Hegel’s Science of Logic
§ 1443
1. I-P-U is the general schema of the determinate syllogism. Individuality unites with universality through particularity; the individual is not universal immediately, but through the medium of particularity; and conversely the universal similarly is not immediately individual but descends to individuality through particularity. These determinations confront each other as extremes and are united in a different third term. Each is determinateness; in this they are identical; this their general determinateness is particularity. But they are no less extremes against this particularity than against each other, because each is present in its immediate determinateness.
§ 1444
The general significance of this syllogism is that the individual, which as such is infinite self-relation and therefore would be merely inward, emerges by means of particularity into existence as into universality, in which it no longer belongs merely to itself but stands in an external relationship; conversely the individual, in separating itself into its determinateness as a particularity, is in this separation a concrete individual and, as the relation of the determinateness to itself, a universal, self-related individual, and consequently is also truly an individual; in the extreme of universality it has withdrawn from externality into itself. In the first syllogism, the syllogism's objective significance is only superficially present, since in it the determinations are not yet posited as the unity which constitutes the essence of the syllogism. It is still subjective in so far as the abstract significance possessed by its terms is not thus isolated in and for itself but only in subjective consciousness. Moreover, the relationship of individuality, particularity and universality is as we have seen the necessary and essential form-relationship of the determinations of the syllogism; the defect consists not in this determinateness of the form, but in the fact that under this form each single determination is not at the same time richer. Aristotle has confined himself rather to the mere relationship of inherence, in stating the nature of the syllogism as follows: When three terms are related to one another in such a manner that one extreme is in the whole of the middle term and this middle term is in the whole of the other extreme, then these two extremes are necessarily united in a conclusion. What is expressed here is more the mere repetition of the like relationship of inherence between one extreme to the middle term, and again between the middle term and the other extreme, than the determinateness of the three terms to one another. Now since the syllogism rests on the stated relative determinateness of the terms, it is immediately evident that other relationships of the terms which are given by the other figures can only have validity as syllogisms of the understanding [Verstandesschlusse] in so far as they can be reduced to that original relationship; they are not different species of figures which stand alongside the first; on the contrary, on the one hand, in so far as they purport to be correct syllogisms, they rest solely on the essential form of the syllogism in general, which is the first figure; on the other hand, in so far as they deviate from it they are transformations into which that first abstract form necessarily passes, thereby further determining itself and advancing to totality. We shall presently see what this process is.
§ 1445
I-P-U is thus the general schema of the syllogism in its determinateness. The individual is subsumed under the particular, and the latter under the universal; therefore the individual too is subsumed under the universal. Or the particular inheres in the individual and the universal in the particular; therefore the universal also inheres in the individual. From one aspect, namely, in relation to the universal, the particular is subject; in relation to the individual it is predicate; or, in relation to the former it is an individual, in relation to the latter it is a universal. Because the two determinatenesses are united in it, the extremes are linked together by this their unity. The therefore appears as the conclusion that has taken place in the subject, a conclusion deduced from subjective insight into the relationship between the two immediate premises. As subjective reflection enunciates the two relations of the middle term to the extremes as particular and indeed immediate judgments or propositions, the conclusion as the mediated relation is also, of course, a particular proposition, and the consequently or therefore is the expression of the fact that it is the mediated one. But this therefore is not to be regarded as an external determination in this proposition, as if it had its ground and seat only in subjective reflection; on the contrary, it is grounded in the nature of the extremes themselves whose relation again is expressed as a mere judgment or proposition only for the purpose of, and by means of, abstractive reflection, but whose true relation is posited as the middle term. That therefore I is U is a judgment, is a merely subjective circumstance; the very meaning of the syllogism is that this is not merely a judgment, that is, not a relation effected by the mere copula or the empty is, but one effected by the determinate middle term which is pregnant with content.
§ 1446
Consequently, to regard the syllogism merely as consisting of three judgments, is a formal view that ignores the relationship of the terms on which hinges the sole interest of the syllogism. It is altogether a merely subjective reflection that splits the relation of the terms into separate premises and a conclusion distinct from them: ®
All men are mortal,
Gaius is a man,
Therefore he is mortal.
§ 1447
At the approach of this kind of syllogism we are at once seized with a feeling of boredom; this stems from that unprofitable form which by means of the separate propositions presents a semblance of difference that immediately dissolves in the fact itself. It is mainly this subjective shape that gives the syllogistic process the appearance of a subjective makeshift, to which reason or understanding resorts when it cannot cognise immediately. The nature of things, the rational element, certainly does not set to work by first framing for itself a major premise, the relation of a particularity to a subsistent universal, and then secondly, picking up a separate relation of an individuality to the particular, out of which thirdly and lastly a new proposition comes to light. This syllogistic process that advances by means of separate propositions is nothing but a subjective form; the nature of the fact is that the differentiated Notion determinations of the fact are united in the essential unity.
§ 1448
This rationality is not a makeshift; on the contrary, in contrast to the immediacy of relation that still obtains in the judgment, it is the objective element; and the former immediacy of cognition is rather the merely subjective element, whereas the syllogism is the truth of the judgment. Everything is a syllogism, a Universal that through particularity is united with individuality; but it is certainly not a whole consisting of three propositions. ®
§ 1449
2. In the immediate syllogism of the understanding the terms have the form of immediate determinations; it is now to be considered from this aspect according to which they are a content. It may thus be regarded as the qualitative syllogism, just as the judgment of existence has the same aspect of qualitative determination. Accordingly the terms of this syllogism, like the terms of that judgment, are individual determinatenesses, the determinateness through its self-relation being posited as indifferent to the form and hence as content. The individual is any immediate concrete object; particularity a single one of its determinatenesses, properties or relationships; universality again a still more abstract, more individual determinateness in the particular' Since the subject as immediately determined is not yet posited in its Notion, its concretion is not reduced to the essential Notion determinations; its self-related determinateness is, therefore, an indeterminate, infinite multiplicity. In this immediacy the individual possesses an infinite number of determinatenesses which belong to its particularity, each of which therefore may constitute a middle term for it in a syllogism. But through any other middle term it is united with another universal; through each of its properties it stands in a different connection and context of existence. Further, the middle term is also a concrete in comparison with the universal; it contains several predicates itself, and the individual in turn can be united through the same middle term with several universals. In general, therefore, it is completely contingent and arbitrary which of the many properties is adopted for the purpose of connecting it with a predicate; other middle terms are transitions to other predicates, and even the same middle term may by itself be a transition to various predicates, for as a particular against the universal it contains several determinations.
§ 1450
But not only is an indefinite number of syllogisms equally possible for one subject and not only is any single syllogism contingent in respect of its content, but these syllogisms that concern the same subject must also pass over into contradiction. For difference in general, which in the first instance is an indifferent diversity, is no less essentially opposition. The concrete is no longer something belonging merely to the sphere of Appearance, but is concrete through the unity in the Notion of the opposites that have become determined as moments of the Notion. Now when, in accordance with the qualitative nature of the terms in the formal syllogism, the concrete is grasped in respect of a single one of the determinations belonging to it, the syllogism assigns to it the predicate corresponding to this middle term; but as from another side the opposite determinateness is inferred, this shows the former conclusion to be false, although its premises by themselves and equally its inference are quite correct. If from the middle term, that a wall has been painted blue, it is inferred that therefore the wall is blue, this is a correct inference; yet in spite of this syllogism the wall can be green if it has also been painted over yellow, from which latter circumstance taken by itself it would follow that it was yellow. If from the middle term of sense-nature it is inferred that man is neither good nor evil, because neither the one nor the other can be predicated of sense, the syllogism is correct but the conclusion is false; because as man is a concrete being, the middle term of spirituality is equally valid. From the middle term of the gravitation of the planets, satellites and comets towards the sun, it correctly follows that these bodies fall into the sun; but they do not fall into it, because each is no less its own centre of gravity, or, as it is said, they are impelled by centrifugal force. Similarly, from the middle term of sociality we can deduce the community of goods among citizens, but from the middle term of individuality, if it is pursued with equal abstractness, there follows the dissolution of the state, as has happened for example in the Holy Roman Empire from holding to the latter middle term. It is justly held that there is nothing so inadequate as a formal syllogism of this kind, since it is a matter of chance or caprice which middle term is employed. No matter how elegantly a deduction of this kind has run its course through syllogisms, however fully its correctness may be conceded it still leads to nothing of the slightest consequence, for the fact always remains that there are still other middle terms from which the exact opposite can be deduced with equal correctness. The Kantian antinomies of reason amount to nothing more than that from a notion first one of its determinations is laid down as basis, and then with equal necessity, the other. In these cases this inadequacy and contingency of a syllogism must not merely be shifted on to the content, as though these defects were independent of the form and the latter alone were the concern of logic. On the contrary, it lies in the form of the formal syllogism that the content is such a one-sided quality; it is determined to this one-sidedness by the said abstract form. ®
§ 1451
It is, namely, one single quality of the many qualities or determinations of a concrete object or Notion because according to the form it is not supposed to be anything more than such an immediate, single determinateness. The extreme of individuality, as abstract individuality, is the immediate concrete, consequently the infinite or indeterminable manifold; the middle term is the equally abstract particularity, consequently a single one of these manifold qualities, and similarly the other extreme is the abstract universal. Therefore it is essentially on account of its form that the formal syllogism is wholly contingent as regards its content; and contingent, not in the sense that it is contingent for the syllogism whether this or another object be submitted to itlogic abstracts from this content-but in so far as a subject forms the basis it is contingent what kind of content determinations the syllogism shall infer from it.
§ 1452
3. The determinations of the syllogism are determinations of content in so far as they are immediate, abstract, and reflected into themselves. But their essential nature, on the contrary, is that they are not such mutually indifferent, intro-reflected determinations, but determinations of form; as such they are essentially relations. These relations are first, those of the extremes to the middle term-relations that are immediate, the propositiones praemissae, and, to name them, the propositio major, the relation of the particular to the universal, and the propositio minor, the relation of the individual to the particular. Secondly we have the relation of the extremes to one another, which is the mediated relation, conclusio. The immediate relations, the premises, are propositions or judgments in general, and contradict the nature Of the syllogism according to which the different Notion determinations are not related immediately, but their unity should also be posited; the truth of the judgment is the syllogism. All the less can the premises remain immediate relations, since their content consists of immediately distinguished determinations and they are therefore not immediately identical in and for themselves-unless they are pure, identical propositions, that is empty tautologies that lead to nothing.
§ 1453
Accordingly, it is commonly demanded of the premises that they shall be proved, that is, that they likewise shall be presented as conclusions. Consequently, the two premises yield two further syllogisms. But these two new syllogisms in turn yield between them four premises which demand four new syllogisms; these have eight premises whose eight syllogisms in turn yield for their sixteen premises sixteen syllogisms, and so on in a geometrical progression to infinity.
§ 1454
Thus there here comes to view again the progress to infinity which appeared before in the lowlier sphere of being, but which was no longer to be expected in the domain of the Notion, of the absolute reflection-into-self out of the finite, in the region of free infinitude and truth. It was shown in the sphere of being that whenever we find the spurious infinite that runs away into a progression, we are faced with the contradiction of a qualitative being and an impotent ought-to-be that goes out and away beyond it; the progression itself is the repetition of the demand for unity in opposition to the qualitative, and of the persistent relapse into the limitation which is inadequate to that demand. Now in the formal syllogism the immediate relation or the qualitative judgment is the basis, and the mediation of the syllogism that which is posited as the higher truth over against it. The unending process of proving the premises does not resolve this contradiction but only perpetually renews it and is the repetition of one and the same original defect. The truth of the infinite progression consists, on the contrary, in the sublation of the progression itself and the form which is already determined by it as defective. This form is that of mediation as I-P-U. The two relations I-P and P-U are to be mediated relations; if this is effected in the same way the defective form I-P-U will merely be duplicated, and so on to infinity. P has to I also the form determination of a universal, and to U the form determination of an individual, because these relations are, in general, judgments. Therefore they require mediation; but mediation in the form just mentioned only results in the re-appearance of the relationship that was to have been sublated.
§ 1455
The mediation must therefore be effected in another manner. For the mediation of P-U, we have I; accordingly the mediation must take the form P-I-U. To mediate I-P, we have U; this mediation therefore becomes the syllogism I-U-P.
When we examine this transition in the light of its Notion, we find in the first place that the mediation of the formal syllogism is, as has been shown, contingent in respect of its content. The immediate individual has in its determinatenesses an indeterminable number of middle terms, and these in turn have a similar number of determinatenesses in general; so that it depends entirely on an external caprice or simply on some external circumstance and contingent determination, with what kind of a universal the subject of the syllogism shall be united. As regards content, therefore, the mediation is not anything necessary or universal; it is not grounded in the Notion of the fact; on the contrary, the ground of the syllogism is something external to it, that is, something immediate; but among the Notion-determinations the immediate is the individual. As regards the form, the mediation likewise has for its presupposition the immediacy of the relation; therefore the mediation is itself mediated, and mediated by the immediate, that is, the individual. More precisely, through the conclusion of the first syllogism, the individual has become the mediating factor. The conclusion is I-U; the individual is thereby posited as a universal In one premise, the minor I-P, it is already present as a particular; hence it is that in which these two determinations are united. Or we may say that the conclusion in and for itself enunciates the individual as a universal, and that too not in an immediate manner but through mediation; consequently as a necessary relation. The simple particularity was the middle term; in the conclusion this particularity is posited in its developed form as the relation of the individual and universality. But the universal is still a qualitative determinateness, a predicate of the individual; the individual in being determined as a universal is posited as the universality of the extremes or as the middle term; by itself it is the extreme of individuality, but because it is now determined as a universal it is at the same time the unity of the two extremes.
§ 1456
1. The truth of the first qualitative syllogism is that something is united with a qualitative determinateness as a universal, not in and for itself but through a contingency or in an individuality. In such a quality, the subject of the syllogism has not returned into its Notion, but is apprehended only in its externality; immediacy constitutes the ground of the relation and consequently the mediation; thus the individual is in truth the middle term.
§ 1457
But further, the syllogistic relation is the sublation of the immediacy; the conclusion is not an immediate relation but relation by means of a third term; it contains therefore a negative unity; consequently the mediation is now determined as possessing within itself a negative moment.
§ 1458
In this second syllogism the premises are: P-I and I-U; only the first of these premises is still immediate; the second, I-U, is already mediated, namely by the first syllogism. The second syllogism therefore presupposes the first, just as conversely the first presupposes the second. Here the two extremes are distinguished as particular and universal; thus the latter still keeps its place; it is predicate. But the particular has changed places; it is subject, or posited in the determination of the extreme of individuality, just as the individual is posited in the determination of middle term, or of particularity. Both are therefore no longer the abstract immediacies that they were in the first syllogism. However, they are not yet posited as concretes; in standing in the place of the other, each is posited in its own determination and at the same time, though only externally, in the determination of the other.
§ 1459
The specific and objective meaning of this syllogism is that the universal is not in and for itself a determinate particular — for on the contrary it is the totality of its particulars — but is one such of its species through the medium of individuality; the rest of its species are excluded from it by the immediate externality. On the other hand the particular likewise is not immediately and in and for itself the universal, but the negative unity strips it of its determinateness and thereby raises it into universality. The individuality stands in a negative relationship to the particular in so far as it is supposed to be its predicate; it is not predicate of the particular.
§ 1460
2. But in the first instance the terms are still immediate determinatenesses; they have not of themselves developed into any objective significance; the altered position which two of them occupy is the form, which is as yet only external to them. Therefore they are still, as in the first syllogism, simply a mutually indifferent content-two qualities that are connected, not in and for themselves, but by means of a contingent individuality.
§ 1461
The syllogism of the first figure was the immediate syllogism or, otherwise expressed, the syllogism in its Notion as abstract form which has not yet realised itself in its determinations. The transition of this pure form into another figure is on the one hand the beginning of the realisation of the Notion, in that the negative moment of mediation, and thereby a further determinateness of form, is posited in the initially immediate qualitative determinateness of the terms. But this is at the same time an alteration of the pure form of the syllogism; the latter no longer completely corresponds to its pure form, and the determinateness posited in its terms differs from the original form determination. Regarded merely as a subjective syllogism proceeding in an external reflection, it counts as a species of syllogism which ought to correspond to the genus, namely to the general schema I-P-U. But to begin with it does not correspond to this; its two premises are P-I, or I-P, and I-U; hence the middle term is in both cases subsumed or in both cases the subject, in which accordingly the two other terms inhere. It is therefore not a middle term, for this should on the one hand subsume or be predicate, and on the other hand be subsumed or be subject, or one of the terms should inhere in it while it itself inheres in the other. The true meaning of the fact that this syllogism does not correspond to the general form of the syllogism, is that the general form has passed over into this syllogism since the truth of that form consists in its being a subjective and contingent connecting of the terms. If the conclusion in the second figure (that is, without taking advantage of the restriction about to be mentioned, which converts it into something indeterminate) is correct, then it is so because it is so on its own account, not because it is the conclusion of this syllogism. But the same is the case with the conclusion of the first figure; it is this, its truth, that is posited by the second figure. In the view that holds the second figure to be merely one species, the necessary transition of the first into this second form is overlooked and the former is adhered to as the true form. Consequently, if in the second figure (which from ancient custom is quoted without further reasons as the third) we are likewise supposed to have a syllogism correct in this subjective sense, then it would have to be conformable to the first; hence, since one premise I-U has the relationship of the subsumption of the middle term under one extreme, then it would have to be possible to give the other premise, P-I, the opposite relation to that which it has and to subsume P under I. But such a relation would be the sublation of the determinate judgment I is P, and could only occur in an indeterminate, in a particular judgment; consequently the conclusion in this figure can only be particular. The particular judgment, however, as remarked above, is both positive and negative — a conclusion to which for that very reason no great value can be attached. Since too the particular and universal are the extremes, and are immediate, mutually indifferent determinatenesses, their relationship is itself indifferent; either can be taken at choice as major or minor term and therefore, too, either premise can be taken as major or minor.
§ 1462
3. The conclusion, being positive as well as negative, is thus a relation indifferent to these determinatenesses and hence a universal relation. More precisely, the mediation of the first syllogism was in itself a contingent one; in the second syllogism this contingency is posited. Hence it is the self-sublating mediation; the mediation has the determination of individuality and immediacy; what is united by this syllogism must on the contrary be in itself and immediately identical; for this middle term, immediate individuality, is determined in an infinitely manifold and external manner. In it, therefore, is rather posited the self-external mediation. But the externality of the individuality is universality; the above mediation by means of the immediate individual points beyond itself to its other form, and the mediation s therefore effected by the universal. In other words, what is to be united by the second syllogism must be conjoined immediately; the immediacy on which this syllogism is based cannot bring about a definite conclusion. The immediacy to which it points is the opposite to its own-the sublated first immediacy of being-therefore the universal that is reflected into itself, or the implicit, abstract universal.
§ 1463
From the point of view we have just considered, the transition of this syllogism was an alteration like transition in the sphere of being, because the qualitative element, that of immediate individuality, lies at its base. But according to the Notion, individuality unites the particular and universal in so far as it sublates the determinateness of the particular, and this presents itself as the contingency of this syllogism; the extremes are not united by their determinate relation which they have for a middle term; this term is therefore not their determinate unity, and the positive unity which still attaches to the middle term is only abstract universality. But with the positing of the middle term in this determination, which is its truth, we have another form of the syllogism.
§ 1464
1. This third syllogism no longer has any immediate premise; the relation I-U has been mediated by the first syllogism, the relation P-U by the second. Hence it presupposes the first two syllogisms; but conversely, they both presuppose it, and in general each presupposes the other two. In this figure, therefore, the determination of the syllogism is in general completed. What this reciprocal mediation precisely contains is this, that each syllogism, although by itself mediation, is none the less not in its own self the totality of the mediation but contains an immediacy whose mediation lies outside it.
§ 1465
The syllogism I-U-P, regarded in itself, is the truth of the formal syllogism; it expresses that the mediation of the formal syllogism is the abstractly universal mediation, and that the extremes are not contained in the middle term according to their essential determinateness but only according to their universality; and that therefore what was supposed to be mediated in it is precisely what is not brought into unity. Here then is made explicit in what the formalism of the syllogism consists; its terms have an immediate content which is indifferent to the form, or, what is the same thing, they are determinations of form which have not yet reflected themselves into determinations of content.
§ 1466
2. The middle term of this syllogism is indeed the unity of the extremes, but a unity in which abstraction is made from their determinateness; it is the indeterminate universal. But since this universal is at the same time distinguished as abstract from the extremes as determinate, it is itself still a determinate relatively to them, and the whole is a syllogism whose relation to its Notion has now to be considered. The middle term, as the universal, is the subsuming term or predicate to both its extremes, and does not occur once as subsumed or as subject. In so far, therefore, as it is supposed to correspond, as a species of syllogism, to the syllogism, it can do so only on condition that when one relation I-U, already possesses the proper relationship, the other relation U-P also possesses it. This occurs in a judgment in which the relationship of subject and predicate is indifferent, in a negative judgment. In this way the syllogism becomes legitimate, but the conclusion necessarily negative.
§ 1467
Thus it is now also indifferent which of the two determinations of this proposition is taken as predicate and which as subject; and in the syllogism, whether it is taken as extreme of individuality or of particularity, therefore as minor or major term. Since on the common assumption it depends on this which of the premises is to be major and which minor, this too has become a matter of indifference. This is the ground of the ordinary fourth figure of the syllogism, a figure unknown to Aristotle and which in any case is concerned with a wholly empty and pointless distinction. In it the immediate position of the terms is the reverse of their position in the first figure. Since subject and predicate of the negative conclusion in the formal treatment of the judgment do not have the definite relationship of subject and predicate, but either can take the place of the other, it is indifferent which term is taken as subject and which as predicate; therefore equally indifferent which premise is taken as major and which as minor. This indifference, aided as it is by the determination of particularity (especially when it is observed that this can be taken in the comprehensive sense) makes this fourth figure a sheer futility.
§ 1468
3. The objective significance of the syllogism in which the universal is the middle term, is that the mediating element, as unity of the extremes, is essentially a universal. But since the universality is in the first instance only qualitative or abstract universality, it does not contain the determinateness of the extremes; their conjunction, if it is to be effected, must similarly have its ground in a mediation lying outside this syllogism and is in respect of this latter just as contingent as in the case of the preceding forms of the syllogism. But now since the universal is determined as the middle term, and the determinateness of the extremes is not contained in it, this middle term is posited as a wholly indifferent and external one.
§ 1469
As the immediate result of this bare abstraction, we obtain, of course, a fourth figure of the syllogism, namely that of the relationless syllogism U-U-U, which abstracts from the qualitative difference of the terms and consequently has for its determination their merely external unity, namely their equality.
§ 1470
1. The mathematical syllogism runs: if two things or determinations are equal to a third, they are equal to each other. Here the relationship of inherence or subsumption of the terms is extinguished.
§ 1471
The mediating factor is a third in general, but it has absolutely no determination whatever as against its extremes. Each of the three can therefore equally well be the third, mediating term. Which one is to be used for this purpose, and which of the three relations, therefore, are to be taken as immediate and which as mediated, depends on external circumstances and other conditions, namely on which two of them are the immediately given terms. But this determination does not concern the syllogism itself and is completely external.
§ 1472
2. The mathematical syllogism ranks as an axiom in mathematics, as an absolutely self-evident, primitive proposition, that neither admits nor requires any proof, that is any mediation, and neither presupposes anything else nor can be deduced from anything else. If its prerogative of being immediately self-evident is looked at more closely, it will be seen that it lies in the formalism of this syllogism which abstracts from all qualitative distinction of the terms and only takes up their quantitative equality or inequality. But for this very reason it is not without presupposition or unmediated; the quantitative determination, which is the only thing in it taken into account, is only through abstraction from qualitative difference and from the determinations of the Notion. Lines, figures, posited as equal to one another, are understood only in terms of their magnitude; a triangle is affirmed to be equal to a square, but not as triangle to square, but only in regard to magnitude, etc. Similarly, the Notion and its determinations do not enter into this syllogising; there is no comprehending at all [that is, in terms of the Notion] in this process; and understanding too has not even the formal, abstract determinations of the Notion before it. The self-evidence of this syllogism, therefore, rests merely on the fact that its thought content is so meagre and abstract.
§ 1473
3. But the result of the syllogism of existence is not merely this abstraction from all Notional determinateness; the negativity of the immediate, abstract determinations which emerged from it has yet another positive side, namely that the abstract determinateness has had its other posited in it and thereby has become concrete.
§ 1474
In the first place, the syllogisms of existence all mutually presuppose one another and the extremes united in the conclusion are only genuinely and in and for themselves united in so far as they are otherwise united by an identity that has its ground elsewhere; the middle term, as it is constituted in the syllogisms we have considered, is supposed to be their Notion unity, but is only a formal determinateness that is not posited as their concrete unity. But this presupposed element of each of those mediations is not merely a given immediacy in general, as in the mathematical syllogism, but is itself a mediation, namely, for each of the two other syllogisms. Therefore what we truly have before us is not mediation based on a given immediacy, but mediation based on mediation. Hence this is not the quantitative mediation that abstracts from the form of mediation, but rather the mediation that relates itself to mediation, or the mediation of reflection. The circle of reciprocal presupposing that these syllogisms unite to form with one another is the return of this act of presupposition into itself, which herein forms a totality, and thus the other to which each individual syllogism points is not placed through abstraction outside the circle but embraced within it.
§ 1475
Further, from the side of the individual determinations of the form it has been seen that in this entirety of the formal syllogisms each individual term has in turn taken the place of middle term. This was determined as particularity; subsequently, through the dialectical movement it determined itself as individuality and universality. Similarly, each of these determinations occupied in turn the places of the two extremes. The merely negative result is the extinction of the qualitative form determinations in the merely quantitative, mathematical syllogism. But what we truly have here is the positive result, that mediation is not effected through an individual qualitative determinateness of form, but through the concrete identity of the determinations. The defect and formalism of the three syllogistic figures considered above consists just in this, that an individual determinateness of this kind was supposed to constitute their middle term. Mediation has thus determined itself as the indifference of the immediate or abstract form determinations and as positive reflection of one into the other. The immediate syllogism of existence has thereby passed over into the syllogism of reflection.
Remark: The Common View of the Syllogism
§ 1476
In the account here given of the nature of the syllogism and its various forms, passing reference has also been made to what in the usual consideration and treatment of syllogisms constitutes the main interest, namely, how a correct conclusion may be obtained in each figure; however, in those references only the main point has been indicated, and those cases and complexities which arise when the distinction of positive and negative judgments, together with the quantitative determination — especially particularity — is also dragged in, have been passed over. Some remarks on the ordinary view and mode of treatment of the syllogism in logic will be in place here. It is a familiar fact that this doctrine was elaborated into such finely drawn distinctions that its so-called subtleties have been the object of universal aversion and disgust. The natural understanding in asserting itself in every department of mental and spiritual culture against the unsubstantial forms of reflection, also turned against this artificial knowledge of the form of reason and supposed itself able to dispense with such a science on the ground that it performed the individual operations of thought specified therein naturally and spontaneously without any special instruction. In point of fact, if a precondition of rational thinking were the laborious study of syllogistic formulae, mankind would in that respect be in the same sorry plight as they would be, as already remarked in the Preface in another respect, if they could not walk or digest without having studied anatomy and physiology. Granting that the study of these sciences may not be without profit for the regulation of one's diet, we must undoubtedly credit the study of the forms of reason with an even more important influence on the correctness of thinking. But without going into this aspect of the matter which concerns the education of subjective thinking and therefore, strictly speaking, pedagogics, it must be admitted that the study which has for its subject matter the modes and laws of operation of reason, must in its own self be of the greatest interest-of an interest at least not inferior to an acquaintance with the laws of nature and of her particular forms. If it is not thought a small matter to have discovered some sixty species of parrots, one hundred and thirty-seven species of veronica, etc., much less ought it to be thought a small matter to discover the forms of reason; is not a figure of the syllogism something infinitely superior to a species of parrot or veronica?
§ 1477
Therefore, though contempt for the knowledge of the forms of reason must be regarded as sheer barbarism, equally we must admit that the ordinary presentation of the syllogism and its particular formations is not a rational cognition, not an exposition of them as forms of reason, and that syllogistic wisdom by its own worthlessness has brought upon itself the contempt which has been its lot. Its defect consists in its simply stopping short at the understanding's form of the syllogism in which the Notion determinations are taken as abstract, formal terms. It is all the more inconsequent to cling to these determinations as abstract qualities, since in the syllogism it is their relations that constitute the essential feature, and inherence and subsumption already imply that the individual, because the universal inhe.-es in it, is itself a universal, and the universal, because it subsumes the individual, is itself an individual; more exactly, the syllogism expressly posits this very unity as middle term, and its determination is precisely mediation, that is, the Notion determinations no longer, as in the judgment, have for basis their mutual externality, but rather their unity. It is thus the Notion of the syllogism that declares the imperfection of the formal syllogism in which the middle term is fixedly held, not as unity of the extremes but as a formal, abstract determination qualitatively distinct from them. The treatment is rendered still less meaningful by the fact that also relations or judgments in which even formal determinations become indifferent, as in negative and particular judgments, and which therefore approximate to propositions, are still regarded as perfect relationships. Now since the qualitative form I-P- U is generally accepted as the ultimate and absolute, the dialectical treatment of the syllogism no longer operates; the remaining syllogisms are consequently regarded not as necessary alterations of that first form but as species. In this case, it is indifferent whether the first formal syllogism itself is regarded only as a kind of species alongside the rest, or as genus and species at the same time; the latter occurs when the other syllogisms are reduced to the first. If this reduction is not expressly effected, yet the basis is always the same formal relationship of external subsumption expressed by the first figure.
§ 1478
This formal syllogism is the contradiction that the middle term which is supposed to be the determinate unity of the extremes does not appear as this unity but as a determination qualitatively distinct from those extremes whose unity it is supposed to be. Because the syllogism is this contradiction, it is in its own nature dialectical. Its dialectical movement exhibits it in each of the moments of the Notion, so that not only the above relationship of subsumption or particularity, but equally essentially negative unity and universality are moments in the union of the extremes. In so far as each of these by itself is equally only a one-sided moment of particularity, they are likewise imperfect middle terms, but at the same time they constitute the developed determinations of the middle term; the entire course through the three figures presents the middle term in each of these determinations, and the true result that emerges from it is that the middle is not an individual Notion determination but the totality of them all.
§ 1479
The defect of the formal syllogism, therefore, does not lie in the form of the syllogism — on the contrary, this is the form of rationality — but in the fact that the form appears only as an abstract and therefore notionless form. It has been shown that the abstract determination, on account of its abstract relation-to-self, can equally be regarded as content; this being so, the formal syllogism merely serves to show that a relation of a subject to a predicate follows or does not follow only from this middle term. Nothing is gained in having proved a proposition by a syllogism of this kind; on account of the abstract determinateness of the middle term, which is a Notionless quality, other middle terms can just as well be given from which the opposite follows; in fact, from the same middle term opposite predicates may in turn be deduced by further syllogisms. Besides being of little service, the formal syllogism is also a very simple affair; the numerous rules which have been invented are tiresome not only because they contrast so strongly with the simple nature of the fact but also because they relate to cases where the formal worth of the syllogism is furthermore diminished by the external form determination, above all, particularity (especially as for this purpose it must be taken in a comprehensive sense), and where even in respect of form nothing but completely worthless results are deduced. However, the most merited and most important aspect of the disfavour into which syllogistic doctrine has fallen is that this doctrine is such a long-drawn out, notionless occupation with a subject matter whose sole content is the Notion itself. The numerous syllogistic rules remind one of the procedure of arithmeticians who similarly give a host of rules about arithmetical operations, all of which rules presuppose that one has not the Notion of the operation. But numbers are a notionless material and the operations of arithmetic are an external combining or separating of them, a mechanical procedure-indeed, calculating machines have been invented which perform these operations; whereas it is the harshest and most glaring of contradictions when the form determinations of the syllogism, which are Notions, are treated as a notionless material.
§ 1480
The extreme example of this irrational treatment of the Notion determinations of the syllogism is surely Leibniz's subjection of the syllogism to the calculus of combinations and permutations and has reckoned thereby how many positions of the syllogism are possible — that is, with respect to the distinctions of positive and negative, of universal, particular indeterminate and singular judgments; 2,048 such combinations are found to be possible of which, after the exclusion of the useless figures, twenty-four useful figures remain. Leibniz makes much of the usefulness of the analysis of combinations for ascertaining not only the forms of the syllogism but also the combinations of other concepts. The operation by which this is ascertained is the same as that by which it is calculated how many combinations of letters are possible in an alphabet, how many throws are possible in a game of dice, how many kinds of play with an ombre card, etc. Here therefore we find the determinations of the syllogism put in the same class with the points of the die and the ombre card, the rational is taken as a dead and non-rational thing, and the characteristic feature of the Notion and its determinations as spiritual essences to relate themselves and through this relating to sublate their immediate determination, is ignored. This Leibnizian application of the calculus of combinations and permutations to the syllogism and to the combination of other notions, differed from the notorious Art of Lully solely in being more methodical on the arithmetical side, but for the rest, they were both equally meaningless. Connected with this was a pet idea of Leibniz, embraced by him in his youth, and in spite of its immaturity and shallowness not relinquished by him even in later life, the idea of a characteristica universalis of notions-a language of symbols in which each notion would be represented as a relation proceeding from others or in its relation to others as though in rational combinations, which is essentially dialectical, a content still retained the same determinations that it possesses when fixed in isolation.
§ 1481
Ploucquet's calculus has undoubtedly got hold of the most consistent method by which the relationship of the syllogism is made capable of being subjected to a calculus. It rests upon the abstraction from the difference of relationship, from the difference of individuality, particularity and universality in the judgment and upon strict adherence to the abstract identity of subject and predicate whereby they are in a mathematical equality — in a relation which converts the syllogising process into a completely meaningless and tautological formulation of propositions. In the proposition: the rose is red, the predicate is not to denote red in general but only the specific red of the rose; in the proposition: all Christians are men, the predicate is to denote only those men who are Christians; from this proposition and the proposition: the Jews are not Christians, there follows the conclusion (which did not particularly commend this syllogistic calculus to Mendelssohn) therefore the Jews are not men (namely, not those men that the Christians are). Ploucquet states as a consequence of his discovery: 'posse etiam rudes mechanics totam logicam doceri, uti pueri arithmeticam docentur, ita quidem, ut nulla formidine in ratiociniis suis errands torqueri, vel fallaciis circumveniri possint, si in calculo non errant.' This recommendation, that by means of the calculus the whole of logic can be mechanically brought within reach of the uneducated, is surely the worst thing that can be said of a discovery bearing on the presentation of the science of logic.