Kant and Russell on Leibniz’ Existential Assertions

Leibniz believed in a God that has the power to create beings and whose existence could be a priori demonstrated. Kant (KrV, A 592-602/B 620-630) objected that similar demonstrations all presuppose the false claim that existence is a real property. Russell (London and New York: Routledge, 1992) added that if existence were a real property Leibniz should have concluded that God does not actually have the power to create anything at all. First, I show that Leibniz’ conception of existence is incompatible with the one that Russell presupposes. Subsequently, I argue that on Leibniz’ conception of existence Russell’s objection is immediately undermined.


Introduction
Leibniz believed that it can be a priori demonstrated that existence is part of the essence of God, or in other words, that God necessarily exists. Throughout his life, Leibniz put forward several such demonstrations (sometimes known as ontological arguments). Famously, Kant argued that all of these relied on a conceptual mistake concerning the predicate of existence. As is well-known, Kant believed that existential assertions were synthetic a posteriori. No property at all is predicated when an existential assertion is made, asserting the existence of something being simply tantamount to saying that that very something, together with all of its predicates, is there. Therefore, Leibniz could not legitimately argue that an entity defined as God had to have the property of existence.
Not much more than a century later, Russell (1992) examined Leibniz' theory of existential predication and found it inconsistent with his (i.e. Leibniz') theory of creation. According to Russell, Leibniz would have ambiguously characterised existence.
When considering the existence of a created substance, Leibniz would have endorsed a Kant-style account. Yet, when considering the existence of God, Leibniz would have accepted that existence is an actual property. Thus, Russell formulates the following dilemma. Leibniz has to agree that there is just one notion of existence and this is either an actual property or not. Suppose, Russell argues, that existence is not an actual property. It immediately follows that every a priori demonstration of God's existence is unsound. Leibniz is thus forced to concede that existence is an actual property, but this answer is not any better to him. Surely Leibniz can now claim that the property of existence is part of the essence of God. However, Leibniz would also be committed to the view that existence, being an actual property, is part of the essence of every existing substance. But if every existing substance is essentially existing, then God does not have the power to create anything at all. Hence, Russell concludes, 'either creation is self-contradictory, or, if existence is not a predicate, the ontological argument is unsound' (Russell 1992, p. 219).
The main goal of this paper is to confront Russell's confutation of Leibniz. First, Russell's assumption that there is just one notion of existence and this is either an actual property or not is not easily available to him. For I argue that Leibniz' conception of existence is more fine-grained. Specifically, my contention is that Leibniz' conception of existence is twofold. One thing is the existence of God, one thing is the existence of created things; in Leibniz, the two have different characterisations. 1 Then, Russell is certainly right in pointing out that Leibniz characterised existence as an actual property of God, though not of created substances. However, Russell's claim that creation is incompatible with the ontological argument misfires. Indeed, it is exactly by means of such a twofold conception of existence that Leibniz 1 An anonymous referee, to whom I am very grateful for pressing me on this point, pointed out that the view I am attributing to Leibniz appears to be a particular case of the doctrine, widely held amongst early-modern philosophers, that words do not signify as usual when applied to God-witness for instance Descartes' discussion of substance in Sect. 51 of his Principles of Philosophy (AT VIII,24). I propose that Leibniz' equivocal use of existence be understood in light of this doctrine, which Leibniz endorsed on several occasions. For example, consider Sect. 48 of the Monadology, where Leibniz compares the perceptive and appetitive faculties of created entities with the power and will of God. He observes that 'in God these attributes are absolutely infinite or perfect, whereas in created monads or entelechies they are only limitations, proportional to the perfection they possess (my italics)' ((G VI, 615) (Leibniz 1998, p. 274)). More interestingly, in his 1692 Animadversiones in Partem Generalem Principio Cartesianorum (G IV, 350-406), Leibniz discusses Descartes' account of substance as formulated in Sect. 51 of the Principles and, as far as I understand him, took a possibly even more radical stance than Descartes. For, the latter had derived the equivocality of substance from the fact that a substance is a thing that only needs itself to exist. Then, since there is only one substance so-characterised (God), any other thing must be a substance in a different sense-indeed, '[i]n the case of all other substances, we perceive that they can exist only with the help of God's concurrence' ((AT VIII, 24) (Descartes 1985, p. 210)). There is no doubt that Leibniz agreed with Descartes that God existed in such a way that did not need anything else to exist, and so this gives us one sense of substance. Yet not only, Leibniz argues, does a created substance need God to exist, it also 'needs accidents even much more' ((G IV, 364) (Leibniz 1989b, p. 389)) given that, he continues, 'I do not know whether the definition of substance as that which needs for its existence only the concurrence of God [Descartes' definition] fits any created substance known to us' (ibidem). I conclude that if Descartes defended an equivocal conception of substance, all the more did Leibniz, whose conception of existence should be understood in light of the doctrine that words do not preserve their usual meaning when applied to God. could consistently accept both creation and the ontological argument. Hence, on this twofold account of existence, a rejection of Russell's dilemma immediately ensues. The upshot is that if Russell's existential monism is true, then Russell's dilemma is conclusive; however, if Leibniz' existential pluralism is true, then Russell's dilemma fails. Of course, one could still argue that Leibniz could not consistently have creation and the ontological argument. What my paper shows is that it is crucial for such a claim to be established under existentially plural assumptions.
In the first section ('Leibniz 1676. The Compossibility of All Perfections'), I will illustrate one of Leibniz' ontological arguments; subsequently, I will reformulate Kant's worries about these proofs and present Russell's dilemma more extendedly ('Kant, Existence and Russell's Dilemma'); then, I show that Leibniz endorsed a twofold conception of existence and that this defeats Russell's dilemma ('Leibniz Vindicated'); finally, I sum up what has been achieved in this paper ('Conclusion').

Leibniz 1676. The Compossibility of All Perfections
The goal of this section is to present Leibniz' 1676 ontological argument, to be found in his essay Quod Ens Perfectissimum Existit (QEPE henceforth,GP VII,(261)(262). I first say a little bit about the role of Descartes behind QEPE, and then turn to Leibniz' argument. Thirty five years before QEPE, in 1641, Descartes had published his Meditations. The fifth of those included an a priori demonstration of the existence of God, defined there as the most perfect being (Ens Summe Perfectum) ((AT VII, 65-71) (Descartes 1996, 46-49)). The argument goes as follows: since we can think of the being defined as the most perfect being, and since existence is a perfection, the most perfect being exists. Or so thought Descartes. In QEPE, Leibniz finds Descartes' reasoning defective. Surely we can define many entities, such as the biggest positive natural number or the set of all non-selfmembered sets; such definitions seem perfectly intelligible. But this is hardly a proof that they define something possible. For example, given that the concept of biggest positive natural number is provably inconsistent, an object defined in this way is arguably not possible.
We can find a clear formulation of this thought in Leibniz' later essay Meditationes de Cognitione, Veritate, et Ideis from 1684 (MCVI henceforth), where he argues that: [W]e cannot safely use definitions for drawing conclusions unless we know first that they are real definitions [my italics], that is, that they include no contradictions, because we can draw contradictory conclusions from notions that include contradictions, which is absurd. To clarify this 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.
[A]t first glance we might seem to have the idea of a fastest motion, for we certainly understand what we say; but yet we certainly have no idea of impossible things. [...] And so, one should not omit any necessary premise, and all premises should have been either previously demonstrated or at least assumed as hypotheses, in which case the conclusion is also hypothetical. Those who carefully observe these rules will easily protect themselves against deceptive ideas ((GP IV 424-426) (Leibniz 1989a, pp. 25-27)).
First, in this passage from MCVI, Leibniz invokes the distinction between real and nominal definitions. An example of nominal definition is the definition of the fastest motion, which Leibniz thought to be contradictory. Given that its definiendum cannot be real, such a definition is merely verbal or nominal. On the other hand, a definition is real when it has no such consequences, that is, when no contradictions follow from it; so its definiendum might very well be real. 2 Secondly, in the final part of the quotation, we can find the following remark regarding the premises of an argument. A premise can be either assumed or demonstrated. In the first case, the conclusion of the argument must be conditional to the assumed premise. In the second case, i.e., when a premise is demonstrated, Leibniz seems to suggest the uncontroversial fact that we can safely rely on that premise to draw a true conclusion.
With this in mind, we can sum up Leibniz' criticism of Descartes in the following terms. Descartes believed to have demonstrated the existence of a most perfect being from the assumptions that such an entity is possible and that existence is a perfection. In particular, Descartes held that the most perfect being's possibility is secured by the fact that we can think of a being defined in those terms. But unbeknownst to us, definitions may sometimes pick out inconsistent concepts and, therefore, be merely nominal. To establish that they are not, a proof of consistency is needed, showing that their definiendum is possible. Descartes failed to offer any such proof for the most perfect being, and Leibniz argued in MCVI that the conclusion of an argument including an unproven premise has to be conditional to (the truth of) such a premise. Hence, what Descartes has actually demonstrated is only that if the most perfect being is possible, then a most perfect being exists.
I can finally turn to QEPE. Leibniz shared with Descartes both the belief that existence is a perfection, and the conditional claim that if the most perfect being is possible, then the most perfect being exists. In essence, QEPE aims to bring to completion Descartes' unfinished business, that is, demonstrate the antecedent of that conditional. What follows is a reconstruction of how Leibniz tried to get to that result.
The opening proposition of QEPE attempts to define what a perfection is. We are thus given the following definition (GP VII 261).
Definition 1 A perfection is a simple quality which is positive and absolute, that is, that expresses without limitations what it does express.
A precise exegesis of this definition is far beyond the aims of the present paper. Nevertheless, what the definition aims to define has at least one straightforward understanding. First, we are told that a perfection is a quality. As such, its appropriate logical category seems to be that of predicates. For after all, Leibniz took existence to be an example of perfection; and such a perfection in Leibniz' view belonged to a being, namely the most perfect one. But the expression 'a being' falls under the category of terms, and terms denote individuals. Therefore, we can assume that by 'quality' Leibniz had in mind something that individuals have, and this requires that a quality be denoted by a predicate.
Moreover, Leibniz characterises a perfection as a quality that is simultaneously simple, positive and absolute. However, since the only characteristic of perfections that QEPE relies on is simplicity, I will only focus on it and leave aside considerations concerning positiveness and absoluteness. 3 Here is an intuitive way to characterise simplicity, based on Leibniz' remarks in QEPE. Call P an arbitrary perfection. Then, P cannot be analysed into simpler qualities; if it could, it would not be a perfection. For instance, on the Aristotelian definition of man as a rational animal, being a man is not a perfection. For call A, B and C the qualities of being a human being, being rational and being an animal respectively. Then, since Ax = Def Bx&Cx, A can be analysed into simpler qualities. As a consequence, it cannot be a perfection.
Leibniz next assumes that the most perfect being, i.e. the being that has every perfection, is not possible. If it is not possible that there is a such a being, then presumably this is because at least two perfections contradict each other. Call and two such perfections and consider the following proposition.

Proposition 1 Necessarily, and contradict each other.
The analysis of Proposition 1 will eventually take us to Leibniz' proof of the most perfect being's possibility. There are two criteria, Leibniz argues, to establish that something is necessarily the case. Only when a truth meets either of those criteria, is it a necessary truth. The first criterion is self-evidence. The truth of the statement 'necessarily, it is not the case that Madrid is in Spain and not in Spain', for example, does not require a proper justification, it is simply self-evident to any rational being. The second criterion is the reduction of necessary truths to identity statements. What Leibniz probably had in mind here is explained in his 1689 essay De Contingentia (DC hereafter), where he argues that Necessary truths are those that can be demonstrated through an analysis of terms, so that in the end they become identities, just as in algebra an equation expressing an identity ultimately results from the substitution of values ((A VI 4, 1649) (Leibniz 1989a, p. 28)).
Here is an example, borrowed from Look (2013), of a necessary truth so characterised. Consider the statement '2 + 2 = 4'. By the reiterated analysis of each of the terms placed at both sides of the identity sign, we will come to the conclusion that '1 + 1 + 1 + 1 = 1 + 1 + 1 + 1'. A necessary truth, according to Leibniz, is one that can be reduced to a true identity statement such as '1 + 1 + 1 + 1 = 1 + 1 + 1 + 1'.
Does Proposition 1 meet either of the two criteria? Surely Proposition 1 is not self-evident. Indeed, Descartes for example believed the negation of it (and so did Leibniz, by the way). Moreover, one cannot analyse the constituents of Proposition 1 into simpler parts. Indeed, since perfections are simple, and cannot be analysed any further. As a consequence, Proposition 1 cannot be reduced to a true identity statement. Hence the second criterion is also not met. It follows that Proposition 1 is not a necessarily true statement. Hence, it is not necessarily the case that and contradict each other. But this is tantamount to saying that it is possible that and do not contradict each other. And since and are just arbitrarily taken, this conclusion will hold for every pair of perfections, which corresponds to the desired conclusion. Since all perfections are compossible, a being having all of them (a most perfect being) is possible. It is now sufficient to assume that existence is a perfection to get the following argument.
1. A being that has every perfection (the most perfect being) is possible. 2. Existence is a perfection. ∴ A most perfect being exists.
A few words to sum up the content of this section. So far, I explained how Leibniz justified premise 1 of the above argument. This also helped to clarify some notions of Leibniz' philosophical vocabulary, such as the distinction between nominal and real definitions, as well as Leibniz' understanding of concepts which are still part of the current philosophical toolkit, such as that of necessarily true statement. It is now time to examine what problems, according to Kant and Russell, Leibniz incurred by accepting premise 2, that is, existence is a perfection.

Kant, Existence and Russell's Dilemma
The plan of this section is illustrated throughout this paragraph. In the previous section, we encountered Leibniz' definition of perfections as simple, positive and absolute qualities. I only tried to give an account of simplicity and left aside any attempt to characterise positiveness and absoluteness. Moreover, I argued that because Leibniz considers perfections as qualities, and because qualities are subsumed under the logical category of predicates, perfections are in turn denoted by predicates. Obviously, there might be predicates that do not denote perfections. Yet, any perfection, if Leibniz' argument has some meaningfulness, must be denoted by a predicate. Now, Leibniz took existence to be a perfection (premise 2). Should we draw the conclusion that the existence predicate denotes a corresponding quality, namely that of existing? In the Critique of Pure Reason, Kant famously argued that we should not, and so rejected both premise 2 and the ontological argument as such (KrV, A 592-602/B 620-630). On the other hand, Russell (1992) took a much more radical stance on the matter. Not only did he agree with Kant that the existence predicate does not denote a corresponding quality, but also argued that if it did Leibniz' system of metaphysics as a whole would be inconsistent. In this section, I will first very quickly rehearse why Kant (KrV, A 592-602/B 620-630) denied that the existence predicate denotes an actual property. Subsequently, I turn to Russell's inconsistency claim.
Kant's rejection of the ontological argument, as already noted, entirely depended on the rejection of premise 2. More precisely, Kant did not find fault with premise 2 in itself, but rather with an allegedly false claim that such a premise would imply. Indeed, by assuming that perfections are qualities, then, if qualities are denoted by predicates, it follows the existence predicate denotes in turn the quality of existence. This last claim, according to Kant, is false. (An aside. From now on, I will replace the expression 'quality' by the more familiar expression 'property'. In doing so, I assume that the meaning of those expressions is the same. Both denote something that can be predicated of things. End of the aside.) A property, Kant argued, can be either analytic or synthetic. 4 We cannot think of a body without also thinking of it as having a spatial extension. This means that the definition of body implies the property of being spatially extended. As such, being spatially extended is an analytic property of bodies. We can more formally rephrase the definition of analytic property as follows. Given a concept B (e.g. body) and a property E (e.g. being spatially extended), E is an analytic property of B if and only if an individual cannot be a B without also being an E. However, we can certainly think of a body without necessarily thinking that it is green. Of course some leaves are green, so some bodies are indeed green. However, the table at which I am now working is brown. Hence, some bodies are not green. This means that the definition of body does not imply the property of being green. As such, being green is a synthetic property of bodies. The definition of synthetic property can then be rephrased as follows. Given a concept B (e.g. body) and a property G (e.g. being green), G is a synthetic property of B if and only if an individual can be a B and yet fail to be a G.
Kant argued that existence is a synthetic property of every concept. We might very well have a definition of a certain concept C, but that does not say in itself that there are Cs. To know that there are Cs, we need some more information. Suppose I want to know whether in the park in front of my house there is a statue of Richard Wagner. What I want to know is whether something that falls under a certain concept (i.e. being a Richard Wagner statue located in the park in front of my house) also falls under the concept 'to exist'. There is just one way, Kant would argue, to unravel the mystery. Go to the park, walk through it, and see whether you can bump into a Richard Wagner statue. If you can, and only if you can, you are entitled to assert that there exists a Richard Wagner statue in the park in front of your house. This means that in absence of empirical evidence, any existential assertion about a certain thing will always be illegitimate. 5 Accordingly, an immediate consequence of this account is that existential assertions can only be legitimately formulated a posteriori. As such, from a Kantian point of view, Leibniz could not legitimately argue that the concept 'most perfect being' entailed the existence of a most perfect being. For presumably Leibniz never stumbled upon any such being. Hence, he lacked the empirical evidence necessary to assert that a most perfect being exists, and so should not have put forward such a claim.
All this tells us that according to Kant there would be something very special about the existence predicate. For presumably, if a statue has the property of being dedicated to Richard Wagner, the statue should exhibit certain characteristics corresponding to that property. For example, the statue of Richard Wagner located in the park in front of my house contains a bust of the German composer and a small plaque in his memory. However, if a statue of Wagner has the property of existence, what characteristics corresponding to such a property should the statue have? Let Kant ((KrV, A 600/B 628) (Kant 1998, p. 567)) reply to that question.
[W]hen I think a thing, through whichever and however many predicates I like (even in its thoroughgoing determination), not the least bit gets added to the thing when I posit in addition that this is.
'Nothing' is therefore Kant's answer. No property at all is predicated when an existential assertion is made. Thus, for instance, when I say that there is a statue of Richard Wagner in the park in front of my house, all I am actually saying is that a certain object, with such and such predicates, simply is there. I can touch it, I can see it, I can have many sorts of causal interactions with it; but no real property corresponds to that object's existence, apart from its being there in front of me.
The effects of Kant's position on the history of modern philosophy can hardly be overestimated. Vilkko and Hintikka (2006, p. 359) summarise them in the following terms: After Kant, existence was left homeless. It found a home in the algebra of logic in which the operators corresponding to universal and particular judgments were treated as duals, and universal judgments were taken to be relative to some universe of discourse. Because of the duality, existential quantifier expressions came to express existence. The orphaned notion of existence thus found a new home in the existential quantifier.
Surely, if existence could find a new home in the existential quantifier, this was possible thanks to the contribution (among others) of Bertrand Russell. Like Kant, Russell (1905) believed that the existence predicate did not denote a genuine property of individuals. 6 That is to say, it would be wrong to divide all individuals into two classes, those who exist and those who do not. This mistake would lead anyone to contradict themselves when making a negative existential claim. For it might seem prima facie reasonable to say that unicorns are non-existing individuals. Yet, it is also uncontroversially true that there are no unicorns. Hence, if unicorns are non-existing individuals, then, in order to deny that there are unicorns one would be committed to the inconsistent claim that there are things that are unicorns and these things are not there. Instead, if existence is taken to be a universal property of a given domain of individuals, these problems do not arise. To say that there are no unicorns now becomes tantamount to saying that that the concept 'unicorn' has no instance in that domain, and the expression 'individual' becomes a shortcut for the expression 'existing individual'. Existence is thus taken to represent the instantiation of a concept with respect to a domain of (existing) individuals. Again, there seems to be something special to the existence predicate. Whilst the predicate 'is a statue of Richard Wagner' denotes a property of one such statue, the job that the predicate 'exists' does, according to Russell, is just to tell us that the predicate 'is a statue of Richard Wagner' has an instance. 7 It is no wonder, then, that Russell too found fault with Leibniz' premise 2 (existence is a perfection) and so rejected the ontological argument. 8 However, according to Russell, much more is actually at stake. Indeed, Russell (1992, pp. 218-219) argues that Leibniz should have given up premise 2 not only because it is false, but also because such a premise would contradict other metaphysical claims that Leibniz himself had endorsed. Specifically, premise 2 would contradict Leibniz' own theory of creation.
What follows is how Russell came to that conclusion. If the ontological argument is sound, then this has three immediate consequences. First, that existence is a real property; second, that existence is part of the essence of God; third, that God cannot fail to exist. Hence, an existential assertion about God is necessarily true. Instead, an existential assertion about a created individual is just contingently true. This has three immediate consequences. First, that a created individual might fail to exist; second, that existence is not a property belonging to that individual's essence; third, that existence is no property at all. So, from the ontological argument, it follows that existence is a real property; and from there being created individuals it follows that existence is not a real property. It follows that the ontological argument is inconsistent with the notion of creation. 6 Although it needs to be observed that Russell did not hold a similar view when he wrote his Critical Exposition of the Philosophy of Leibniz (1900). For example, it is rejected in his 1903 book Principles of Mathematics (Russell 2009, pp. 482-483). Thanks to an anonymous referee for calling my attention on this. 7 The distinction that is often invoked here is between a first-order property, (being a statue of Richard Wagner), and a second-order property (existence). Being a statue of Richard Wagner is a first-order property in that it is a property of individuals, i.e. the individual being the statue of Richard Wagner. Existence on the other hand is a second-order property in that it would be a property of properties-in this case, of the property of being a statue of Richard Wagner. See on this Nelson (2012). 8 Russell (1905, p. 491) explicitly rejects Leibniz' QEPE argument. However, one might now wonder why did Russell claim that from the fact that an individual's existence is contingent it would follow that existence would not be a real property. Russell (1992, p. 219) appeals to Leibniz' theory of substances. Famously, Leibniz argued that each individual substance is associated to a complete concept (GP IV,. A complete concept is the set of absolutely all the predicates (past, present and future) that pertain to a given substance. For instance, Julius Caesar crossed the Rubicon in 49 BC. Of course he might have failed to do so. But he could not have failed to do so without also ceasing to be Julius Caesar. Crossing the Rubicon is an essential property of Julius Caesar. For Julius Caesar's complete concept includes the property of crossing the Rubicon. Hence, a Julius Caesar that has not crossed the Rubicon could not actually exist. Similarly, Julius Caesar existed. Hence, by analogous reasoning, if existence were a real property Julius Caesar could not fail to exist. But this means that existence is an essential property of Julius Caesar. As such, Julius Caesar's existence is not a contingent fact, and so God would have no power to bring Julius Caesar into existence. Julius Caesar would simply exist in virtue of his own essence, i.e. necessarily. Thus, Russell claims, the upshot is that if Leibniz wants to argue that there are created substances (which exist contingently), he also has to give up the claim that existence is a real property.
Nevertheless, and this is the crucial passage of Russell's point, if Leibniz were to concede that existence is not a real property, then the ontological argument could not be conclusive anymore. For then, if existence were not a real property, existence would not be a perfection either. But if existence is not a perfection, then premise 2 (existence is a perfection) is falsified. And this would make the ontological argument unsound. Either way, therefore, problems are forthcoming for Leibniz. Below is a schematic reformulation of Russell's dilemma. On the one hand, Leibniz can maintain that existence is part of God's essence. R1 Therefore, existence is a real property of everything (i.e. God and created substances). R2 But, if existence is a real property, then it is part of the essence (complete concept) of any existing substance. R3 Hence, existence is an essential property of every existing individual. R4 If so, God would not have the power to create anything, for all substances would exist simply in virtue of their essence (complete concept).
Otherwise, Leibniz could maintain that God has the power to create substances.
R5 Hence, existence is not a real property of anything (i.e. God and created substances). R6 For the ontological argument to be sound, it is necessary that existence is a real property. R7 Therefore, the ontological argument is unsound.
To sum up, Leibniz' ontological argument would be inconsistent with there being created substances. Is it really so? This question is to be addressed in the next section, where I will argue that it is not.

Leibniz Vindicated
We have just learned from Russell that Leibniz held mutually inconsistent metaphysical beliefs. Indeed, the ontological argument would result in a confutation of the notion of creation, and creation would make the ontological argument unsound. After all, the ontological argument presupposes that existence is a real property, and creation that it is not. Thus, Russell argues, Leibniz should have picked out just one of the two options and rejected the other.
In the first part of this section ('Leibniz' Twofold Conception of Existence'), I argue that what Russell actually spotted is no contradiction at all. Leibniz simply used two distinct notions of existence: one pertaining to created substances and one specifically pertaining to God. My contention is that Leibniz believed that existence was a real property of God, but denied that existence was a real property of created substances. Moreover, on this twofold conception of existence, Russell's dilemma can immediately be refuted. Giving a refutation of Russell's dilemma based on such a twofold conception of existence will be the goal of the second part of this section ('A Refutation of Russell's Dilemma').

Leibniz' Twofold Conception of Existence
That Leibniz believed existence to be a real property of the essence of God is a widely acknowledged fact that need not be justified. I will now argue that Leibniz believed existence was not a real property of created substances. 9 Should this turn out to be right, this would also be sufficient to show that Leibniz took existence to be a twofold notion. However, more can be said about this story. Indeed, Leibniz explicitly and literally admitted to have endorsed a twofold conception of existence. This happens in a 1678 letter to Henning Huthmann (A II 1, 585-591). The most straightforward interpretation of this letter will shed light on Leibniz' prima facie inconsistent conception of existence. When Leibniz characterised existence as a real property, he certainly circumscribed his attention to the existence of God; and when he denied that existence was a real property, he meant to refer to the existence of created beings. Eventually, it will become evident that by 'existence' Leibniz clearly meant two distinct things, namely the existence of God and the existence of all the rest.
A creatable person, before their creation is decided, already includes in their complete possible concept [notione sua plena possibili] all the things that can be said of them if they were created, and among those a certain series of graces [gratiarum] and free actions. Thus, God does not decide to give a grace or other things to that person with respect to their essence [quoad essentiam], but with respect to their existence [quoad existentiam]. That is, God decides to create a creature whose complete concept includes that certain series of graces and free actions, although this series is postulated by God not necessarily but in accordance with the nature of the thing.
In this passage from DLCR, Leibniz gives us a metaphysical account of what it means for an individual (specifically, a human being) to be created. First, Leibniz invokes the notion of complete concept which, recall, Leibniz defined as the set of absolutely all predicates that can be ascribed to an individual. Then, Leibniz considers the complete concept of a possible, but not yet existing, human being. The complete concept of such a human being will include a number of properties. However, what happens when a not yet existing human being comes into existence? In particular, did Leibniz believe that the complete concept of a not yet existing human being, when coming into existence, acquires the new property of existence? If we say yes, then we would incur an awkward situation. Indeed, in the above passage, Leibniz argues that a not yet existing human being's complete concept includes 'all the things that can be said of them if they were created'. Thus, suppose that existence is a real property that human beings acquire when they are created. It follows that it can be said of a created human being that they possess the property of existence. However, it also follows that a not yet created human being possesses the property of existence, as the complete concept of such a human being includes everything that can be said of them if they were created. This is odd, as Leibniz should then admit that a not yet created human being exists.
This unpleasant conclusion depends on the hypothesis that Leibniz believed existence to be a real property of individuals. Then, this leaves room for two options: either Leibniz held such a belief and was not aware of the oddity that it incurs, or indeed he did not believe here that existence was a real property of created individuals. The above quotation from DLCR includes evidence for the latter claim. Indeed, we are here told that 'God does not decide to give a grace or other things to [a] person with respect to their essence [quoad essentiam], but with respect to their existence [quoad existentiam]'. Suppose that God decides to give a grace a to an existing person. For instance, suppose that God decides to give a person the grace of surviving a car crash. Call the person t and the property of surviving the car crash C. Leibniz says that we should not conclude that, at a certain time after t's creation, God decides to add the new property C to the complete concept of t-that is, C is not given to t quoad essentiam. For if so, then we should admit that t's complete concept before creation would not have a property that t's complete concept after creation does have. Leibniz, as we have seen, rules this option out as he states that a not yet existing human being's complete concept 'includes all the things that can be said of them if they were created'. Thus, C had always been included in t's complete concept, and God just makes it possible that C is actually instantiated by t-that is, C is given to t quoad existentiam. All this also shows us that Leibniz could not have taken existence to be a real property of a created individual. Indeed, we just saw that an individual's complete concept before creation has to be identical to that individual's complete concept after creation. Then, surely, an individual could not be given the property of existence at creation, as in this case the individual would have two non-identical complete concepts, namely one before creation and one after creation. However, an individual could not be given the property of existence before creation either. Indeed, we saw above that this case would imply that a not yet created individual would already exist, which is clearly odd. Therefore, whenever existence is treated as a real property, troubles are forthcoming. Thus, this situation makes it implausible that Leibniz believed existence was a real property of created individuals.
All this happens in 1686. In 1677, Leibniz had put forward some claims on existence using an expression that we already found in QEPE. This time, though, Leibniz' claims seem to contradict QEPE. Thus Leibniz in his essay De Existentia (DE henceforth (AVI 4, 1354) (Adams 1994 (165)).
[I]t is true indeed that what exists is more perfect than the non-existent, but it is not true that existence itself is a perfection [verum non est, ipsam existentiam esse perfectionem], since it is only a certain comparative relation of perfections among themselves In QEPE (1676), we are told that existence is a perfection (premise 2); and in the above passage from DE (1677), we are told that this is not the case.
As an aside, recall that in QEPE Leibniz gives us a definition of perfections. According to such a definition, a perfection would be a quality with some distinctive characteristics (simplicity, positiveness, absoluteness). On the account assumed throughout the paper, there would be no conceptual difference between qualities and properties. Both seem to be something that can be predicated of individuals. As such, in this section, the expressions 'quality' and 'property' will keep being used interchangeably.
So in 1676 according to Leibniz existence would be a perfection, that is, a property with the above distinctive characteristics (QEPE). However, in 1677, according to Leibniz, this would no longer be case: existence is not a perfection (DE). Unfortunately, if we were to take the intuitive line that in 1677 Leibniz simply revised his beliefs on existence, then we would be faced with a difficulty.
Indeed, in January 1678, Leibniz was responding to Henning Hutmann, who not long before had asked Leibniz to provide him with a proof that the existence of God follows from God's essence (A II 1, 584). From the critical edition of the Berlin Academy, we learn that Leibniz' response to Huthmann included three manuscripts, called L 1 , L 2 and L 3 (A II 1, 585). Each of those includes a different variant of the proof. Maybe not all of them eventually reached Henning Huthamnn. However, we nonetheless notice something very interesting. In each of these manuscripts, Leibniz refers to existence either directly as a real property (L 1 ), or as a perfection (L 2 and L 3 ).
In L 1 , Leibniz observes that ' [i]f we define God as Ens a Se, or being from whose essence existence follows, or necessary being, it follows that if God is possible God also exists in fact' (A II 1, 586, my translation). In L 2 , we are told that 'God is also defined as a most perfect being [Ens Perfectissimum] and in particular existence is one of the perfections; so here one can rightly say that God is the being whose essence includes existence' (A II 1, 587, my translation). The L 3 variant includes two subvariants . In the first of these, we find a premise to the extent that 'existence in fact [existentia actualis] is a certain perfection in kind' (A II 1, 589, my translation); in the second variant, a premise is stated according to which 'the essence of God includes necessary existence [necessitatem existendi involvit]' (A II 1, 589, my translation). Given these remarks, it seems clear that in 1678 Leibniz believed existence to be a perfection and, therefore, a real property.
Above I mentioned that if we claimed that in 1677 Leibniz abandoned the idea that existence is a real property, we would be faced with a difficulty. Indeed, the Letter to Huthmann shows that in 1678 Leibniz surely believed that existence was a real property. Thus, the difficulty is that if we are ready to admit that in 1677 Leibniz changed his mind for the first time, then, we should also be committed to the view that in 1678 Leibniz changed his mind for the second time. The scenario that this view would yield is therefore the following. In 1676, Leibniz believed that existence was a perfection-and, hence, a real property (QEPE)-then, in 1677, he rejected such a claim (DE); finally, in 1678, he changed his mind again and returned to his old view that existence is a perfection-and, hence, a real property (Leibniz' Letter to Huthmann).
If this were not intricate enough, we must also take into account that, as we have already seen at the beginning of this section, in 1686 Leibniz clearly seems to rule out that existence is a real property (DLCR). That would mean that in 1686 Leibniz changed his mind for the third time. There would even be room for a fourth revision, as in the Monadology (1714), we read what follows ((GP VI, 614) (Leibniz 1998, pp. 273-274)).
it is clear that if there is any reality among essences or possibilities [...] that reality must be grounded in something actually existent; therefore it must be grounded in the existence of the necessary being, in whom essence includes existence [my italics], that is, for whom being possible is sufficient for being actual.
Leibniz would have eaten his own words again. For according to Leibniz here, existence is part of the essence of God. Therefore, existence would now be a real property, against Leibniz' own 1686 remarks from DLCR. Now, if we seriously took the view that Leibniz genuinely denied his own beliefs all these times on whether existence was a real property, then we would not clearly be able to determine which position he actually held. This situation is barely believable. How could Leibniz contradict himself so explicitly, so many times, and on such a crucial aspect of his own system? My claim is that Leibniz held both views, namely that existence was a real property and that it was not. The contradiction arising here is merely illusory. The view on existence that I am ascribing to Leibniz is not inconsistent in that Leibniz took existence as a twofold notion. The first notion of existence corresponds to the existence of God; here existence is understood as a real property (of God only). The second notion of existence corresponds to the existence of created beings; here existence is not understood as a real property (of created beings). As far as I understand Leibniz, the texts mentioned in this section should be understood in the following manner. In QEPE, the Letter to Huthmann and the Monadology Leibniz deployed the first notion of existence, i.e. the existence of God. In DE and DLCR Leibniz resorted to the second notion of existence, i.e. the existence of all the rest. Thus, when Leibniz does not characterise existence as a real property, e.g. in DE and DLCR, he is just using one possible notion of existence; and when Leibniz characterises existence as a real property, e.g. in QEPE, the Letter to Huthmann and the Monadology, he is actually using another notion.
If my understanding is correct, then we would be in the presence of a deep philosophical commitment. As such, there would be, I think, two things to establish. First, that somewhere in his texts Leibniz effectively considers the possibility that existence would be a twofold notion; second, that a twofold conception of existence fits the wider picture of Leibniz' metaphysics. I will start with (what I think is) the easy task, namely surveying the presence of Leibniz' references to existence as a twofold notion.
There  [Ens Necessarium]. This privilege of God's essence certainly does not seem to be despised. The existence of contingent things does not follow from their essence or possibility, but from God's will or, which is the same, from the universal harmony of things.
Four years later, in the opening lines of De Contingentia ((A VI 4, 1649) (Leibniz (1989a, 28)), Leibniz says that: Existence does not differ from essence in God, or, what is the same thing, it is essential for God to exist. Whence God is a necessary being. Creatures are contingent, that is, their existence does not follow from their essence.
Surely these passages establish one thing, namely that Leibniz used to distinguish between the following two cases. The existence of God, that can be deduced from his own essence, and the existence of created beings, that cannot be deduced from their own essence. However, this does not say in itself that Leibniz thought of existence as a twofold notion. For all we know, Leibniz could merely be thinking that the existence of God was necessary and that the existence of created beings was contingent. On this modal account, existence does not split into two different notions. Of course, the existence of God and that of a created being would generate two different modal truths; specifically, the existence of God would result in a necessary truth, whereas the existence of a created being would result in a contingent truth. However, this account would not result in a dichotomy between two notions of existence.
To see this dichotomy emerge more clearly, we must consider again Leibniz' Letter to Huthmann. In the L 3 manuscript of such a letter, Leibniz argues as follows ((A II 1, 590) (Leibniz 2006, p. 185)).
[I]n the region of eternal truths, or in the realm of ideas that exist objectively, there subsist unity, the circle, power, equality, heat, the rose, and other realities or forms or perfections [my italics], even if no individual beings exist, and these universals were not thought about; so also there is found, among other forms or objective realities, actual existence [actualis existentia], not as is found in the world or in examples [non ut in Mundo et in exemplis reperitur, my italics], but as some kind of universal form, which, if it is inseparably connected with some other form in the realm of ideas, results in a being necessarily existing in fact.
This passage tells us four facts. First, that perfections subsist independently of someone thinking of them. As such, perfections do not owe their ontological status to any contingent being ('perfections [...] [subsist] even if no individual beings exist'). Secondly, we are told that actual existence is one of these perfections ('[among the perfections] is found actual existence'). Thirdly, and this is the crucial point of the passage, we are told that although actual existence is a perfection, this is not the kind of existence shared by created things ('not as is found in the world or in examples'). Finally, Leibniz says that when actual existence is connected with some other perfection it 'results in a being necessarily existing in fact'.
The importance of the third point is hard to exaggerate. In the Letter to Huthmann, Leibniz says that existence understood as a perfection is not the kind of existence that is found in the world. 10 This claim is consistent with what Leibniz had written one year before the Letter to Huthmann, in the previously quoted De Existentia, where he had denied that the existence of created beings is a perfection altogether. But then, the crux of the question is how to understand the dichotomy between existence as a perfection and existence as found in worldly things. For, although in De Existentia Leibniz had denied that the existence of created beings is a perfection, we have seen that he nonetheless conceded that 'it is true indeed that what exists is more perfect than the non-existent'. And if so, is not Leibniz taking the existence of created beings as a real property characterising them?
In De Existentia, Leibniz considers, along with this option, another one. Indeed, he argues that '[I]t seems to be true that existence is a certain degree of reality; or certainly that it is some relation to degrees of reality' ((A VI 4, 1354) (Adams 1994, 165)). The option that existence is a degree of reality is immediately rejected ('Existence is not a degree of reality, however' (ibidem)). We would accordingly expect an acceptance of the other disjunct, and indeed Leibniz argues that 'Existence will therefore be the superiority of the degrees of reality of one thing over the degrees of reality of an opposed thing' (ibidem). This remark is immediately followed by an explanation, 'That is, that which is more perfect than all things mutually incompatible exists, and conversely what exists is more perfect than the rest' (ibidem).
The account given by Leibniz here, I submit, has one straightforward interpretation. Existence is first presented as the 'superiority of the degrees of reality of one thing over the degrees of reality of an opposed thing'. And this assertion, Leibniz argues, means that the existence of a thing is equivalent to its being more perfect than any other way in which it could have been ('that which is more perfect than all things mutually incompatible exists, and conversely what exists is more perfect than the rest'). But if existence is superiority of the degrees of reality of one thing, existence appears to be not much a property of individuals, but rather of the way they are, that is, of their properties. For example, the actual Julius Caesar (who crossed the Rubicon) existed because his properties made for a more perfect being than the properties of a possible Julius Caesar who did not cross the Rubicon (and who is accordingly incompatible with the actual Julius Caesar). In other words, to say that (the actual) Julius Caesar existed reduces to saying that his properties were more perfect than the properties of any merely possible Julius Caesar. But the predicate 'more perfect than' here denotes a property of properties (specifically, a relation among properties); indeed, Leibniz appears to define 'x exists' as 'the properties of x are more perfect than the properties of any other possible x'. Thus, I conclude that, on Leibniz' account, the existence of created beings is best understood not as a (first-order) property that they possess, but rather as a property of their properties; and that is, as a second-order property.
Thus, if we read the Letter to Huthmann in light of Leibniz' remarks in De Existentia, as I propose, the sense in which Leibniz contrasted existence as a perfection with existence as found in worldly instances is clear. On the one hand, existence as a perfection is a real attribute of God. On the other hand, the existence of created beings is neither a perfection nor a property that they possess in any sense, but it is simply their being more perfect than they could have alternatively been. And since an analysis of this claim entails that the existence of created beings is a second-order property, the consequence I derive is that Leibniz could only be taking existence to be a twofold notion.
If what documented so far would not make this fact evident enough, let us observe another passage from L 3 ((A II 1, 591) (Leibniz 2006, 186)): [A]nyone who understands its [i.e. God's] specific nature [my italics] (whatsoever it may ultimately be) will also understand, from the essence alone, i.e. a priori, that it has within it necessity of existence, or the inseparability of essence and existence.
Existence, indeed necessity of existence, is a real property being part of God's essence. However, Leibniz here also explains why a second notion of existence would be available. Indeed, the information that existence is a real property of God's essence comes from understanding 'its specific nature'. A correct understanding of God's nature will eventually reveal, according to Leibniz here, that God is a necessary being. Similarly, a correct understanding of my own nature will eventually reveal that I a am not a necessary being. In other words, my existence fails to be part of my own essence, for otherwise I could not fail to exist, which means that I would be a necessary being. This consequence is clearly rejected by Leibniz in many occasions. For instance, in the already mentioned De Contingentia, Leibniz very clearly says that 'Creatures are contingent, that is, their existence does not follow from their essence' ((A VI 4, 1649) (Leibniz (1989a, p. 28)). It is nonetheless true, though, that I exist. Hence, no matter how exactly Leibniz characterised the existence of created beings like me, he certainly did not take it to be a real property of their essence. This finally gives us the second notion of existence that Leibniz deployed. Indeed if there are created, contingent beings their existence is certainly not a real property. This is Leibniz' second notion of existence.
That it is possible to a priori demonstrate the existence of God is usually taken to be a highly controversial claim. Some philosophers have endorsed it, while some others have rejected it. What in this section I tried to do is provide the conceptual framework that would enable Leibniz to hold such a belief while also holding that there are created beings. On the condition that the word 'existence' has not just one but two meanings, the two claims are perfectly compatible. Would Leibniz be ready to split existence into two notions? I provided evidence to the extent that not only would Leibniz not find fault with this view, but also that he clearly articulates it in his writings-and sometimes even formulates it clearly, as we saw in the Letter to Huthmann.
We now have to show that such a conception of existence can defeat the dilemma presented in the previous section and vindicate Leibniz from Russell's inconsistency accusation. This will be the goal of the next section.

A Refutation of Russell's Dilemma
It is now time to show that with the twofold conception of existence just presented, Leibniz would have been in a position to immediately undermine Russell's dilemma.
Recall, the first horn of the dilemma (R1-R4) starts with the assumption that existence is a real property being part of God's essence. R1 Therefore, existence is a real property of everything (i.e. God and created substances). R2 If existence is a real property, then it is part of the essence (complete concept) of any existing substance. R3 Hence, existence is an essential property of every existing individual. R4 If so, God would not have the power to create anything, for all substances would exist simply in virtue of their essence (complete concept).
On Leibniz' twofold conception of existence, one cannot argue that because existence is a real property of God (initial assumption), existence is also a real property of created substances (R1). That is, of course Leibniz took existence to be a real property of God. However, that does not mean that existence is also a real property of created beings. In fact, as we saw, Leibniz appeared to deny that much. Thus, on Leibniz' twofold conception of existence, R1 is false and the first horn of Russell's dilemma accordingly unsound. The second horn of the dilemma (R5-R7) starts with the assumption that God has the power to create beings. R5 Hence, existence is not a real property of anything (i.e. God and created substances). R6 For the ontological argument to be sound, it is necessary that existence is a real property. R7 Therefore, the ontological argument is unsound. Given Leibniz' twofold conception of existence, his response would go as follows. It is of course true that if God can create something (initial assumption), then existence is not a real property of what God has created (second half of R5). However, that does not mean that existence is also not a real property of God (first half of R5). In fact, as we saw, Leibniz appeared to deny that much. Thus, on Leibniz' twofold conception of existence, R5 is false and the second horn of Russell's dilemma accordingly unsound. Now that Russell's dilemma has been proven faulty under Leibniz' twofold conception of existence, we can finally state the conclusions reached in this paper.

Conclusion
Leibniz believed both that God has the power to create beings and that the existence of God could be a priori demonstrated (by so-called ontological arguments). Kant argued against such arguments, as they would imply the false claim that existence is a real property. Russell shared Kant's view and added that if existence were a real property Leibniz could not even legitimately hold that God has the power to create beings. That poses a dilemma for Leibniz' system, for if existence is not a real property, then the ontological argument cannot be sound. In this paper, I argued that Russell's dilemma crucially relies on an assumption that Leibniz was not prepared to accept, namely that there is only one notion of existence and this is either a real property or not. Moreover, I argued that on Leibniz' account of existence Russell's dilemma can be easily defeated. Thus, I first presented one of Leibniz' ontological arguments. Then, I illustrated Kant and Russell's positions in more detail. Finally, I turned to my two claims. First, that in Leibniz not just one but two distinct notions of existence are at work (one characterising created beings, and one uniquely characterising God). Second, that such a twofold conception of existence falsifies Russell's dilemma. This does not necessarily establish that Leibniz could consistently accept creation and the ontological argument. What the paper establishes is only that, in order to be conclusive, an inconsistency claim in this respect should always presuppose Leibniz' twofold conception of existence.