11

LEIBNIZ

Leibniz (1646–1716) was one of the supreme intellects of all time, but as a human being he was not admirable. He had, it is true, the virtues that one would wish to find mentioned in a testimonial to a prospective employee: he was industrious, frugal, temperate, and financially honest. But he was wholly destitute of those higher philosophic virtues that are so notable in Spinoza. His best thought was not such as would win him popularity, and he left his records of it unpublished in his desk. What he published was designed to win the approbation of princes and princesses. The consequence is that there are two systems of philosophy which may be regarded as representing Leibniz: one, which he proclaimed, was optimistic, orthodox, fantastic, and shallow; the other, which has been slowly unearthed from his manuscripts by fairly recent editors, was profound, coherent, largely Spinozistic, and amazingly logical. It was the popular Leibniz who invented the doctrine that this 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 Leibniz whom Voltaire caricatured as Doctor Pangloss. It would be unhistorical to ignore this Leibniz, but the other is of far greater philosophical importance.

Leibniz was born two years before the end of the Thirty Years' War, at Leipzig, where his father was professor of moral philosophy. At the university he studied law, and in 1666 he obtained a Doctor's degree at Altdorf, where he was offered a professorship, which he refused, saying he had 'very different things in view'. In 1667 he entered the service of the archbishop of Mainz, who, like other West German princes, was oppressed by fear of Louis XIV. With the approval of the archbishop, Leibniz tried to persuade the French king to invade Egypt rather than Germany, but was met with a polite reminder that since the time of St Louis the holy war against the infidel had gone out of fashion. His project remained unknown to the public until it was discovered by Napoleon when he occupied Hanover in 1803, four years after his own abortive Egyptian expedition. In 1672, in connection with this scheme, Leibniz went to Paris, where he spent the greater part of the next four years. His contacts in Paris were of great importance for his intellectual development, for Paris at that time led the world both in philosophy and in mathematics. It was there, in 1675–6, that he invented the infinitesimal calculus, in ignorance of Newton's previous but unpublished work on the same subject. Leibniz's work was first published in 1684, Newton's in 1687. The consequent dispute as to priority was unfortunate, and discreditable to all parties.

Leibniz was somewhat mean about money. When any young lady at the court of Hanover married, he used to give her what he called a 'wedding present', consisting of useful maxims, ending up with the advice not to give up washing now that she had secured a husband. History does not record whether the brides were grateful.

In Germany, Leibniz had been taught a neo-scholastic Aristotelian philosophy, of which he retained something throughout his later life. But in Paris he came to know Cartesianism and the materialism of Gassendi, both of which influenced him; at this time, he said, he abandoned the 'trivial schools', meaning scholasticism. In Paris he came to know Malebranche and Arnauld the Jansenist. The last important influence on his philosophy was that of Spinoza, whom he visited in 1676. He spent a month in frequent discussions with him, and secured part of the Ethics in manuscript. In later years he joined in decrying Spinoza, and minimized his contacts with him, saying he had met him once, and Spinoza had told some good anecdotes about politics.

His connection with the House of Hanover, in whose service he remained for the rest of his life, began in 1673. From 1680 onwards he was their librarian at Wolfenbüttel, and was officially employed in writing the history of Brunswick. He had reached the year 1009 when he died. The work was not published till 1843. Some of his time was spent on a project for the reunion of the Churches, but this proved abortive. He travelled to Italy to obtain evidence that the Dukes of Brunswick were connected with the Este family. But in spite of these services he was left behind at Hanover when George I became king of England, the chief reason being that his quarrel with Newton had made England unfriendly to him. However, the Princess of Wales, as he told all his correspondents, sided with him against Newton. In spite of her favour, he died neglected.

Leibniz's popular philosophy may be found in the Monadology and the Principles of Nature and of Grace, one of which (it is uncertain which) he wrote for Prince Eugene of Savoy, Marlborough's colleague. The basis of his theological optimism is set forth in the Théodicée, which he wrote for Queen Charlotte of Prussia. I shall begin with the philosophy set forth in these writings, and then proceed to his more solid work which he left unpublished.

Like Descartes and Spinoza, Leibniz based his philosophy on the notion of substance, but he differed radically from them as regards the relation of mind and matter, and as regards the number of substances. Descartes allowed three substances, God and mind and matter; Spinoza admitted God alone. For Descartes, extension is the essence of matter; for Spinoza, both extension and thought are attributes of God. Leibniz held that extension cannot be an attribute of a substance. His reason was that extension involves plurality, and can therefore only belong to an aggregate of substances; each single substance must be unextended. He believed, consequently, in an infinite number of substances, which he called 'monads'. Each of these would have some of the properties of a physical point, but only when viewed abstractly; in fact, each monad is a soul. This follows naturally from the rejection of extension as an attribute of substance; the only remaining possible essential attribute seemed to be thought. Thus Leibniz was led to deny the reality of matter, and to substitute an infinite family of souls.

The doctrine that substances cannot interact, which had been developed by Descartes' followers, was retained by Leibniz, and led to curious consequences. No two monads, he held, can ever have any causal relation to each other; when it seems as if they had, appearances are deceptive. Monads, as he expressed it, are 'windowless'. This led to two difficulties: one in dynamics, where bodies seem to affect each other, especially in impact; the other in relation to perception, which seems to be an effect of the perceived object upon the percipient. We will ignore the dynamical difficulty for the present, and consider only the question of perception. Leibniz held that every monad mirrors the universe, not because the universe affects it, but because God has given it a nature which spontaneously produces this result. There is a 'pre-established harmony' between the changes in one monad and those in another, which produces the semblance of interaction. This is obviously an extension of the two clocks, which strike at the same moment because each keeps perfect time. Leibniz has an infinite number of clocks, all arranged by the Creator to strike at the same instant, not because they affect each other, but because each is a perfectly accurate mechanism. To those who thought the pre-established harmony odd, Leibniz pointed out what admirable evidence is afforded of the existence of God.

Monads form a hierarchy, in which some are superior to others in the clearness and distinctness with which they mirror the universe. In all there is some degree of confusion in perception, but the amount of confusion varies according to the dignity of the monad concerned. A 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 which is what is called the soul of the man of whose body it forms part. This monad is dominant, not only in the sense of having clearer perceptions than the others, but also in another sense. The changes in a human body (in ordinary circumstances) happen for the sake of the dominant monad: when my arm moves, the purpose served by the movement is in the dominant monad, i.e. my mind, not in the monads that compose my arm. This is the truth of what appears to common sense as the control of my will over my arm.

Space, as it appears to the senses, and as it is assumed in physics, is not real, but it has a real counterpart, namely the arrangement of the monads in a three-dimensional order according to the point of view from which they mirror the world. Each monad sees the world in a certain perspective peculiar to itself; in this sense we can speak, somewhat loosely, of the monad as having a spatial position.

Allowing ourselves this way of speaking, we can say that there is no such thing as a vacuum; every possible point of view is filled by one actual monad, and by only one. No two monads are exactly alike; this is Leibniz's principle of the 'identity of indiscernibles'.

In contrasting himself with Spinoza, Leibniz made much of the free will allowed in his system. He had a 'principle of sufficient reason', according to which nothing happens without a reason; but when we are concerned with free agents, the reasons for their actions 'incline without necessitating'. What a human being does always has a motive, but the sufficient reason of his action has no logical necessity. So, at least, Leibniz says when he is writing popularly, but, as we shall see, he had another doctrine which he kept to himself after finding that Arnauld thought it shocking.

God's actions have the same kind of freedom. He always acts for the best, but He is not under any logical compulsion to do so. Leibniz agrees with Thomas Aquinas that God cannot act contrary to the laws of logic, but He can decree whatever is logically possible, and this leaves Him a great latitude of choice.

Leibniz brought into their final form the metaphysical proofs of God's existence. These had a long history; they begin with Aristotle, or even with Plato; they are formalized by the scholastics, and one of them, the ontological argument, was invented by St Anselm. This argument, though rejected by St Thomas, was revived by Descartes. Leibniz, whose logical skill was supreme, stated the arguments better than they had ever been stated before. That is my reason for examining them in connection with him.

Before examining the arguments in detail, it is as well to realize that modern theologians no longer rely upon them. Medieval theology is derivative 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 one whose appeal is intellectual: His existence solves certain puzzles which otherwise would create argumentative difficulties in the understanding of the universe. This Deity who appears at the end of a piece of reasoning, like the proof of a proposition in geometry, did not satisfy Rousseau, who reverted to a conception of God more akin to that of the Gospels. In the main, modern theologians, especially such as are Protestant, have followed Rousseau in this respect. The philosophers have been more conservative; in Hegel, Lotze, and Bradley arguments of the metaphysical sort persist, in spite of the fact that Kant professed to have demolished such arguments once for all.

Leibniz's arguments for the existence of God are four in number; they are (1) the ontological argument, (2) the cosmological argument, (3) the argument from eternal truths, (4) the argument from the pre-established harmony, which may be generalized into the argument from design, or the physico-theological argument, as Kant calls it. We will consider these arguments successively.

The ontological argument depends upon the distinction between existence and essence. Any ordinary person or thing, it is held, on the one hand exists, and on the other hand has certain qualities, which make up his or its 'essence'. Hamlet, though he does not exist, has a certain essence; he is melancholy, undecided, witty, etc. When we describe a person, the question whether he is real or imaginary remains open, however minute our description may be. This is expressed in scholastic language by saying that, in the case of any finite substance, its essence does not imply its existence. But in the case of God, defined as the most perfect Being, St Anselm, followed by Descartes, maintains that essence does imply existence, on the ground that a Being who possesses all other perfections is better if He exists than if He does not, from which it follows that if He does not He is not the best possible Being.

Leibniz neither wholly accepts nor wholly rejects this argument; it needs to be supplemented, so he says, by a proof that God, so defined, is possible. He wrote out a proof that the idea of God is possible, which he showed to Spinoza when he saw him at the Hague. This proof defines God as the most perfect Being, i.e. as the subject of all perfections, and a perfection is defined as a 'simple quality which is positive and absolute, and expresses without any limits whatever it does express'. Leibniz easily proves that no two perfections, as above defined, can be incompatible. He concludes: 'There is, therefore, or there can be conceived, a subject of all perfections, or most perfect Being. Whence it follows also that He exists, for existence is among the number of the perfections.'

Kant encountered this argument by maintaining that 'existence' is not a predicate. Another kind of refutation results from my theory of descriptions. The argument does not, to a modern mind, seem very convincing, but it is easier to feel convinced that it must be fallacious than it is to find out precisely where the fallacy lies.

The cosmological argument is more plausible than the ontological argument. It is a form of the First-Cause argument, which is itself derived from Aristotle's argument of the unmoved mover. The First-Cause argument is simple. It points out that everything finite has a cause, which in turn had a cause, and so on. This series of previous causes cannot, it is maintained, be infinite, and the first term in the series must itself be uncaused, since otherwise it would not be the first term. There is therefore an uncaused cause of everything, and this is obviously God.

In Leibniz the argument takes a somewhat different form. He argues that every particular thing in the world is 'contingent', that is to say, it would be logically possible for it not to exist; and this is true, not only of each particular thing, but of the whole universe. Even if we suppose the universe to have always existed, there is nothing within the universe to show why it exists. But everything has to have a sufficient reason, according to Leibniz's philosophy; therefore the universe as a whole must have a sufficient reason, which must be outside the universe. This sufficient reason is God.

This argument is better than the straightforward First-Cause argument, and cannot be so easily refuted. The First-Cause argument rests on the assumption that every series must have a first term, which is false; for example, the series of proper fractions has no first term. But Leibniz's argument does not depend upon the view that the universe must have had a beginning in time. The argument is valid so long as we grant Leibniz's principle of sufficient reason, but if this principle is denied it collapses. What exactly Leibniz meant by the principle of sufficient reason is a controversial question. Couturat maintains that it means that every true proposition is 'analytic', i.e. such that its contradictory is self-contradictory. But this interpretation (which has support in writings that Leibniz did not publish) belongs, if true, to the esoteric doctrine. In his published works he maintains that there is a difference between necessary and contingent propositions, that only the former follow from the laws of logic, and that all propositions asserting existence are contingent, with the sole exception of the existence of God. Though God exists necessarily, He was not compelled by logic to create the world; on the contrary, this was a free choice, motivated, but not necessitated, by His goodness.

It is clear that Kant is right in saying that this argument depends upon the ontological argument. If the existence of the world can only be accounted for by the existence of a necessary Being, then there must be a Being whose essence involves existence, for that is what is meant by a necessary Being. But if it is possible that there should be a Being whose essence involves existence, then reason alone, without experience, can define such a Being, whose existence will follow from the ontological argument; for everything that has to do only with essence can be known independently of experience—such at least is Leibniz's view. The apparent greater plausibility of the cosmological as opposed to the ontological argument is therefore deceptive.

The argument from the eternal truths is a little difficult to state precisely. Roughly, the argument is this: Such a statement as 'it is raining' is sometimes true and sometimes false, but 'two and two are four' is always true. All statements that have only to do with essence, not with existence, are either always true or never true. Those that are always true are called 'eternal truths'. The gist of the argument is that truths are part of the contents of minds, and that an eternal truth must be part of the content of an eternal mind. There is already an argument not unlike this in Plato, where he deduces immortality from the eternity of the ideas. But in Leibniz the argument is more developed. He holds that the ultimate reason for contingent truths must be found in necessary truths. The argument here is as in the cosmological argument: there must be a reason for the whole contingent world, and this reason cannot itself be contingent, but must be sought among eternal truths. But a reason for what exists must itself exist; therefore eternal truths must, in some sense, exist, and they can only exist as thoughts in the mind of God. This argument is really only another form of the cosmological argument. It is, however, open to the further objection that a truth can hardly be said to 'exist' in a mind which apprehends it.

The argument from the pre-established harmony, as Leibniz states it, is only valid for those who accept his windowless monads which all mirror the universe. The argument is that, since all the clocks keep time with each other without any causal interaction, there must have been a single outside Cause that regulated all of them. The difficulty, of course, is the one that besets the whole monadology: if the monads never interact, how does any one of them know that there are any others? What seems like mirroring the universe may be merely a dream. In fact, if Leibniz is right, it is merely a dream, but he has ascertained somehow that all the monads have similar dreams at the same time. This, of course, is fantastic, and would never have seemed credible but for the previous history of Cartesianism.

Leibniz's argument, however, can be freed from dependence on his peculiar metaphysic, and transformed into what is called the argument from design. This argument contends that, on a survey of the known world, we find things which cannot plausibly be explained as the product of blind natural forces, but are much more reasonably to be regarded as evidences of a beneficent purpose.

This argument has no formal logical defect; its premisses are empirical, and its conclusion professes to be reached in accordance with the usual canons of empirical inference. The question whether it is to be accepted or not turns, therefore, not on general metaphysical questions, but on comparatively detailed considerations. There is one important difference between this argument and the others, namely, that the God whom (if valid) it demonstrates need not have all the usual metaphysical attributes. He need not be omnipotent or omniscient; He may be only vastly wiser and more powerful than we are. The evils in the world may be due to His limited power. Some modern theologians have made use of these possibilities in forming their conception of God. But such speculations are remote from the philosophy of Leibniz, to which we must now return.

One of the most characteristic features of that philosophy is the doctrine of many possible worlds. A world is 'possible' if it does not contradict the laws of logic. There are an infinite number of possible worlds, all of which God contemplated before creating the actual world. Being good, God decided to create the best of the possible worlds, and He considered that one to be the best which had the greatest excess of good over evil. He could have created a world containing no evil, but it would not have been so good as the actual world. That is because some great goods are logically bound up with certain evils. To take a trivial illustration, a drink of cold water when you are very thirsty on a hot day may give you such great pleasure that you think the previous thirst, though painful, was worth enduring, because without it the subsequent enjoyment could not have been so great. For theology, it is not such illustrations that are important, but the connection of sin with free will. Free will is a great good, but it was logically impossible for God to bestow free will and at the same time decree that there should be no sin. God therefore decided to make man free, although he foresaw that Adam would eat the apple, and although sin inevitably brought punishment. The world that resulted, although it contains evil, has a greater surplus of good over evil than any other possible world; it is therefore the best of all possible worlds, and the evil that it contains affords no argument against the goodness of God.

This argument apparently satisfied the Queen of Prussia. Her serfs continued to suffer the evil, while she continued to enjoy the good, and it was comforting to be assured by a great philosopher that this was just and right.

Leibniz's solution of the problem of evil, like most of his other popular doctrines, is logically possible, but not very convincing. A Manichæan might retort that this is the worst of all possible worlds, in which the good things that exist serve only to heighten the evils. The world, he might say, was created by a wicked demiurge, who allowed free will, which is good, in order to make sure of sin, which is bad, and of which the evil outweighs the good of free will. The demiurge, he might continue, created some virtuous men, in order that they might be punished by the wicked; for the punishment of the virtuous is so great an evil that it makes the world worse than if no good men existed. I am not advocating this opinion, which I consider fantastic; I am only saying that it is no more fantastic than Leibniz's theory. People wish to think the universe good, and will be lenient to bad arguments proving that it is so, while bad arguments proving that it is bad are closely scanned. In fact, of course, the world is partly good and partly bad, and no 'problem of evil' arises unless this obvious fact is denied.

I come now to Leibniz's esoteric philosophy, in which we find reasons for much that seems arbitrary or fantastic in his popular expositions, as well as an interpretation of his doctrines which, if it had become generally known, would have made them much less acceptable. It is a remarkable fact that he so imposed upon subsequent students of philosophy that most of the editors who published selections from the immense mass of his manuscripts preferred what supported the received interpretation of his system, and rejected as unimportant essays which prove him to have been a far more profound thinker than he wished to be thought. Most of the texts upon which we must rely for an understanding of his esoteric doctrine were first published in 1901 or 1903, in two works by Louis Couturat. One of these was even headed by Leibniz with the remark: 'Here I have made enormous progress.' But in spite of this, no editor thought it worth printing until Leibniz had been dead for nearly two centuries. It is true that his letters to Arnauld, which contain a part of his more profound philosophy, were published in the nineteenth century; but I was the first to notice their importance. Arnauld's reception of these letters was discouraging. He writes: 'I find in these thoughts so many things which alarm me, and which almost all men, if I am not mistaken, will find so shocking, that I do not see of what use a writing can be, which apparently all the world will reject.' This hostile opinion no doubt led Leibniz, thenceforth, to adopt a policy of secrecy as to his real thoughts on philosophical subjects.

The conception of substance, which is fundamental in the philosophies of Descartes, Spinoza, and Leibniz, is derived from the logical category of subject and predicate. Some words can be either subjects or predicates; e.g. I can say 'the sky is blue' and 'blue is a colour'. Other words—of which proper names are the most obvious instances—can never occur as predicates, but only as subjects, or as one of the terms of a relation. Such words are held to designate substances. Substances, in addition to this logical characteristic, persist through time, unless destroyed by God's omnipotence (which, one gathers, never happens). Every true proposition is either general, like 'all men are mortal', in which case it states that one predicate implies another, or particular, like 'Socrates is mortal', in which case the predicate is contained in the subject, and the quality denoted by the predicate is part of the notion of the substance denoted by the subject. Whatever happens to Socrates can be asserted in a sentence in which 'Socrates' is the subject and the words describing the happening in question are the predicate. All these predicates put together make up the 'notion' of Socrates. All belong to him necessarily, in this sense, that a substance of which they could not be truly asserted would not be Socrates, but some one else.

Leibniz was a firm believer in the importance of logic, not only in its own sphere, but as the basis of metaphysics. He did work on mathematical logic which would have been enormously important if he had published it; he would, in that case, have been the founder of mathematical logic, which would have become known a century and a half sooner than it did in fact. He abstained from publishing, because he kept on finding evidence that Aristotle's doctrine of the syllogism was wrong on some points; respect for Aristotle made it impossible for him to believe this, so he mistakenly supposed that the errors must be his own. Nevertheless he cherished throughout his life the hope of discovering a kind of generalized mathematics, which he called Characteristica Universalis, by means of which thinking could be replaced by calculation. 'If we had it,' he says, 'we should be able to reason in metaphysics and morals in much the same way as in geometry and analysis.' 'If controversies were to arise, there would be no more need of disputation between two philosophers than between two accountants. For it would suffice to take their pencils in their hands, to sit down to their slates, and to say to each other (with a friend as witness, if they liked): Let us calculate.'

Leibniz based his philosophy upon two logical premisses, the law of contradiction and the law of sufficient reason. Both depend upon the notion of an 'analytic' proposition, which is one in which the predicate is contained in the subject—for instance, 'all white men are men'. The law of contradiction states that all analytic propositions are true. The law of sufficient reason (in the esoteric system only) states that all true propositions are analytic. This applies even to what we should regard as empirical statements about matters of fact. If I make a journey, the notion of me must from all eternity have included the notion of this journey, which is a predicate of me. 'We may say that the nature of an individual substance, or complete being, is to have a notion so completed that it suffices to comprehend, and to render deducible from it, all the predicates of the subject to which this notion is attributed. Thus the quality of king, which belongs to Alexander the Great, abstracting from the subject, is not sufficiently determined for an individual, and does not involve other qualities of the same subject, nor all that the notion of this prince contains, whereas God, seeing the individual notion or haecceity of Alexander, sees in it at the same time the foundation and the reason of all the predicates which can be truly attributed to him, as e.g. whether he would conquer Darius and Porus, even to knowing a priori (and not by experience) whether he died a natural death or by poison, which we can only know by history.'

One of the most definite statements of the basis of his metaphysic occurs in a letter to Arnauld:

'In consulting the notion which I have of every true proposition, I find that every predicate, necessary or contingent, past, present, or future, is comprised in the notion of the subject, and I ask no more.... The proposition in question is of great importance, and deserves to be well established, for it follows that every soul is as a world apart, independent of everything else except God; that it is not only immortal and so to speak impassible, but that it keeps in its substance traces of all that happens to it.'

He goes on to explain that substances do not act on each other, but agree through all mirroring the universe, each from its own point of view. There can be interaction, because all that happens to each subject is part of its own notion, and eternally determined if that substance exists.

This system is evidently just as deterministic as that of Spinoza. Arnauld expresses his horror of the statement (which Leibniz had made): 'That the individual notion of each person involves once for all everything that will ever happen to him.' Such a view is evidently incompatible with the Christian doctrine of sin and free will. Finding it ill received by Arnauld, Leibniz carefully refrained from making it public.

For human beings, it is true, there is a difference between truths known by logic and truths known by experience. This difference arises in two ways. In the first place, although everything that happens to Adam follows from his notion, if he exists, we can only ascertain his existence by experience. In the second place, the notion of any individual substance is infinitely complex, and the analysis required to deduce his predicates is only possible for God. These differences, however, are only due to our ignorance and intellectual limitations; for God, they do not exist. God apprehends the notion of Adam in all its infinite complexity, and can therefore see all true propositions about Adam as analytic. God can also ascertain a priori whether Adam exists. For God knows His own goodness, from which it follows that He will create the best possible world; and He also knows whether or not Adam forms part of this world. There is therefore no real escape from determinism through our ignorance.

There is, however, a further point, which is very curious. At most times, Leibniz represents the Creation as a free act of God, requiring the exercise of His will. According to this doctrine, the determination of what actually exists is not effected by observation, but must proceed by way of God's goodness. Apart from God's goodness, which leads Him to create the best possible world, there is no a priori reason why one thing should exist rather than another.

But sometimes, in papers not shown to any human being, there is a quite different theory as to why some things exist and others, equally possible, do not. According to this view, everything that does not exist struggles to exist, but not all possibles can exist, because they are not all 'compossible'. It may be possible that A should exist, and also possible that B should exist, but not possible that both A and B should exist; in that case, A and B are not 'compossible'. Two or more things are only 'compossible' when it is possible for all of them to exist. Leibniz seems to have imagined a sort of war in the Limbo inhabited by essences all trying to exist; in this war, groups of compossibles combine, and the largest group of compossibles wins, like the largest pressure group in a political contest. Leibniz even uses this conception as a way of defining existence. He says: 'The existent may be defined as that which is compatible with more things than is anything incompatible with itself'. That is to say, if A is incompatible with B, while A is compatible with C and D and E, but B is only compatible with F and G, then A, but not B, exists by definition. 'The existent,' he says, 'is the being which is compatible with the most things.'

In this account, there is no mention of God, and apparently no act of creation. Nor is there need of anything but pure logic for determining what exists. The question whether A and B are compossible is, for Leibniz, a logical question, namely: Does the existence of both A and B involve a contradiction? It follows that, in theory, logic can decide the question what group of compossibles is the largest, and this group consequently will exist.

Perhaps, however, Leibniz did not really mean that the above was a definition of existence. If it was merely a criterion, it can be reconciled with his popular views by means of what he calls 'metaphysical perfection'. Metaphysical perfection, as he uses the term, seems to mean quantity of existence. It is, he says, 'nothing but the magnitude of positive reality strictly understood'. He always argues that God created as much as possible; this is one of his reasons for rejecting a vacuum. There is a general belief (which I have never understood) that it is better to exist than not to exist; on this ground children are exhorted to be grateful to their parents. Leibniz evidently held this view, and thought it part of God's goodness to create as full a universe as possible. It would follow that the actual world would consist of the largest group of compossibles. It would still be true that logic alone, given a sufficiently able logician, could decide whether a given possible substance would exist or not.

Leibniz, in his private thinking, is the best example of a philosopher who uses logic as a key to metaphysics. This type of philosophy begins with Parmenides, and is carried further in Plato's use of the theory of ideas to prove various extra-logical propositions. Spinoza belongs to the same type, and so does Hegel. But none of these is so clear cut as Leibniz in drawing inferences from syntax to the real world. This kind of argumentation has fallen into disrepute owing to the growth of empiricism. Whether any valid inferences are possible from language to non-linguistic facts is a question as to which I do not care to dogmatize; but certainly the inferences found in Leibniz and other a priori philosophers are not valid, since all are due to a defective logic. The subject-predicate logic, which all such philosophers in the past assumed, either ignores relations altogether, or produces fallacious arguments to prove that relations are unreal. Leibniz is guilty of a special inconsistency in combining the subject-predicate logic with pluralism, for the proposition 'there are many monads' is not of the subject-predicate form. To be consistent, a philosopher who believes all propositions to be of this form should be a monist, like Spinoza. Leibniz rejected monism largely owing to his interest in dynamics, and to his argument that extension involves repetition, and therefore cannot be an attribute of a single substance.

Leibniz is a dull writer, and his effect on German philosophy was to make it pedantic and arid. His disciple Wolf, who dominated the German universities until the publication of Kant's Critique of Pure Reason, left out whatever was most interesting in Leibniz, and produced a dry professorial way of thinking. Outside Germany, Leibniz's philosophy had little influence; his contemporary, Locke, governed British philosophy, while in France Descartes continued to reign until he was overthrown by Voltaire, who made English empiricism fashionable.

Nevertheless, Leibniz remains a great man, and his greatness is more apparent now than it was at any earlier time. Apart from his eminence as a mathematician and as the inventor of the infinitesimal calculus, he was a pioneer in mathematical logic, of which he perceived the importance when no one else did so. And his philosophical hypotheses, though fantastic, are very clear, and capable of precise expression. Even his monads can still be useful as suggesting possible ways of viewing perception, though they cannot be regarded as windowless. What I, for my part, think best in his theory of monads is his two kinds of space, one subjective, in the perceptions of each monad, and one objective, consisting of the assemblage of points of view of the various monads. This, I believe, is still useful in relating perception to physics.

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