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§4. Leibniz ’s Predicate-in-Subject Theory of Truth
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
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. .
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
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
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
of sufﬁcient reason, and the distinction between necessary and contingent
(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,”
(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.
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
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. .