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LEIBNIZ, GOTTFRIED WILHELM (b. Leipzig,
Germany, 1 July 1646; d. Hannover, Germany,
14 November 1716), mathematics, philosophy, metaphysics.
LEIBNIZ: Physics, Logic, Metaphysics
sunt or compatibilia sunt). For example,
one of the propositions of the calculus states:
A - B = C holds exactly, if and only if A = B + C,
and B and C are incompatible; in modern notation
a ? b = c ? (b ? a) ? a = (b ? c) ? (b
|
c).
(The condition b ? a is implicit in Leibniz' use of the
symbol —).
If we disregard a few syntactical details and observe
that Leibniz' work gives an approximation to a
complete interpretation of the elements of a logical
calculus (including the rules for transformation), we
see here for the first time a formal language and thus
an actual successful example of a characteristica
universalis. It is true that Leibniz did not make
sufficient distinction between the formal structure of
the calculus and the interpretations of its content;
for example, the beginnings of the calculus are immediately
considered as axiomatic and the rules of
transformation are viewed as principles of deduction.
Yet it is decisive for Leibniz' program and our
appreciation of it that he succeeded at all, in his
logical calculus, in the formal reconstruction of
principles of deduction concerning concepts.
General Logic.
Leibniz' general logical investigations
play just as important a role in his systematic philosophy
as does his logical calculus. Most important are
his analytical theory of judgment, the theory of
complete concepts on which this is based, and the
distinction between necessary and contingent propositions.
According to Leibniz' analytical theory of
judgment, in every true proposition of the subject-predicate
form, the concept of the predicate is contained
originally in the concept of the subject (praedicatum
inest subiecto). The inesse relation between
subject and predicate is indeed the converse of the
universal-affirmative relation between concepts, long
known in traditional syllogisms (B inest omni A is the
converse of omne A est B).23 Although it is thus taken
for granted that subject-concepts are completely
analyzable, it suffices, in a particular case, that a
certain predicate-concept can be considered as
contained in a certain subject-concept. Fundamentally,
there is a theory of concepts according to which concepts
are usually defined as combinations of partials,
so that analysis of these composite concepts (notiones
compositae) should lead to simple concepts (notiones
primitivae or irresolubiles). With these, the characteristica
universalis could then begin again. When the
predicate-concept simply repeats the subject-concept,
Leibniz speaks of an identical proposition; when this
is not the case, but analysis shows the predicate-concept
to be contained implicitly in the subject-concept,
he speaks of a virtually identical proposition.
The distinction between necessary propositions or
truths of reason (vérités de raisonnement) and
contingent propositions or truths of fact (vérités de
fait) is central to Leibniz' theory of science. As
contingent truths, the laws of nature are discoverable
by observation and induction but they have their
rational foundation, whose investigation constitutes
for Leibniz the essential element in science, in
principles of order and perfection. Leibniz replaces
the classical syllogism, as a principle of deduction, by
the principle of substitution of equivalents to reduce
composite propositions to identical propositions.
Contingent propositions are defined as those that are
neither identical nor reducible through a finite
number of substitutions to identical propositions.
All contingent propositions are held by Leibniz to
be reducible to identical propositions through an
infinite number of steps. Only God can perform these
steps, but even for God, such propositions are not
necessary (in the sense of being demanded by the
principle of contradiction). Nevertheless, contingent
propositions, in Leibniz' view, can be known a priori
by God and, in principle, also by man. For Leibniz,
the terms a priori and necessary are evidently not
synonymous. It is the principle of sufficient reason
that enables us (at least in principle) to know contingent
truths a priori. Consequently, the deduction of
such truths involves an appeal to final causes. On the
physical plane, every event must have its cause in an
anterior event. Thus we have a series of contingent
events for which the reason must be sought in a
necessary Being outside the series of contingents. The
choice between the possibles does not depend on
God's understanding, that is to say, on the necessity
of the truths of mathematics and logic, but on his
volition. God can create any possible world, but, being
God, he wills the best of all possible worlds. Thus the
contingent truths, including the laws of nature, do not
proceed from logical necessity but from a moral
necessity.
Methodological Principles.
Logical calculi and the
notions mentioned under “General Logic” belong to
a general theory of foundations that also encompasses
certain important Leibnizian methodological principles.
The principle of sufficient reason (principium rationis
sufficientis, also designated as principium nobilissimum)
plays a special role. In its simplest form it is phrased
“nothing is without a reason” (nihil est sine ratione),
which includes not only the concept of physical
causality (nihil fit sine causa) but also in general the
concept of a logical antecedent-consequent relationship.
According to Leibniz, “a large part of metaphysics
[by which he means rational theology],
physics, and ethics” may be constructed on this