Version: v1, Published online: 1998

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## 10. Logic and language

From his youth, Leibniz dreamed of constructing a perfect, logical language, ‘a certain alphabet of human thoughts that, through the combination of the letters of this alphabet and through the analysis of the words produced from them, all things can both be discovered and judged’. This programme, which Leibniz called the ‘universal characteristic’, gets its first expression in the very early work, Dissertatio de arte combinatoria (Dissertation on the art of combinations) (1666). But it is most fully developed later, from the mid-1670s into the 1680s.

Leibniz’s programme had two parts. First, one must assign characteristic numbers to all concepts that show how they are built up out of simpler concepts. Leibniz tried a number of schemes for this, but one strategy was to assign simple concepts prime numbers, and then assign complex concepts the product of the characteristic numbers of its constituent simple concepts. The second part of the programme was then to find simple mechanical rules for the truth of propositions in terms of the characteristic numbers of their constituent concepts. Leibniz’s fundamental rule in his Universal Characteristic was the principle discussed above in connection with his metaphysics: a predicate is true of a subject if and only if its concept is contained in the concept of the subject. If the concepts in question can be expressed numerically, then Leibniz thought that the rule can be given a mathematical form as well, and the truth of a proposition could be established by a simple arithmetical calculation. Leibniz’s project in these writings was to show how this basic intuition about truth could be extended to propositions that are not in simple subject-predicate form. He also sought to extend the programme to formalize the validity of the standard inferences in Aristotelian logic. Even if he could not assign definite characteristic numbers to particular concepts, Leibniz tried to show that for certain configurations of premises and conclusions, if the premises are true (on his definition of truth), then so too must be the conclusion.

The programme was very ambitious; it if were successful, it would allow the truth or falsity of any proposition, necessary or contingent, to be determined by calculation alone. However, it soon dawned on Leibniz that the idea of finding all the conceptual dependencies necessary to express the contents of notions numerically was utopian in the extreme, particularly given the doctrine of infinite analysis of contingent truths Leibniz came to in the late 1680s. This realization still left in place the more modest programme of validating patterns of inference. But even this more modest programme turned out to be beyond Leibniz’s ability to bring to completion, and after the early 1690s he seems to have given up trying to make it work, although he returned to it from time to time.

But even though this particular programme collapsed, the idea of formalism was quite basic to Leibniz’s thought. Part of the reaction against the Aristotelian philosophy of the schools was an attack on formal logic. Descartes, Locke and others in the seventeenth century argued that we all have an innate ability to recognize truth, what was often called intuition, and that we should cultivate that capacity, and not waste our time learning formal rules. While Leibniz certainly agreed that we do have the innate capacity to grasp certain truths, he still thought that formalism is very important ((Leibniz to Elisabeth of Bohemia, 1678). Much of our reasoning is ‘blind’ or symbolic, Leibniz thought, conducted through the manipulation of symbols without having a direct hold on the ideas that underlie the symbols. For that reason we must have clear and unambiguous symbol systems, and strict rules for manipulating them (Meditations).

This view is evident in the papers on the Universal Characteristic. But it also underlies another project of the same period, the differential and integral calculus, one of Leibniz’s greatest accomplishments, worked out by 1676 and made public from 1684. Though others before him had solved many of the particular problems his calculus could solve, problems relating to tangents, areas, volumes and so on, Leibniz invented a simple notation, still used in the calculus (‘d’ to represent the operation of differentiation, and ‘∫’ to represent the operation of infinite summation (integration)), and worked out a collection of simple rules for applying these operations to equations of different kinds. In this way, Leibniz was able to produce simple algorithms for solving difficult geometrical problems ‘blindly’, by manipulating certain symbols in accordance with simple rules.

Another issue closely connected with Leibniz’s logic is that of relations. In the Primae veritates (First truths)
[1689] 1989: 32) Leibniz wrote: ‘*There are no purely extrinsic denominations* [that is, purely relational properties], denominations which have absolutely no foundation in the very thing denominated.…And consequently, whenever the denomination of a thing is changed, there must be a variation in the thing itself’. In this way, all relations must be, in some sense, grounded in the non- relational properties of things. But it is not clear that Leibniz held that relations had to be reducible to non-relational predicates of things. In one example he gives, he paraphrased ‘Paris is the lover of Helen’ by the following proposition: ‘Paris loves, and by that very fact [*eo ipso*] Helen is loved’. While this certainly relates the relation ‘*A* loves *B*’ to two propositions that have the form of simple subject-predicate propositions (‘*A* loves’ and ‘*B* is loved’), it should be noted that the predicates in question (‘loves’ and ‘is loved’) would seem to be implicitly relational; whether this is an accidental feature of the example Leibniz chose or a clue to Leibniz’s views is a question of some dispute. Furthermore, it is important not to ignore that which connects the two propositions (‘and by that very fact’), without which one cannot say that the two non-relational propositions capture the relation ‘*A* loves *B*’ (Leibniz 1966: 14). Other texts suggest that individuals properly speaking have non-relational properties, and that the relations between things are something imposed by the mind onto the world: ‘My judgement about relations is that paternity in David is one thing, sonship in Solomon another, but that the relation common to both is a merely mental thing whose basis is the modifications of the individuals’ (Leibniz to Des Bosses, 21 April 1714). But in saying that the relations between individuals are ‘merely mental’, Leibniz does not necessarily mean to dismiss them. He wrote: ‘God not only sees individual monads and the modifications of every monad whatsoever, but he also sees their relations, and in this consists the reality of relations and of truth’ (Letter to Des Bosses, 5 February 1712).

In addition to formal languages, Leibniz was also keenly interested in the study of natural languages. Like many of his contemporaries, he was interested in the controversies over the question of the Adamic language, the language spoken in Eden and from which all modern languages supposedly derive. This, among other motivations, led him to the empirical study of different languages and the etymology of words (see Universal language).

Garber, Daniel. Logic and language. Leibniz, Gottfried Wilhelm (1646–1716), 1998, doi:10.4324/9780415249126-DA052-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/biographical/leibniz-gottfried-wilhelm-1646-1716/v-1/sections/logic-and-language-2.

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