Stony Brook University

The principles of contradiction and identity, as we will see, are also the foundation of
demonstrations. But these principles are true because all contradictions are impossible.
That is, there is no possible world in which a contradictory proposition is true. 9 An
identity (whose form ‘A = A’) is also said to be true since no further reason, other than
non-contradiction, can be given for its truth. 10 Therefore, a proposition is said to be
necessarily true when an analysis of its terms terminates in an identical proposition or is
shown to be not contradictory. This is also what it means to demonstrate a proposition in
a mathematical or geometrical sense. Below, we will see an example of mathematical
demonstration (2 + 2 = 4).
As just stated, the principles of necessary truths are contradiction and identity.
But the principle of contingent truths is the principle of sufficient reason. Stated another
way, propositions whose analysis of terms does not terminate but proceeds to infinity are
called contingent truths. The containment of terms never results in an identity, and thus
contingent propositions cannot be demonstrated in the strict sense of necessity.
Epistemologically, this means that we finite knowers can never have absolute certainty of
the reason for the connection between subject and predicate (or the containment of
predicate in the subject); although, through experience we can be more certain. The
connection is nonetheless metaphysically guaranteed, since it is known by God, who
“grasps” the connection “in one intuition” (see quotations below). For example, the
proposition, “Caesar crossed the Rubicon” is true if the concept ‘crossing the Rubicon’ is
contained in the concept of ‘Caesar’. This proposition is in fact true, but it is not a
necessary truth because its contrary ‘Caesar did not cross the Rubicon’ is not a logical
contradiction. The latter proposition is therefore logically possible, i.e., logically
contingent. A most important point, however, is that not only does God see the
connection, but God determines the connection. That is, the fact of Caesar having crossed
the Rubicon is a truth determined by God, who nevertheless, as Leibniz insists, freely
chose this predicate to be included in the concept of Caesar.
These distinctions are of enormous consequence for Leibniz’s philosophy,
especially for his argument for divine and human freedom, issues about which there is
much debate. 11 But here we need be concerned only with clarifying the use he makes of
nor false” (NE 362).
9 To be precise, Leibniz nowhere talks of “possible worlds” in reference to necessary truths. He only says
that necessary truths are those whose contraries are impossible. This implies that there is no possible world
in which they are true—although Leibniz does not say this. Nor does Leibniz say ‘necessary truths are true
in all possible worlds’ although this is true. Here is an example of what he does say, in Corresondence with
Clarke, 5 th letter sec. 10 p. 57: “Absolute and metaphysical necessity depend upon the other great principle
of our reasonings, that of essences; that is, the principle of identity or contradiction: for, what is absolutely
necessary, is the only possible way, and its contrary implies a contradiction.” See Margaret Wilson for the
claim that Leibniz does not speak of possible worlds in reference to necessary truths.
10 This is to say that the principles of identity and contradiction are indemonstrable, as I will explain below.
11 For example, it may be thought that the distinction between necessary and contingent truths does not
really work, since supposedly contingent propositions, such as “Caesar crossed the Rubicon,” really turn
out to be necessary. According to the “complete concept” of a subject, if a subject lacks any one of his or
her predicates, then that subject would be a different subject altogether; thus all predicates are essential
(i.e., necessary) to the concept of the subject. But there is debate about whether Leibniz must be committed
to the “complete concept” theory, whether it really does commit him to spinizistic necessity. Another point
of controversy involves the criterion for God’s choice of the best possible world and whether the criterion
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