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Leibniz: Logic and Harmony

The Uses of Logic

Life and Works
. . Logic & Truth
. . Individual Substances
. . Sufficient Reason
. . Space & Time
. . Best of All Possible
. . Freedom
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The last of the great Continental Rationalists was Gottfried Wilhelm Leibniz. Known in his own time as a legal advisor to the Court of Hanover and as a practicing mathematician who co-invented the calculus, Leibniz applied the rigorous standards of formal reasoning in an effort to comprehend everything. A suitably sophisticated logical scheme, he believed, can serve as a reliable guide to the ultimate structure of reality.

But Leibniz published little of his philosophical work during his own lifetime. For an understanding of the technical logical foundations of his system, we must rely upon letters and notebooks which became available only centuries later and upon the aphoristic summary of its results in La Monadologie (Monadology) (1714). His Discours de Metaphysique (Discourse on Metaphysics) (1686) and Théodicée (Theodicy) (1710) present to the general public more popular expositions of Leibniz's central themes. Our strategy will be to begin with the logical theories and work outward to the more accessible doctrines.

True Propositions

The basis for Leibniz's philosophy is pure logical analysis. Every proposition, he believed, can be expressed in subject-predicate form. What is more, every true proposition is a statement of identity whose predicate is wholly contained in its subject, like "2 + 3 = 5." In this sense, all propositions are analytic for Leibniz. But since the required analysis may be difficult, he distinguished two kinds of true propositions: (Monadology 33)

Truths of Reason are explicit statements of identity, or reducible to explicit identities by a substitution of the definitions of their terms. Since a finite analysis always reveals the identity-structure of such truths, they cannot be denied without contradiction and are perfectly necessary.

Truths of Fact, on the other hand, are implicit statements of identity, the grounds for whose truth may not be evident to us. These truths are merely contingent and may be subject to dispute, since only an infinite analysis could show them to be identities.

Anything that human beings can believe or know, Leibniz held, must be expressed in one or the other of these two basic forms. The central insight of Leibniz's system is that all existential propositions are truths of fact, not truths of reason. This simple doctrine has many significant consequences.

Complete Individual Substances

Consider next how this logic of propositions applies to the structure of reality itself for Leibniz. The subject of any proposition signifies a complete individual substance, a simple, indivisible, dimensionless being or monad, while the predicate signifies some quality, property, or power. Thus, each true proposition represents the fact that some feature is actually contained in this substance.

Each monad is a complete individual substance in the sense that it contains all of its features—past, present, and future. Because statements of identity are timeless, the facts they express perpetually obtain. (Thus, for example, I am the person whose daughter was born in 1982 and the person who now develops this web site and the person who will vacation in Manitoba next summer; since each of these predicates can be truly affirmed of me, each of these features is contained in me.) Everything that was, is, or will ever be true of any substance is already contained in it. (Monadology 22)

Moreover, each monad is a complete individual substance in the sense that its being is utterly independent of everything else. Because statements of identity are self-contained, any apparent relation between substances must actually be a matching pair of features that each possesses alone. (Thus, for example, I happen to have the property of being Aaron's father, and Aaron happens to have the property of being my son, but these are two facts, not one.) Hence, on Leibniz's view, there can be no interaction between substances, each of which is purely active. Monads are "windowless." (Monadology 7)

Where Spinoza saw the world as a single comprehensive substance like Descartes's extended matter, then, Leibniz supposed that the world is composed of many discrete particles, each of which is simple, active, and independent of every other, like Descartes's minds or souls. The rationalists' common reliance upon mathematical models of reasoning led to startlingly different conceptions of the universe. Yet the rationality, consistency, and necessity within each system is clear.

Logical Principles

Another way of summing up the structure of the universe on Leibniz's view is by reviewing the great logical principles from which all truths are said to flow:

The Principle of Contradiction generates the truths of reason, each of which states the connection between an individual substance and one of its finite number of essential features. (Monadology 31) It would be a contradiction to deny any of these propositions, since the substance would not be what it is unless it had all of these features. Truths of reason, then, are not influenced by any contingent fact about the world; they are true "in all possible worlds." Thus, for example, "Garth Kemerling is a human being" would be necessarily true even if my parents had been childless.

The Principle of Sufficient Reason generates the truths of fact, each of which states the connection between an existing individual substance and one of its infinitely many accidental features or relations. (Monadology 32) The sufficient reason for the truth of each of these propositions is that this substance does exist as a member of the consistent set of monads which constitutes the actual world. Truths of fact, then, depend upon the reciprocal mirroring of each existing substance by every other. Thus, for example, "Garth Kemerling is an oldest child" is contingently true only because my parents had no children before I was born.

The Principle of the Identity of Indiscernibles establishes the fact that, within the set of monads that constitutes any possible world, no two can be exactly alike. (Monadology 9) If, on the contrary, there were two distinct but perfectly identical substances, Leibniz argued, then there could be no sufficient reason for each to occupy its own location rather than that of the other. More positively, since each monad mirrors the entire structure of the world, each must reflect a unique set of relations to every other.

Finally, the Principle of the Plenum (or principle of plenitude) affirms that the actual world, considered as a set of monads, is as full as it can possibly be. Since there is no genuine interaction among distinct substances, there would be no sufficient reason for the non-existence of any monad that would be consistent with the others within a possible world. Hence, anything that can happen will; every possibility within this world must be actualized. The world in which we live, then, is but one among the infinitely many possible worlds that might have existed. What makes this one special?

Space and Time

Since we experience the actual world as full of physical objects, Leibniz provided a detailed account of the nature of bodies. As Descartes had correctly noted, the essence of matter is that it is spatially extended. But since every extended thing, no matter how small, is in principle divisible into even smaller parts, it is apparent that all material objects are compound beings made up of simple elements. But from this Leibniz concluded that the ultimate constitutents of the world must be simple, indivisible, and therefore unextended, particles—dimensionless mathematical points. So the entire world of extended matter is in reality constructed from simple immaterial substances, monads, or entelechies.

In fact, Leibniz held that neither space nor time is a fundamental feature of reality. Of course individual substances stand in spatial relation to each other, but relations of this sort are reducible in logic to the non-relational features of windowless monads. In exactly the same way, temporal relations can be logically analyzed as the timeless properties of individual monads. Space and time are unreal, but references to spatial location and temporal duration provide a convenient short-hand for keeping track of the relations among the consistent set of monads which is the actual world.

What is at work here again is Leibniz's notion of complete individual substances, each of which mirrors every other. A monad not only contains all of its own past, present, and future features but also, by virtue of a complex web of spatio-temporal references, some representation of every other monad, each of which in turn contains . . . . In a universe of windowless mirrors, each reflects any other, along with its reflections of every other, and so on ad infinitum. It is for this reason that an infinite analysis would be required to reveal the otherwise implicit identity at the heart of every truth of fact. In order fully to understand the simple fact that my eyes are brown, one would have to consider the eye-color of all of my ancestors, the anatomical structure of the iris, my personal opthalmological history, the culturally-defined concept of color, the poetical associations of dark eyes, etc., etc., etc.; the slightest difference in any one of these things would undermine the truth of this matter of fact. Existential assertions presuppose the reality of just this one among all possible worlds as the actual world.

The Best of All Possible Worlds

Both in the Monadology and at the more popular level of presentation that characterizes the Discourse on Metaphysics, Leibniz (like Descartes) resolved some of the most thorny philosophical problems by reference to god. God (alone) exists necessarily, and everything else flows from the divine nature. Limited only by contradiction, god first conceives of every possible world—the world with just one monad; the worlds with exactly two monads; those with three, with seventeen, with five billion, etc. Then god simply chooses which of them to create.

Of course even god must have a sufficient reason for actualizing this world rather than any other. The most direct advantage of this world is that (as the plenum principle requires) it is the fullest. That is, more things exist and/or more events actually take place in this world than in any other consistent set of interrelated monads. In a more lofty tone, Leibniz declared that a benevolent god would choose to create whatever possible world contained the smallest amount of evil; hence (in a phrase that would later be mocked by Voltaire) this is "the best of all possible worlds," according to Leibniz. Nothing about it could be changed without making things worse rather than better on the whole.

Similarly, the existence of a benevolent god can be used to account for the smooth operation of a universe that consists of indefinitely many distinct individual substances, none of which have any causal influence over any other. (Monadology 51) A crucial element of god's creative activity, Leibniz held, is the establishment of a "pre-established harmony" among all existing things. Like well-made clocks that have been synchronized, wound, and set in motion together, the monads that make up our world are independent, self-contained, purely active beings whose features coincide without any genuine interaction among them.

One special case of this pre-established harmony, of course, accounts for the apparent interaction of mind and body in a human being as nothing more than the perfect parallelism of thier functions. In fact, the human mind is just the dominant member of a local cluster of monads which collectively constitute the associated human body. (Monadology 63) Neither has any real effect on the other, but these monads are most clearly reflected in each others' foreground. Thus, in both sensation and volition, the divinely-ordained coincidence of bodily movements and mental thoughts creates an illusion of genuine causal influence.

Knowledge and Freedom

The possibility human knowledge emerges more clearly from a slightly more technical account of Leibniz's position. All monads have the capacity for perception of the external world in the sense that, as complete individual substances, each of them contains as properties unconscious images of its spatio-temporal relations to everything else. (Monadology 19) These innate ideas constitute the unique point of view from which any monad may be said to represent the world as a whole.

But Leibniz held that some monads—namely, the souls of animals and human beings—also have conscious apperception in the sense that they are capable of employing sensory ideas as representations of physical things outside themselves. And a very few monads—namely, spirits such as ourselves and god—possess the even greater capacity of self-consciousness, of which genuine knowledge is the finest example. Although Leibniz himself did not draw the inference directly, notice that if a cluster of dimensionless monads can make up an extended body, it might be equally possible for a cluster of unconscious monads to constitute a thinking thing.

What Leibniz did claim is that we have the free will required for moral responsibility even though all of our future actions are already contained in us (along with the future of the entire actual world). Any awareness of those contingent future actions would follow from the principle of sufficient reason only upon an infinite analysis of my nature. Hence, since I lack knowledge of what I will do tomorrow, it will seem to me as if I act freely when I do it. Like space and time, freedom is a benevolent illusion that adequately provides for life in an uncertain world.

Concluding note on the Rationalists

Descartes, Spinoza, and Leibniz illustrate well the range of diverse outcomes that may result from an effort to understand the world through a priori knowledge. If their systems of thought seem implausibly remote from the world of ordinary experience, it may help to remember that modern science leads to a similar result. Once we grant that the reality of things may be quite different from the way they appear to us, only the internal coherence of the scheme of thought makes much difference. Next we'll look at modern philosophers who were more determined to make sense out of the materials provided in everyday life.

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