LEIBNIZ, GOTTFRIED WILHELM (b. Leipzig, Germany, 1 July 1646; d. Hannover, Germany, 14 November 1716), mathematics, philosophy, 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