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Leibniz's Modal Metaphysics


First published Fri May 23, 2008

In the main article on Leibniz, it was claimed that Leibniz's philosophy can be seen as a reaction to the Cartesian theory of corporeal substance and the necessitarianism of Spinoza and Hobbes. This entry will address this second aspect of his philosophy. In the course of his writings, Leibniz developed an approach to questions of modality—necessity, possibility, contingency—that not only served an important function within his general metaphysics, epistemology, and philosophical theology but also has continuing interest today. Indeed, it has been suggested that 20th-century developments in modal logic were either based on Leibnizian insights or at least had a Leibnizian spirit.



  • 1. Individuals and Worlds

  • 2. The Nature of Modality

  • 3. Leibnizian Essentialism

  • 4. Human Freedom: Certainty without Necessity

  • 5. Why This World?

  • Bibliography

    • Primary Sources for Leibniz with Abbreviations

    • Secondary Sources

  • Other Internet Resources

  • Related Entries


1. Individuals and Worlds


In order to explain Leibniz's modal metaphysics—the metaphysics of necessity, contingency, and possibility—we must look first at the foundation of Leibniz's system more generally: his conception of an individual substance. In §8 of the Discourse on Metaphysics, Leibniz presents his classic picture, writing:

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 (A VI iv 1540/AG 41).

In other words, each individual substance has a complete individual concept (CIC), which contains (or from which are deducible) all predicates true of it past, present, and future. Leibniz asks his reader to consider the case of Alexander the Great. On his view, God can, as it were, look at the complete individual concept of Alexander and see that he conquered Darius and Porus, that he was the student of Aristotle, that his armies would march into India, and so on. For our purposes, it will suffice to think of the CIC as the essence of an individual substance and to think of God as able to survey all essences of all individual substances. (The issue is, in fact, vexed; for insightful presentations of views rivaling that presented here, see Sleigh 1990 or Cover and Hawthorne 1999.) Further, according to Leibniz, one of the consequences of this view of the nature of an individual substance is that no two substances can be qualitatively identical and differ numerically. In other words, the Principle of the Identity of Indiscernibles (PII) follows from this conception of the nature of substance, and PII entails that, for any possible world, there is at most one instance of a CIC.

Individual substances, of course, are parts of, or rather, members of a world. In other words, a world is a set of individual substances; or, as Leibniz puts in the opening line of On the Ultimate Origination of Things, a world is a “collection of finite things.” (G VII 302/AG 149) More specifically, Leibniz tells Bourguet, “the universe is only a certain kind of collection of compossibles; and the actual universe is the collection of all possible existents, that is, of those things that form the richest composite.” (G III 573) In saying that a world is a set of compossible things, however, Leibniz is saying that a world is a kind of collection of things that God could bring into existence. For not even God can bring into existence a world in which there is some contradiction among its members or their properties. But this opens up the question: Just what is meant by a contradiction in the case of the members of a world?

Let us say that two or more substances are compossible if and only if there is no contradiction between the predicates derivable from their CICs. For example, consider the two individuals, Don and Ron. One of Don's properties is being the tallest man (at time t); one of Ron's properties is also being the tallest man (at time t). Naturally, Don and Ron cannot then inhabit the same world. On the other hand, each could have the property being over 2 meters tall and be members of the same world. Now in Leibniz's fully developed metaphysics this example might not be considered a good one, since it is most likely the case that Leibnizian individuals are not to be thought of as constituted by such relational properties. Rather than thinking of compossibility in terms of the properties of substances, it might be easier to think of it in terms of the perceptions of substances. Certainly, Don and Ron cannot be said to be members of the same world if Don perceives the moon landing of Apollo 11 on July 20, 1969 and Ron does not (or would not)—and not simply because he does not have a television set, but because on his July 20, 1969, there is no United States of America, let alone a space program. From this, it should also be clear that the compossibility of substances in a world is another manifestation of Leibniz's thesis of the universal harmony of perceptions of substances.

A possible world, however, is not simply a set of compossible individuals. According to Leibniz, a possible world also entails certain laws of nature. As Leibniz says to Arnauld in a letter from 14 July 1686,

I think there is an infinity of possible ways in which to create the world, according to the different designs which God could form, and that each possible world depends on certain principal designs or purposes of God which are distinctive of it, that is, certain primary free decrees (conceived sub ratione possibilitatis) or certain laws of the general order of this possible universe with which they are in accord and whose concept they determine, as they do also the concepts of all the individual substances which must enter into this same universe. (G II 51/L 333)

Leibniz's basic idea here should conform closely with our intuitions. Imagine God's considering a set of individuals {a, b, c, d}. From God's point of view he could choose to actualize this world with one set, L*, of laws of nature or with another set of laws, L**. And this choice represents a choice between two possible worlds. This is similar to our saying that we would have a different world if the gravitational constant were different. Now, this way of speaking is not exactly right, for on Leibniz's view, the different sets of laws would ultimately produce different properties and perceptions with the individual substances. As a result, the individuals in the worlds governed by L* and L** would strictly speaking be different.

The reason Leibniz mentions the different laws governing the different possible worlds is that the systems of laws and their effects serve as criteria by which God chooses a world. We know that, for Leibniz, God chooses the “best of all possible worlds.” In §6 the Discourse on Metaphysics, however, we learn that this means that God chooses the world that is simplest in hypotheses (or laws) and richest in phenomena. Thus, while one might be tempted to see natural laws as derivative of the actual properties and perceptions of individual substances, in fact they are objects of God's choice.

When Leibniz speaks of a possible world, he means a set of compossible, finite things that God could have brought into existence if he were not constrained by the goodness that is part of his nature. The actual world, on the other hand, is simply that set of finite things that is instantiated by God, because it is greatest in goodness, reality and perfection. Naturally, the fact that we are here experiencing this world—the actual world—means that there is at least one possible world. So are there others?

Yes, indeed, there are. At least Leibniz thinks so. In his view, as we saw above, there are an infinite number of possible worlds—worlds that God did not see fit to bring into existence. Now, given that Leibniz's safe claim is that “[t]here are as many possible worlds as there are series of things that can be conceived that do not imply a contradiction,” (Grua 390) it might still be the case that there is only one possible world—only one set of essences that implies no contradiction. If we accept the claim that a possible world is simply any set of compossible individuals (i.e. a set of individuals whose properties or perceptions do not contradict each other), then there is a rather trivial way to show that there are an infinite number of possible worlds. Namely, we just have an infinity of one-object worlds: thus W1 contains a blue, 8lb. bowling ball; W2, a red, 10lb. bowling ball; W3, a pepperoni pizza; W4, a bicycle; and so on. We could even imagine worlds with two, three, or more objects.

Now, Leibniz's reasons for proposing an infinity of possible of worlds are somewhat different. In “On Contingency” (1689?), Leibniz makes the following claim:

One must certainly hold that not all possibles attain existence, otherwise one could imagine no novel that did not exist in some place and at some time. Indeed, it does not seem possible for all possible things to exist, since they get in one another's way. There are, in fact, an infinite number of series of possible things. Moreover, one series certainly cannot be contained within another, since each and every one of them is complete. (A VI iv 1651/AG 29)

The point here is that there are unactualized possibles as exemplified in works of fiction. In other words, a work of fiction represents some way the world could be. But, insofar as these possibles conflict among themselves, it is clear that not all possibles are compossible. Rather, as we saw above, sets of compossible individuals are Leibniz's possible worlds. If we add to this picture the idea that each individual has a complete individual concept—that is, a determinate concept—which is sufficient to distinguish it from every other possible world, then we see that possible individuals are tied to determinate possible worlds.

But what is the character of these non-actualized possibles? The most natural assumption is that they are like individual substances as characterized by Leibniz in Discourse on Metaphysics §8, that is, as having complete individual concepts. In other words, all individuals (actual or merely possible) have determinate essences. In the case of merely possible individuals, however, their being is contained solely in the divine mind. Consider Leibniz's claim in the Theodicy §189: “In the region of the eternal verities are found all the possibles.” (G VI 229/H 246) And, in §44 of the Monadology, in the middle of his argument for God's necessary existence, Leibniz says, “if there is reality in essences or possibles, or indeed, in eternal truths, this reality must be grounded in something existent and actual, and consequently, it must be grounded in the existence of the necessary being, in whom essence involves existence, that is, in whom possible being is sufficient for actual being.” (G VI 614/AG 218) In other words, there will be determinate essences of some substances that are actualized in the world as well as determinate essences of substances that exist solely in the divine mind.

By way of contrast, Spinoza argues in the Ethics (Ip33) that there are no unactualized possibles. In the case of a story depicting another “reality,” we have only partial and confused knowledge of the world depicted. If we were to have true knowledge of the way all the characters and their actions extend backward and forward in time, we could ultimately arrive at some contradiction. Moreover, Spinoza holds that everything that is truly possible will be expressed at some point as God or nature expresses its infinite essence. That is, since God is a substance of infinite attributes expressed in infinite ways, everything that is possible finds existence at some point in this world. Of course, this difference between Leibniz and Spinoza is largely attributable to Leibniz's adherence to what Spinoza finds so wrong: the anthropomorphic conception of God and the attendant distinction between divine intellect and will, which is what allows for God's supposed contemplation of non-existent possibles.

In Meaning and Necessity, Rudolf Carnap suggests that a Leibnizian possible world is represented by his state-descriptions: a class of sentences containing, for every atomic sentence, either it or its negation. In other words, for each possible world there is a series of propositions that will fully describe that world. The idea of a possible world as being such a set of sentences or propositions leads naturally to the notion of a “world-book”—a term employed by Alvin Plantinga (The Nature of Necessity). Lest such devices seem anachronistic, it should be pointed out that, in the fascinating conclusion to the Theodicy, Leibniz speaks in a similar way; indeed, Leibniz's parable is something that one might imagine has come straight from Jorge Luis Borges (cf. “The Library of Babel”). The story concerns Theodorus, a high priest present when Sextus Tarquinius complains to Jupiter about his fate. (Sextus was the son of the last king of Rome, whose crime of raping Lucretia so incensed Brutus that he led a revolt that ultimately abolished the monarchy and began the Roman Republic.) Theodorus is moved by Sextus' complaint and is sent to receive instruction at the temple of Pallas Athena where he is shown “the palace of the fates.” According to Athena, “Here are representations not only of that which happens but also of all that which is possible. Jupiter, having surveyed them before the beginning of the existing world, classified the possibilities into worlds, and chose the best of all.” (Theodicy §414: G VI 363/H 370) She continues,

I have only to speak, and we shall see a whole world that my father might have produced, wherein will be represented anything that can be asked of him; and in this way one may know also what would happen if any particular possibility should attain unto existence. And whenever the conditions are not determinate enough, there will be as many such worlds differing from one another as one shall wish, which will answer differently the same question, in as many ways as possible… But if you put a case that differs from the actual world only in one single definite thing and in its results, a certain one of those determinate worlds will answer you. These worlds are all here, that is, in ideas. I will show you some, wherein shall be found, not absolutely the same Sextus as you have seen (that is not possible, he carries with him always that which he shall be) but several Sextuses resembling him, possessing all that is already in him imperceptibly, nor in consequence all that shall yet happen to him… (Theodicy §414: G VI 363/H 370–71)

Theodorus is led into one of the halls of the palace and, observing a great volume of writings in the hall, asks what it is. “It is,” Athena tells him, “the history of this world which we are now visiting…; it is the book of fates.” (Theodicy §415: G VI 363/H 371–72) And, finally, we have an argument for the infinity of worlds wrapped in a fantasy tale:

The halls rose in a pyramid, becoming even more beautiful as one mounted towards the apex, and representing more beautiful worlds. Finally they reached the highest one which completed the pyramid, and which was the most beautiful of all: for the pyramid had a beginning, but one could not see its end; it had an apex, but no base; it went on increasing to infinity. That is (as the Goddess explained) because amongst an endless number of possible worlds there is the best of all, else would God not have determined to create any; but there is not any one which has not also less perfect worlds below it: that is why the pyramid goes on descending to infinity. (Theodicy §416: G VI 364/H 372)

Leibniz makes clear here his basic view of God's obligation to choose the best world possible, as well as his view that, if there were not a single best world, no world at all would have been brought into existence. At the same time, the argument for the infinity of possible worlds given here is rather comical: (granted, there must be one best world) for any world you can imagine (or find in the pyramid), there is one that is worse.

The world-books are on permanent display in the divine intellect. This is what Leibniz means when he says that they reside in the realm of eternal verities. But one thing should be absolutely clear: Leibniz is no modal realist à la David Lewis. Although Leibniz claims that there are an infinity of possible worlds, he does not mean that an infinity of possible worlds exist in the same way as this (actual) world does, or that an infinity of worlds are running as it were parallel to this one, or that “actual” is an indexical expression like “here” and “now.” The existence claim commits Leibniz only to the existence of different ways the world could be and to the shelving of these world-books in the infinite and eternal library of the divine mind. As he puts it in the same letter to Arnauld, “there is no other reality in pure possibles than the reality they have in the divine understanding.” (G II 45/AG 75) But it is crucial to Leibniz's position that there be one and only one actual world, the best of all possible worlds. As we have seen, God is as it were obligated to bring this one world into existence. And, we have also seen that, according to Leibniz, if there were no uniquely best world, then God would not have brought any world into existence. (And God could not simply play Eenee, Meenee, Mainee, Mo! with two worlds; he must, according to the Principle of Sufficient Reason, have a ground for his decision.) Further, there cannot be two worlds that are absolutely equal in terms of their degrees of perfection, for God, by virtue of his omnipotence and omniscience, must be able to determine some distinction between the worlds. And if God had actualized more than one world and they had been of different degrees of perfection, then God would have brought into existence that which is less perfect than possible (a violation of the requirements of divine benevolence). (Concerning the reasons for this world, more below.)





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