Freedom and Possibility

1 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. Every four-point ellipsis.... indicates the omission of a brief passage that seems to present more difficulty than it is worth. First launched: September 2004 Last amended: July 2007 Freedom and Possibility By 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 won t act if equal contrary conditions obtain. That s one way for a person to not-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 necessarygiven-what-god-wills.) So the principle of contradiction is the basis for all truths about possibilities or essences, and all truths about a thing s 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 don t, shouldn t be sought in x s definition alone. If x s definition did explain its existence, its nonexistence would imply a contradiction; and those other things wouldn t be possible, contrary to our hypothesis. For the reason why x exists and those others don t, 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. My over-riding thought here is a notion of possibility and necessity according to which some things are not necessary and don t 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) doesn t take away our freedom. This also shows what distinguishes God s 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 can t be demonstrated from that definition that God makes whatever is most perfect, for there s nothing contradictory in the proposition that he doesn t. If there were, it wouldn t 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 let s 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, A s existing rather than B s 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 A s 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 God s 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 can t will voluntarily. That is, it can t 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 God s 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 doesn t will is not possible, and I deny that. For things remain possible, even if God doesn t select them. Given that God doesn t will x to exist, it is still possible for x to exist, because x s nature is such that x could exist if God were to will it to exist. You will object : But God can t will it to exist. 2

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. We ll 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 doesn t 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 doesn t imply a contradiction, even if it follows from the harmony of things that a pentagon can t 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 don t 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 can t be pictured.... But there is a great difference between (1) problems that are insoluble because a solution requires imaginary roots and (2) problems that are insoluble because of their absurdity. An example of (2): 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 (1): Find a number x such that x 2 + 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 wouldn t follow that the existence of an incommensurable line implies a contradiction, even if because of the principle of perfection God couldn t have made such a line. 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 3

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 doesn t will to exist. I deny that something that isn t going to exist is thereby impossible in itself. So the proposition What God doesn t will to exist doesn t 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 won t be. So why should we argue about circumstances that do not arise? I don t 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 God s loving himself. But (they hold) with respect to creatures God does not decide with necessity.... 4

1 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. Every four-point ellipsis.... indicates the omission of a brief passage that seems to present more difficulty than it is worth. First launched: September 2004 Last amended: July 2007 Meditations on Knowledge, Truth, and Ideas By 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 isn t 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 can t 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 Aristotle s 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 coin s 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. Vivid knowledge is either confused or clear. It is confused when I can t 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 can t explain what red is to a blind man; and we can t 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 Leibniz s 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 can t 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 [French for I don t know what ]. 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 can t 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 can t list the marks whereby you recognize those marks. For example, someone s knowledge of gold may be clear yet inadequate: he knows that heaviness, colour, solubility in aqua fortis etc. are the marks of gold, but he can t 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 don t 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 don t 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 don t always think about the nature of a side, or of equality, or of thousandfoldness....; in place of such thoughts, 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 needn t 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 can t bear in mind all of its component notions at the same time, and this forces us into symbolic thinking. 2

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 don t have components. This shows that it s 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 haven t explained any of them. Some people hold that we can t 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 doesn t involve ideas. When we settle for this blind thinking, and don t pursue the resolution of notions far enough, we may have a thought that harbours a contradiction that we don t 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 can t safely draw conclusions from definitions unless we know first that they are real definitions, that is, that they don t 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 nail s 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 don t, because we can t have an idea of something impossible. Similarly, the fact that we think about a most perfect being doesn t 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. 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 that s my answer to Hobbes, who claimed that truths are arbitrary because they depend on nominal definitions. What he didn t take into account was that a definition s being real is not something we decide, and that not just any notions can be joined to one another. Nominal 3

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. Something s 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 something s 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 thing s 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 shan t 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 don t 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 can t see that they have used this principle well. [Leibniz is referring to a principle of Descartes s 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 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. 4

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 shan t 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 God s ideas with him - or whether we have our own ideas. The view that we see everything in God, though recently made famous through Malebranche s defence of it, is an old opinion, and properly understood it shouldn t 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 God s 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 mind s state. (2) Don t 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 thing s 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 can t clearly attend to each one separately, so that it doesn t 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 don t notice this, and invent something new - the colour green - for ourselves. 5

1 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 Last amended: July 2007 Contingency By 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 doesn t 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 sides and 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 = x 2 + 2xy + y 2 if we put 2 for x and 3 for y we get (2 + 3) 2 = 2 2 + 2(2 3) + 3 2 which comes to 25 = 4 + 12 + 9 which 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 can t be reduced to the principle of contradiction. If they could, they wouldn t 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 don t 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.

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; that s 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 Leibniz s compact formulation in ways that can t 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 doesn t 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 (F F) = 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 can t 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 can t 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 best necessary? 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 exist necessary? I answer that the former proposition is not necessary: God always chooses the best because he decrees that that s what he ll 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 God s pleasure (as we say), but that is never an alternative to its being done for a reason. Since we can t 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 2

general principle, 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 couldn t think up any story that wasn t actually true somewhere at some time! Anyway, it doesn t seem possible for all possible things to exist, because they would get in one another s 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 proposition s existing, apparently meaning its being true.] Assuming that this proposition: The proposition P that has the greater reason for being true is true is necessary, we must see whether it then follows that P itself is necessary. It isn t. 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 : A proposition is true if and only if there is greater reason for it to be true than for it to be false. God always acts with the highest wisdom. But from these one can t demonstrate any proposition of the form Contingent proposition P has greater reason for being true than for being false or of the form Contingent proposition P is in conformity with divine wisdom. So it doesn t 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 doesn t 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 don t assert it. I say only that it seems safer to attribute to God the most 3

perfect way possible of operating. When it comes to creatures, one can t be as sure as we can 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 - can t be demonstrated. 4

1 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 Last amended: July 2007 First Truths By G. W. Leibniz First truths are the ones that assert something of itself or deny something of its opposite. For example, A is A A is not not-a If it is true that A is B, then it is false that A isn t B (i.e. false that A is not-b) Everything is as it is Everything is similar or equal to itself Nothing 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 doesn t 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, or A 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. 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 isn t. 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 couldn t be proved a priori, that is, a truth that couldn t 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 - where we start from something symmetrical - 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 can t be two individual things that differ in number alone, i.e. that don t 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 isn t grounded in any non-relational property. It might seem to us that a thing s 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 Leibniz s corresponding Latin) mark the fact that he wavers between making this a point about the properties and relations a 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 2

3 the top of page 2, this applies to relational predicates as well as qualitative ones, i.e. it applies to seemingly extrinsic as well as to obviously 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, how the sentence continues: considered in the realm of possibility by withdrawing the mind from the divine decree for creating him, the underlying line of thought: 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. and is seen there by God. 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 didn t decide that Peter would sin or 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, rather than other possible things, should come into existence. Don t think that Peter s eventual salvation occurs without the help of God s 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 God s 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 can t 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. 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,