Leibniz’s tract on combinations contained discussion o a urther logical topic that he himsel considered so important that he mentioned it in retrospect as one o the main reasons why this otherwise immature work was still valuable. Tis topic was ‘the alphabet o human thoughts’ – an idea that was linked to a line o thinking that can be ound with numerous philosophers rom Plato to Descartes. It entailed, roughly, that everything we know and think can be broken down into smaller units. Just as sentences are composed o words, and words consist o letters or speech sounds, so is the thought expressed by a sentence composed o ideas. Ideas themselves can be analysed into components, which in their turn can be analysed. But analysis cannot go on or ever; it must stop where it reaches primitives: simple notions that have no component parts. Leibniz assumed that a class o such ideas exists, and this is what he called ‘the alphabet o human thoughts’. He made it a guiding principle or many o his logical investigations.
He derived the idea rom his study o logic textbooks when he was still a boy. He noticed that such textbooks were usually organized according to increasing levels o complexity: they started with the treatment o terms, presenting them as belonging to a hierarchical scheme o categories called genus and species, proceeded to propositions, i.e. combinations o terms, and concluded with syllogisms, i.e. combinations o propositions. But it did not seem to be transparent how the levels were connected; in particular, it was unclear how complex terms, i.e. propositions, were connected to syllogisms, since propositions were not divided into categories as simple terms were. On urther thought, he realized that the connection between the level o terms and the level o propositions was unclear as well, and that the categories o simple terms also needed revision.
Tus, in the tract on combinations he proposed to reorganize the theory o terms altogether, using combinatorial principles or a new arrangement, so that terms were classified on the basis o the number o terms o which they are composed. Tere would be a first class o primitive notions, a second class
Leibniz 109
containing concepts resulting rom combinations o two primitives, a third class o concepts in which three primitives are involved, and so on. Supposing that such a classification is given, it would become possible to produce a list o all the terms that could be a predicate in a true proposition with the given term as subject, as well as a list o terms which could be the subject o a true proposition with the given term as predicate. o illustrate this, let us arbitrarily assume that
there are only our primitive notions: a, b, c, d. Ten there are our classes in all, containing the ollowing concepts:
class I: a, b, c, d
class II : ab, ac, ad, bc, bd, cd class III : abc, abd, acd, bcd class IV: abcd
For every complex term, it can be determined which terms can be predicated o it so as to orm a true, universal proposition. For example, the term ‘ab’ has ‘a’, ‘b’ and ‘ab’ as such predicates, or ‘ab is a’, ‘ab is b’, and ‘ab is ab’ are all true propositions. I the same term ‘ab’ occurs as a predicate, the combination with the terms ‘ab’, ‘abc’, ‘abd’, and ‘abcd’ in subject position yields a true proposition, as or example ‘abc is ab’ is true. Leibniz also gave rules or determining the subjects and predicates o a given term in particular and negative propositions. On analysis, these rules turn out to be inconsistent (Kauppi 1960: 143–4; Maat 2004: 281ff.). Leibniz probably saw this eventually; he did not pursue this particular idea o computing the number o subjects and predicates in later writings. But he did not give up the general principles on which it relied.
Tese principles are, first, that all concepts result ultimately rom the combination o simpler concepts and that the entire edifice o knowledge is based upon, and reducible to a set o primitive unanalysable concepts, the alphabet o human thoughts. Secondly, it is assumed that in a true affi rmative proposition the predicate is contained in the subject, as shown in the examples ‘ab is a’ and ‘abc is ab’. Tis latter principle is what is called the intensional view o the proposition. On this view, a proposition expresses a relation between concepts, such that in a true proposition ‘All A is B’ the concept o the subject A contains the concept o the predicate B. According to the alternative, extensional, view, a proposition expresses a relation between classes or sets; a proposition such as ‘All As are Bs’ is true i the class denoted by B contains the class denoted by A. Leibniz preerred the intensional view, or several reasons. First, it fitted quite well with the classification o concepts according to combinatorial complexity. Tus, the truth o the proposition ‘man is rational’ can be proved by
Te History o Philosophical and Formal Logic
110
replacing the term ‘man’ by its definition, consisting o supposedly simpler concepts, namely ‘rational animal’. Afer this substitution, the proposition reads ‘rational animal is rational’, which shows on the surace that the concept o the predicate ‘rational’ is identical to part o the concept o the subject ‘man’. Tis type o proo relied on two principles that Leibniz later ormulated explicitly in a number o papers. First, the law o substitutivity, according to which two terms
or expressions are the same i they can be substituted or the other without affecting truth. Secondly, the ‘law o identity’, expressed as ‘A is A’, or in some equivalent orm, which became an indispensable axiom in his logical calculi. A second reason why Leibniz preerred the intensional view is that he did not want logic to be dependent on the existence o individuals. As he put it in a paper o 1679, he rather considered ‘universal concepts, i.e. ideas, and their combinations’, apparently because in this way he was dealing with propositions that can be true or alse regardless o whether individuals exist to which the concepts it contains are applicable.
A choice between the intensional and the extensional view o the proposition was not o particular importance in Leibniz’s view, because he believed that the two were ultimately equivalent, and that results obtained within the intensional perspective could be expressed in extensional terms, and vice versa, by ‘some kind o inversion’ (P 20). He noted that the extension and intension o terms are systematically related: the more individuals a term is applicable to, i.e. the greater its extension, the ewer parts its concept contains, i.e. the smaller its intension. For example, the term ‘animal’ has greater extension but smaller intension than the term ‘man’. Evaluating the proposition ‘every man is an animal’ can be done in two equivalent ways: either by checking whether the intension o ‘animal’ is contained in the intension o ‘man’, or by checking whether the extension o ‘man’ is contained in the extension o ‘animal’. Tis relationship between the intension and extension o terms has been called the ‘law o reciprocity’. It may seem a questionable principle, because it entails that i two terms have the same intension, they also have the same extension. Now this seems to be alse or numerous pairs o terms, such as terms that are coextensive (have the same extension) but differ in meaning (have different intension) such as ‘creature with a heart’ and ‘creature with a kidney’. However, Leibniz wished to consider not just individuals that in act exist, but all possible individuals, assuming that i two terms differ in intension, there would be a possible individual to which only one o the two terms applies. Conversely, i such an individual is impossible, this proves that the two terms have the same intension afer all.
Leibniz 111
Leibniz’s views concerning knowledge (as ultimately derived rom simple notions) and truth (as consisting in a relation between concepts and provable through analysis by means o definitions) were intimately connected with a scheme he first sketched in his tract on combinations, and which he pursued or the rest o his lie. Tis was the construction o an artificial language that he believed would have extremely beneficial effects: it would be a reliable tool or
thinking, so that by its means human knowledge and science could be enhanced at a pace previously unheard o.
Quite a ew seventeenth-century philosophers were preoccupied with searching or means to advance science. Tey usually sought the solution in developing new methods o research. Leibniz, by contrast, believed that the construction o a new symbolism was crucial or the improvement o science. He was not altogether unique in this respect, as schemes or artificial or ‘philosophical’ (i.e., roughly, scientific) languages were widespread in the period, with their authors ofen claiming ar-reaching advantages or their inventions. wo o the most ully developed schemes o this kind were created in England, by Dalgarno (1661) and Wilkins (1668), who claimed that the languages they had constructed were more logical and more philosophical than existing languages. Leibniz studied their schemes careully, and used the vocabularies and grammar o their languages in preliminary studies or his own project. Nevertheless, he was convinced that neither Dalgarno nor Wilkins had perceived what a truly logical and philosophical language could accomplish. Such a language would not only be a means or communication, and not just an instrument or the accurate representation o knowledge, both o which characteristics were realized to some extent by the English artificial languages, but it would first o all be a tool or making new discoveries, and or checking the correctness o inerences. Tis logical language amously made it possible to decide controversies by translating conflicting opinions into it; or this purpose, participants in a debate would sit down together and say ‘let’s calculate’ (G VII 200). Tus, Leibniz’s scheme included the construction o an encyclopedia encompassing all existing knowledge, as well as a ully general logical calculus that would comprise all sorts o logical inerences, including but not limited to syllogistic ones. Leibniz worked towards the achievement o these goals throughout his lie. He produced numerous lists o definitions, which were apparently intended or use in the encyclopedia. He also drafed many versions o a logical calculus, and he wrote a number o papers on rational grammar, investigating how existing languages express logical relationships. Te next two sections provide a brie description o some o the logical calculi, and o rational grammar, respectively.
Te History o Philosophical and Formal Logic
112