Copyright Jonathan Bennett

[Brackets] enclose editorial explanations. Small dots enclose material that has been added, but can be read as though it were part of the original text. Occasional bullets, and also indenting of passages that are not quotations, are meant as aids to grasping the structure of a sentence or a thought. Four ellipses . . . . indicate the omission of a brief passage that seems to present more difficulty than it is worth.First launched: September 2004

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Freedom and PossibilityBy G. W. Leibniz

In God everything is spontaneous. It can hardly be doubted that in every human person there is the freedom to do what he wills to do. A volition is an attempt to act of which we are conscious. An act necessarily follows from a volition to do it and the ability to do it. When all the conditions for willing to do something are matched by equally strong conditions against willing to do it, no volition occurs. Rather there is indifference [here = equilibrium]. Thus, even if someone accepts that all the conditions requisite for acting are in place, he wont act if equal contrary conditions obtain. Thats one way for a person to to act on reasons that he has. Here is another: a person may be unmoved by reasons through sheer forgetfulness, i.e. by turning his mind away from them. So it is indeed possible to be unmoved by reasons. Unless this proposition is accepted: There is nothing without reason. That is: In every true proposition there is a connection between the subject and the predicate, i.e. every true proposition can be proved a priori. There are two primary propositions: one is the principle of necessary things, that

whatever implies a contradiction is false, and the other is the principle of contingent things, that

whatever is more perfect or has more reason is true. All truths of metaphysics - indeed all truths that are absolutely necessary, such as those of logic, arithmetic, geometry, and the like - rest on the former principle, for someone who denies one of those truths can always be shown that his denial implies a contradiction. All contingent truths rest on the latter principle. (I mean truths that are in themselves contingent. They may be necessary-given-what-God-wills.) So the principle of contradiction is the basis for all truths about possibilities or essences, and all truths about a things impossibility or its necessity (that is, the impossibility of its contrary). And the principle of perfection is the basis for all truths about contingent things, that is, about what exists. God is the only being whose existence is not contingent. The reason why some particular contingent thing x exists, and other possible things dont, shouldnt be sought in xs definition alone. If xs definition did explain its existence, its nonexistence would imply a contradiction; and those other things wouldnt be possible, contrary to our hypothesis. For the reason why x exists and those others dont, we must look to how x compares with the others; the reason is that x is more perfect than the others that are its rivals for existence.

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My over-riding thought here is a notion of possibility and necessity according to which some things are not necessary and dont actually exist but nevertheless are possible. It follows from this that a reason that always brings it about that a free mind chooses one thing rather than another (whether that reason derives from the perfection of a thing, as it does in God, or from our imperfection) doesnt take away our freedom. This also shows what distinguishes Gods free actions from his necessary actions. Here is one example of each kind of action. It is necessary that God loves himself, for that can be demonstrated from the definition of God. But it cant be demonstrated from that definition that God makes whatever is most perfect, for theres nothing contradictory in the proposition that he doesnt. If there were, it wouldnt be possible for him to make something less perfect, and that is contrary to the hypothesis that there are non-existent possibles. Moreover, this conclusion derives from the notion of existence, for only the most perfect exists. Let there be two possible things, A and B, such that necessarily one and only one of them exists; and lets assume that A is more perfect than B. Then we can certainly explain why A should exist rather than B - this is a basis for us to predict which of the two will exist. Indeed, As existing rather than Bs doing so can be demonstrated, by which I mean that it can be rendered certain from the nature of the case. Now, if being certain were the same as being necessary then it would also be necessary for A to exist. But As existence has merely what I call hypothetical necessity, meaning that

it is necessary that: if God always chooses what is most perfect, then A exists.That is to be distinguished from the proposition that

it is necessary that: A exists.If it were absolutely and not just hypothetically necessary that A exists, then B - which we have stipulated cannot exist if A exists - would be absolutely impossible, i.e. would imply a contradiction, which is contrary to our stipulation that A and B are both possible. So we must hold that anything that has some degree of perfection is possible, and anything that is more perfect than its opposite actually exists - not because of its own nature but because of Gods general resolve to create the more perfect. Perfection (or essence) is an urge for existence; it implies existence, not necessarily but through there not being a more perfect thing that prevents it from existing. All truths of physics are of this sort; for example, when we say that a body persists in the speed with which it begins, we mean . . . if nothing gets in its way. God produces the best - not necessarily, but because he wills to do so. If you ask Does God will by necessity? I ask you to explain what you mean by necessity, spelling it out in detail so as to make clear what exactly you are asking. For example, you might be asking:

Does God will by necessity or does he will freely?that is:

Does God will because of his nature or because of his will?My answer to that is of course that God cant will voluntarily. That is, it cant be the case that whenever God wills to do something, it is because he has willed to will to do that thing; because that would involve willing to will . . . to infinity. Rather, we must say that it is Gods nature that leads him to will the best. So he wills by necessity? you say, implying that I am demeaning God. I reply with St. Augustine that such necessity is blessed. But surely it follows from this that things exist by necessity. How so? Because the nonexistence of what God wills to exist implies a contradiction? I deny that this proposition is absolutely true. It entails that what God doesnt will is not possible, and I deny that. For things remain possible, even if God doesnt select them.

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Given that God doesnt will x to exist, it is still possible for x to exist, because xs nature is such that x could exist if God were to will it to exist. You will object: But God cant will it to exist. Granted; yet x remains possible in its nature even if it is not possible with respect to the divine will, since we have defined as possible in its nature anything that in itself implies no contradiction, even if its coexistence with God can in some way be said to imply a contradiction. Well need to use unambiguous meanings for words if we are to avoid every kind of absurd locution. I start with the meaning I give to possible. I say:

a possible thing is something with some essence or reality, that is, something that can be clearly understood.

For an illustrative example, let us pretend that nothing exactly pentagonal ever did or will exist in nature. A pentagon would nevertheless remain possible. However, if we are to maintain that pentagons are possible, we should give some reason why no pentagon ever did or will exist. The reason is simply the fact that the pentagon is incompatible with other things that got into existence ahead of it because they include more perfection, i.e. involve more reality, than it does. Returning to your previous line of attack, you will say: So according to you it is necessary that the pentagon doesnt exist. I agree, if what you mean is that

The proposition No pentagon ever did or will exist is necessary. But what you say is false if it is understood to mean that

The timeless proposition No pentagon exists is necessary,because I deny that this timeless proposition can be demonstrated. The pentagon is not absolutely impossible, and doesnt imply a contradiction, even if it follows from the harmony of things that a pentagon cant find a place among real things. The following argument is valid (its second premise is the one we have been pretending to be true):

If a pentagon exists, it is more perfect than other things.A pentagon is not more perfect than other things.Therefore, a pentagon does not exist.

But the premises dont imply that it is impossible for a pentagon to exist. This is best illustrated by analogy with imaginary roots in algebra, such as -1. For -1 does involve some notion, though it cant be pictured . . . . But there is a great difference between

problems that are insoluble because a solution requires imaginary rootsand

problems that are insoluble because of their absurdity.An example of the latter: Find a number which multiplied by itself is 9, and which added to 5 makes 9. Such a number implies a contradiction, for it must be both 3 and 4, implying that 3 = 4, a part equals the whole. An example of the former: Find a number x such that x2 + 9 = 3x. Someone trying to solve this could certainly never show that the solution would imply any such absurdity as that the whole equals its part, but he could show that such a number cannot be designated because the only solutions to the equation are imaginary roots. To accompany the pentagon example, I now offer another one, in which I use a real line to mean a line that really bounds some body. If God had decreed that there should be no real line that was incommensurable with other real lines, it wouldnt follow that the existence of an incommensurable line implies a contradiction, even if because of the principle of perfection God couldnt have made such a line.

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All this removes the difficulties about the foreknowledge of future contingents. For God, who foresees the future reasons or causes for some things to exist and others not to, has certain foreknowledge of future contingents through their causes. He formulates propositions about them that are

necessary, given that the state of the world has been settled once and for all,that is,

necessary, given the harmony of things.But the propositions about future contingents are not necessary in the absolute sense, as mathematical propositions are. This is the best answer to the difficulty about how, if future contingents are not necessary, God can have foreknowledge of them. It involves us in saying that it is possible for the imperfect rather than the more perfect to exist. You may object: It is impossible for something to exist that God doesnt will to exist. I deny that something that isnt going to exist is thereby impossible in itself. So the proposition What God doesnt will to exist doesnt exist should be accepted as true, but its necessity should be denied.* * * *[Near the end of this paper Leibniz has an incomplete sentence which he probably meant to turn into something saying:] The only existential proposition that is absolutely necessary is God exists.* * * *[Early in the paper, Leibniz mentions indifference or equilibrium. He wrote the following note in the margin about that:] If complete indifference is required for freedom, then there is scarcely ever a free act, since I think it hardly ever happens that everything on both sides is equal. For even if the reasons happen to be equal, the passions wont be. So why should we argue about circumstances that do not arise? I dont think examples can be found in which the will chooses - that is, where it arbitrarily breaks a deadlock by just choosing - because there is always some reason for choosing one alternative rather than the other. The followers of Aquinas place freedom in the power of the will, which stands above every finite good in such a way that the will can resist it. And so, in order to have indifference of will, they seek indifference of intellect. They think that necessity is consistent with freedom in God - for example the free necessity of Gods loving himself. But (they hold) with respect to creatures God does not decide with necessity. . . .

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Copyright Jonathan Bennett

[Brackets] enclose editorial explanations. Small dots enclose material that has been added, but can be read as though it were part of the original text. Occasional bullets, and also indenting of passages that are not quotations, are meant as aids to grasping the structure of a sentence or a thought. Four ellipses . . . . indicate the omission of a brief passage that seems to present more difficulty than it is worth.First launched: September 2004

Meditations on Knowledge, Truth, and IdeasBy G. W. Leibniz

Controversies are boiling these days among distinguished men over true and false ideas. This is an issue of great importance for recognizing truth - an issue on which Descartes himself is not altogether satisfactory. So I want to explain briefly what I think can be established about the distinctions and criteria that relate to ideas and knowledge. [Here and in the title, knowledge translates cognitio, which means something weaker than knowledge strictly so-called, involving certainty and guaranteed truth, for which the Latin word is scientia]. Here is the skeleton of what I have to say. Knowledge is either

dim or vivid;vivid knowledge is either

confused or clear;clear knowledge is either

inadequate or adequate;and adequate knowledge is either

symbolic or intuitive.Knowledge that was at the same time both adequate and intuitive would be absolutely perfect. [Here and throughout, vivid translates clarus. (The more usual rendering as clear is no better from a dictionary point of view, and makes much worse sense philosophically because it has Leibniz saying that knowledge can be at once clear and confused.) This use of vivid points to dim as the better translation of the contrasting term obscurus, and liberates clear for use in translating distinctus.] A dim notion is one that isnt sufficient for recognizing the thing that it represents - i.e. the thing that it is a notion of. Example: I once saw a certain flower but whenever I remember it I cant bring it to mind well enough to recognize it, distinguishing it from other nearby flowers, when I see it again. Another kind of example: I have dim notions when I think about some term for which there is no settled definition - such as Aristotles entelechy, or his notion of cause when offered as something that is common to material, formal, efficient and final causes. [For a coin, these causes would be, respectively, the metal of which the coin is composed, the coins shape, weight etc., the force of the die against the hot metal, and the commercial purpose for which the coin was made. Leibniz implies that these seem not to be four species of a single genus.] And a proposition is dim if it contains a dim notion as an ingredient. Accordingly, knowledge is vivid if it gives me the means for recognizing the thing that is represented.

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Vivid knowledge is either confused or clear. It is confused when I cant list, one by one, the marks that enable me to differentiate the represented thing from other things, even though the thing has such marks into which its notion can be resolved [= analysed, broken down into its simpler constituents]. And so we recognize colours, smells, tastes, and other particular objects of the senses vividly enough to be able to distinguish them from one another, but only through the simple testimony of the senses, not by way of marks that we could list. Thus we cant explain what red is to a blind man; and we cant give anyone a vivid notion of things like red except by leading him into the presence of the thing and getting him to see, smell, or taste the same thing we do, or by reminding him of some past perception of his that is similar. This is so even though the notions of these qualities are certainly composite and can be resolved - after all, they do have causes. [Perhaps Leibnizs thought is that the complexity of the causes must be matched by the complexity of the caused quality, and thus by the complexity of the complete notion of it.] Similarly, we see that painters and other skilled craftsmen can accurately tell well-done work from what is poorly done, though often they cant explain their judgments, and when asked about them all they can say is that the works that displease them lack a certain je-ne-sais-quoi. But a clear notion is like the one an assayer has of gold - that is, a notion connected with listable marks and tests that are sufficient to distinguish the represented thing from all other similar bodies. Notions common to several senses - like the notions of number, size, and shape - are usually clear. So are many notions of states of mind, such as hope and fear. In brief, we have a clear notion of everything for which we have a nominal definition (which is nothing but a list of sufficient marks). Also, we have clear knowledge of any indefinable notion, since such a notion is basic, something we start with; it cant be resolved into marks or simpler constituents, as it has none; so it has to serve as its own mark, and be understood through itself. An inadequate notion is what you have when

the notion is clear, meaning that you understand vividly the individual marks composing it, but

the grasp of some or all of those marks is (though vivid) confused, because you cant list the marks whereby you recognize those marks.

For example, someones knowledge of gold may be clear yet inadequate: he knows that heavi-ness, colour, solubility in aqua fortis etc. are the marks of gold, but he cant produce a list of the marks whereby he recognizes heaviness, yellowness, and all the others. When every ingredient of a clear notion is itself clearly known - that is, when the analysis of the original notion has been carried to completion - then our knowledge of it is adequate. (I dont know whether humans have any perfectly adequate knowledge, though our knowledge of numbers certainly comes close.) Symbolic notions are ones in which words stand in for thoughts. We dont usually grasp the entire nature of a thing all at once, especially one whose analysis is long; so in place of thoughts about the things themselves we use thoughts about signs. In our thought we usually omit the explicit explanation of what a sign means, knowing or believing that we have the explanation at our command and could produce it on demand. Thus, when I think about a chiliagon [pronounced kill-ee-a-gon], that is, a polygon with a thousand equal sides, I dont always

think about the nature of a side, or of equality, or of thousandfoldedness . . . .; in place of such thoughts,

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in my mind I use the words side, equal and thousand.The meanings of these words appear only dimly and imperfectly to my mind, but I remember that I know what they mean, so I decide that I neednt explain them to myself at this time. This kind of thinking is found in algebra, in arithmetic, and indeed almost everywhere. I call it blind or symbolic thinking. When a notion is very complex, we cant bear in mind all of its component notions at the same time, and this forces us into symbolic thinking. When we can keep them all in mind at once, we have knowledge of the kind I call intuitive. (Actually, I treat this as a matter of degree; so I should have said: insofar as we can keep all that in mind at once, to that extent our knowledge is intuitive.) Whereas our thinking about composites is mostly symbolic, our knowledge of a clear basic notion has to be intuitive. That is because symbolic knowledge involves letting words stand in for components of a notion, and basic notions dont have components. This shows that its only if we use intuitive thinking that we have ideas in our minds, even when we are thinking about something we know clearly. We often mistakenly believe that we have ideas of things in our mind, assuming that we have already explained to ourselves some of the terms we are using, when really we havent explained any of them. Some people hold that we cant understand what we are saying about a thing unless we have an idea of it; but this is false or at least ambiguous, because we can have understanding of a sort even when our thinking is blind or symbolic and doesnt involve ideas. When we settle for this blind thinking, and dont pursue the resolution of notions far enough, we may have a thought that harbours a contradiction that we dont see because it is buried in a very complex notion. At one time I was led to consider this point more clearly by an old argument for the existence of God . . . . that Descartes revived. The argument goes like this:

Whatever follows from the idea or definition of a thing can be predicated of the thing. God is by definition the most perfect being, or the being nothing greater than which can be thought. Now, the idea of the most perfect being includes ideas of all perfections, and amongst these perfections is existence. So existence follows from the idea of God. Therefore existence can be predicated of God, which is to say that God exists.

But this argument shows only that if God is possible then it follows that he exists. For we cant safely draw conclusions from definitions unless we know first that they are real definitions, that is, that they dont include any contradictions. If a definition does harbour a contradiction, we can infer contradictory conclusions from it, which is absurd. My favourite illustrative example of this is the fastest motion, which entails an absurdity. I now show that it does:

Suppose there is a wheel turning with the fastest motion. Anyone can see that if a spoke of the wheel came to poke out beyond the rim, the end of it would then be moving faster than a nail on the rim of the wheel. So the nails motion is not the fastest, which is contrary to the hypothesis.

Now, we certainly understand the phrase the fastest motion, and we may think we have an idea corresponding to it; but we dont, because we cant have an idea of something impossible. Similarly, the fact that we think about a most perfect being doesnt entitle us to claim that we have an idea of a most perfect being. So in the above demonstration - the one revived by Descartes - in order properly to draw the conclusion we must show or assume the possibility of a most perfect being. It is indeed true - nothing truer! - that we do have an idea of God and that a most perfect being is possible, indeed, necessary. But that argument is not sufficient for drawing the conclusion, and Aquinas rejected it.

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So we have a line to draw between nominal definitions, which contain only marks that distinguish the thing from other things, and real definitions, from which the thing can be shown to be possible. And thats my answer to Hobbes, who claimed that truths are arbitrary because they depend on nominal definitions. What he didnt take into account was that a definitions being real is not something we decide, and that not just any notions can be joined to one another. Nominal definitions are insufficient for perfect knowledge [scientia] except when the possibility of the thing defined is established in some other way. Near the start of this paper I listed four classifications of ideas, now at last we come to a fifth - true and false. It is obvious what true and false ideas are: an idea is true when it is a possible notion, and false when it includes a contradiction. Somethings possibility can be known either a priori or a posteriori. The possibility of a thing is known a priori when we resolve a notion into its requisites, i.e. into other notions that are known to be possible and to be compatible with one another, and that are required if the notion is to apply. [These requisita could be components of the notion: closed is a component of circular, and could be called a logical requisite for somethings being circular. In the very next sentence, however, Leibniz also brings in causal requisites.] This happens, for instance, when we understand how a thing can be produced, which is why causal definitions are more useful than others. A things possibility is known a posteriori when we know through experience that it actually exists, for what did or does actually exist is certainly possible! And, indeed, whenever we have adequate knowledge we also have a priori knowledge of possibility: if an analysis is brought to completion with no contradiction turning up, then certainly the analysed notion is possible. For men to produce a perfect analysis of their notions would be for them to reduce their thoughts to basic possibilities and unanalysable notions, which amounts to reducing them to the absolute attributes of God - and thus to the first causes and the ultimate reason for things. Can they do this? I shant venture to settle the answer to that now. For the most part we are content to have learned through experience that certain notions are real [here = possible], from which we then assemble others following the lead of nature. All this, I think, finally lets us understand that one should be cautious in claiming to have this or that idea. Many people who use this glittering title idea to prop up certain creatures of their imagination are using it wrongly, for we dont always have an idea corresponding to everything we consciously think of (as I showed with the example of greatest speed). People in our own times have laid down the principle:

Whatever I vividly and clearly perceive about a thing is true, i.e. can be said of the thing;but I cant see that they have used this principle well. [Leibniz is referring to a principle of Descartess that is almost always translated in English as Whatever I clearly and distinctly perceive . . ..] For people who are careless in judgment often take to be vivid and clear what is really dim and confused in their minds. So this axiom is useless unless (1) explicitly stated criteria for vividness and clarity are introduced, and (2) we have established the truth of the ideas that are involved - in my sense, in which an idea is true if and only if it is possible, i.e. could have instances. Furthermore, the rules of common logic - which geometers use too - are not to be despised as criteria for the truth of assertions: for example, the rule that nothing is to be accepted as certain unless it is shown by careful testing or sound demonstration - a sound demonstration being one that follows the form prescribed by logic. Not that we always need arguments to be in syllogistic order as in the Aristotelian philosophy departments . . . .; but the argument must somehow reach

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its conclusion on the strength of its form. Any correct calculation provides an example of an argument conceived in proper logical form. Such an argument should not omit any necessary premise, and all premises should have been previously demonstrated - or else have been assumed as hypotheses, in which case the conclusion is also hypothetical. Someone who carefully observes these rules will easily protect himself against deceptive ideas. The highly talented Pascal largely agrees with this in his excellent essay On the Geometrical Mind . . . . The geometer, he says, must define all terms that are slightly obscure and prove all truths that are slightly dubious. But I wish he had made precise the line beyond which a notion or statement is no longer even slightly obscure or dubious. Most of what matters regarding this can be gathered from careful attention to what I have said above; and I shant go further into it now, because I am trying to be brief. Before finishing, I offer three further remarks, only loosely connected with one another, but all having to do with ideas. (1) There has been controversy over whether we see everything in God - that is, perceive the world by sharing Gods ideas with him - or whether we have our own ideas. The view that we see everything in God, though recently made famous through Malebranches defence of it, is an old opinion, and properly understood it shouldnt be rejected completely. But the point I want to make here is that even if we did see everything in God, we would still also have to have our own ideas - not little sort-of copies of Gods ideas, but states of our mind corresponding to the thing we perceived in God. For when go from having one thought to having another, there has to be some change in our mind - some alteration of our minds state. (2) Dont think that in these changes of state the previous ideas are entirely wiped out. In fact, the ideas of things that we are not now actually thinking about are in our mind now, as the figure of Hercules is in a lump of marble. In God, on the other hand, all ideas are always actually engaged in his thought: he must have not only an actually occurrent idea of absolute and infinite extension but also an idea of each shape - a shape being merely a modification of absolute extension [meaning that a things having a certain shape is just its being extended in a certain way]. (3) A final point: when we perceive colours or smells, all that we really perceive - all! - are shapes and of motions; but they are so numerous and so tiny that our mind in its present state cant clearly attend to each one separately, so that it doesnt notice that its perception is composed purely of perceptions of minute shapes and motions. This is like what happens when we perceive the colour green in a mixture of yellow powder and blue powder. All we are sensing is yellow and blue, finely mixed, but we dont notice this, and invent something new - the colour green - for ourselves.

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Copyright Jonathan Bennett

[Brackets] enclose editorial explanations. Small dots enclose material that has been added, but can be read as though it were part of the original text. Occasional bullets, and also indenting of passages that are not quotations, are meant as aids to grasping the structure of a sentence or a thought. First launched: September 2004

ContingencyBy G. W. Leibniz

In God existence is the same as essence; or - the same thing put differently - it is essential for God to exist. So God is a necessary being, a being who exists necessarily. Created things are contingent, i.e. their existence doesnt follow from their essence. Necessary truths are ones that can be demonstrated through an analysis of terms, so that they end up as identities. For example, square analyses into figure that is plane, closed, equilateral, and has four sides. Apply this analysis to the necessary truth

A square has four sidesand you get

A figure that is plane, closed, equilateral, and has four sides has four sides,which is an identity. Similarly, in algebra when in a correct equation you substitute values for the variables you get an identity. For example, in the equation

(x + y)2 = x2 + 2xy + y2if we put 2 for x and 3 for y we get

(2 + 3)2 = 22 + 2(23) + 32which comes to

25 = 4 + 12 + 9which comes to 25 = 25,which is an identity. Thus, necessary truths depend upon the principle of contradiction, which says that the denial of an identity is never true. Contingent truths cant be reduced to the principle of contradiction. If they could, they wouldnt be contingent, and everything would be necessary and nothing would be possible except what actually exists. Nevertheless, since we say that both God and creatures exist and that necessary propositions and some contingent ones are true, there must be a notion of existence and one of truth that can be applied both to what is contingent and what is necessary. What is common to every truth, in my view, is that one can always give a reason for a true proposition unless it is an identity. In necessary propositions the reason necessitates, whereas in contingent ones it inclines. Identical propositions are, as I have said, the rock-bottom reasons for all necessary truths; we dont have reasons why they are true. And it seems to be common to things that exist, whether necessarily or contingently, that there is more reason for their existing than there is for any others to exist in their place. Every true universal affirmative proposition, whether necessary or contingent, has some connection between subject and predicate. In identities this connection is self-evident; in other propositions it has to be brought out through the analysis of terms.

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This little-known fact reveals the distinction between necessary and contingent truths. It is hard to grasp unless one has some knowledge of mathematics, because it goes like this. When the analysis of a necessary proposition is continued far enough it arrives at an identical equation; thats what it is to demonstrate a truth with geometrical rigour. But the analysis of a contingent proposition continues to infinity, giving reasons (and reasons for the reasons (and reasons for those reasons . . .)), so that one never has a complete demonstration. There is always an underlying complete and final reason for the truth of the proposition, but only God completely grasps it, he being the only one who can whip through the infinite series in one stroke of the mind. [This paragraph expands Leibnizs compact formulation in ways that cant be flagged by dots. For more on incommensurables, see pages 4-5 of his Dialogue on human freedom.] I can illustrate this with a good example from geometry and numbers. In necessary propositions, as I have said, a continual analysis of the predicate and the subject can eventually get us to the point where we can see that the notion of the predicate is in the subject. For a numerical analogue of this, consider the process of getting an exact comparison between two numbers: we repeatedly divide each until we arrive at a common measure. For example, wanting to compare 24.219 with 12.558, we find that each can be divided by 3 then by 13 then by 23, giving us the more graspable relationship of 27 to 14. But that doesnt work with an incommensurable pair of numbers such as any whole number and 2: as Euclid has demonstrated, there is no fraction F (however tiny) such that (FF) = 2. We can work along a series of fractions, squaring as we go, and get ever nearer to 2, but it is mathematically impossible for us to end the series by finding a fraction whose square exactly equals 2. Still, there is a proportion or relation between (say) 3 and 2; we cant express it exactly in terms of fractions, but we know that it exists: 3 is a certain determinate definite amount larger than 2. I offer this as analogous to the situation with contingent truths: in them there is a connection between the terms - i.e. there is truth - even if that truth cant be reduced to the principle of contradiction or necessity through an analysis into identities. Here are two questions that can be asked about the necessity of certain propositions. Is this proposition:

God chooses the bestnecessary? Or is it one - indeed, the first - of his free decrees? Again, is this proposition:

Whatever exists, there is a greater reason for it to exist than for it not to existnecessary? I answer that the former proposition is not necessary: God always chooses the best because he decrees that thats what hell do. It follows that the latter proposition is not necessary either: there is always a greater reason for the existence of an actual thing than for any possible rival to it, but only because God has freely decided always to choose the best. It is certain that there is a connection between subject and predicate in every truth. So the truth of Adam, who sins, exists requires that the possible notion of Adam, who sins involves something by virtue of which he is said to exist. It seems that we must concede that God always acts wisely, i.e. in such a way that anyone who knew his reasons would know and worship his supreme justice, goodness, and wisdom. And it seems that God never acts in a certain way just because it pleases him to act in this way, unless there is a good reason why it is pleasing. Thus, something may be done at Gods pleasure (as we say), but that is never an alternative to its being done for a reason. Since we cant know the true formal reason for the existence of any particular thing, because that would involve an infinite series of reasons, we have to settle for knowing contingent truths a posteriori, i.e. through experience. But we must at the same time hold the general principle,

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implanted by God in our minds and confirmed by both reason and experience, that nothing happens without a reason, as well as the principle of opposites, that of rival possibilities the one for which there is more reason always happens. (I said confirmed by experience, but treat that cautiously. I meant only that experience confirms the principle to the extent that we can discover reasons through experience.) And just as God decreed that he would always act in accordance with true reasons of wisdom, so too he created rational creatures in such a way that they act in accordance with prevailing or inclining reasons - reasons that are true or, failing that, seem to them to be true. Unless there were such a principle as this one about reasons, there would be no principle of truth in contingent things, because to them the principle of contradiction is certainly irrelevant. Not all possibles come to exist - we have to accept that, because if it were false you couldnt think up any story that wasnt actually true somewhere at some time! Anyway, it doesnt seem possible for all possible things to exist, because they would get in one anothers way. There are, in fact, infinitely many series of possible things, no one of which can be contained within any other, because each of them is complete. From the following two principles, the others follow:

1. Whatever God does bears the mark of perfection or wisdom.2. Not every possible thing comes to exist.

To these one can add:3. In every true universal affirmative proposition the predicate is in the subject, i.e. there is a connection between predicate and subject.

[In this next paragraph, Leibniz wrote of a propositions existing, apparently meaning its being true.] Assuming that this proposition:

The proposition P that has the greater reason for being true is trueis necessary, we must see whether it then follows that P itself is necessary. It isnt. If by definition a necessary proposition is one whose truth can be demonstrated with geometrical rigour, then indeed it could be the case that these two propositions are demonstrable and thus necessary:

Every truth and only a truth has greater reason.God always acts with the highest wisdom.

But from these one cant demonstrate any proposition of the form Contingent proposition P has greater reason for being true than has contingent proposition not-P or of the form Contingent proposition P is in conformity with divine wisdom. So it doesnt follow from the above two displayed propositions that any contingent proposition P is necessary. Thus, although one can concede that it is necessary for God to choose the best, or that the best is necessary, it doesnt follow that P is necessary, where P is something that has been chosen; for there is no demonstration that P is the best. This can be put in terms of the technical distinction between necessity of the consequence and necessity of the consequent - that is, between P necessarily follows from Q and P is itself necessary. Assuming that the best is necessarily chosen, we have

From P is the best it follows necessarily that P is true,but we do not have

Necessarily P is true,because we have no demonstration that P is the best. Though I have been exploring the implications of the thesis that necessarily God always chooses the best, I dont assert it. I say only that it seems safer to attribute to God the most perfect way possible of operating. When it comes to creatures, one cant be as sure as we can

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with God that they will act in accordance with even the most obvious reason; with respect to creatures, this proposition - that they will always so act - cant be demonstrated.

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Copyright Jonathan Bennett

[Brackets] enclose editorial explanations. Small dots enclose material that has been added, but can be read as though it were part of the original text. Occasional bullets, and also indenting of passages that are not quotations, are meant as aids to grasping the structure of a sentence or a thought. Bold type is used where Leibniz used italics, apparently for emphasis.First launched: September 2004

First TruthsBy G. W. Leibniz

First truths are the ones that assert something of itself or deny something of its opposite. For example,

A is AA is not not-AIf it is true that A is B, then it is false that A isnt B (i.e. false that A is not-B)Everything is as it isEverything is similar or equal to itselfNothing is bigger or smaller than itself

and others of this sort. Although they may have a rank-ordering among themselves, they can all be lumped together under the label identities. Now, all other truths are reducible to first ones through definitions, that is, by resolving notions into their simpler components. Doing that is giving an a priori proof - a proof that doesnt depend on experience. From among the axioms that are accepted by mathematicians and by everyone else, I choose as an example this:

A whole is bigger than its part, orA part is smaller than the whole.

This is easily demonstrated from the definition of smaller or bigger together with the basic axiom, that is, the axiom of identity. Here is a definition of smaller than:

For x to be smaller than y is for x to be equal to a part of y (which is bigger).This is easy to grasp, and it fits with how people in general go about comparing the sizes of things: they take away from the bigger thing something equal to the smaller one, and find something left over. With that definition in hand, here is an argument of the sort I have described:

1. Everything is equal to itself .................................................... (axiom of identity)2. A part is equal to itself ......................................................................... (from 1)3. A part is equal to a part of the whole .................................................... (from 2)4. A part is smaller than the whole ....... (from 3 by the definition of smaller than).

Because all truths follow from first truths with the help of definitions, it follows that in any true proposition the predicate or consequent is always in the subject or antecedent. It is just this - as Aristotle observes - that constitutes the nature of truth in general, or the true-making connection between the terms of a statement. In identities the connection of the predicate with the subject (its inclusion in the subject) is explicit; in all other true propositions it is implicit, and has to be shown through the analysis of notions; a priori demonstration rests on this.

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This is true for every affirmative truth - universal or particular, necessary or contingent - and it holds when the predicate is relational as well as when it isnt. And a wonderful secret lies hidden in this, a secret that contains the nature of contingency, i.e. the essential difference between necessary and contingent truths, and removes the difficulties concerning the necessity - and thus the inevitability - of even those things that are free. These considerations have been regarded as too simple and straightforward to merit much attention; but they do deserve attention because many things of great importance follow from them. One of their direct consequences is the received axiom Nothing is without a reason, or There is no effect without a cause.If that axiom were false, there would be a truth that couldnt be proved a priori, that is, a truth that couldnt be resolved into identities, contrary to the nature of truth, which is always an explicit or implicit identity. Thus, if the axiom were false, my account of truth would be false; which is why I say that (the truth of) the axiom follows from (the truth of) my account. It also follows that when there is a perfect balance or symmetry in a physical set-up there will also be a balance or symmetry in what follows from it. Stated more abstractly: when there is symmetry in what is given, there will be symmetry in what is unknown. This is because any reason for an asymmetry in the unknown must derive from the givens., and in the case as stated there is no such reason. An example of this is Archimedes postulate at the beginning of his book on statics, that if there are equal weights on both sides of a balance with equal arms, everything is in equilibrium. There is even a reason for eternal truths. Suppose that the world has existed from eternity, and that it contains nothing but little spheres; for such a world we would still have to explain why it contained little spheres rather than cubes. From these considerations it also follows that In nature there cant be two individual things that differ in number alone,i.e. that dont differ in any of their qualities, and differ only in being two things rather than one. For where there are two things it must be possible to explain why they are different - why they are two, why it is that x is not y - and for that explanation we must look to qualitative differences between the things. St. Thomas said that unembodied minds never differ by number alone - that is, no two of them are qualitatively exactly alike; and the same must also be said of other things, for we we never find two eggs or two leaves or two blades of grass that are exactly alike. So exact likeness is found only in notions that are incomplete and abstract. In that context things are considered only in a certain respect, not in every way - as, for example, when we consider shapes alone, ignoring the matter that has the shape. And so it is justifiable to consider two perfectly alike triangles in geometry, even though two perfectly alike triangular material things are not found anywhere. Gold and other metals, also salts and many liquids, are taken to be homogeneous, which implies that two portions of gold could be qualitatively exactly alike. This way of thinking and talking is all right if it is understood as referring only to differences that our senses can detect; but really none of these substances is strictly homogeneous. [Leibniz is about to use the phrase purely extrinsic denomination. This means purely relational property, meaning a relational property that isnt grounded in any non-relational property. It might seem to us that a things spatial relations to other things constitute such an extrinsic denomination: the thing could be moved without being in anyway altered in itself. That is what Leibniz is going to deny. The word denomination (and Leibnizs corresponding Latin) mark the fact that he wavers between making this a point about the properties and relations a

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thing can have, and the linguistic expressions that can be used in talking about a thing. Although basically an external denomination is meant to be a relational property, Leibniz sometimes writes as though it were a relational predicate.] It also follows that

There are no purely extrinsic denominations- that is, denominations having absolutely no foundation in the denominated thing. For the notion of the denominated subject must contain the notion of the predicate; and, to repeat what I said at the top of page 2, this applies to relational predicates as well as qualitative ones, i.e. it applies to extrinsic as well as to intrinsic denominations. So whenever any denomination of a thing is changed, there must be an alteration in the thing itself. The complete notion of an individual substance contains all its predicates - past, present, and future. If a substance will have a certain predicate, it is true now that it will, and so that predicate is contained in the notion of the thing. Thus, everything that will happen to Peter or Judas - necessary events and also free ones - is contained in the perfect individual notion of Peter or Judas, and is seen there by God. [The next two sentences expand a condensed clause of Leibnizs.] To grasp how the concept of the complete notion of Judas is being used here, think of it as the complete total utterly detailed specifications for Judas, viewed as a possibility without any thought of whether God has chosen to make the possibility actual. That is the notion that God employed when deciding to make Judas actual: he pointed to the possibility Judas and said Let him come into existence, which means that he pointed to that complete notion and said Let that be actualized. This makes it obvious that out of infinitely many possible individuals God selected the ones he thought would fit best with the supreme and hidden ends of his wisdom. Properly speaking, he didnt decide that

Peter would sinor that

Judas would be damned.All he decreed was that two possible notions should be actualized - the notion of

Peter, who would certainly sin (but freely, not necessarily)and the notion of

Judas, who would suffer damnation - which is to decree that those two individuals should come into existence rather than other possible things. Dont think that Peters eventual salvation occurs without the help of Gods grace, just because it is contained in the eternal possible notion of Peter. For what that complete notion of Peter contains is the predicate achieves salvation with the help of Gods grace. [Leibniz says, puzzlingly, that the complete notion contains this predicate sub notione possibilitatis = under the notion of possibility. That seems to say where in the complete notion the predicate will be found - Look it up in the file labelled Possibility, as it were - but that cant be right.] Every individual substance contains in its complete notion the entire universe and everything that exists in it - past, present, and future. [The next sentence is stronger than what Leibniz wrote, but it seems to express what he meant.] That is because: for any given things x and y, there is a true proposition about how x relates to y, if only a comparison between them. And there is no purely extrinsic denomination, which implies that every relational truth reflects non-relational truths about the related things. I have shown this in many ways, all in harmony with one another.

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Indeed, all individual created substances are different expressions of the same universe and of the same universal cause, namely God. But the expressions vary in perfection, as do different pictures of the same town drawn or painted from different points of view. Every individual created substance exercises physical action and passion on all the others. Any change made in one substance leads to corresponding changes in all the others, because the change in the one makes a difference to the relational properties of the others. For example, a pebble on Mars becomes colder, so that you move from having the property

...has spatial relation R to a pebble that is at 5Cto having the property

...has spatial relation R to a pebble that is at 2C;and, because there are no purely extrinsic denominations, that change in your relational properties will be backed by a change in your intrinsic properties. This fits with our experience of nature. In a bowl filled with liquid, a movement of the liquid in the middle is passed on out to the edges, becoming harder and harder to detect the further it gets from the centre but never being wiped out altogether. Well, the whole universe is just such a bowl! Strictly speaking, one can say that no created substance exercises a metaphysical action or influence on anything else. [Leibniz is saying that no real causal force or energy passes from one substance to another. Influence here translates the Latin influxus [= in-flow], which reflects one view about what would have to happen for one substance to act on another: according to this view, when the hot poker heats the water, some of its heat literally passes from one to the other; when a man falls against a wall and knocks it down, some his motion passes to the wall. The basic idea is that of an accident - a property-instance - travelling from one substance to another. The pokers heat is an accident in this sense; it is to be distinguished from the poker (an individual substance) and from heat (a universal property); it is the-present-heat-of-this-particular-poker, an individualized property. Leibniz is sceptical about the transfer of accidents from one thing to another, but since he thinks that substances dont act on one another, he doesnt mind implying that if they did act on one another it would have to be by the transfer of accidents.] For one thing, there is no explanation of how something -an accident - could pass from one thing into the substance of another; but Ill let that pass. I have already shown that there is no work for inter-substance causation to do, because all a things states follow from its own complete notion. What we call causes are, speaking with metaphysical strictness, only concurrent requirements. This too is illustrated by our experience of nature. For bodies really rebound from others through the force of their own elasticity, and not through the force of other things, even if a body other than x is required in order for xs elasticity to be able to act. Assuming that soul and body are distinct, from the foregoing we can explain their union, without appealing to the popular but unintelligible idea of something in-flowing from one to the other, and without the hypothesis occasional causes, which appeals to God as a kind of puppet-master. [Leibniz says Deus ex machina - a God who comes on-stage by being winched down from the ceiling of the theatre. The phrase occasional causes refers to the view that minds cant literally act on bodies, and that when I will to raise my arm that act of my mind is the prompt or occasion for God to raise my arm.] For Gods wisdom and workmanship enabled him to set up the soul and the body, at the outset, in such a way that from the first constitution or notion of each of them everything that happens in it through itself corresponds perfectly to everything that happens in the other through itself, just as if something - some accident - passed from one to the other. This hypothesis of mine (which I call the hypothesis of

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concomitance) is true for all substances in the whole universe, but it cant be sensed in all of them as it can in the case of the soul and the body. There is no vacuum. For if there were empty space, two different parts of it could be perfectly similar and congruent and indistinguishable from one another. Thus, they would differ in number alone - differ in being two, but not in any other way - which is absurd. One can also prove that time is not a thing, in the same way as I just did for space, namely arguing that if time were a thing there could be stretches of empty time, i.e. time when nothing happens; and two parts of such empty time would be exactly alike, differing only in number, which is absurd. There is no atom, which means that any body could be split. In fact, every body, however small, is actually subdivided. Because of that, each body, while it constantly changes because it is acted on by everything else in the universe in ways that make it alter, also preserves all the states that have been impressed on it in the past and contains in advance all that will be impressed on it in the future. You might object:

Your view that every body is affected by every other body, and that each body contains information about all its past and all its future states, could be true even if there were atoms. It could be that other bodies affect an atom by making it move in certain ways and by changing its shape, and these are effects that the atom can receive as a whole, without being divided.

I reply that not only must there be effects produced in an atom from all the impacts of the universe upon it, but also conversely the state of the whole universe must be inferable from the states of the atom - the cause must be inferable from the effect. However, any given motion of an atom and any given shape could have come about through different impacts, so there is no way to infer from the present shape and motion of the atom what effects have been had upon it. And there is a different objection to atoms, independent of my metaphysics, namely the fact that one couldnt explain why bodies of a certain smallness couldnt be further divided - that is, there couldnt be an explanations of why there are any atoms. From this it follows that every particle in the universe contains a world of an infinity of creatures. However, the continuum is not divided into points, because points are not parts but boundaries; nor is it divided in all possible ways, because the contained creatures are not all separately there. Its just that a series of divisions could go on ad infinitum separating some from others at each stage. But no such sequence separates out all the parts, all the contained creatures, because every division leaves some of them clumped together - just as someone who bisects a line leaves clumped together some parts of it that would be separated if the line were trisected. There is no determinate shape in actual things, for no determinate shape can be appropriate for infinitely many effects. So neither a circle, nor an ellipse, nor any other definable line exists except in the intellect; lines dont exist until they are drawn, and parts dont exist until they are separated off. Extension and motion, are not substances, but true phenomena (like rainbows and reflections). The same holds for bodies, to the extent that there is nothing to them but extension and motion. For there are no shapes in reality, and if we think about bodies purely as extended, each of them is not one substance but many. Something unextended is required for the substance of bodies. Without that there would be no source for the reality of phenomena or for true unity. There is always a plurality of bodies, never just one (so that really there isnt a plurality either, because a many must consist of many

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ones). Cordemoy used a similar line of thought as an argument for the existence of atoms. But since I have ruled out atoms, all that remains as a source of unity is something unextended, analogous to the soul, which they once called form or species. Corporeal substance cant come into existence except through creation or go out of existence except through annihilation, because once a corporeal substance exists it will last for ever, since there is no reason for it not to do so. Any body may come apart - its parts may come to be scattered - but this has nothing in common with its going out of existence. Therefore, animate things dont come into or go out of existence, but are only transformed.

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Copyright Jonathan Bennett[Brackets] enclose editorial explanations. Small dots enclose material that has been added, but can be read as though it were part of the original text. Occasional bullets, and also indenting of passages that are not quotations, are meant as aids to grasping the structure of a sentence or a thought. The division into sections is Leibnizs; the division of some sections into paragraphs is not. Leibniz wrote brief summaries of the 37 sections of this work, but did not include them in the work itself. Some editors preface each section with its summary, but that interrupts the flow. In this version the summaries are given at the end.First launched: July 2004 Amended: November 2004

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Discourse on Metaphysicsby G.W. Leibniz

1. The most widely accepted and sharpest notion of God that we have can be expressed like this: God is an absolutely perfect being;

but though this is widely accepted, its consequences havent been well enough thought out. As a start on exploring them, let us note that there are various completely different ways of being perfect, and that God has them all, each in the highest degree. We also need to understand what a perfection is. Here is one pretty good indicator: a property is not a perfection unless there is a highest degree of it; so number and shape are not perfections, because there cannot possibly be a largest number or a largest thing of a given shape - that is, a largest triangle, or square, or the like. But there is nothing impossible about the greatest knowledge or about omnipotence [here = greatest possible power]. So power and knowledge are perfections, and God has them in unlimited form. It follows that the actions of God, who is supremely - indeed infinitely - wise, are completely perfect. This is not just metaphysical perfection, but also the moral kind. His moral perfection, so far as it concerns us, amounts to this: the more we come to know and understand Gods works, the more inclined we shall be to find them excellent, and to give us everything we could have wished.2. Some people - including Descartes - hold that there are no rules of goodness and perfection in the nature of things, or in Gods ideas of them, and that in calling the things God made good all we mean is that God made them. I am far from agreeing with this. If it were right, then God would not have needed after the creation to see that they were good, as Holy Scripture says he did, because he already knew that the things in question were his work. In saying this - And God saw everything that he had made, and, behold, it was very good (Genesis 1:31) - Scripture treats God as like a man; but its purpose in doing this appears to be to get across the point that a things excellence can be seen by looking just at the thing itself, without reference to the entirely external fact about what caused it. Reinforcing that point is this one: the works must bear the imprint of the workman, because we can learn who he was just by inspecting them. I have to say that the contrary opinion strikes me as very dangerous, and as coming close to the view of the Spinozists that the beauty of the universe, and the goodness we attribute to Gods works, are merely the illusions of people who conceive God as being like themselves. Furthermore, if you say as Descartes did that things are good not because they match up to objective standards of goodness, but only because God chose them, you will unthinkingly destroy all Gods love and all

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his glory. For why praise him for what he has done, if he would be equally praiseworthy for doing just the opposite? Where will his justice and wisdom be,

if there is only a kind of despotic power, if reasons place is taken by will, and if justice is tyrannically defined as what best pleases the most powerful?

[Leibniz here relies on his view that it is through reason that we learn what things are good.] And another point: it seems that any act of the will presupposes some reason for it - a reason which naturally precedes the act - so that Gods choices must come from his reasons for them, which involve his knowledge of what would be good; so they cant be the sources of the goodness of things. That is why I find it weird when Descartes says that the eternal truths of metaphysics and geometry, and therefore also the rules of goodness, justice, and perfection, are brought about by Gods will. Against this, they seem to me to be results of his understanding, and no more to depend on his will than his intrinsic nature does.3. Nor could I ever accept the view of some recent philosophers who have the nerve to maintain that Gods creation is not utterly perfect, and that he could have acted much better. This opinion, it seems to me, has consequences that are completely contrary to the glory of God. Just as a lesser evil contains an element of good, so a lesser good contains an element of evil. To act with fewer perfections than one could have done is to act imperfectly; showing an architect that he could have done his work better is finding fault with it. Furthermore, this opinion goes against holy scriptures assurance of the goodness of Gods works. That goodness cant consist simply in the fact that the works could have been worse; and here is why. Whatever Gods work was like, it would always have been good in comparison with some possibilities, because there is no limit to how bad things could be. But being praiseworthy in this way is hardly being praiseworthy at all! I believe one could find countless passages in the holy scriptures and the writings of the holy fathers that support my opinion, and hardly any to support the modern view to which I have referred - a view that I think was never heard of in ancient times. It has arisen merely because we are not well enough acquainted with the general harmony of the universe and of the hidden reasons for Gods conduct; and that makes us recklessly judge that many things could have been improved. Furthermore, these moderns argue - subtly but not soundly - from the false premiss that however perfect a thing is, there is always something still more perfect. They also think that their view provides for Gods freedom, through the idea that if God is free, it must be up to him whether he acts perfectly or not; but really it is the highest freedom to act perfectly, in accordance with sovereign reason. For the view that God sometimes does something without having any reason for his choice, besides seeming to be impossible, is hardly compatible with his glory. Suppose that God, facing a choice between A and B, opts for A without having any reason for preferring it to B. I see nothing to praise in that, because all praise should be grounded in some reason, and in this case we have stipulated that there is none. By contrast, I hold that God does nothing for which he does not deserve to be glorified.4. The love that we owe to God, above all things, is based (I think) on our grasp of the great truth that God always acts in the most perfect and desirable way possible. For a lover looks for satisfaction in the happiness or perfection of the loved one and of his actions. Friendship is wanting the same things and not-wanting the same things. And I think it will be hard to love God properly without being disposed to want what he wants, even if one had the power to get something different. Indeed, those who are not satisfied with what God does strike me as being

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like malcontent subjects whose set of mind is not much different from a rebels. These principles lead me to maintain that loving God requires a certain attitude to everything that happens to us through his will: not merely accepting it patiently because one has no alternative, but being truly satisfied with it. I am saying this about the past; for we shouldnt be quietists about the future, stupidly waiting with folded arms for what God will do, as in the fallacy of the argument for idleness (as the ancients called it). So far as we can judge what God wants, in a general way, we should act in accordance with that, doing our very best to contribute to the general good, and in particular to adorning and perfecting the things that concern us - what is close to us, within reach (so to speak). The outcome may show that in a particular instance God didnt want our good will to have its effect, but it doesnt follow that he didnt want us to do what we did. On the contrary, as he is the best of masters, he never asks more than the right intention, and it is up to him to know when and where good intentions should succeed.5. So it is enough to be sure of this about God: that he does everything for the best, and that nothing can harm those who love him. But to know in detail his reasons for ordering the universe as he has, allowing sin, and granting his saving grace in one way rather than another, is beyond the power of a finite mind, especially one that has not yet attained the delight of seeing God. Still, some general remarks can be made about how God goes about governing things. Thus, we can liken someone who acts perfectly to an expert geometer who knows how to find the best construction for a problem; to a good architect who exploits the location and the budget for his building to the best advantage, not allowing anything nasty, or less beautiful than it could be; to a good head of a household, who manages his property so that no ground is left uncultivated or barren; to a clever special-effects technician in the theatre, who produces his effect by the least awkward means that can be found; or to a learned author, who gets the largest amount of subject-matter into the smallest space he can. Now, minds are the most perfect of all things, occupying the least space and thus providing the least hindrance to one another because they dont take up space at all; and their perfections are virtues. That is why we should be sure that the happiness of minds is Gods principal aim, which he carries out as far as the general harmony will permit. Ill say more about this later. The simplicity of Gods ways relates to the means he adopts, while their variety, richness or abundance relate to ends or effects. These should be in balance with one another, as the money for putting up a building has to be balanced against its desired size and beauty. Admittedly, whatever God does costs him nothing - even less than it costs a philosopher or scientist to invent theories out of which to build his imaginary world - for God can bring a real world into existence merely by decreeing it. But in the exercise of wisdom by God or a scientist there is something analogous to the cost of a building, namely the number of independent decrees or theories that are involved. For Gods creative activity to be economical is for it to involve very few separate decrees; for a scientific theory to be economical in its means is for it to have very few basic principles or axioms. Reason requires that multiplicity of hypotheses or principles be avoided, rather as the simplest system is always preferred in astronomy.6. Gods wishes or actions are usually divided into the ordinary and the extraordinary. But we should bear in mind that God does nothing that isnt orderly. When we take something to be out of the ordinary, we are thinking of some particular order that holds among created things. We do not, or ought not to, mean that the thing is absolutely extraordinary or disordered, in the sense of being outside every order; because there is a universal order to which everything conforms.

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Indeed, not only does nothing absolutely irregular ever happen in the world, but we cannot even feign [= tell a consistent fictional story about] such a thing. Suppose that someone haphazardly draws points on a page, like people who practise the ridiculous art of fortune-telling through geometrical figures. I say that it is possible to find a single formula that generates a geometrical line passing through all those points in the order in which they were drawn. And if someone drew a continuous line which was now straight, now circular, now of some other kind, it would be possible to find a notion or rule or equation that would generate it. The contours of anyones face could be traced by a single geometrical line governed by a formula. But when a rule is very complex, what fits it is seen as irregular. So one can say that no matter how God had created the world, it would have been regular and in some general order. But God chose the most perfect order, that is, the order that is at once simplest in general rules and richest in phenomena - as would be a geometrical line whose construction was easy yet whose properties and effects were very admirable and very far-reaching. These comparisons help me to sketch some imperfect picture of divine wisdom, and to say something that might raise our minds to some sort of conception, at least, of what cannot be adequately expressed. But I dont claim that they explain this great mystery of creation on which the whole universe depends.7. Now, because nothing can happen that isnt orderly, miracles can be said to be as orderly as natural events. The latter are called natural because they conform to certain subordinate rules - ones that are not as general and basic as Gods fundamental creative decrees - which we call the nature of things. This Nature is only a way in which God customarily goes about things, and he can give it up if he has a reason for doing so - a reason that is stronger than the one that moved him to make use of these subordinate maxims in the first place. General acts of the will are distinguished from particular ones. Using one version of this distinction, we can say that God does everything according to his most general will, which conforms to the most perfect order that he has chosen; but that he also has particular wills, which are exceptions (not to the most general of Gods laws, which regulates the whole order of the universe, and to which there are no exceptions, but) to the subordinate maxims I have mentioned, the ones that constitute Nature. Any object of Gods particular will is something he can be said to want. But when it comes to the objects of his general will - such as are actions of created things (especially rational ones) which God chooses to allow - we cannot say that God wants them all, and must make a distinction. (1) If the action is intrinsically good, we can say that God wants it, and sometimes commands it, even if it doesnt happen. (2) But an action may be intrinsically bad, and only incidentally good because later events - especially ones involving punishment and reparations - correct its wickedness and make up for the bad with some to spare, so that eventually there is more perfection overall than if this bad thing had not been done. In a case like that we must say that God allows the action but not that he wants it, even though he goes along with it because of the laws of Nature that he has established and because he sees how to derive from it a greater good.8. It is quite hard to distinguish Gods actions from those of created things. Some believe that God does everything, and others suppose that he only conserves the force he has given to created things, allowing them to decide in what directions the force shall be exercised. We shall see later on what truth there is in each of these. Now since actions and passions properly belong to individual substances (when there is an action there is something, some subject, that acts), I have to explain what such a substance is. This much is certain: when several predicates are attributed to

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the same subject, and this subject is not attributed to any other, it is called an individual substance. For example, we call John a substance because we can attribute to him honesty, intelligence, and so on; but we dont call his honesty a substance because, although we can attribute predicates to it (His honesty is charming, and surprising) we can attribute it to something else, namely to John. In contrast, John cannot be attributed to anything else. But that explanation is only nominal - all it does is to relate our calling a thing a substance to other facts concerning what we say about it. Beyond that, we need to think about what it is for something to be truly attributed to a certain subject - e.g. what it is for honesty to be a property of John. Now it is certain that all true predication is founded in the nature of things, and when a proposition is not identical, that is, when the predicate is not explicitly included in the subject as in The man who governs Somalia governs Somalia, it must be implicitly included in it. This is what philosophers call in-esse [being-in] when they say that the predicate is in the subject. So the notion of the subject term must always include that of the predicate, so that anyone who understood the subject notion perfectly would also judge that the predicate belongs to it. We can therefore say that the nature of an individual substance or of a complete being is to have a notion so complete that it is sufficient to include, and to allow the deduction of, all the predicates of the subject to which that notion is attributed. An accident, on the other hand, is a being whose notion doesnt involve everything that can be attributed to the subject to which that notion is attributed. Thus Alexander the Greats kinghood is an abstraction from the subject, leaving out much detail, and so is not determinate enough to pick out an individual, and doesnt involve the other qualities of Alexander or everything that the notion of that prince includes; whereas God, who sees the individual notion or thisness of Alexander, sees in it at the same time the basis and the reason for all the predicates that can truly be said to belong to him, such as for example that he would conquer Darius and Porus, even to the extent of knowing a priori (and not by experience) whether he died a natural death or by poison - which we can know only from history. Furthermore, if we bear in mind the interconnectedness of things, we can say that Alexanders soul contains for all time traces of everything that did and signs of everything that will happen to him - and even marks of everything that happens in the universe, although it is only God who can recognise them all.9. Several considerable paradoxes follow from this, amongst others that it is never true that two substances are entirely alike, differing only in being two rather than one. It also follows that a substance cannot begin except by creation, nor come to an end except by annihilation; and because one substance cant be destroyed by being split up, or brought into existence by the assembling of parts, in the natural course of events the number of substances remains the same, although substances are often transformed. Moreover, each substance is like a whole world, and like a mirror of God, or indeed of the whole universe, which each substance expresses in its own fashion - rather as the same town looks different according to the position from which it is viewed. In a way, then, the universe is multiplied as many times as there are substances, and in the same way the glory of God is magnified by so many quite different representations of his work. It can even be said that each substance carries within it, in a certain way, the imprint of Gods infinite wisdom and omnipotence, and imitates him as far as it is capable of doing so. For it expresses (though confusedly) everything that happens in the universe - past, present, and future - and this is a little like infinite perception or knowledge. And as all the other substances express this one in their turn, and adapt themselves to it - that is, they are as they are because it is as it is - it can be said to have power over all the others, imitating the creators omnipotence.

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10. The ancients, as well as many able teachers of theology and philosophy a few centuries ago - men accustomed to deep thought, and admirable in their holiness - seem to have had some knowledge of the things I have been saying, and to have been led by that to introduce and defend substantial forms. These are much sneered at today, but they are not so far from the truth, nor so ridiculous, as the common run of our new philosophers suppose. I agree that these forms have no work to do in explaining particular events, and thus no role in the details of physics. That is where our scholastics [= mediaeval Christian philosophers influence by Aristotle, Thomas Aquinas being the most famous example] went wrong, and the physicians of the past followed them into error: they thought they could invoke forms and qualities to explain the properties of bodies, without bothering to find out how the bodies worked - like settling for saying that a clocks form gives it a time-indicative quality, without considering what all that consists in - that is, without considering what mechanisms are involved. Actually, that might be all the clocks owner needs to know, if he leaves the care of it to someone else. But this misuse and consequent failure of forms shouldnt make us reject them. Metaphysics needs a knowledge of them, so much so that without that knowledge - I maintain - we couldnt properly grasp the first principles of metaphysics, and couldnt raise our minds to the knowledge of immaterial natures and the wonders of God. However, important truths need not be taken into account everywhere. A geometer need not worry about the famous labyrinth of the composition of the continuum [that is, the puzzles that arise from the idea that a line has no smallest parts]; and the huge difficulties to be found in trying to reconcile free will with Gods providence need not trouble a moral philosopher, still less a lawyer or politician; for the geometer can do all his proofs, and the politician can complete his plans, without getting into those debates, necessary and important though they are in philosophy and theology. In the same way a physicist can explain his experiments - sometimes using simpler experiments he has already made, sometimes proofs in geometry and mechanics - without needing to bring in general considerations belonging to another sphere. And if he does go outside his sphere, and appeal to Gods co-operation, or to some soul or spiritual force or other thing of that kind, he is talking nonsense, just as much as someone who drags large-scale reflections about the nature of destiny and our freedom into an important practical deliberation. Indeed men often enough unthinkingly make this mistake, when they let the idea of what is fated to happen tangle their thoughts, and sometimes are even deterred by that idea from some good decision or some important precaution.11. I know I am putting forward a considerable paradox in claiming to rehabilitate the ancient philosophy, in a way, and to re-admit substantial forms when they have been all but banished. But perhaps you wont just brush me off if you realize that I have thought a lot about the modern philosophy, that I have spent much time on experiments in physics and proofs in geometry, and that for a long time I was sure that these entities [substantial forms] are futile. Eventually I had to take them up again - against my will, as though by force - after my own researches made me recognize that thinkers these days do less than justice to St. Thomas and to other great men of his time, and that the views of scholastic philosophers and theologians contain much more good stuff than people suppose, provided they are used relevantly and in their right place. I am convinced, indeed, that if some exact and thoughtful mind took the trouble to clarify and digest their thoughts, in the way the analytic geometers do, he would find them to be a treasure-house of important and completely demonstrable truths.

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12. Picking up again the thread of our reflections, I believe that anyone who thinks about the nature of substance, as I have explained it above, will find that there is more to the nature of body than extension (that is, size, shape, and motion), and that we cant avoid attributing to body something comparable with a soul, something commonly called substantial form - though it has no effect on particular events, any more than do the souls of animals, if they have souls. It can be proved, indeed, that the notion of size-shape-movement is less sharp and clear than we imagine, and that it includes an element that belongs to imagination and the senses, as do - to a much greater degree - colour, heat, and other such qualities, which we can doubt are really there in the nature of external things. That is why qualities of such kinds could never constitute the basic nature of any substance. Moreover, if there is nothing but size-shape-movement to make a body the thing that it is, then a body can never persist for more than a moment because bodies constantly gain and lose tiny bits of matter. However, the souls and substantial forms of bodies other than ours are quite different from our thinking souls. Only the latter know their own actions; and they dont naturally go out of existence, but last for ever and always retain the foundation of the knowledge of what they are. This is what makes them alone liable to punishment and reward, and what makes them citizens of the republic of the universe, of which God is the monarch. It also follows that all other creatures must serve them. I shall say more about that later.13. The foundations that I have laid down give rise to a big problem, which I must try to solve before moving on. I have said that the notion of an individual substance involves, once and for all, everything that can ever happen to it; and that by looking into that notion one can see in it everything that will ever be truly sayable of the substance, just as we can see in the nature of a circle all the properties that are deducible from it. But this seems to destroy the difference between contingent and necessary truths, to rule out human freedom, and to imply that all the events in the world - including our actions - are governed by an absolute fate. To this I reply that we have to distinguish what is certain from what is necessary. Everyone agrees that future contingents are assured, because God foresees them; but we dont infer from this that they are necessary. You may say:

But if some conclusion can be infallibly deduced from a definition or notion, it is necessary. And you contend that everything that happens to a person is already included implicitly in his nature or notion, just as a circles properties are contained in its circle; so you are still in trouble.

I shall now resolve this problem completely. To that end, I remark that there are two kinds of connection or following-from. One is absolutely necessary, and its contrary implies a contradiction; such deduction pertains to eternal truths, such as those of geometry. The other is necessary not absolutely, but only ex hypothesi, and, so to speak, accidentally. It doesnt bring us to It is necessary that P, but only to Given Q, it follows necessarily that P. This is contingent in itself, and its contrary does not imply a contradiction. This second kind of connection is based not purely on ideas and on Gods understanding alone, but also on his free decrees, and on the history of the universe. Let us take an example. Since Julius Caesar will become the permanent dictator and master of the Republic, and will overthrow the freedom of the Romans, these actions are comprised in his perfect or complete notion; because we are assuming that it is the nature of such a perfect notion of a subject to include everything, so that the predicate can be contained in the subject. It could be put like this: it is not because of that notion or idea that Caesar will perform the action, since that notion applies to him only because God knows everything. You may

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object: But his nature or form corresponds to that notion, and since God has imposed this character or nature or form on him, from then on he must necessarily act in accordance with it. I could reply to that by bringing up the case of future contingents: they have as yet no reality except in Gods understanding and will, yet since God has given them that form in advance, they will nevertheless have to correspond to it. So I could counter-attack by challenging you to choose between two options, each of which you will find uncomfortable: either (1) say that future contingents are really necessary, and not contingent, or (2) say that God does not know them in advance. But I prefer to resolve difficulties rather than excusing them by likening them to other similar ones; and what I am about to say will throw light on both of the above problems. Applying now the distinction between different kinds of connection, I say that whatever happens in accordance with its antecedents is assured but is not necessary; for someone to do the contrary of such an assured outcome is not impossible in itself, although it is impossible ex hypothesi - that is, impossible given what has gone before. For if you were capable of carrying through the whole demonstration proving that this subject (Caesar) is connected with this predicate (his successful power-grabbing enterprise), this would involve you in showing that Caesars dictatorship had its foundation in his notion or nature, that a reason can be found the