Page 509 - Encyclopedia of Philosophy of Language

plex language expressions are to be derived from their primitive parts. This combination of doctrines makes Leibniz a precursor of the twentieth-century's logical atomism of Wittgenstein and of Russell (for a more thorough discussion see Ishiguro 1990).
Perhaps it should be stressed that Leibniz's phil- osophy of language is so intimately connected with his ontology, his theory of knowledge, and his logic, that it can only be properly understood as an integral part of his complete philosophical system. (Rec- ommended reading is Leibniz 1981, Book III: Of
Words.)
4. Logic
The first two millennia of formal logic can be sum- marized in terms of three giant steps and a tragedy. First, Aristotle created logic and introduced the syl- logism. Then Leibniz turned the subject into math- ematics. Finally, Frege invented the predicate calculus. The tragedy is that the second step remained unnoticed. This made it possible for Kant to remark in 1800 that the discipline of logic was apparently finished, as there had been no essential developments since Aristotle. Another consequence was that Boole, by redoing the second step, became recognized as the founder of mathematical logic.
In numerous unfinished and often fragmentary manuscripts Leibniz put down his attempts to develop a completely formalized system of logic. The algebra- ization of logic was a cornerstone of the project of the Characteristica Universalis and the Calculus Rati- ocinator. The possibility of reducing reasoning to mere calculation rested on the premise that logical prin- ciples could be laid down in algebraic equations, thereby making them suitable for mathematical manipulation.
Adhering to the tradition of syllogistic logic, Leib- niz attempted to render syllogistic reasoning as equa- tional reasoning in an algebra of classes, much in the way George Boole would propose later. For Leibniz, the basic sentence form is: 'A is B,' where the term A is called the subject and B the predicate. Terms can be conjoined: if, e.g., A stands for 'red,' and B stands for 'flower,' then the term AB would stand for 'red flower.' The subject-predicate sentence 'A is B' will now be true, according to Leibniz, if the predicate is contained in the subject, that is, if for some term Y thereispositedtheequationA=BY.Leibnizusesone basic principle of proof; equal terms may be sub- stituted for each other. A simple application is a deri- vation of 'A is C' from the premises 'A is B' and 'B is C: if A=BY and B=CZ, then A=CZY. Note the conjoining is here assumed to be associative; Leibniz leaves this assumption implicit in his notation. In his manuscripts Leibniz has several tentative axio- matizations of his logical system, involving typical Boolean equations such as AB= BA and AA= A. It
is important to remark that Leibniz regards his terms as intensional, although he is aware of the possibility of an extensional reading of the same formalism as well.
A few other advances in logic deserve mention. Through the work of the German logician Jungius, Leibniz was aware of the fact that relations caused problems in logical analysis. Accordingly, Leibniz took pains to demonstrate that arguments involving relational expressions could always be reduced to arguments dealing only with sentences in subject- predicate form.
It is well-known that Leibniz introduced the con- cept of 'possible world' and argued that from the perfection of God it follows that humans must live in the best one. But he also used the notion of possible world in his analysis of necessary and contingent truths, the necessary ones being those holding in all possible worlds. He recognized the issue of cross- world identity, reducing it to similarity of individual concepts.
Finally a word on Leibniz's famous principle of the identity of indiscernables. According to Ishiguro (1990) it is often confused with another principle of Leibniz: that of'substitutivity salva veritate.' The mat- ter is rather subtle, but approximates to this. The first principle says that if what is true of A is true of B and vice versa, then A and B are the same. It offers a constraint on ontology. In contrast, substitutivity salva veritate acts as a metalogical principle: terms that can be substituted for each other in any proposition without affecting its truth value, are equal.
See also: Logic: Historical Survey; Possible Worlds. Bibliography
Recommended further reading on Leibniz's logic: Leibniz 1966; Kneale and Kneale 1961; Ishiguro 1990. Arens H 1969 Sprachwissenschaft: der Gang ihrer Entwick-
lung von der Antike bis zur Gegenwart. 2nd edn. Alber,
Freiburg and Munich
Ishiguro H 1990 Leibniz's Philosophy of Logic and Language,
2nd edn. Cambridge University Press, Cambridge Kneale W, Kneale M 1961 The Development of Logic. Oxford
University Press, Oxford
Leibniz G W 1966 Logical Papers. Parkinson G H R (ed.)
Oxford University Press, Oxford
Leibniz G W 1969 Philosophical Papers and Letters. 2nd edn.
Loemker L E (ed.) Reidel, Dordrecht
Leibniz G W 1981 New Essays on Human Understanding.
Remnant P, Bennett J (eds.) Cambridge University Press,
Cambridge
Locke J 1975 An Essay Concerning Human Understanding.
Nidditch P H (ed.) Clarendon Press, Oxford
MacDonald R G 1984 Leibniz. Oxford University Press,
Oxford
Robins R H 1990/4 Short History of Linguistics, 3rd edn.
Longman, London
Leibniz, Gottfried Wilhelm
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