensionalities, C-3 together with CCTPL and C-1+2 entails a lemma concerning true attribution:

Assume λ(a):

With CCTPL and C-1+2 we infer by reasoning with coextensionalities

As an instance of LEM-A we have

(2) and (3) together entail

which, given LEM-D, is equivalent to α*(a) = ‘"x: Axa → Dxa’.

This successful logical reconstruction of Leibniz’s reasoning in §8 validates Rodriguez-Pereyra’s interpretative insights that, first, the Aristotelian nominal definition of substance serves as one crucial premise alongside the conceptual containment theory of truth and, secondly, that by ‘the subject’ in his real definition Leibniz does not refer to the candidate substance itself but to all instances of its (complete) concept. I know of no valid accurate reconstruction that ignores these two points.

If the conclusion of Leibniz’s argument is compatible with duplicate substances, so that the Aristotelian premise must be amended accordingly, what becomes of the Identity of Indicscernibles (IoI) concerning substances? A methodological point to note is that important as it is, a logical reconstruction of an argument can only be one part of an interpretation. In addition to providing the logical skeleton, the descriptive terms involved need to be interpreted. In particular, it has been left open so far what exactly is in the range of the variable ‘f’ for predicables. Rodriguez-Pereyra may still be right that Leibniz’s simple point is that predicables such as being identical to Alexander are in that range, that therefore identities result from purely qualitative individual concepts and that from this the principle in question follows. However, those premises “would not be granted by a denier of the Identity of Indiscernibles” (Rodriguez-Pereyra 2014, 60).

My impression is that in the context of §8 Leibniz has a deeper motivation for IoI. Why would a denier of the principle not grant that purely qualitative concepts determine identity? Because she maintains that a and b can be distinct and nevertheless qualitatively perfectly alike. Most plausibly she would hold that a and b are distinct in virtue of primitive individuators: a and b contain different primitive individual cores in addition to their qualitative properties or universals (such as D. M. Armstrong’s “thin particulars”, 1989, 94–96, or Duns Scotus’s “haecceties”, see Adams 1979, 9), or a and b simply are their own primitive individuators (see Adams 1979 on “primitive thisness”). By establishing his conclusion that the nature of being a substance is to have a complete (qualitative) concept that contains all predicates truly attributable to each of its instances Leibniz rejects all such views of non-qualitative individuation. This, I suggest, is the core point of his real definition of substance-hood.

7 Nominal vs Real Definition of Substance-Hood: Lowest Species qua Substances Block Endless Chains of Attributions

The interpretative question most closely connected to the argumentative transition from the initial Aristotelian definition to the complete concept definition of substance is: why exactly is Leibniz discontent with ADS, why does he call it merely verbal, and how does he take CCS to overcome this problem and to provide a real definition of substance?

First, when Leibniz arrives at his result that “the nature of an individual substance […] is to have a notion so complete that […]”, his aim is not to say something in general about the individual nature of each single substance, although it is plausible that a substance’s complete concept captures this substance’s individual nature. Instead, his ambition is to reveal the nature of what it is to be an individual substance in general, i.e. the general nature of substance-hood.

Secondly, his official distinction between nominal and real definitions says that the first merely suffice to distinguish the kind of things in question, but without providing insight into the possibility of such things, while real definitions do establish the possibility of the kind of things defined (DM §24, 1567–1569). From the fact that Leibniz turns to the question of what it is to be truly attributed to a subject, Rodriguez-Pereyra (2015, 149) infers that Leibniz’s worry concerning the merely nominal Aristotelian definition is “that the idea of true predication or attribution might conceal a contradiction or impossibility”, and he diagnoses that with CCT Leibniz “has done nothing to show that there is no contradiction or impossibility concealed in the idea of a predicate (or concept of a predicate, for that matter) being included in a concept”. However, Leibniz’s worry is not, or need not be, that the idea of true attribution might be contradictory per se. His worry is only that this idea might generate a contradiction within the Aristotelian definition of substance, because it may exclude ultimate subjects in the sense of entities that can only be predicated of themselves. One may also wonder whether in a Leibnizian context the thought that the idea of true predication is impossible per se is coherent in the first place. Quite independently from CCT as a general account of truth, an idea or concept is impossible just in case two contradictory predicates F and non-F are both contained in it. Apparently, this definition presupposes that conceptual containment can occur.

Thirdly, the most prominent context in which Leibniz insists on the importance of real definitions is the ontological proof of the existence of God. Criticising the Cartesian revitalization of the Anselmian proof, he maintains that before existence can be predicated of the most perfect being, it must be shown that such a being is possible, i.e. that the notion of a most perfect being does not imply a contradiction. There is strong evidence that Leibniz’s worry concerns the maximality feature of the concept in question: how can we exclude that the underlying notion of different degrees of reality allows for beings with higher and higher degrees without end, in which case the idea of a most perfect being would turn out to be inconsistent?11 For his analogical illustration in the same passage concerns a maximality notion that, he argues, does contain a contradiction:

[…] I usually use the example of the fastest motion, which entails an absurdity. For let us suppose some wheel turning with the fastest motion. Everyone can see that any spoke of the wheel extended beyond the edge would move faster than a nail on the rim of the wheel. Therefore the nail’s motion is not the fastest, contrary to the hypothesis.

Leibniz, Meditations on Knowledge, Truth, and Ideas, 588–589

Similarly, in DM §23 he in the immediate context of discussing the ontological proof mentions the inconsistencies of “the highest degree of speed” and “the greatest number” (1567).12

By analogy, my proposal is that Leibniz’s more concrete reason for worrying about the merely nominal Aristotelian definition is that it concerns a maximality notion, viz. the notion of an ultimate subject of predication in the sense that this subject can only be predicated of itself. Consider a sequence of true attributions such as: living being is attributed to animal, animal is attributed to mammal, mammal is attributed to human being…. The Aristotelian assumes that such a sequence typically terminates in a true attribution concerning an individual substance: … and human being is attributed to Socrates. But what if the relation of attribution behaves like the notion of one motion being faster than another, so that there are always true attributions of given items to other, ever more specific items without end?

This account of Leibniz’s worry about the Aristotelian ‘merely verbal’ definition explains the particular importance of a clarification of the nature of true predication from Leibniz’s perspective. For him, the relation of true attribution is, in a specifiable way, the possible troublemaker with regard to the very possibility of substances: its nature must be uncovered in order to rule out that it generates endless chains of attributions. On the basis of CCT and the resulting real definition CCS Leibniz takes himself to be able to do so, because he finds perfect candidates for the role of endpoints of chains of attributions, viz. lowest species. They satisfy the definiens of CCS, as there are no even more specific species that could fall under a lowest species’ concept and have predicates not already contained in that concept. And they smoothly fit into the concept containment account of predication, as there is nothing more to them than an exhaustively conceptualisable determinate intrinsic qualitative nature. For thus is the impact of the account of truth in terms of a conceptual mirroring: substances are not merely completely, but exhaustively conceptualisable by an ideal being; there is strictly nothing more to them than what is reflected in their concepts; since concepts can only reflect qualitative aspects (as Leibniz seems to presuppose), substances can be nothing more than perfectly determinate qualitative units. Thus, I conjecture that the Infimae Species thesis is not supposed to merely follow from CCS, but is virtually equivalent to it (cf. Rutherford 1995, 136).

I suggest that it is in this context that Leibniz’s deeper reason for embracing IoI in DM emerges: his central result is that individual substances are lowest species, and duplicate lowest species make no sense.13 By focussing on conceptual containment Leibniz embraces a model of true predications that is tailor-made for predicative relations between universals, such as that humans are rational. Plausibly, the content of and the in-esse relations between universals can be mirrored faithfully by the qualitative content of and the containment relations between concepts. By insisting on the containment model also for predications concerning individuals Leibniz assimilates individuals to universals, identifying them with the limiting case of perfectly determinate species, with the consequence that there can be no two individuals that are perfectly alike.14

Mates maintains that for Leibniz not the individual substance itself but its complete concept is a lowest species (1986, 65). But in DM §9 the doctrine clearly pertains to substances rather than to their concepts; to quote from the original: “que ce que S. Thomas asseure sur ce point des anges ou intelligences (quod ibi omne individuum sit species infima) est vray de toutes les substances” (1541). Mates is clearly motivated by his nominalist interpretation, according to which only individuals are real. Yet Leibniz’s point in DM and related writings could precisely be that by not being further specifiable, by not allowing for duplicates and by necessitating a multitude of less specific features, lowest species are individual enough to count as substances—and that no other candidates than lowest species satisfy CCT, which at the time of DM he defends as non-negotiable.

Rodriguez-Pereyra observes that CCS is unable to distinguish individual substances from their individual essences or substantial forms (2015, 158). A possible conclusion could be that in the deepest metaphysical analysis the Leibniz of DM is committed to a Platonist view to the effect that an individual substance does not so much have, but is an individual, perfectly determinate essence, nature, or form. To be sure, there are passages in which Leibniz distinguishes between a substance and the essence or nature of a substance, e.g. DM §16, 1555, where, however, he seems to identify a substance’s essence with its complete concept. Of course my proposal is not that substances are essences in the sense of concepts or ideas in God’s mind (cf. Adams 1995, 178 on this use); the relevant sense of ‘essence’ is that of a unit of reality (in the sense of realitas, not actuality), of positive qualitative content. Also, in DM §23 Leibniz appears to be referring to minds (“esprits”), and hence arguably to substances, as immaterial natures (“natures immaterielles”, also “natures incorporelles”, 1566; see Adams 1995, 268 on conclusive passages in the letters to Arnauld). And when immediately after introducing his doctrine of substance he confesses the re-introduction of substantial forms in DM §§10–12, the natural reading is that he is thereby referring to individual substances themselves as such forms rather than to something abstract those substances have.

I am not suggesting that such a Platonist metaphysics of substances is defensible, nor that it is without tensions with other assertions in DM. One question would be how a lowest species qua substance relates to a temporal series of states, given that species appear to be eternal and unchangeable entities par excellence. An even more pressing question is what distinguishes existing individual essences from mere possibilia, since Leibniz sharply distinguishes between what results from God’s understanding and what is caused by his free will (§2, 1533; §13, 1549), and correspondingly between essences and existents (“les essences” vs. “les existences”, §26, 1570). However, those distinctions seem compatible with a view according to which existing things are essences that have a special status.15 One possible view is that mere possibilia have only virtual or intentional being as contents of ideas in the mind of God, while existing things are perfectly specific essences that are real results of divine emanation rather than merely virtual. Another is that all essences have a kind of being in virtue of God’s understanding but that due to God’s free act of will only some specific essences enjoy the status (cf. Adams 1995, 160) of existence. Real or existing essences would still be just that, essences in the sense of determinate units of positive qualitative content, rather than essences combined with or realised in a primitive, non-qualitative particularity. If Mercer (2001, chapter 7) is right about Leibniz’s anti-Aristotelian and anti-Cartesian turn in the early 1670s, then at the time of DM accounts are no longer available to him according to which a concrete individual results from an essence, nature or form combining with a primitive portion of matter or any other kind of primitive inert particularity.16

8 Conclusion

After minor repair of one premise, Leibniz’s argument in DM §8 could be reconstructed as a valid inference in first-order predicate logic with two substantial premises and the conclusion that every substance is distinguished by a complete concept containing every predicate of every instance of that concept. Let me close by suggesting an overall view of Discours §8: After his dismissal of Cartesian and post-Cartesian Spinozist as well as occasionalist metaphysics, defending a tenable account of a world of created individual substances is of central importance for the Leibniz of the middle years. Leibniz embraces the Aristotelian definition of substance, which distinguishes substances as the endpoints of chains of predications, as giving a correct role description. Yet a functional definition by itself does not establish that anything can and in fact does play the role described. The predication relation might generate chains of predications without end, with the chains at best metaphysically sustained by one single, monistic substance—the threat of Spinozism. Leibniz’s ambitious endeavour is to infer from the Aristotelian functional definition an account of the real nature of the presumed role player, the metaphysical status of substance-hood, by adding his concept containment account of true predication as a second premise. The result is a rather Platonic identification of individual substances with lowest species, from which the Identity of Indiscernibles for substances follows because duplicate species make no sense. It suggests an interpretation to the effect that created substances do not so much have but are individual essences, natures or forms emanating from the infinite, most perfect divine substance. Leibniz thereby advances a mono-categorial foundational metaphysics, which admits no dualism of form and matter or quality and particularity, but maintains that all there is is qualitative forms of various perfections and specifications.17

Acknowledgement

I am very grateful to Nick Haverkamp for helping with the logical reconstruction and to Stefan Seit for discussing my ideas on Leibniz in a joint seminar. I am indebted to two anonymous referees for extraordinarily detailed and helpful comments and suggestions.

Bibliography

Primary Texts

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  • Leibniz, G. W. Notiones Generales. In: AA VI, 4, Part A, 550557.

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1

Page numbers from 1529 to 1588 refer to DM in Sämtliche Schriften und Briefe (henceforth = AA) VI, 4. Part B. §8 is rather short; quotations without individual references are from that paragraph.

2

I will not take a stance on the logic-centred Russell-Couturat interpretation. See Adams (1994, 71, 75) for the view that CCS is motivated by Leibniz’s views about the structure of concrete individual substances rather than the other way around and for scepticism about the logic-centred interpretation; similarly Mercer (2001, 450).

3

I will be using the translations from Ariew & Garber (henceforth A&G). For DM §§8–9 see A&G, 40–42.

4

To be fair, he also writes that Leibniz “appears to regard the claim that individuals have complete concepts as offering a deeper analysis of Aristotle’s definition of substance rather than a replacement for it” (Jolley 2005, 47).

5

G. H. R. Parkinson (1965, 124–125) reports on Leibniz’s proceeding correctly when he writes: “Leibniz says that [Aristotle’s] answer is not satisfactory, and that the explanation given is only ‘nominal’. This […] seems to mean […] that something has been left out—namely an account of what it is to be truly ascribed to a certain subject. This deficiency Leibniz proceeds to remedy […]”. See also the quote from Jolley in the previous footnote and Woolhouse (1982, 45).

6

For further references see Rodriguez-Pereyra (2015, 159, n. 1).

7

On Leibniz’s distinction between full and complete notions see Rodriguez-Pereyra (2015, 155–158). I will not take up Rodriguez-Pereyra’s (2015, 154–155) distinction between two senses in which entities can be said to have concepts. I suppose concepts in my reconstruction function in his (predicative) sense (b).

8

See Rodriguez-Pereyra (2015, 149, 160 n. 12, 13) for different formulations of the containment theory in Leibniz’s writings, some of which do not make the difference between subjects/predicates and their concepts, or not explicitly. As Rodriguez-Pereyra (2015, 149–150) notes, when Leibniz presents the mirroring of an ontic in-esse by a conceptual containment as his account of predication, he may be seen as advancing a coalescence of (or as conflating) the not in another entity-(or independence) criterion and the not said of another entity-criterion for primary substances, which Aristotle kept apart. By distinguishing an ontic from a conceptual level I do not mean that all items at the former level exist, but merely that they have being in Leibniz’s wide sense; see Rodriguez-Pereyra (2015, 150–151) on the (non-)existence of accidents in particular.

9

Two open formulas are coextensional iff the universal closure of their biconditional is true.

10

This consideration is relevant to the interpretation of the Vestiges-and-marks and the Expression thesis, but also to Leibniz’s account of contingency. As for Expression, Leibniz does not maintain that every piece of information about the universe is a constituent part of Alexander’s concept, but merely that in his soul, which is descriptively reflected by his complete concept, there are “traces of everything that happens in the universe”. And at the end of §13 he seems to be suggesting that Caesar’s future predicates, such as crossing the Rubicon, cannot be detected in Caesar’s complete concept as constituent parts, but can only be read off this concept on the basis of “what is or appears to be best” (1549). See Adams (1994, 13–14) on the distinction between a “relatively narrow” and a richer construal of individual concepts and Adams (30–34) on §13.

11

According to DM §1 a perfection is a form or nature that has a maximal degree (cf. Adams 1995, 113–119). Yet this very notion may prove inconsistent, and with it the notion of a being that has all perfections to the highest degree. To be sure, Leibniz is also concerned with the mutual compatibility of different perfections, see Adams (1995, 141–148).

12

May Leibniz instead be worried that a perfect being would have to be omnipotent and there may be a contradiction in the concept of an omnipotent being, e.g. the contradiction manifested by the paradox of the stone, as an anonymous referee has suggested? It does not seem so, because in DM §1 he states confidently that “the greatest knowledge and omnipotence do not involve any impossibility”, in contrast to “the greatest of all numbers” (1533). In both passages from the Meditationes and DM §23 it is in the immediate context of the problem of the perfect being that Leibniz, by analogy, refers to the inconsistency of the concept of the fastest motion and the greatest number, so the maximality is clearly a central issue.

13

Cf. Notiones Generales, written in the years immediately before DM, where he infers the Lowest Species doctrine as an immediate consequence from the application of the concept containment theory to individuals: “Nam cum dicimus Alexander est robustus, nihil aliud volumus quam robustum in Alexandri notione contineri […] Hinc porro sequitur Singularia esse revera Species infimas […]” (553); cf. Rodriguez-Pereyra (2014, 66).

14

I am indebted to an anonymous referee for suggesting this way of presenting the interpretation.

15

On the relationship between essence and existence see Adams (1994, 157–176); see (42–46), (63–65) on existence.

16

See DM §20 for Leibniz’s reverence to Plato, §27 for his preference for Plato over Aristotle, §14 for his employment of the neo-Platonist notion of emanation. See Mercer (2001) for the Platonist origins of Leibniz’s metaphysics, cf. Mercer (2008, 237): “at the very center of Leibniz’s philosophy, is the Platonist theory of emanation”. Prima facie, such a Platonist reading is opposed to D. Garber’s (1985) Aristotelian interpretations of Leibniz’s metaphysics in his middle years; see Adams’ critique (1994, 324–338). However, a Platonist deep metaphysics is compatible with Aristotelian conceptions at a derivative level. For an early qualitativist interpretation of Leibniz see R. Adams (1979, 9): “Leibniz held […] that the thisness of each particular is a suchness. ‘Singulars’ [are] the most specific members of the system of kinds.” For a defence of such a purely “generalist” interpretation of Leibniz see Møller-Nielsen (2015).

17

The general problem of categorial monism is that elements of one category need an element of some other in order to combine to form the complex world as a whole. For Leibniz the relationship of emanation and/or the status of existence may be such further elements. For a categorial dualism of particulars and predicables see Busse (2018).

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