Stony Brook University

In sum, this says that a proposition is demonstrated when an analysis of its terms shows
that the proposition is reducible to a definition, an identity, or an experiential
(observation) statement. But some distinctions need to be made. First, only propositions
whose reductions are terminable are demonstrations of a necessary truth. Second,
identities (propositions of the form A = A) are indemonstrable, meaning that they are
known to be true either by virtue of non-contradiction, or by intuitive certainty. Third,
axioms and postulates are also identities, and although well-established, should be and
can be demonstrated. 25 Fourth, a definition, once shown to be possible, is also a necessary
truth, although it is not an identity in the same sense that propositions of the form ‘A = A’
are identities. To show that a definition is possible, its terms must be analyzed down to
primitive notions. However, Leibniz is often not clear on which notions are primitive. I
will come back to these points about definitions in Section 4.
In any case, the main result is this: A proposition is demonstrated when, through a
transitively-linked chain of identity statements, i.e., definitions, all of its terms are
reduced to primitives, identities, or experiential (observational) statements. Now, in the
account above it sounds as if a demonstration begins with an identity; but in practice this
almost never happens. A demonstration begins with a proposition not known to be an
identity, and then through a chain of substitutable definitions, either an express identity is
discovered (e.g., ‘4 = 4’), or a “basic” definition (which is a type of identity) is reached.
This is what he means by resolving all truths into definitions and identical propositions.
In other words, a proposition not immediately known to be an identity may be shown to
be one by reducing its terms to an identity. When this is done, the proposition is shown to
be an a priori necessary truth. As Leibniz says, however, a demonstration may also
include “observations” or experimata. Observations are not necessary truths, but are
contingent. Therefore, a demonstration that includes an observation is not an a priori
demonstration, but is a posteriori.
What we are interested in knowing is how Leibniz applies his demonstrative
method to make moral claims. Toward this end, I provide examples of demonstrations
contrarium implicet contradictionem, qui est verus atque unicus character impossibilitatis. Porro ut
impossibili respondet necessarium, ita propositioni contradictionem implicanti respondet identica, nam ut
primum impossibile in propositionibus est haec: A non est A, ita primum necessarium in propositionibus
est haec: A est A. Unde solae identicae sunt indemonstrabiles, Axiomata autem omnia, quanquam
plerumque ita clara ac facilia sint ut demonstratione non indigeant, sunt tamen demonstrabilia vel ideo quia
demum terminis intellectis (id est substituendo definitionem in definiti locum) patet ea esse necessaria seu
contrarium implicare in terminis. . . . Propositiones autem identicas necessarias esse constat, sine omni
terminorum intellectu sive resolutione, nam scio A esse A, quicquid demum intelligatur per A. Omnes
autem propositiones quarum veritatem ex terminorum demum resolutione et intellectu patere necesse est,
demonstrabiles sunt per eorum resolutionem, id est per definitionem. Hinc patet, Demonstrationem esse
catenam definitionum [my emphasis]. Nam in demonstratione alicujus propositionis non adhibentur nisi
definitiones, axiomata (ad quae hoc loco postulata reduco), theoremata jam demonstrata et experimenta.
Cumque theoremata rursus demonstrata esse debeant, et axiomata omnia exceptis identicis demonstrari
etiam possint, patet denique omnes veritates resolvi in definitiones, propositiones identicas et experimenta
(quanquam veritates pure intelligibiles experimentis non indigeant) et perfecta resolutione facta apparere,
quod catena demonstrandi ab identicis propositionibus vel experimentis incipiat, in conclusionem desinat,
definitionum autem interventu principia conclusioni connectantur, atque hoc sensu dixeram
Demonstrationem esse catenam definitionum.” Note that ‘experimenta’ means empirical observations.
25 Leibniz has in fact demonstrated some axioms, for example from Euclid’s Elementa, Book I, axiom 2: “if
equals are added to equals, then the result is an equality” (A.6.4.507).
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