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

    proposition.24 Viewed methodologically, this means that, in the principle of sufficient reason, there is a teleological as well as a causal principle; the particular import of the proposition is that both principles may be used in the same way for physical processes and human actions.

Defending the utility of final causes in physics (in opposition to the view of Descartes), Leibniz explained that these often provided an easier path than the more direct method of mechanical explanation in terms of efficient causes.25 Leibniz had himself in 1682 used a variation of Fermat's principle in an application of his method of maxima and minima to the derivation of the law of refraction. Closely associated with the principle of sufficient reason is the principle of perfection (principium perfectionis or melioris). In physics, this principle determines the actual motion from among the possible motions, and in metaphysics leads Leibniz to the idea of “the best of all possible worlds.” The clearest expression of Leibniz' view is to be found in his Tentamen anagogicum,26 written in about 1694, where he remarks that the least parts of the universe are ruled by the most perfect order. In this context, the idea of perfection consists in a maximum or minimum quantity, the choice between the two being determined by another architectonic principle, such as the principle of simplicity. Since the laws of nature themselves are held by Leibniz to depend on these principles, he supposed the existence of a perfect correlation between physical explanations in terms of final and efficient causes.

In relation to Leibniz' analytical theory of judgment and his distinction between necessary and contingent propositions, the principle of sufficient reason entails that, in the case of a well-founded connection between, for example, physical cause and physical effect, the proposition that formulates the effect may be described as a logical implication of the proposition that formulates the cause. Generalized in the sense of the analytical theory of truth and falsehood that Leibniz upholds, this means: “nothing is without a reason; that is, there is no proposition in which there is not some connection between the concept of the predicate and the concept of the subject, or which cannot be proved a priori.”27 This logical sense of the principle of sufficient reason contains also (in its formulation as a principium reddendae rationis) a methodological postulate; propositions are not only capable of being grounded in reasons (in the given analytical manner) but they must be so grounded (insofar as they are formulated with scientific intent).28

In addition to the principle of sufficient reason, the principle of contradiction (principium contradictionis) and the principle of the identity of indiscernibles (principium identitatis indiscernibilium) are especially in evidence in Leibniz' logic. In its Leibnizian formulation, the principle of contradiction, ?(A ? ?), includes the principle of the excluded middle, A ? ?A (tertium non datur): “nothing can be and not be at the same time; everything is or is not.”29 Since Leibniz' formulation rests on a theory according to which predicates, in principle, can be traced back to identical propositions, he also classes the principle of contradiction as a principle of identity. The principle of the identity of indiscernibles again defines the identity of two subjects, whether concrete or abstract, in terms of the property that the mutual replacement of their complete concepts in any arbitrary statement does not in the least change the truth value of that statement (salva veritate). Two subjects s1 and s2 are different when there is a predicate P that is included in the complete concept S1 of s1 but not in the complete concept S2 of s2, or vice versa. If there is no such predicate, then because of the mutual replaceability of both complete concepts S1 and S2, there is no sense in talking of different subjects. This means, however, that the principle of the identity of indiscernibles, together with its traditional meaning (“there are no two indistinguishable subjects”30), is synonymous with the definition of logical equality (“whatever can be put in place of anything else, salva veritate, is identical to it”31).

Metaphysics (Logical Atomism).

Since the investigations of Russell and Couturat, it has become clear that Leibniz' theory of monads is characterized by an attempt to discuss metaphysical questions within a framework of logical distinctions. On several occasions, however, Leibniz himself remarks that dynamics was to a great extent the foundation of his system. For example, in his De primae philosophiae emendatione et de notione substantiae, Leibniz comments that the notion of force, for the exposition of which he had designed a special science of dynamics, added much to the clear understanding of the concept of substance.32 This suggests that it was the notion of mechanical energy that led to the concept of substance as activity. Again, it is in dynamics, Leibniz remarks, that we learn the difference between necessary truths and those which have their source in final causes, that is to say, contingent truths,33 while optical theory, in the form of Fermat's principle, pointed to the location of the final causes in the principle of perfection.34 Even the subject-predicate logic itself, which forms the rational foundation of Leibniz' metaphysics, seems to take on a biological image, such as the growth of a plant from a seed, when Leibniz writes to De Volder that the present state of a substance must