Page 298 - Encyclopedia of Philosophy of Language

Language and Logic
possible to reform language so as to make it an instru- ment of instruction and knowledge rather than an obstacle. For example, it was noted (the example is not Leibniz's own) that English speakers have to learn the word 'cow' in order to be able to refer to these creatures, but that the effort put into this gives the learner little more return than the piece of knowledge that 'cow' (this particular sound pattern) refers to cows. How much better it would be if, in learning the word, one also learned some information about cows. That is, why not redesign language so that it forms reflected scientific knowledge of the world? In this way, when someone went to the trouble of learning the language, he would as well learn, from the symbolism itself, a lot about cows.
The 'universal character' of Leibniz had the above as one of its goals. This goal required, improbably, that knowledge be complete or nearly so, but that did not seem an obstacle to Leibniz and others. But Leibniz's project had a feature that distinguished it from other such projects—a feature that is important for the development of logic. For Leibniz thought that the construction of the universal character could go hand in hand with a mechanization of reasoning. That is, he thought that one could develop a set of rules which could be mechanically applied to the premises of any argument (formulated in the universal charac- ter) so that the conclusion would then arise by a cal- culation not unlike that in algebra. Most famously, he thought that disputes could be settled not by the usual wrangling, but by participants sitting down to calculate. In aid of this project, Leibniz investigated syntax or purely formal structure, hoping to develop a general theory of reasoning by this means. In con-
ception if not in execution, his efforts in this direction bear a striking resemblance to projects that came later, and which culminated in the formal systems of logic we have today.
In spite of his innovative thinking about logic and language, Leibniz did not advance the subject in a detailed way, nor did he have much direct influence on his contemporaries. Part of the problem here is that Leibniz stubbornly resisted any move away from the categorical form of sentence at the heart of the syllogistic. He saw that there were special problems about the logic of relations, but he did not carry through his mechanization of reason so as to deal with them.
There is no space to discuss their work in detail, but in their different ways, Bernard Bolzano (1781-1848) in Prague and John Stuart Mill (1806-73) in Britain made important contributions to the philosophy of logic and language. It is not easy to place Bolzano in the history of the subject, inasmuch as he drew inspiration from a wide variety of sources and was distinctlyoriginal.However,itisnotunreasonableto see him as continuing in the broad tradition of Leib- niz. On the other hand, Mill's System of Logic (1843)
was clearly within the empiricist tradition, and can be seen as an attempt to provide a clear home for logic, including formal logic, within that tradition. This lat- ter task was necessary, for, as already shown, empiri- cists from Hobbes and Locke onwards tended to view logic as of no particular philosophical or scientific importance. However, it should be pointed out that neither Bolzano nor Mill made any significant con- tribution to the formal methods of logic.
See: Leibniz, G W.
6.2 The Algebra of Logic
The period after Leibniz saw great advances in various branches of mathematics, from analysis to number theory, algebra and geometry. One particularly important feature of these advances came ultimately to have crucial importance for the development of logic. That feature is usually called 'generality.' With the increasing sophistication of mathematics, it came to be appreciated that results in one branch of the quickly ramifying subject could have important conse- quences for other branches. Moreover, the way in which these consequences were often realized was not so much by direct methods, as by a process math- ematicians call 'generalization.' In this method, one takes certain features of one subject matter and thinks of them in a sufficiently abstract way to allow them to be used of a completely different subject matter. This sort of thinking might be illustrated with the very simple example in the development of the number series. If one persists in thinking of the numbers as related directly to objects that one needs to count, then negative, rational, real, and complex numbers will remain forever beyond one's ken. But if one comes to have a conception of a number series as a kind of purely formal order generated by certain operations on its elements, it will be reasonable to think of there being different sorts of numbers related to different sorts of operation.
In 1847, George Boole published Mathematical Analysis of Logic. The basic idea behind this work was that it was possible to apply algebraic laws to elements which were not themselves numbers. What was important to Boole were the relationships between the elements rather than their nature. More- over, in using algebraic relationships between these undefined elements, Boole saw no reason to keep the relationships fixed to those obtaining between numbers. For example, Boole came to think of the equation '* = 1' as capable of expressing the claim that the proposition x is true, and 'x = 0' as expressing the falsity of x. This allows a nice analogy between multiplication of 1's and O's and conjunction: 'prop- osition x and proposition y* functions in this way just like'x.>>.'However,sincethecomplexproposition*x or y will have the value false when x and y are both false, and true otherwise, it seems at first impossible
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