Philosophy of the Modern Era

Leibniz

Leibniz (1646—1716) was one of the greatest minds of all time, but as a human being, he is difficult to admire. It is true that he possessed the virtues one would seek in a prospective employee: he was industrious, frugal, temperate, and honest in financial matters. However, he lacked the lofty philosophical virtues so characteristic of Spinoza. His finest thoughts would not have made him popular, and he left the manuscripts containing these ideas unpublished. What he did publish was aimed at gaining the approval of princes and monarchs. As a result, two philosophical systems can be identified as representing Leibniz’s views: one, which he openly professed, was optimistic, orthodox, fantastical, and superficial; the other, which has been gradually unearthed from his manuscripts by more recent editors, was profound, clear, remarkably logical, and bore a close resemblance to the philosophy of Spinoza. It was the popular Leibniz who invented the theory that this world is the best of all possible worlds (to which F. H. Bradley added the sardonic comment, "and everything in it is a necessary evil"); it was this same Leibniz that Voltaire caricatured as Dr. Pangloss. To ignore this Leibniz would be historically inaccurate, but the other Leibniz played a far more significant role in philosophy.

Leibniz was born two years before the end of the Thirty Years' War in Leipzig, where his father was a professor of moral philosophy. He studied law at university and earned his doctorate at Altdorf in 1666, where he was offered a professorship, which he declined, stating that he "had entirely different things in mind." In 1667, he entered the service of the Archbishop of Mainz, who, like other Western European princes, lived in constant fear of Louis XIV. With the archbishop’s approval, Leibniz attempted to convince the French king to invade Egypt rather than Germany, but he was politely reminded that holy wars against infidels had gone out of fashion since the time of Louis the Saint. His project remained unknown to the wider public until it was discovered by Napoleon when he seized Hanover in 1803, four years after his own failed Egyptian campaign. In 1672, in connection with this plan, Leibniz went to Paris, where he spent most of the next four years. His acquaintances in Paris greatly influenced his intellectual development, as Paris at that time was the leading center of both philosophy and mathematics. It was there, in 1675—1676, that he created the calculus of infinitesimals, unaware of Newton’s prior but unpublished work on the same subject. Leibniz’s work was published in 1684, while Newton’s appeared in 1687. The subsequent dispute over priority was unbecoming and shameful for both sides.

Leibniz was somewhat frugal. When a young lady of the Hanoverian court married, he would typically give her what he called a “wedding gift,” consisting of useful advice that concluded with a recommendation not to neglect bathing now that she had secured a husband. History does not record whether the brides were pleased with this gift.

In Germany, Leibniz was taught the neo-scholastic Aristotelian philosophy, some aspects of which he retained throughout his life. But in Paris, he became acquainted with Cartesian philosophy and the materialism of Gassendi, both of which influenced him; he said that it was during this period that he abandoned the "unscientific schools," by which he meant scholasticism. In Paris, he also met Malebranche and the Jansenist Arnauld. Finally, the philosophy of Spinoza, whom he visited in 1676, exerted a significant influence on him. He spent a month in frequent debates with Spinoza and received a manuscript copy of part of the Ethics. Later, Leibniz joined the persecution of Spinoza and downplayed his acquaintance with him, claiming they had only met once and that Spinoza had merely told a few witty political anecdotes.

In 1673, his relationship with the Hanoverian court began, and from then on, he served it for the rest of his life. From 1680 onward, he was the librarian at Wolfenbüttel, and he was officially tasked with writing a history of the Brunswick dynasty, which he completed only up to the year 1005. His work was not published until 1843. He also devoted some time to a failed project for the reunification of the churches. He traveled to Italy to find evidence that the Dukes of Brunswick were related to the Este family. Despite these services, he was left behind in Hanover when George I became King of England, largely because of his quarrel with Newton, which had turned England against him. Nevertheless, as he informed all his correspondents, the Princess of Wales supported him against Newton. Despite her favor, however, Leibniz died in obscurity.

Leibniz's popular philosophy is laid out in the Monadology and the Principles of Nature and Grace, one of which (it is unclear which) he wrote for Prince Eugene of Savoy, a colleague of the Duke of Marlborough. The basis of his theological optimism is presented in the Theodicy, written for Queen Charlotte of Prussia. I will begin with the philosophy outlined in these works and then move on to his more fundamental works, which he left unpublished.

Like Descartes and Spinoza, Leibniz founded his philosophy on the concept of "substance," but he radically differed from them in his views on the relationship between mind and matter and on the number of substances. Descartes posited three substances: God, mind, and matter; Spinoza admitted only one substance, God. For Descartes, extension was the essence of matter; for Spinoza, both extension and thought were attributes of God. Leibniz, however, believed that extension could not be an attribute of substance. His reasoning was that extension involves multiplicity and therefore can only belong to an aggregate of substances; each individual substance must be unextended. Hence, he believed in an infinite number of substances, which he called "monads." Each monad shared certain qualities with a physical point, but only in abstract consideration, for in actuality, each monad was a soul. This naturally followed from the rejection of extension as an attribute of substance; the only remaining essential attribute appeared to be thought. Thus, Leibniz arrived at a denial of the reality of matter and replaced it with infinite assemblies of souls.

The theory that substances cannot interact, developed by Descartes’s followers, was adopted by Leibniz and led to curious conclusions. He believed that no two monads could ever have causal relations with one another; when it appears that they do, it is merely an illusion. Monads, as he put it, "have no windows." This leads to two difficulties: one arises from dynamics, where bodies seem to influence one another, particularly in collisions; the other stems from perception, which seems to be the effect of the perceived object on the perceiver. For now, we will set aside the difficulties related to dynamics and focus on the issue of perception. Leibniz believed that each monad reflects the universe not because the universe acts upon it, but because God endowed it with a nature that spontaneously produces this result. There exists a "pre-established harmony" between changes in one monad and changes in another, creating the appearance of interaction. Clearly, this is an extension of the theory of two clocks, which strike the same time at the same moment because they work synchronously. Leibniz posited an infinite number of clocks, all set by God to strike the same time simultaneously, not because they influence one another, but because each is a perfectly accurate mechanism. For those who found the pre-established harmony peculiar, Leibniz pointed out how elegantly it proved the existence of God.

Monads form a hierarchy, with some rising above others in the clarity and distinctness with which they reflect the universe. All monads possess a degree of confusion in their perceptions, but the amount of confusion varies according to the dignity of the monad. The human body is entirely composed of monads, each of which is a soul and each of which is immortal, but there is one dominant monad, representing what we call the human soul, a part of whose body it is. This monad dominates not only in having clearer perceptions than the others, but also in another sense. Changes in the human body (under ordinary circumstances) occur for the sake of the dominant monad: when my hand moves, the purpose of the movement is found in the dominant monad, that is, in my soul, not in the monads that compose my hand. Here lies the truth behind the common-sense notion that my will controls my hand.

Space, as it appears to the senses and as it is understood in physics, does not exist, but it has a real counterpart, namely the arrangement of monads in a three-dimensional order according to the perspective from which they reflect the world. Each monad sees the world from a particular viewpoint unique to itself; in this sense, we can somewhat arbitrarily speak of monads as having spatial locations.

Following this line of reasoning, we can say that there is no such thing as empty space. Every possible point of view is occupied by one and only one actually existing monad. No two monads are exactly alike. This is Leibniz's principle of the "identity of indiscernibles."

In contrast to Spinoza, Leibniz allows for the presence of free will in his system. He introduced the "principle of sufficient reason," which asserts that nothing happens without a reason; but when dealing with free agents, the causes of their actions are "inclination without necessity." Every human act is motivated, but there is no logical necessity in the sufficient reason for its occurrence. At least, this is what Leibniz states when writing for a broad audience, though, as we shall see, he has another theory, one he concealed after learning that Arnauld was disturbed by it.

The same kind of freedom applies to the actions of God. God always acts for the best, but no logic compels Him to act in this way. Leibniz agrees with Thomas Aquinas that God cannot act contrary to the laws of logic, but He may command anything that is logically possible, granting Him the greatest freedom of choice.

Leibniz presents metaphysical proofs for the existence of God in their final form. These proofs have a long history; their beginnings can be traced to Aristotle or even Plato. They were formulated in their complete form by the scholastics, and one of them—the ontological proof—was developed by St. Anselm. Though rejected by St. Thomas, this proof was revived by Descartes. Leibniz, having reached the highest mastery of logic, formulated these proofs better than anyone before him. This is why I consider them in connection with him.

Before examining the proofs in detail, it is important to understand that modern theologians no longer rely on them. Medieval theology is derived from the Greek intellect. The God of the Old Testament is a God of power. The God of the New Testament is also a God of love; but the God of the theologians, from Aristotle to Calvin, is a God whose appeal lies in His intellect: His existence resolves certain puzzles that would otherwise raise contentious questions in our understanding of the universe. This God, who appears at the end of an argument like the conclusion of a geometric proof, did not satisfy Rousseau, who returned to a conception of God closer to the evangelical vision. Modern theologians, especially Protestants, largely follow Rousseau in this regard. Philosophers, however, have been more conservative; metaphysical proofs continued to exist for thinkers like Hegel, Lotze, and Bradley, despite Kant’s assertion that such proofs had been refuted once and for all.

Leibniz offers four proofs for the existence of God: 1) the ontological proof, 2) the cosmological proof, 3) the proof from eternal truths, and 4) the proof from pre-established harmony, which can be summarized as the proof from design, or as Kant calls it, the physico-theological proof. We will examine these proofs in turn.

The ontological proof is based on the distinction between existence and essence. It is held that any ordinary person or thing, on the one hand, exists, and on the other hand, has certain qualities that constitute its "essence." Hamlet, though he does not exist, has a certain essence: he is melancholic, indecisive, witty, and so on. When we describe a person, no matter how detailed the description, the question of whether they are real or imagined remains open. In scholastic terms, this is expressed by saying that for any finite substance, its essence does not include its existence. But in the case of God, defined as the most perfect being, St. Anselm, followed by Descartes, argues that essence includes existence on the grounds that a being which possesses all other perfections is more perfect if it exists than if it does not. From this, it follows that if He does not exist, He is not the most perfect of possible beings.

Leibniz neither fully accepts nor entirely rejects this argument, as he suggests that it should be supplemented by a proof that God, so defined, is possible. He provided a detailed argument demonstrating that the idea of God is possible; it was this idea that he introduced to Spinoza when he met with him in The Hague. This proof defines God as the most perfect being, that is, as the subject of all perfections, and perfection is defined as "a simple quality, which is positive and absolute, and expresses without limitation all that it expresses." Leibniz easily demonstrates that no two such perfections, as defined above, are incompatible. He concludes, "Therefore, there exists, or at least it is possible to conceive, a subject of all perfections or the most perfect Being. From this, it also follows that It exists, since existence is included among perfections."

Kant objected to this argument, claiming that "existence" is not a predicate. Another form of refutation arises from my theory of descriptions. To modern readers, the argument seems unconvincing, but it is easier to recognize that it must be flawed than to precisely determine where the flaw lies.

The cosmological proof appears more plausible than the ontological one. It is a form of the argument from the first cause, which itself originates from Aristotle's proof of the unmoved mover. The first cause argument is simple. It states that everything finite has a cause, which in turn has a cause, and so on. But this chain of prior causes cannot, it is argued, be infinite, and the first cause in the chain must be uncaused, for otherwise, it would not be the first cause. Therefore, there exists an uncaused cause of everything, which is evidently God.

Leibniz’s version of the proof takes a slightly different form. He argues that every individual thing in the world is "contingent," meaning that it is logically possible for it not to exist, and this is true not only of each individual thing but also of the entire universe. Even if we allow that the universe has existed eternally, nothing within it explains why it exists. But according to Leibniz’s philosophy, everything must have a sufficient reason, so the universe as a whole must have a sufficient reason, which lies outside it. This sufficient reason is God.

This proof is better than the unconvincing proof from the first cause and is not so easily refuted. The first cause argument rests on the assumption that every series must have a first member, which is false—for example, the series of proper fractions has no first member. But Leibniz's proof does not depend on the view that the universe must have had a beginning in time. The proof is valid insofar as we accept Leibniz’s principle of sufficient reason, but if we deny this principle, the argument collapses. What exactly Leibniz means by the principle of sufficient reason is a matter of debate. Couturat claims that it means every true proposition is "analytic," that is, its denial is self-contradictory. However, this interpretation (which relies on Leibniz’s unpublished manuscripts), even if correct, pertains to Leibniz’s esoteric theory. In his published works, he asserts that there is a distinction between necessary and contingent propositions, that only the former follow from the laws of logic, and that all propositions asserting existence are contingent, except the proposition asserting the existence of God. Although God exists necessarily, His creation of the world was not compelled by logical necessity; rather, it was a free choice, motivated but not coerced by His goodness.

It is clear that Kant was right in saying that this proof depends on the ontological proof. If the existence of the world can only be explained by the existence of a necessary Being, then there must be a Being whose essence includes existence, for this is precisely what is meant by a necessary Being. But if such a Being, whose essence includes existence, is possible, then reason alone, without experience, can infer its existence from the ontological proof, for anything pertaining solely to essence can be known independently of experience—at least, this is Leibniz’s view. Thus, the apparent superiority of the cosmological proof over the ontological proof is illusory.

Formulating the argument from eternal truth with precision is somewhat challenging. Broadly speaking, the argument proceeds as follows: a statement like "it is raining" is sometimes true and sometimes false, but "two and two make four" is always true. All propositions that concern only essence and not existence are either always true or never true. Those that are always true are called "eternal truths." The essence of the argument is that truths form part of the content of minds, and an eternal truth must therefore be part of the content of an eternal mind. Plato presents something similar to this argument when he derives the immortality of ideas from their eternal nature. However, in Leibniz’s version, the argument takes a more developed form. He believes that the ultimate ground for contingent truths must be found in necessary truths. The structure of the argument mirrors that of the cosmological argument: for the contingent world, there must be a ground, and that ground cannot itself be contingent; it must be sought among eternal truths. But the ground for what exists must itself exist; therefore, eternal truths must in some sense exist, and they can exist only as thoughts in the mind of God. Indeed, this argument is merely another form of the cosmological proof. It is, however, vulnerable to the objection that truth is unlikely to "exist" in the mind of the one who reasons.

The argument from pre-established harmony, as formulated by Leibniz, holds only for those who accept his windowless monads that reflect the universe. The proof asserts that just as multiple clocks, with no causal interaction, show the same time, there must be a single external Cause that regulates them all. Of course, this introduces a difficulty inherent throughout his monadology: if monads never interact, how can any one of them know that others exist? What seems to be a reflection of the universe could merely be a dream. In fact, if Leibniz is right, it is precisely a dream, but one in which all monads somehow dream similar dreams at the same time. This is, of course, a fantastical notion and would never seem plausible had it not been for the earlier history of Cartesian thought.

Leibniz’s argument, however, can be detached from his particular metaphysics and transformed into what is called the argument from design. This proof claims that, when observing the known world, we find things that cannot plausibly be explained as the product of blind forces of nature, but are far more reasonably seen as evidence of benevolent purposes.

This proof does not suffer from formal logical flaws: its premises are empirical, and its conclusion is reached in accordance with the usual rules of empirical inference. Therefore, the question of whether to accept it or not pertains not to general metaphysical concerns, but to relatively concrete considerations. There is one significant difference between this proof and others, namely that the God whose existence it proves (if the argument holds) need not possess all the traditional metaphysical attributes. He need not be omnipotent or omniscient; He only needs to be far wiser and more powerful than we are. The evils in the world may be explained by His limited power. Some modern theologians use the possibilities this proof provides to craft their own conceptions of God. However, such theories are far removed from the philosophy of Leibniz, to which we must now return.

One of the most distinctive features of Leibniz’s philosophy is his doctrine of many possible worlds. A world is "possible" if it does not contradict the laws of logic. There is an infinite number of possible worlds, each of which God contemplated before creating the actual world. Being good, God chose to create the best of all possible worlds, and He judged that the best must be the one in which good outweighs evil to the greatest degree. He could have created a world without evil, but it would not have been as good as the world that actually exists. This is why great good is logically connected with some evil. Take the most commonplace example: a sip of cold water on a hot day when you are suffering from thirst can bring you such great pleasure that you might think the suffering of thirst was worth it, because without it, the subsequent enjoyment would not have been as great. Theologically, the connection between sin and free will is more important. Free will is a great good, but it is logically impossible for God to grant free will and at the same time command that there be no sin. Therefore, God decided to make man free, even though He foresaw that Adam would eat the apple and that sin would inevitably lead to punishment. In the resulting world, although there is evil, the preponderance of good over evil is greater than in any other possible world; hence, it is the best of all possible worlds, and the evil within it is not an argument against the goodness of God.

This argument evidently satisfied the queen of Prussia. Her serfs suffered the evils while she enjoyed the good, and it was of great convenience to her to be assured by a great philosopher that this was just and proper.

Leibniz’s resolution of the problem of evil, like most of his other well-known theories, is logically possible, but not particularly convincing. The Manichaeans might object that this is the worst of all possible worlds, where the good things that exist only serve to heighten the evil. They might say that the world was created by a malicious demiurge, who allowed free will—a good—in order to bring about sin, which is evil, and where evil outweighs the good of free will. The demiurge, they might continue, created a few virtuous people so that the wicked could punish them, for the punishment of the virtuous is such a great evil that it makes the world worse than if no good people had existed at all. I do not advocate this view and consider it fanciful, but I assert that it is no more fanciful than Leibniz’s theory. People want to believe that the universe is good, and they are indulgent toward weak arguments proving this, while weak arguments proving that it is bad are scrutinized with great severity. In truth, of course, the world is partly good and partly bad, and no "problem of evil" arises unless this obvious fact is denied.

Now I turn to the esoteric philosophy of Leibniz; here we will find the grounds for much that seems arbitrary or fantastical in his public doctrines, and I proceed to an interpretation of his teachings, which (had they become widely known) would have made his philosophy significantly less palatable. It is noteworthy that his influence on subsequent generations of philosophers was such that most editors, when selecting from the vast mass of his manuscripts, favored those pieces that confirmed the accepted interpretation of his system, and dismissed as irrelevant the essays that demonstrated he was a far deeper thinker than he wished to be perceived. Most of the texts on which we must rely for an understanding of his secret philosophy were first published in 1901 or 1903 in two works by Louis Couturat. One of these was even titled by Leibniz himself, accompanied by the remark: "Here I have made great progress." Yet, despite this, no editor thought it fit for publication for nearly two centuries after his death. To be sure, his letters to Arnauld, in which his deeper philosophy is partially contained, were published in the 19th century; and I was the first to note their significance. The reception Arnauld gave to these letters was discouraging. He writes: "I find in these thoughts so many troubling assertions, which, if I am not mistaken, will offend almost all people, that I see no benefit in labors that will evidently be rejected by the whole world." Without doubt, this hostile opinion later led Leibniz to conceal his true thoughts on philosophical matters.

The concept of substance, foundational in the philosophy of Descartes, Spinoza, and Leibniz, is derived from the logical categories of subject and predicate. Some words can be either subjects or predicates; for example, I can say "the sky is blue" and "blue is a color." Other words, of which proper names provide the most obvious examples, are never predicates, but only subjects or terms of a relation. Such words are intended to designate substances. Substances, in addition to this logical characteristic, persist in existence unless destroyed by Divine omnipotence (which, one may assume, will never happen). Every true proposition is either general, such as "all men are mortal," in which case it asserts that one predicate is contained in another; or particular, such as "Socrates is mortal," in which case the predicate is contained in the subject, and the quality designated by the predicate is part of the concept of the substance designated by the subject. Whatever happens to Socrates can be expressed by a proposition in which "Socrates" is the subject and the words describing what happens are the predicate. All these predicates, taken together, constitute the "concept" of Socrates. Everything necessarily belongs to him in the sense that a substance to which they could not be truly attributed would not be Socrates, but something else.

Leibniz firmly believed in the importance of logic not only within its own domain but also as the foundation of metaphysics. He devoted significant effort to mathematical logic and achieved considerable results, which would have had great impact if he had published them. In that case, Leibniz would have been the founder of mathematical logic, making it known a century and a half earlier than it actually became. However, he refrained from publishing because he discovered that Aristotle's theory of the syllogism was incorrect in some respects. His reverence for Aristotle prevented him from trusting his own conclusions, leading him to erroneously believe he had made a mistake. Nevertheless, throughout his life, he cherished the hope of discovering a kind of generalized mathematics, which he called Characteristica Universalis, through which thinking could be replaced by calculation. "If we had this," he said, "we would be able to reason in metaphysics and morality just as we do in geometry and mathematical analysis." "If contradictions arose, there would be no more need for disputes between two philosophers than between two accountants, for they would only need to take up pencils, sit down with their slates, and say to one another (with a friendly witness present, if desired): Let us calculate."

Leibniz based his philosophy on two logical principles: the law of contradiction and the principle of sufficient reason. Both are connected to the notion of an "analytic" judgment, which is a judgment in which the predicate is contained within the subject, for example: "All white men are men." The law of contradiction asserts that all analytic propositions are true. The principle of sufficient reason (though only in esoteric philosophical systems) asserts that all true propositions are analytic. This applies even to what we regard as empirical statements about actual reality. If I travel, the concept of me must forever include the concept of this journey, which is my predicate. "Thus, the nature of an individual substance, or complete being, consists in having such a full and complete concept that it encompasses and allows the derivation of all the predicates of the subject to which it applies... Thus, the quality of being a king, which belongs to Alexander the Great, taken separately from the subject, does not possess sufficient individual determinacy and does not include the other qualities of the same subject or what is contained in the concept of that ruler, whereas God, seeing the individual concept or haecceity of Alexander, sees within it the reason and cause of all the predicates that can truly be affirmed of Alexander (such as that he defeated Darius and Porus), and could even know a priori (rather than through experience) whether he died a natural death or from poison, which we can only learn from history."

One of the most definitive statements regarding the foundations of Leibniz's metaphysics appears in his letter to Arnauld:
"Considering the concept I have of every true judgment, I have discovered that every predicate, whether necessary or contingent, pertaining to the past, present, or future, is contained in the concept of the subject, and I ask no more than that. ... The judgment in question is of great importance and deserves to be well proven, for it follows that every soul represents a separate world, independent of everything except God, that it is not only immortal and, so to speak, imperishable, but that it contains within its substance the traces of everything that happens to it."

He further explains that substances do not act upon each other but instead harmoniously reflect the universe, each from its own point of view. There can be no interaction because everything that happens to each substance is part of its own concept and is eternally determined if that substance exists.

Clearly, this system is as deterministic as Spinoza's. Arnauld expressed horror at Leibniz’s assertion that "the individual concept of each person contains once and for all everything that will ever happen to them." Such a view is evidently incompatible with the Christian doctrine of sin and free will. Seeing how deeply troubled Arnauld was by all of this, Leibniz, out of caution, refrained from publishing his work.

Yet, for human beings, there is a distinction between truths known through logic and those known through experience. This distinction manifests in two ways. First, although everything that happens to Adam follows from his concept if he exists, we can only confirm his existence empirically. Second, the concept of any individual substance is infinitely complex, and the analysis required to derive its predicates can only be accomplished by God. However, these differences arise solely from our ignorance and intellectual limitations; for God, they do not exist. God perceives Adam's concept in all its infinite complexity and thus can regard all true judgments about Adam as analytic. God can also know a priori whether Adam exists. Since God knows His own goodness, from which it follows that He will create the best of all possible worlds, He also knows whether Adam is part of that world. Therefore, there is no real escape from determinism through our ignorance.

However, there is another curious perspective. For the most part, Leibniz presents the creation of the world as a free act of God, requiring the exercise of His will. According to this theory, what actually exists is determined not by observation but through God's goodness. Apart from God's goodness, which compels Him to create the best of all possible worlds, there is no a priori reason why one thing rather than another should exist.

But occasionally, in Leibniz’s unpublished works, we find entirely different theories regarding why certain things exist while others, equally possible, do not. According to this view, everything that exists strives to exist, but not all possible things can exist because not all of them are "compossible." It may be possible for A to exist and possible for B to exist, but impossible for both A and B to exist together; in that case, A and B are not "compossible." Two or more things are only "compossible" when they can all exist. It seems that Leibniz imagined something akin to a battle in the antechamber of creation, inhabited by entities, each striving to exist: in this struggle, groups of "compossible" things join forces, and the largest groups prevail, much like the most powerful coalitions triumph in political conflict. Leibniz even uses this concept as a definition of existence. He says: "The existing can be defined as that which is compatible with more things than any of its incompatibles." In other words, if A is incompatible with B, and A is compatible with C, D, and E, while B is compatible only with F and G, then A, not B, exists by definition. "The existing," he says, "is that being which is compatible with the greatest number of things."

In this case, there is no mention of God, and apparently no act of creation. There is no need for anything but pure logic to determine what exists. The question of whether A and B are compossible is, for Leibniz, a logical question—does the existence of both A and B involve a contradiction? It follows that theoretically, logic alone could determine which group of compossibilities is the largest, and therefore, that group would exist.

However, perhaps Leibniz did not actually mean this as a definition of existence. If it were simply a criterion, it could be reconciled with his well-known views through what he called "metaphysical perfection." The term "metaphysical perfection" in his usage seems to mean the quantity of existence. This, he says, "is nothing other than the amount of positive reality taken in the strict sense." He always argued that God created as much as possible, and this was one of his reasons for rejecting the idea of empty space. There is a general belief (which I have never understood) that it is better to exist than not to exist; on this basis, children are urged to be grateful to their parents. Leibniz evidently held this view and considered it part of God's grace to create the fullest universe. It follows that the actual world must consist of the largest group of compossibilities. And it would still be true that logic alone, given a sufficiently capable logician, could decide whether a given possible substance will exist or not.

Leibniz, in his unpublished reflections, stands as the finest example of a philosopher who employs logic as the key to metaphysics. This kind of philosophy begins with Parmenides and continues with Plato's use of the theory of ideas to establish various non-logical propositions. The same philosophical tradition encompasses Spinoza's system, as well as Hegel's. However, none of these figures managed to reason from syntax to the actual world with the same clarity as Leibniz. This form of argumentation later fell into disrepute with the rise of empiricism. Whether any effective inferences can be drawn from language to non-linguistic facts is a question on which I refrain from offering dogmatic judgments. Yet it is undeniable that the conclusions found in Leibniz and other a priori philosophers are not effective, for they are all grounded in faulty logic. The subject-predicate logic, accepted by all such philosophers in the past, either entirely ignored relations or employed spurious arguments to claim that relations were unreal. Leibniz, in particular, was guilty of a unique inconsistency: combining subject-predicate logic with pluralism, as the proposition "there exist many monads" is not a judgment of subject-predicate form. To be consistent, a philosopher who believes that all judgments take this form must be a monist, like Spinoza. Leibniz, however, rejected monism largely due to his commitment to dynamics and his argument that extension presupposes repetition and thus cannot be an attribute of a single substance.

Leibniz is a dry writer, and his influence on German philosophy rendered it pedantic and arid. His disciple, Wolff, whose theories dominated German universities until the appearance of Kant’s Critique of Pure Reason, stripped Leibniz’s philosophy of everything that was truly interesting and created a dry, professorial mode of thought. Outside of Germany, Leibniz’s philosophy had little impact; his contemporary Locke directed British philosophy, and in France, Descartes reigned until Voltaire, who made English empiricism fashionable, supplanted him.

Nevertheless, Leibniz remains a towering figure, and his greatness is now more apparent than ever before. Beyond his significance as a mathematician and the creator of the calculus of infinitesimals, he was the founder of mathematical logic, a field whose importance he understood long before anyone else did. His philosophical hypotheses, though fantastical, are remarkably clear and can be expressed with precision. Even his monads, despite their supposed lack of windows, might still be useful as conceptual tools for understanding perception. What I find most valuable in his theory of monads is his division of space into two kinds: subjective space, in the perception of each monad, and objective space, consisting of the aggregate of viewpoints from various monads. This, I believe, remains useful in linking perception with physics.