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

    theories of concepts and judgments, he discusses the possibility of transforming rules of inference into schematic deductive rules. Within this framework there is also a complementary ars iudicandi, a mechanical procedure for decision making. However, the thought of gaining scientific propositions by means of a calculus of concepts derived from the ars combinatoria and a mechanical procedure for decision making remained lodged in a few attempts at the formation of the “alphabet.” Leibniz was unable to complete the most important task for this project, namely, the proof of its completeness and irreducibility, nor did he consider this problem in his plans for the scientia generalis, a basic part of which was the “alphabet,” the characteristica universalis in the form of a mathématique universelle.

The scientia generalis exists essentially only in the “tables of contents,” which are not internally consistent terminologically and thus admit of additions at will. Nevertheless, it is clear that Leibniz was thinking here of a structure for a general methodology, consisting, on the one hand, of partial methodologies concerning special sciences such as mathematics, and on the other hand, of procedures for the ars inveniendi, such as the characteristica universalis; taken together, these were probably intended to replace traditional epistemology as a unified conceptual armory. This was by no means impracticable, at least in part. For example, the analytical procedures in which arithmetical transformations occur independently of the processes to which they refer, employed by Leibniz in physics, may be construed as a partial realization of the concept of a characteristica universalis.

Formal Logic.

Leibniz produced yet another proof of the feasibility of his plan for schematic operations with concepts. Besides the infinitesimal calculus, he created a logical calculus (calculus ratiocinator, universalis, logicus, or rationalis) that was to lend the same certainty to deductions concerning concepts as that possessed by algebraic deduction. Leibniz stands here at the very beginning of formal logic in the modern sense, especially in relation to the older syllogistics, which he succeeded in casting into the form of a calculus. A number of different steps may be distinguished in his program for a logical calculus. In 1679 various versions of an arithmetical calculus appeared that permitted a representation of a conjunction of predicates by the product of prime numbers assigned to the individual predicates. In order to solve the problem of negation—needed in the syllogistic modes—negative numbers were introduced for the nonpredicates of a concept. Every concept was assigned a pair of numbers having no common factor, in which the factors of the first represented the predicates and the factors of the second represented the nonpredicates of the concept. Because this arithmetical calculus became too complex, Leibniz replaced it in about 1686 by plans for an algebraic calculus treating the identity of concepts and the inclusion of one concept in another. The components of this calculus were the symbols for predicates, a, b, c, ... (termini), an operational sign - (non), four relational signs ?, ?, =, ? (represented in language by est, non est, sunt idem or eadem sunt, diversa sunt) and the logical particles in vernacular form. To the rules of the calculus (principia calculi)—as distinct from the axioms (propositiones per se verae) and hypotheses (propositiones positae) which constitute its foundation —belong the principles of implication and logical equivalence and also a substitution formula. Among the theses (propositiones verae) that can be proved with the aid of the axioms and hypotheses, such as a ? a (reflexivity of the relation ?) and a ? b et b ? c implies a ? c (transitivity of ?), is the proposition a ? b et d ? c implies ad ? bc. This was called by Leibniz the “admirable theorem” (praeclarum theorema) and appears again, much later, with Russell and Whitehead.21

Leibniz extended this algebraic calculus in various ways, first with a predicate-constant ens (or res), which may be understood as a precursor of the existential quantifier, and secondly with the interpretation of the predicates as propositions instead of concepts. Inclusion between concepts becomes implication between propositions and the new predicate-constant ens appears as the truth value (verum), intensionally designated as possibile. These discourses were concluded in about 1690 with two calculi22 in which a transition is made from an (intensional) logic of concepts to a logic of classes. The first, originally entitled Non inelegans specimen demonstrandi in abstractis (a “plus-minus calculus”), is a pure calculus of classes (a dualization of the thesis of the original algebraic calculus) in which a new predicate-constant nihil (for non-ens) is introduced. The second calculus (a “plus calculus”) is an abstract calculus for which an extensional as well as an intensional interpretation is expressly given. Logical addition in the “plus-minus calculus” is symbolized by +. In the “plus calculus,” logical addition, as well as logical multiplication in the intensional sense, is symbolized by ?, while the relational sign = (sunt idem or eadem sunt) is replaced by ? and the sign ? by non A ? B. Furthermore, subtraction appears in the “plus-minus calculus,” symbolized by - or ?, and also the relation of incompatibility (incommunicantia sunt) together with its negation (communicantia