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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
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