Although Leibniz viewed syllogisms as only part o the grander logic he envisaged, he devoted considerable time and effort to investigating syllogistic theory. Te first results o this are ound in one o the ew pieces on logic he ever published, in certain passages o A Dissertation on the Art o Combinations
(DAC , 1666, G IV 27–104), written as a student thesis when he was nineteen years old. Tis tract was organized around a large number o ‘uses’ or applications o the mathematical theory o combinations and permutations. Although the uses pertained to various fields, ranging rom law to geometry and logic, Leibniz described all these uses as instances o ‘the logic o invention’. It is in one o the uses o this combinatorial logic o invention that a systematic treatment o syllogistics is ound. Te problem Leibniz set himsel was to determine the number o valid types o syllogisms. In answering it, he proceeded by steps, employing combinatorics and traditional rules or the syllogism in turn.
Te actual procedure he used was not the most simple, and a short description o it may serve to give an impression o Leibniz’s tract on combinations. Leibniz started out by distinguishing our possible quantities a proposition may have, namely universal, particular, singular and indefinite. He combined these in groups o three, as in each syllogism three propositions occur. Next, he sorted out which o the sixty-our resulting combinations may give rise to a valid syllogism, using such rules as ‘rom pure particulars nothing ollows’, which lef thirty-two ‘useul’ moods. Subsequently, he combined these with the useul moods with respect to the two qualities, affi rmative (A) and negative (N), o which there are only three, namely AAA (both premises and the conclusion affi rmative), NAN, and ANN , resulting in thirty-two times three, which equals ninety-six useul moods. He then applied rules that apply to specific figures. O the ninety-six useul moods, eight turned out to be valid in none o the our figures. Tis lef eighty-eight useul moods, which Leibniz urther reduced by giving up the distinction o our quantities he had started out with. He now
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equated singular propositions to universal ones, and indefinite propositions to particular ones, so that ultimately only universal and particular propositions remained. Tis rather roundabout procedure resulted in the identification o twenty-our types o valid syllogisms, neatly divided into six types in each o the our figures.
Tat combinatorics has a role to play in determining the number o valid
syllogisms was as obvious in Leibniz’s time as it is now. For example, a similar, though different procedure can be ound, in the Port Royal logic (Arnauld and Nicole 1662, part 3, ch. 4). However, there was no consensus on the exact number o valid syllogisms when Leibniz wrote this. Contemporary authors held different views, some wishing to exclude subalternate moods such as Barbari and Celaront (Arnauld and Nicole part 3, ch. 3, rule 2, corollary 4), and others denying that the ourth figure should be regarded as a genuine figure at all (Sanderson 1672 part 3, ch. 4; Wallis 1687, part 3, ch. 9). Leibniz insisted, also in later writings, that the ourth figure is as legitimate as the other three.
Leibniz returned to syllogistics several times in later years, since he was interested in providing it with a solid theoretical oundation. Among his papers is one that is devoted to the so-called reduction o syllogistic moods. It was a topic already treated by Aristotle, who was concerned to show that all valid syllogistic orms are reducible to the our ‘perect’ syllogisms o the first figure that were aferwards labeled Barbara, Celarent, Darii and Ferio, whose validity he assumed to be sel-evident. Leibniz proposed a similar way o proving the validity o syllogistic orms, but systematizing the procedure. He first showed that the our perect syllogisms derive their certainty rom an axiom o ‘no less geometrical certainty than i it were said that that which contains a whole contains a part o the whole’ (P 106), and which was known rom scholastic times as ‘the dictum de omni et nullo’. It says that whatever is affi rmed or denied o the members o a class is also affi rmed or denied o the members o a subclass o that class. Now this is what is expressed, as ar as the affi rmative part goes, by Barbara and Darii, or the whole or part o a subclass respectively, and similarly in the negative case by Celarent and Ferio. As a next step, Leibniz proved subalternation by means o Darii, and the ‘identical’ statement ‘some A is A’, assumed to be sel-evidently true, as ollows: ‘Every A is B, some A is A, thereore, some A is B.’ He proved the negative case in a similar way by means o Ferio: ‘No A is B, some A is A, thereore some A is not B’. Once subalternation was proved, two urther moods o the first figure could be derived: Barbari and Celaro, in which a particular conclusion replaces, by subalternation, the universal ones o Barbara and Celarent, respectively.
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Having thus established six moods in the first figure, Leibniz derived rom each o these a valid mood in the second, and a valid mood in the third figure, using a single principle, which he called ‘regress’. It was traditionally also known as ‘reductio per impossibile’, and was used by Aristotle. It works by assuming that the conclusion o a valid syllogism is alse and demonstrating that on this assumption one o the premises must be alse as well. Tus, assuming that the
conclusion o a syllogism in Barbara is alse and its major premise is true, it ollows that the minor must be alse. Replacing the alse propositions by their respective contradictories and interchanging the minor and the conclusion results in a valid syllogism o the second figure, Baroco: rom ‘Every C is D, every B is C, thereore every B is D’ (Barbara) results ‘Every C is D, some B is not D (contradictory o the conclusion), thereore some B is not C (contradictory o the minor) (Baroco). Analogously, a syllogism o the third figure Bocardo is derived by assuming the conclusion and the major premise, rather than the minor, o a syllogism in Barbara to be alse. Tis procedure applied to each o the six moods o the first figure, so that all six valid moods in the second, and all six valid moods in the third figure could be derived.
Leibniz’s point in proposing this procedure was to show that a uniorm method could be used or deriving all the valid moods o the second and third figure. Furthermore, he maintained that it was the best method o proo, as it was synthetic rather than analytic, which meant that it contained ‘the method by which they could have been discovered’ (P 110). However, the tidy systematicity o the procedure did not extend to the moods o the ourth figure, the derivation o which requires the principle o conversion (e.g. ‘No A is B, thereore no B is A’) that Leibniz had been able to avoid so ar. In sum, Leibniz systematized the meta- theory o syllogistics, taking the dictum de omni et nullo as an axiom, and using identical propositions, subalternation, the method o regress, which as he noted presupposes the principle o contradiction, and finally conversion as urther principles to prove the validity o syllogistic moods in all our figures. All these principles were traditional, except the use o ‘identical’ propositions, which Leibniz employed in proving principles such as subalternation and conversion that were usually taken or granted without proo. Tis use o identical propositions was, as Leibniz noted, invented by Peter Ramus (Couturat 1901: 8; G IV 55).
A urther example o Leibniz’s efforts to investigate the theoretical basis o syllogistics is ound in a paper entitled ‘A Mathematics o Reason’ (P 95–104). Again, he used insights that had been developed by logicians working beore him, but aiming at a more rigorous treatment. In this case, it was the theory o
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the distribution o terms that Leibniz took as a starting point. Medieval logicians had observed that in categorical propositions some terms are ‘distributed’, and others are ‘undistributed’, meaning that only a distributed term applies to every individual belonging to the class denoted by the term. Tus, in a universal affi rmative proposition such as ‘All horses are animals’ the term ‘horse’ is distributed, whereas the term ‘animal’ is not, since the proposition is about every
individual horse, but not about every individual animal. On the basis o this criterion, it can be established that subject terms in universal propositions and predicate terms in negative propositions are distributed. Te distribution o terms lay at the basis o several rules that could be used as a test or the validity o a syllogism; or example, in every valid syllogism, the middle term should be distributed in at least one o the premises. In ‘A Mathematics o Reason’, Leibniz called distributed terms ‘universal’ and undistributed terms ‘particular’. He discussed the traditional rules concerning distribution, providing a justification or them: or example, i the middle term is particular (i.e. undistributed) in both premises, nothing can be concluded rom the premises, because there is no guarantee that the same individuals are denoted by both occurrences o the middle term in each premise. He also enumerated and explained a series o other rules and observations, such as ‘i the conclusion is a universal affi rmative, the syllogism must be in the first figure’, and ‘in the second figure, the major proposition is universal and the conclusion negative’, all o which he could justiy on the basis o principles and corollaries he had proved first. It is clear, however, that, just as with the reduction o syllogistic moods, Leibniz was working within a traditional ramework, and putting orward results that or the most part were already known.
By contrast, a third example o Leibniz’s concern with syllogistics constituted a distinctive novelty. In a urther attempt to assimilate logic to geometry, he employed diagrams o various sorts in representing the our traditional types o proposition. At first, he used circles to represent the subject and predicate terms, and made the way they did or did not overlap indicate the quality and quantity o the proposition. Similar circles are usually called Venn diagrams today, but it was Leibniz who introduced their use as a graphical representation o terms and their interrelations. A second type o diagram that Leibniz devised consisted o parallel horizontal lines representing terms, with their overlap in a vertical direction indicating how the terms were related. For example, a universal affi rmative proposition was represented by a horizontal line symbolizing the predicate term, while the subject term was symbolized by a shorter parallel line drawn under it and nowhere extending beyond the longer line. Tus, it could be
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read off rom the diagram that the subject term and the predicate term coincided in part. By adding a third line, an entire syllogism could be represented by this means, and the validity o an inerence rom the premises was made apparent by the resulting diagram.