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showing that the proposition to be demonstrated is an identical proposition, the certainty
of which requires no further demonstration—nor can one be given. The certainty does not
rest on “self-evidence” as Locke claims, since the proposition depends on more-basic
truths. This is a model of demonstration in the strictest sense. No other type of
demonstration comes this close to following the strict requirements of demonstration, i.e.,
reducing all terms and propositions to identities.
The next two demonstrations are representative of a more frequent type of
demonstration having to do with theological matters, one for God’s existence and the
other for the immortality of the soul. The importance of these demonstrations, in addition
to the method they illustrate, is that Leibniz often alludes to demonstrations for God and
immortality as required to make moral doctrine truly demonstrable. 28 The first
demonstration is from Dissertatio de Arte Combinatoria (1666). 29 While it is formally
similar to his mathematical demonstrations, the meaning of the terms is assumed rather
than analyzed. To illustrate its form, without regard for the content (at the moment), the
demonstration may be schematized like so:
Proposition to be demonstrated: God exists.
Hypotheses: (Praecognita)
Definitions, 1, 2, 3.
Postulate, 4.
Axioms, 5, 6, 7, etc.
Observation, 9.
Proof: (by Ekthesis)
Propositions 1, 2, 3, etc., derived from the hypotheses.
Conclusion. Q.E.D.
For the demonstration to be strictly “geometrical,” the definitions themselves must be
necessary truths. However, this is assumed rather than demonstrated. For example,
Definition 1 says, “God is an incorporeal substance of infinite power,” and Definition 2
says, “I call substance whatever moves or is moved.” 30 These definitions are not here (nor
anywhere else) shown to be necessary truths—indeed, this definition of ‘substance’ does
not survive Leibniz’s own later formulation. But for the conclusion to be necessarily true
(and Leibniz thinks it is) Leibniz must, according to his own requirements, show that
these definitions are demonstrably certain (i.e., necessary).
The most important difference between this demonstration and the mathematical
one is the admission here of an experiential or observational premise, number 9: “there is
a moving body.” 31 As we saw in his letter to Conring above, Leibniz says that a
demonstration can include “experimata” or observations. However, we also saw that
observations are truths of fact, dependent on the immediacy of feeling, and thus are not
28 This claim was made for example in the Codex, A 4.5.63: “Ut vero universali demonstratione conficiatur,
omne honestum esse utile, et omne turpe damnosum, assumenda est immortalitas animae, et rector universi
DEUS.” It is also made in the Nouveaux essais, as we will see.
29 “Demonstratio Existentiae Dei” (LL 73-4; GP 4.32-33).
30 G.4.32: “1. Definitio 1: Deus est Substantia incorporea infinitae virtutis. 2. Def 2. Substantiam autem
voco, quicquid movet aut movetur. 3. Def. 3. Virtus infinita est Potentia principalis movendi infinitum.
Virtus enim idem est quod potentia principalis.”
31 G.4.32: “9. Observatio. Aliquod corpus movetur.”
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