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§4. Leibniz ’s Predicate-in-Subject Theory of Truth

§4. Leibniz ’s Predicate-in-Subject Theory of Truth

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moral perfection, and will, and who creates the world for the best of reasons, not arbitrarily or by logical necessity.

(b) This requires that the actual world must be the best of all possible

worlds; and the created things that make up the world—the complete substances—must be genuinely created things, having their own active forces

and tendencies that move them to act in accordance with their own principles.

As we will discuss next time, Leibniz views created things as moved by

their own active powers, while he thinks that Descartes does not; and he

sees created things not as Spinoza does, as mere attributes and modes of

the one complete substance, but as genuine substances.

. One way to present Leibniz’s view is to think of him as starting from

an idea of what a true proposition is. His basic thought might be said to

be this:

A proposition is true if and only if the concept expressed by its predicate

is contained in the concept expressed by its subject.

Thus Leibniz says: “[T]he predicate is present in the subject; or else I do

not know what truth is.”5

“Necessary and Contingent Truths” (ca. ) has a quite full statement

of Leibniz’s predicate-in-subject theory of truth. In the quotations from this

essay below, the first asserts that all knowledge has an a priori reason for

its truth; the second defines necessary truths as about the essences of things

lying in the divine reason, and the third characterizes contingent truths as

about the existence of things in space and time.

. An affirmative truth is one whose predicate is in the subject;

and so in every true affirmative proposition, necessary or contingent, universal or particular, the notion of the predicate is

in some way contained in the notion of the subject. Moreover,

it is contained in the notion of the subject in such a way that

if anyone were to understand perfectly each of the two notions

just as God understands it, he would by that very fact per5. The Leibniz-Arnauld Correspondence, ed. and trans. H. T. Mason (Manchester: Manchester

University Press, ), letter of July , p. .

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ceive that the predicate is in the subject. From this it follows

that all knowledge of propositions that is in God, whether

it is simple knowledge, concerning the essence of things, or

whether it is knowledge of vision, concerning the existence

of things, or mediate knowledge concerning conditioned existences, results immediately from the perfect understanding of

each term which can be the subject or predicate of any proposition. That is, the a priori knowledge of complexes arises from

the understanding of that which is not complex.

. An absolutely necessary proposition is one which can be resolved [in a finite number of steps] into identical propositions,

or whose opposite implies a contradiction.

. In the case of a contingent truth, even though the predicate

is really in the subject, yet one never arrives at a demonstration or an identity, even though the resolution of each term

is continued indefinitely. In such cases it is only God, who

comprehends the infinite at once, who can see how the one

is in the other, and can understand a priori the perfect reason

for contingency; in creatures this is supplied a posteriori, by


. To explain: take the proposition “Caesar crossed the Rubicon,” in

which the subject is “Caesar.” The proposition is contingent, since it concerns the existence of things: it refers to an individual at a time and place,

namely, the historical individual Caesar. Its subject expresses the complete

individual concept (or notion) of Caesar. (Leibniz holds that a proper name

of an individual substance, e.g., “Caesar,” expresses a complete individual

concept.) The predicate term is “crossed the Rubicon,” which expresses the

complex attribute of crossing the Rubicon.

The proposition expressed by the sentence “Caesar crossed the Rubicon” is true if and only if the concept expressed by the predicate term,

“crossed the Rubicon,” is included in the complete individual concept expressed by the subject term, the proper name “Caesar.”

Here by “included in” Leibniz means included in the complete analysis

of all the attributes that make up the complete individual concept of Caesar.

This complete individual concept, on Leibniz’s view, includes everything

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that characterizes Caesar. It includes the attribute of crossing the Rubicon,

of being the winner of the Battle of Pharsalus, of being dictator of Rome;

and also of being descended after n generations from Adam, of being assassinated in Rome more than a century prior to Saint Paul’s martyrdom there,

and so on. Each substance (or monad), from its own point of view, mirrors

the whole universe, past, present, and future; and therefore it mirrors down

to the last detail the infinitely many aspects of the universe to which it

belongs. As Leibniz even says at one place in his correspondence with Arnauld, “[E]ach possible individual of any one world contains in the concept

of him the laws of his world” (Mason:).

. In subsection  above, I said that one way to present Leibniz’s account

of truth was to begin with his idea of a true proposition as one in which

the concept of the predicate is in the concept of the subject. But another

way to begin is to start with his idea of the complete concept of an individual

thing, or monad (to use his term). To this end, let’s think of a monad as

represented by a complete list of all its properties, not only a complete list at

any moment of time, but a complete list of all its properties for every moment

of time. So when we say something true about a monad, the concept of the

predicate must be in the concept of the subject, that is, it must appear somewhere on the complete (infinite) list that represents the monad.

Starting with this idea of an individual thing, or monad, how may we

imagine that God selects the best of all possible worlds, and why should

this process involve (as we look at it and not as God does) an infinite resolution that never ends? Begin first with what we may call bare monads: these

are possibles (like all possibles) that lie in God’s reason. Their mark is that

their lists of properties do not contain any properties that reflect the properties of any (one or more) other monads. In a world consisting of two bare

monads, say, neither monad reflects the other.

. Next, imagine a world consisting of partially reflecting monads. Recall

from the Monadology § that monads are windowless: that is, complete substances that do not causally affect one another. Nevertheless, they do reflect,

or represent, one another’s properties. Thus a world of partially reflecting

monads is one in which, say, each monad reflects the properties of at least

one or more of the other monads. No monad is completely self-contained,

but some at least are partial in that there are some monads whose properties

they do not reflect. From here, we move to a world of completely reflecting

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monads. From the list of properties of any one monad, we can discern all

the properties of all the other monads. Now, what distinguishes one monad

from another is not that their lists of properties as such are different, but

that the reflecting properties of each monad are indexed by the monad’s

point of view on the universe. Since each monad expresses the universe

from its own point of view, the monads are distinct from one another, as

can be seen from how their lists of reflecting properties are indexed.

Now, take a further step to a world of time-coordinated completely reflecting monads. By this I mean that the reflecting properties of each monad

are listed in the right time sequence, so that the monads reflect one another’s state at the appropriate moment: we thus have a world in which what

Leibniz calls preestablished harmony is satisfied. We imagine, then, that

God looks only at those worlds which satisfy this condition, the reason

being that only the class of these worlds can contain the best of all possible

worlds. Leibniz says in the Monadology (§§, ): “This interconnection . . .

of all created things to each other, and each to all the others, brings it about

that each simple substance has relations that express all the others . . . [and]

each simple substance is a perpetual, living mirror of the universe. . . . And

this is a way of obtaining as much variety as possible, but with the greatest

order possible, that is, it is a way of obtaining as much perfection as possible.”

In this sketch, I have supposed that bare monads and partially reflecting

monads and the rest are possibles lying in the divine reason. The reason

why only a world of time-coordinated and completely reflecting monads

exists lies in God’s free selection of the best of all possible worlds guided

by the principle of perfection. This sketch also lays out why God’s choice

of the best world involves an infinity of comparisons going far beyond the

capacity of any finite intelligence. We can see why, given Leibniz’s idea of

a monad, bare or otherwise, the concept of the predicate of all true propositions about a monad will be contained in the concept of its subject. And

given the preestablished harmony of the best world, not only will the concept of the predicate of crossing the Rubicon be in the complete concept

of Caesar, but also so will the concept of the predicate of being assassinated

in Rome a century before Saint Paul’s martyrdom there.

. With his theory of truth and his conception of monads as mirroring

the universe each from its own point of view, Leibniz explains the principle

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of sufficient reason, and the distinction between necessary and contingent

truths, thus:

(a) Every proposition must be either true or false.

(b) A proposition is true or false in virtue of the relation conceptinclusion (as explained above) between the concepts expressed by its subject

and predicate terms: true when the inclusion relation holds, false otherwise.

(c) Therefore, for every true proposition, of whatever kind, there exists

an a priori proof of it (known to God). This proof takes the form, in general,

of an analysis of the concept expressed by the subject term carried to the

point where this concept is seen to include the concept expressed in the

predicate term. (Discourse:§; and “Necessary and Contingent Truths,”

paragraph ).

(d) In those cases where the a priori proof of the proposition can be

carried out in a finite number of steps, and the last step of the proof reduces

the proposition to an identity (such as A is A, etc.), so that its denial would

violate the Principle of Contradiction, the proposition is a necessary truth.

Its contradictory is impossible. It is true in any world that God might have

created: it is true in all possible worlds, so understood. We can know these

finitely demonstrable truths about the essences of things, about numbers

and geometrical objects, and about moral goodness and justice (“Necessary

and Contingent Truths,” paragraphs  and ).

(e) In those cases where the a priori proof of the proposition cannot be

carried out in a finite number of steps, but rather involves the infinite analysis of an infinitely complex complete individual concept, and even infinitely

many comparisons of other worlds that God might have created, the proposition is contingent and its truth depends on the Principle of the Best. Carrying out the proof would require an analysis taking infinitely many steps,

which can never be completed. God doesn’t see the end of the analysis,

for the analysis has no end. But only God sees the answer—by God’s intuitive vision of the possibles all at once.

§. Some Comments on Leibniz’s Account of Truth

. First, the statement (c) above is one form of the Principle of Sufficient

Reason: it says that every true proposition has an a priori proof of its truth.

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Leibniz often uses this name, “the Principle of Sufficient Reason,” to refer

to various less general principles. The form above is, I think, the most

general form of the principle. Thus he says: “There are two first principles of all reasonings, the principle of contradiction . . . and the principle

that a reason must be given, that is, that every true proposition which is

known per se, has an a priori proof, or that a reason can be given for

every truth, or as is commonly said, that nothing happens without a cause.

Arithmetic and geometry do not need this principle, but physics and

mechanics do.”6

Another quotation brings out how contingent truths depend on God’s

decrees and choice of the best of all possible worlds.

The demonstration of this predicate of Caesar [that he resolved to cross

the Rubicon] is not as absolute as are those of numbers or of geometry,

but presupposes the series of things which God has chosen freely, and

which is founded on the first decree of God, namely to do always what

is most perfect, and on the decree which God has made [in consequence of the first], in regard to human nature, that man will always

do (though freely) what appears best. . . . [E]very truth which is

founded on decrees of this kind is contingent, although it is certain.

(Discourse:§ [Ariew and Garber:])

. A second comment on Leibniz’s account of truth is that today we

use the term “a priori” as an epistemological term. It says something about

how a proposition can be known, namely, that it can be known independently of experience. But this is not Leibniz’s idea of the a priori: when he

says that true propositions have an a priori proof, he means a proof based

on the ultimate reasons for their being true and not false. Clearly Leibniz

does not mean that we (human beings) can know contingent propositions

to be true independent of experience. The proofs he has in mind can be

known only by God, because only God sees by intuitive vision of the possible existences the answer to the requisite infinite analysis.

A further comment, related to the preceding one, is that Leibniz’s con6. Gerhardt, Philosophischen Schriften, VII:, in Leibniz Selections, ed. Philip R. Weiner (New

York: Scribners, ), p. .

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§4. Leibniz ’s Predicate-in-Subject Theory of Truth

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