On Hegel's 'Science of Logic' : A Realm of Shadows - part forty five.

'Argument'

by Elizabeth Bishop, (1911 – 1979) 

Days that cannot bring you near

Or will not,

Distance trying to appear

Something more obstinate,

Argue argue argue with me

Endlessly

Neither proving you less wanted nor less dear.

Distance: Remember all that land

Beneath the plane;

That coastline

Of dim beaches deep in sand

Stretching indistinguishably

All the way,

All the way to where my reasons end?

Days: And think

Of all those cluttered instruments,

One to a fact,

Canceling each other's experience;

How they were

Like some hideous calendar

"Compliments of Never & Forever, Inc."

The intimidating sound

Of these voices

We must separately find

Can and shall be vanquished:

Days and Distance disarrayed again

And gone

Both for good and from the gentle battleground.

Georg Wilhelm Friedrich Hegel, (1770 - 1831), 'The Science of Logic'

Syllogism

Syllogism or inference (Schluss). 'It is evident that the term 'syllogism' is the worst possible translation for the German word Schluss, which does not signify the well-known scholastic technique for reaching a conclusion, but rather the 'issue', the 'unification', the 'reconciliation' of the artificial distinctions of the understanding', declares Eugène Fleischmann. Schluss is the stage of the Logic at which the word rational begins to appear. Rationality 'is not one of the categories to be found within the system of categories of the Logic, but is rather the title for that which is presented by the theory as a whole', explains Deborah Chaffin. Rationality stands for the triune structure of the Notion. Not only is the syllogism rational, but everything rational is a syllogism.

'Thus the syllogism is the completely posited concept; it is, therefore, the rational. – The understanding is taken to be the faculty of the determinate concept which is held fixed for itself by virtue of abstraction and the form of universality. But in reason the determinate concepts are posited in their totality and unity. Therefore, it is not just that the syllogism is rational but that everything rational is a syllogism. Syllogistic inference has long since been ascribed to reason; but, on the other hand, reason in and for itself, and rational principles and laws, are so spoken of that no light is thrown on why the one reason that syllogizes, and the other which is the source of laws and otherwise eternal truths and absolute thoughts, hang together. If the former is supposed to be only a formal reason while the latter is supposed to be the one that generates content, then one would expect on this distinction that precisely the form of reason, the inference, would not be missing in the latter. And yet, the two are commonly held so far apart, the one without mention of the other, that it seems as though the reason of absolute thoughts were ashamed, so to speak, of inferential reason, and the syllogism were listed as also an activity of reason merely as matter of tradition. But surely, as we have just remarked, logical reason must be essentially recognizable, when regarded as formal, also in the reason that is concerned with a content; indeed, no content can be rational except by virtue of the rational form. In this matter we cannot rely on what is commonly said about reason, for common views fail to tell us what we are to understand by reason; this would-be rational wisdom is so busy with its objects that it forgets to pay attention to reason itself but only identifies it by characterizing it through the objects that it is said to have. If reason is supposed to be a cognition that would know about God, freedom, right and duty, the infinite, the unconditional, the suprasensible, or even gives only representations and feelings of such objects, then for one thing these objects are only negative, and for another the original question still stands, what is there in all these objects that makes them rational? – The answer is that the infinitude in them is not the empty abstraction from the finite, is not a universality which is void of content and determination, but is the fulfilled universality, the concept which is determined and is truly in possession of its determinateness, namely, in that it differentiates itself internally and is the unity of its thus intelligible and determined differences. Only in this way does reason rise above the finite, the conditioned, the sensuous, or however one might define it, and is in this negativity essentially replete with content, for as unity it is the unity of determinate extremes. And so the rational is nothing but the syllogism'.

- 'The Science of Logic';

As he explains elsewhere the three figures of the syllogism declare that everything rational is manifested as a triple syllogism. Each one of the members takes in turn the place of the extremes, as well as of the mean which reconciles them. Such is the case with the three branches of philosophy: the Logical Idea, Nature, and the Mind. As we first see them, Nature is the middle term which links the others together. Nature unfolds itself into the two extremes of the Logical Idea and Mind. But Mind is Mind only when it is mediated through nature. Then Mind is the mean. It is Mind which cognizes the Logical Idea in Nature and which thus raises Nature to its essence. In the third place again the Logical Idea becomes the mean; it is the absolute substance both of mind and of nature, the universal and all-pervading principle.

'The objective sense of the figures of the syllogism is generally that everything rational shows itself to be a threefold syllogism, and it does that in such a way that each of its members occupies the position both of an extreme and of the mediating middle. This is the case especially with the three 'members' of philosophical science, i. e., the logical Idea, Nature, and Spirit. Here, it is first Nature that stands in the middle as the member that con-eludes the others. As the immediate totality, Nature unfolds itself in the two extremes of logical Idea and Spirit. Spirit, however, is Spirit by being mediated through Nature. In the second place, Spirit which we know as what is individual and actuating is the middle, and Nature and the logical Idea are the extremes. It is Spirit that knows the logical Idea in Nature, and elevates it to its essence. Equally, in the third place, the logical Idea itself is the middle; it is the absolute substance of Spirit and of Nature, that which is universal and all-pervading. These are the members of the absolute syllogism'.

- 'The Encyclopaedia Logic'

Common chatter about reason, Hegel complains, neglects to define the term. Perhaps thinking of Immanuel Kant, Hegel says that supposedly rational cognition is mostly so busy with its objects that it forgets to cognize reason itself and only distinguishes and characterizes it by the objects that it possesses. Reason is that which recognizes God, freedom, duty, etc. These are Kant's practical objects as described in the 'Critique of Pure Reason'. Left unanswered is, What makes these objects rational? According to Hegel, they are rational because they are triune; their infinitude is not the empty abstraction from the finite, not the universality that lacks content and determinateness, but the universality that is fulfilled or realized, the Notion that is determinate and possesses its determinateness in this true way. And how does the Notion make itself determinate? It differentiates itself within itself and is the unity of these fixed and determinate differences. When triune Notion is before us, reason rises above the finite, conditioned, sensuous, call it what you will, and in this negativity is essentially pregnant with content [inhaltsvoll], for it is the unity of determinate extremes. In Syllogism, Notion is overtly itself, its other, and the unity of self and other.

Syllogism is all about proof. In chapter thirteen, Hegel defined proof as mediated cognition. A proposition that appears as a middle term is a mediated cognition and hence "proved." When a premise is merely given (i.e., "all men are mortal"), proof of that premise is demanded. The premise must be proved by becoming a middle term to two other given premises, which in their turn must also be proven. 'What is being demanded of the syllogism here is something we do not usually ask of our inferences: not just that the conclusion follow from the premisses, but that these too be grounded in necessity', explains Charles Taylor. Eventually, the premises of Syllogism must take their turn in the middle. Only then can all the premises be proved. If rationality is the triune form of the Notion, we must account for the fact that Judgment is tetrachotomous, while Syllogism is trichotomous. Comparing the sequences of Judgment and Syllogism, we find:

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U.S. President Lyndon Baines Johnson and U.S. Senator J. William Fulbright inspect 'Squaring the Circle' a bright red 1963 painting by Richard Anuszkiewicz at the 1965 White House Arts Festival.

'Sofia S', Richard Joseph Anuszkiewicz, (1930 – 2020)

Meanwhile, the initial Syllogism of Existence is not triune. It replays the four Judgments:

In this table, I is Individuality, P is Particularity, and U is Universality. E stands for Syllogism of Existence. R and N will denote Syllogism of Reflection and Necessity respectively. Why is there no Syllogism of the Notion? Why does quadruplicity yield to triunity? 'The precise answer [for omitting notional syllogism] is hard to come by', claims Errol E. Harris. And why, within triune Syllogism, is the initial Syllogism of Existence tetradic? Hegel does not explain the return of triunity, but he explains that Syllogism is the restoration of the Notion in the judgement. Accordingly one might suppose that trichotomy returns because the Notion is already comprehended at the beginning of Syllogism. The Judgment of the Notion is the Syllogism of Existence. The overlap guarantees that, when, Judgment and Syllogism are considered together, six steps are at stake. G.R.G. Mure offers an alternative answer: 'But the puzzle vanishes if we do not look for one-one correspondence, but remember that in the rationality of Syllogism the 'broken-backedness' of the Understanding is mediated and transcended. If we insist on pressing the correspondence we must say, I think, that Syllogism of Necessity 'corresponds to' notional Judgment as well as to Judgement of Necessity'.

The excess in Judgment is swallowed up by the Syllogism of Existence.

The Syllogism of Existence. The progress of the four Syllogisms of Existence is as follows: (1) IPU (E), the classic formal Syllogism, is, like the Judgment of Existence, non-notional. Its unity is strictly external to Syllogism. (2) In PIU (E), Syllogism's inability to prove anything is the universal predicate; similarly, in the Judgment of Reflection, the subject realized its nothingness. All being was in the external predicate. (3) IUP (E) is speculative. It admits that the unity holding Syllogism together is entirely external to itself. All three terms of Syllogism are seen as fundamentally alike. Difference is externally imposed. At this point, externality - the true unity of Syllogism - is raised to the level of Universality (UUU (E)). In fact, UUU (E) is not truly a fourth. It is the analytical truth of the prior three steps; UUU (E) adds nothing that was not entirely true of IUP (E). Appearances to the contrary, there are only three Syllogisms of Existence.

The Understanding begins by holding rigidly to the self-subsistence of the extremes.

'Now the syllogism, like judgment, is at first immediate; as such, its determinations (termini) are simple, abstract determinacies; it is then the syllogism of the understanding. If one stays at this configuration of the syllogism, then its rationality, though present there and posited, is not apparent. The essential element of the syllogism is the unity of the extremes, the middle term that unites them and the ground that supports them. Abstraction, by holding fast to the self-subsistence of the extremes, posits this unity opposite them, as a determinateness with just as fixed an existence of its own, thus grasping it more as a non-unity than as a unity. The expression, 'middle term' (medius terminus), is derived from spatial representation, and has its share of responsibility for why we stop short at the externality of the terms. Now if the syllogism consists in the positing in it of the unity of the extremes, but if this unity is simply taken on the one hand as a particular by itself, and on the other hand as only an external connection, and non-unity is made the essential relation of syllogism, then the reason of the syllogism is of no help to rationality'.

- 'The Science of Logic'

The middle term or copula is the essential feature - the unity of the extremes. The copula pregnant with content is precisely where the Judgement of the Notion left off. At first, the Syllogism of Existence is immediate and formal - not yet concrete. It manifests no internal unity. The extremes are self-identities 'that cannot be comprehended, but only indicated', explains John W. Burbidge. Its overall form is IPUE. 'A particular mediates between an individual and its category', says Burbidge. P is the middle term since it unites immediately within itself the two moments of individuality and universality.

'The syllogism in its immediate form has for its moments the determinations of the concept as immediate. Accordingly, these are the abstract determinacies of form, such as have not yet been developed by mediation into concretion but are only singular determinacies. The first syllogism is thus the one which is strictly formal. The formalism of syllogistic inference consists in stopping short at the form of this first syllogism. The concept, when partitioned into its abstract moments, has singularity and universality for its extremes, and itself appears as the particularity that stands between them. Because of their immediacy, these determinacies only refer to themselves, one and all a single content. Particularity constitutes at first the middle term by uniting within itself, immediately, the two moments of singularity and universality. Because of its determinateness, on the one hand it is subsumed under the universal; on the other hand, the singular with respect to which it possesses universality is subsumed under it. This concretion is at first, however, only a double-sidedness; the middle term, because of the immediacy that affects it in the immediate syllogism, is a simple determinateness, and the mediation which it constitutes is not as yet posited. Now the dialectical movement of the syllogism of existence consists in positing the moments of the mediation that alone constitutes the syllogism'.

- 'The Science of Logic'

Mure suggests that P is the middle term because, following the Judgement of the Notion, Notion must manifest itself, Notion is genus, and what genus manifests is Particularity. Following Aristotle, U subsumes P, and P subsumes I; the Universal descends to individuality through particularity. Incidentally just to confuse matters in the translation I am using I for individual is S for singular:

'S-P-U is the general schema of the determinate syllogism. Singularity connects with universality through particularity; the singular is not universal immediately but by means of particularity; and conversely, universality is likewise not singular immediately but lowers itself to it through particularity. – These determinations stand over against each other as extremes and are one in a third term which is diverse from them. The two are both determinateness; in this they are identical; this, their universal determinateness, is particularity. But they are no less extremes with respect to this particularity than they are to each other, for each is in its immediate determinateness'.

- 'The Science of Logic'

In Syllogism, the predicate (UP) subsumes and the subject (PI) is subsumed. UP is therefore the major premise and PI the minor premise.

IPU. In IPU (E), the Individual is Universal (and vice versa) through the medium of Particularity. The extremes of each of the following forms can be flipped, so that IPU is equally UPI, etc. For example, Hegel will later refer to the second figure of formal syllogism as UIP, though he usually writes PIU.

'The second figure of the formal syllogism, U-S-P, does not correspond to this schema, because the S that constitutes the middle term did not subsume or was not a predicate. In induction this deficiency is eliminated; here the middle term is 'all singulars'; the proposition, U-S, which contains as the subject the objective universal or the genus set apart as an extreme, has a predicate which is of at least equal extension as the subject and is consequently identical with it for external reflection. Lion, elephant, etc., constitute the genus of quadruped; the difference, that the same content is posited once in singularity and again in universality, is thus just an indifferent determination of form – an indifference which in the syllogism of reflection is the posited result of the formal syllogism and is posited here through the equality of extension'.

- The Science of Logic'

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NEU! - Negativland

IPU (E)'s significance is that the abstract individual 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. 'The general meaning of this syllogism is that the singular, which as such is infinite self-reference and consequently would be only an inwardness, emerges through the medium of particularity into existence, into a universality wherein it no longer belongs just to itself but stands in external conjunction; conversely, since in its determinateness the singular sets itself apart as particularity, in this separation it is a concreted term and, because of the self-reference of the determinateness, it is a self-referring universal, and consequently also a true singular; in the extreme of universality the singular has gone from externality into itself. – The objective significance of the syllogism is in this first figure only superficially present at first, for the determinations are not as yet posited in it as the unity which constitutes the essence of the syllogistic inference. The syllogism is still something subjective inasmuch as the abstract meaning which its terms have has no being in and for itself but is rather only in a subjective consciousness, and is thus isolated. – Moreover, as we have seen,78 the relation of singularity, particularity, and universality is the necessary and essential form-relation of the determinations of the syllogism; the deficiency does not rest in the determinateness of the form but in that each single determination is not at the same time richer under it. – Aristotle confined himself rather to the mere relation of inherence by defining the nature of the syllogism as follows: When three terms are so related to each other that the one extreme is in the entire middle term, and this middle term is in the entire other extreme, then these two extremes are necessarily united in the conclusion. What is here expressed is the repetition of the equal relation of inherence of the one extreme to the middle term, and then again of this last to the other extreme, rather than the determinateness of the three terms to each other. – Now since the syllogistic inference rests on this determinateness of the terms to each other, it is immediately apparent that the other relations of terms as are given by the other figures can have validity as inferences of the understanding only to the extent that they let themselves be reduced to that original relation; these other are not diverse species of figures that stand alongside the first but, on the one hand, to the extent that they are assumed to be correct inferences, they rest on the form of syllogistic inference in general; and, on the other hand, to the extent that they deviate from it, they are variant forms into which the first abstract form necessarily passes over and thereby further determines itself and becomes totality. How this occurs, we must now see in greater detail'.

- 'The Science of Logic'

'Annual Edition', 1987 - 1988, Richard Joseph Anuszkiewicz

IPU (g)

Through Particularity, the Individual is a concrete Universal.

IPUE is Aristotelian Syllogism. Hegel paraphrases Aristotle 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. The favourite perfect syllogism holds:

All men are mortal.

Gaius is a man.

Therefore Gaius is mortal.

To replicate this in the form of IPU (E), Hegel rearranges the terms as follows:

AH men are mortal. (PU).

Therefore Gaius is mortal. (P, or IU).

Gaius is a man. (IP).

In this Syllogism, U inheres in P and P inheres in I therefore I inheres in U. The therefore belongs to the middle term. It establishes the truth that P = IU. In the parlance of symbolic logic, this is the hypothetical syllogism. A hypothetical syllogism in symbolic logic bears the form: (1) p → q, (2) q → r, (3) r → s. Hegel's Hypothetical Syllogism is what symbolic logic designates as modus ponens: if p is true, and if p implies q (p. q), then q is true.

The truths of a hypothetical syllogism are strictly conditional because the premises are conditional.

Two diverse judgments are externally joined to form IPU (E). First is PU, where P is subject and U is predicate. Second is IP. Nothing in PU demands conjunction with IP. Men are many things other than mortal; there is no reason why the major and minor premises must be brought together. 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.

'S-P-U is thus the general schema of the syllogism in its determinateness. The singular is subsumed under the particular and the particular under the universal; therefore, the singular is also subsumed under the universal. Or the particular inheres in the singular and the universal in the particular; therefore, the universal also inheres in the singular. With respect to the side of the universal, the particular is the subject; with respect to that of the singular, it is predicate; or as against the one it is singular, as against the other it is universal. Since both these determinations are united in it, by virtue of this unity of determinations the extremes are joined together. The 'therefore' appears as an inference that has taken place in the subject and derives from the subjective insight into the relation of the two immediate premises. Since subjective reflection expresses the two connections of the middle to the extremes as particular and indeed immediate judgments or propositions, the conclusion as the mediated connection is of course also a particular proposition, and the “hence” or the “therefore,” is the expression that it is the one which is mediated. But this 'therefore is not to be regarded as a determination which is external to this proposition, one that would have its ground and seat in subjective reflection, but as grounded rather in the nature of the extremes themselves whose connection is again enunciated as a mere judgment or proposition only for the sake of, and by virtue of, abstractive reflection, but whose true connection is posited as the middle term. – 'Therefore S is A': that this is a judgment is a merely subjective circumstance; that it is not a merely subjective judgment, that is, not a connection drawn through the mere copula or the empty 'is' but one drawn rather through a determinate middle which is replete with content, that is precisely the meaning of the syllogistic inference'.

- 'The Science of Logic'

IP and PU are immediate, externally supplied and not themselves proven or mediated by Syllogism. What is expressed between IP and PU is mere likeness, to which PU and PI are indifferent. Yet once the two extremes are externally conjoined, the therefore is no external determination. It is grounded in the nature of the extremes themselves. The copula 'therefore' is no empty 'is' but is pregnant with content. IPU (E) is not two separate premises and a diverse conclusion. The premises may be separate, but the conclusion follows. But as a mere composition, IPU (E) is flawed. In it, the right extreme (IP) is an immediate I, indifferent to being made part of a Syllogism which reveals one of its Fs. I has infinite Fs that could have been revealed. I is only subjectively brought together with the revealed P.

Rational thinking cannot proceed by IPU. Notional determinations must be essentially united. Anything else is makeshift and still subjective. As Hegel explains elsewhere in ordinary proof reasons given are themselves in need of further reasons, and so on ad infinitum.

'Here we should bear in mind that the dialectical movement likewise has propositions for its parts or elements; the difficulty just indicated seems, therefore, to recur perpetually, and to he inherent in the very nature of philosophical exposition. This is like what happens in ordinary proof, where the reasons given are themselves in need of further reasons, and so on ad infinitum, This pattern of giving reasons and stating conditions belongs to that method of proof which differs from the dialectical movement, and belongs therefore to external cognition. As regards the dialectical movement itself, its element is the one Notion; it thus has a content which is, in its own self, Subject through and through. Thus no content occurs which functions as an underlying subject, nor receives its meaning as a predicate; the proposition as it stands is merely an empty form'.

- 'The Phenomenology of Spirit'

IPU (E) is true only if the premises are true. Accordingly, it is commonly demanded of the premises that they shall be proved, that is, that they likewise shall be presented as conclusions.

'Accordingly, the normal expectation is that the premises will be proved, that is, that they ought likewise to be exhibited as conclusions. The two premises, therefore, yield two further syllogisms. But these two new syllogisms together yield four premises that require four new syllogisms; these have eight premises whose eight conclusions yield in turn sixteen conclusions for their sixteen premises, and so on in a geometrical progression to infinity'.

- 'The Science of Logic'

Each premise requires a further Syllogism, made up of two conjoined premises. These in turn demand two new Syllogisms, and so on in a geometrical progression to infinity. I as an infinite number of middle terms, and each of these middle terms has infinite middle terms. Mediation is, so far, contingent and subjective, not anything necessary or universal.

'Thus we have again the progress to infinity that occurred in the lower sphere of being but we would not expect now in the domain of the concept, the domain of the absolute reflection from the finite to the self, the region of free infinity and truth. It was shown in the sphere of being that whenever the bad infinity that runs away into a progression raises its head, what we have is the contradiction of a qualitative being and of an impotent ought that would transcend it; the progression itself is the repeated demand that there be unity that intervenes to confront the qualitative, and the constant fall back into the limitation which is inadequate to the demand. Now in the formal syllogism the immediate connection or the qualitative judgment is the basis, and the mediation of the syllogism is the higher truth posited over against it. The infinite progression of the proof of the premises does not resolve this contradiction but only perpetually renews it and is the repetition of one and the same original deficiency. – The truth of the infinite progression is rather the sublation of it and of the form which the progression itself has already determined as deficient. – This form is that of the mediation S-P-U. The two connections, S-P and P-U are supposed to be mediated; if this is done in the same manner, only the deficient form S-P-U is replicated, and so on to infinity. With respect to S, P also has the form determination of a universal; and with respect to U that of a singular, for these connections are as such judgments. As such, they are in need of mediation; but in that form of mediation, only the relation that was supposed to be sublated comes up again'.

- 'The Science of Logic'

Spurious Infinity. An impotent ought-to-be certainly has no place in the domain of the Notion. Because IPU (E) yields the Spurious Infinite, mediation must therefore be effected in another manner. The truth of IPU (E) is that immediate Individuality (I) is really the mediator. Gaius is indifferent if external reflection has emphasized his manhood. What really mediates IPU (E) is I - Gaius himself. Furthermore, IPU, the external conjunction of PU and PI, bids us to prove the major (i.e., PU) and minor (PI) premises. Let us take each in turn. Take PU. Here, the excluded mediator is I. If we make I the middle term, we obtain PIU (E). Similarly, IP excludes U, if U mediates, we have IUP. PIU next takes the stage.

PIU. In IPU (E), something [was] united with a qualitative determinateness as a universal, not in and for itself but through a contingency or in an individuality. I was a Spurious Infinity of concrete Universals, each revealed in a completely contingent manner. So far I was apprehended only in it's externality.

'The truth of the first qualitative syllogism is that something is not in and for itself united to a qualitative determinateness which is a universal, but is united to it by means of a contingency or in a singularity. The subject of the syllogism has not returned in such a quality to its concept but is conceived only in its externality; the immediacy constitutes the basis of the connection and hence the mediation; to this extent, the singular is in truth the middle. The truth of the first qualitative syllogism is that something is not in and for itself united to a qualitative determinateness which is a universal, but is united to it by means of a contingency or in a singularity. The subject of the syllogism has not returned in such a quality to its concept but is conceived only in its externality; the immediacy constitutes the basis of the connection and hence the mediation; to this extent, the singular is in truth the middle'.

- 'The Science of Logic'

Accordingly, immediacy (i.e., the unproven) is what truly mediates IPU (E). This realization brings I into the middle spot between P and U. Of course, unprovenness is precisely what Syllogism is not supposed to be. PIU (E) makes the scandal explicit. PIU (E) is to be regarded as a subjective syllogism proceeding in an external reflection.

'The syllogism of the first figure was the immediate syllogism, or again, the syllogism in so far as its concept is an abstract form that has not yet realized itself in all its determinations. The transition of this pure form into another figure is on the one hand the beginning of the realization of the concept, in that the negative moment of the mediation, and thereby one further determinateness of the form, is posited in the originally immediate, qualitative determinateness of the terms. – But, on the other hand, this is at the same time an alteration of the pure form of the syllogism; the latter no longer conforms to it fully, and the determinateness posited in its terms is at variance with that original form determination. – In so far as it is regarded as only a subjective syllogism that runs its course in external reflection, we can then take it as a species of syllogistic inference that should conform to the genus, namely the general schema S-P-U. But it does not at the moment conform to it; its two premises are P-S or S-P and S-U; the middle term is in both cases the one which is subsumed or is the subject in which the two other terms thus inhere – is not therefore a middle term that in one case would subsume or be predicate, and in the other would be subsumed or be subject, or a middle in which one of the terms would inhere but would itself inhere in the other. – The true meaning of this syllogism’s lack of conformity to the general form of the syllogism is that the latter has passed over into it, for its truth consists in being a subjective, contingent conjoining of terms. If the conclusion in this second figure is correct (that is, without recurring to the restriction, to which we shall presently turn, that makes of it something indeterminate), then it is correct because it is so on its own, not because it is the conclusion of this syllogism. But the same is the case for the conclusion of the first figure; it is this, the truth of that first figure, which is posited by the second. – On the view that the second figure is only one species, we overlook the necessary transition of the first figure into this second and stop short at the first as the true form. Hence, if in the second figure (which from ancient custom is referred to, without further ground, as the third) we are equally supposed to find a correct syllogism in this subjective sense, this syllogism would have to be commensurate with the first; consequently, since the one premise S-U has the relation of the subsumption of the middle term under one extreme, then it would have to be possible for the other premise S-P to receive the opposite relation to that which it has, and for P to be subsumed under S. But such a relation would be the sublation of the determinate judgment S is P, and could only occur in an indeterminate judgment, a particular judgment; consequently, the conclusion in this figure can only be particular. But the particular judgment, as we remarked above, is positive as well as negative – a conclusion, therefore, to which no great value can be ascribed. – Since the particular and universal are also the extremes, and are immediate determinacies indifferent to each other, their relation itself is indifferent; each can be the major or the minor term, indifferently the one or the other, and consequently either premise can also be taken as major or minor'.

- 'The Science of Logic'

NEU! - Für immer:

What was supposed to be true is now revealed to be subjective delusion. There is a negative unity - no relation - between IU, on the one hand, and PI on the other. PIU (E) contains two judgments, PI and IU. The second judgment, IU-P, was IPU (E)'s result. IU is therefore already proved (or mediated) by the first Syllogism. In effect, we have learned that all Is are composed. PI, however, remains unproved (i.e., an immediacy). It stands for the proposition that at least one I is not composed. Meanwhile, PIUE and IPUE have in common U as predicate. The subjects I and P have changed places. The significance of this is that P is now posited in the determination of the extreme of individuality and I is likewise posited as Particularity.

'In this second syllogism, the premises are: P-S and S-U; only the first of these premises is still an immediate one; the second, S-U, is already mediated, namely through the first syllogism; the second syllogism thus presupposes the first just as, conversely, the first presupposes the second. – The two extremes are here determined, the one as against the other, as particular and universal. The latter thus retains its place; it is predicate. But the particular has exchanged places; it is subject or is posited in the determination of the extreme of singularity, just as the singular is posited with the determination of the middle term or of particularity. The two no longer are, therefore, the abstract immediacies which they were in the first syllogism. However, they are not yet posited as concrete somethings; in standing in the place of the other, each is thereby posited – in its own determination and at the same time, although only externally – into that of the other'.

- 'The Science of Logic'

I and P are not mere immediacies. On the other hand, they are not yet posited as concretes. Because I=P, an outside force is needed to determine which is really I and which is P. Neither yet bears a relation to its predicate (U). As predicate, U is to be taken as the totality of all the particulars I has. U is genus. Yet U can only be revealed in the Syllogism of Existence as one of the species that make up the genus. For this reason, PIU (E) is the syllogism of mere perception or of contingent existence.

'Induction, therefore, is not the syllogism of mere perception or of contingent existence, like the second figure corresponding to it, but the syllogism of experience – of the subjective gathering together of singulars in the genus, and of the conjoining of the genus with a universal determinateness on the ground that the latter is found in all singulars. It also has the objective significance that the immediate genus has determined itself through the totality of singularity as a universal property and possesses its existence in a universal relation or mark. – But the objective significance of this syllogism, as it was of the others, is at first only its inner concept, and is not as yet posited in it'.

- 'The Science of Logic'

When a species is brought forth as U, all the other species (equally U) are suppressed. Therefore, when I haphazardly reveals itself to be a species of U it equally reveals itself not to be U, and so I stands in a negative relationship to the particular in so far as it is supposed to be its predicate.

'The determinate and objective meaning of this syllogism is that the universal is not in and for itself a determinate particular (it is rather the totality of its particulars) but that it is one of its species through the mediation of singularity; the rest of its species are excluded from it by the immediacy of externality. Likewise the particular is not for its part immediately, and in and for itself, the universal; the negative unity is rather what removes the determinateness from it and thereby raises it to universality. – The singularity thus relates to the particular negatively in so far as it is supposed to be its predicate; it is not the predicate of the particular'.

- 'The Science of Logic'

'Three works from the Spectral Cadmiums portfolio', 1968, Richard Joseph Anuszkiewicz

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PIU (E)

Hegel compares the second figure PIU to the Particular Judgment - which was both positive and negative. In the Particular Judgment, the Singular Judgment (this thing is useful) was revealed to imply that many things are not useful.

P and I are two qualities that are connected, not in and for themselves, but by means of a contingent individuality. That is to say, the predicate (U) is fixed, but either I or P is subject. Only external reflection can determine whether the subject is I or P. Nevertheless, the transition into PIU (E) is the beginning of the realization of the Notion.

'The syllogism of the first figure was the immediate syllogism, or again, the syllogism in so far as its concept is an abstract form that has not yet realized itself in all its determinations. The transition of this pure form into another figure is on the one hand the beginning of the realization of the concept, in that the negative moment of the mediation, and thereby one further determinateness of the form, is posited in the originally immediate, qualitative determinateness of the terms. – But, on the other hand, this is at the same time an alteration of the pure form of the syllogism; the latter no longer conforms to it fully, and the determinateness posited in its terms is at variance with that original form determination. – In so far as it is regarded as only a subjective syllogism that runs its course in external reflection, we can then take it as a species of syllogistic inference that should conform to the genus, namely the general schema S-P-U. But it does not at the moment conform to it; its two premises are P-S or S-P and S-U; the middle term is in both cases the one which is subsumed or is the subject in which the two other terms thus inhere – is not therefore a middle term that in one case would subsume or be predicate, and in the other would be subsumed or be subject, or a middle in which one of the terms would inhere but would itself inhere in the other. – The true meaning of this syllogism’s lack of conformity to the general form of the syllogism is that the latter has passed over into it, for its truth consists in being a subjective, contingent conjoining of terms. If the conclusion in this second figure is correct (that is, without recurring to the restriction, to which we shall presently turn, that makes of it something indeterminate), then it is correct because it is so on its own, not because it is the conclusion of this syllogism. But the same is the case for the conclusion of the first figure; it is this, the truth of that first figure, which is posited by the second. – On the view that the second figure is only one species, we overlook the necessary transition of the first figure into this second and stop short at the first as the true form. Hence, if in the second figure (which from ancient custom is referred to, without further ground, as the third) we are equally supposed to find a correct syllogism in this subjective sense, this syllogism would have to be commensurate with the first; consequently, since the one premise S-U has the relation of the subsumption of the middle term under one extreme, then it would have to be possible for the other premise S-P to receive the opposite relation to that which it has, and for P to be subsumed under S. But such a relation would be the sublation of the determinate judgment S is P, and could only occur in an indeterminate judgment, a particular judgment; consequently, the conclusion in this figure can only be particular. But the particular judgment, as we remarked above, is positive as well as negative – a conclusion, therefore, to which no great value can be ascribed. – Since the particular and universal are also the extremes, and are immediate determinacies indifferent to each other, their relation itself is indifferent; each can be the major or the minor term, indifferently the one or the other, and consequently either premise can also be taken as major or minor'.

- 'The Science of Logic'

The syllogism of the first figure was the immediate syllogism, or again, the syllogism in so far as its concept is an abstract form that has not yet realized itself in all its determinations. The transition of this pure form into another figure is on the one hand the beginning of the realization of the concept, in that the negative moment of the mediation, and thereby one further determinateness of the form, is posited in the originally immediate, Because P is isolated and in negative unity with U, P cannot determine itself. In its indeterminacy, P is raised to abstract Universality (PU). Put in other terms, PIU (E) is mediated only by subjectivity and nothing else. 'But if it is the subjectivity of intellectual association that mediates between the terms, then we have identified a type of inference that is neither a determination, nor something individual and unique, but is common to all intellectual operations. This mediating process is universal', explains Burbidge.

'Original Painting', 1963, Richard Joseph Anuszkiewicz

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PIU (g)

The truth of PIU (E) is P=U. Particularity is now the true Universal. U is now determined to be the true middle term of the second figure. In other words, each of the terms in PIU (E) is an abstract Universal. This leads to the third figure, IUP (E), which is identical to the fourth figure UUU (E). Hegel compares this transition to the transitions from the realm of Being. There, Being-for-self expelled all its content from itself and became Quantity. Something similar now happens. Syllogism, which is supposed to be mediation and proof, is now nothing but immediacy and unproof. All mediation is on the outside. But when such a thing happened in chapter 19 - when isolated Universality became Particularity in Infinite Judgment - Individuality was revealed. As Hegel now puts it, "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 the syllogism.

'From the standpoint of the present consideration, the transition of this syllogism was like the transition of being an alteration, for its base is qualitative; it is the immediacy of singularity. But according to the concept, singularity conjoins the particular and the universal by sublating the determinateness of the particular – and this is what presents itself as the contingency of this syllogistic inference. The extremes are not conjoined by the specific connective which they have in the middle term; this term is not, therefore, their determinate unity, and the positive unity that yet pertains to it is abstract universality. But inasmuch as the middle term is posited in this determination which is its truth, we have another form of the syllogism'.

- 'The System of Logic'

IUP (E) UUU (E)

IUP. Unlike its predecessors, IUP (E) has no immediate premise. IU was the result of the first Syllogism, PU the result of the second. Syllogism is now complete. Reciprocal mediation reigns between the first two Syllogisms and this third one. Each side of IUPE is proven. Yet the two extremes are still mutually indifferent. Universality, which is supposed to be the subsuming predicate is now present on both sides of the third figure. As a result, we cannot tell whether IU or UP is subject (subsumed) or predicate (subsuming). Outside force must decide. Inference is reduced to mere associational logic.

None of the three Syllogisms is the totality. What totalizes (or mediates) is an outside force. The truth of formal Syllogism is notional failure. The terms are determinations of form, not content. Indifference (abstract Universality) is the true ground of the Syllogism of Existence. IUP (E) yields UUU (E) - a fourth figure unknown to Aristotle.

'Consequently, also indifferent is now which of the two determinations of this proposition is taken as predicate or subject, and whether the determination is taken in the syllogism as the extreme of singularity or the extreme of particularity, hence as the minor or major term. Since on the usual assumption which of the premises is supposed to be the major or the minor depends on this distinction, this too has now become a matter of indifference. – This is the ground of the customary fourth figure of the syllogism which was unknown to Aristotle and has to do with an entirely void and uninteresting distinction. In it the immediate position of the terms is the reverse of their position in the first figure; since from the point of view of the formal treatment of judgment the subject and predicate of the negative conclusion do not have the determinate relation of subject and predicate, but each can take the place of the other, it is a matter of indifference which term is taken as subject and which as predicate; and just as indifferent is therefore which premise is taken as the major and which as the minor. – This indifference, to which the determination of particularity also contributes (especially if it is noted that this particularity can be taken in a comprehensive sense), makes of this fourth figure something totally idle'.

- 'The Science of Logic'

UUU expresses the equality and merely external unity of the internal parts. In UUUE, distinguishing between the three terms is a sheer futility. Errol Harris draws a different lesson from Mathematical Syllogism: 'Things which are equal to the same thing are equal to one another. This, says Hegel, is regarded as an axiom in mathematics, but it is really a logical principle demonstrable through dialectical derivation from the Concept ... So Hegel demonstrates in an entirely different way what Frege and Russell have attempted to do, that mathematics is deducible in its entirety from pure logic'. This is to miss the point. Hegel's view of Mathematical Syllogism has a darkovertone for Mathematics works because Quality is suppressed. The lesson of Mathematical Syllogism is mathematical failure. Following Mure and Winfield we need not interpret UUU (E) as an independent step in the Logic. It therefore warrants no schematic drawing. Rather, it represents the truth of IUP (E) - that nothing internal to Syllogism justifies the identification of any one of them as middle term to the other two.

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NEU! - After Eight:

Mathematical Syllogism. UUU (E) represents pure transitivity. Mure writes of UUU: 'One must confess that this leaves it obscure why ... Hegel allows it to appear in the dialectical movement'. The answer is that UUU represents the inability of Syllogism to generate any result whatever without the aid of external reflection. But since nothing is something, Hegel will find some positive content in nothing in the next few sections. This "Mathematical" Syllogism states that if I=U, and if U=P, then I=P. In other words, all the terms are the same. It is impossible to tell which term mediates the other two. Only external force can identify which of the terms is the middle one. Here the relationship of inherence or subsumption of the terms is extinguished.

'The mathematical syllogism goes like this: if two things or two determinations are equal to a third, then they are equal to each other. – The relation of inherence or subsumption of terms is done away with'.

- 'The Science of Logic'

UUU (E) represents an axiom - an absolutely self-evident, primitive proposition - that requires no proof.

'The mathematical syllogism ranks in mathematics as an axiom, as a first self-explanatory proposition which is neither capable nor in need of proof, i.e of any mediation – which neither presupposes anything else nor can be derived from anything else. – If we take a closer look at this prerogative that the proposition claims, of being immediately self-evident, we find that it lies in its formalism, in the fact that it abstracts from every qualitative diversity of determinations and only admits their quantitative equality or inequality. But for this very reason it is not without presupposition or mediation; the quantitative determination, which alone comes into consideration in it, is only by virtue of the abstraction from qualitative differentiation and from the concept determinations. – Lines, figures, posited as equal to each other, are understood only according to their magnitude. A triangle is posited as equal to a square, not however as triangle to square but only according to magnitude, etc. Nor does the concept and its determinations enter into this syllogism; there is in it, therefore, no conceptual comprehension at all; the understanding is also not faced here by even the formal, abstract determinations of the concept. The self-evidence of this syllogism rests, therefore, solely on the indigence and abstractness of its mode of thought'.

- 'The Science of Logic'

Transitivity is the isolation of Quantity taken in abstraction from Quality. But for this very reason it is not without presupposition. Transitivity is only because Quality is not. The self-evidence of this syllogism, therefore, rests merely on the fact that its thought content is so meagre and abstract. 'The syllogistic forms relate abstract concepts that have been isolated through conceptual operations', says Burbidge. In other words, transitivity works because conceptual operations have isolated quantity from quality. In mathematics, 'the process of abstracting is pushed to its limit where all determinate and differentiating characteristics, even that of reference, are excluded, leaving only abstract quantitative identity and the 'axiom' of equality', Burbidge adds.

Hegel also emphasizes that, in a Syllogism, the conclusion is supposed to follow from its premises. But, in the Mathematical Syllogism, the conclusion contradicts the premises. The premises - U, I, and P - were immediacies. The Mathematical Syllogism, however, is nothing but relation, which contradicts the immediacy of the premises. In short, the Syllogism of Existence is deeply false. Abstraction is not, however, the final result of the Syllogism of Existence. Abstraction is not notional. Rather, abstraction posits its other, which is what really holds the Syllogisms of Existence together.

Therefore what we truly have before us is not mediation based on a given immediacy, but mediation based on mediation. The other is not placed through abstraction outside the circle but embraced within it.

'In the first place, the syllogisms of existence all have one another for presupposition, and the extremes conjoined in the conclusion are truly conjoined, in and for themselves, only inasmuch as they are otherwise united by an identity grounded elsewhere; the middle term, as constituted in the syllogisms we have examined, ought to be the conceptual unity of these syllogisms but is in fact only a formal determinateness that is not posited as their concrete unity. But what is thus presupposed by each and every of these mediations is not merely a given immediacy in general, as is the case for the mathematical syllogism, but is itself a mediation, namely of each of the other two syllogisms. Therefore, what is truly present here is not a mediation based on a given immediacy, but a mediation based on mediation. And this mediation is not quantitative, not one that abstracts from the form of mediation, but is rather a self-referring mediation, or the mediation of reflection. The circle of reciprocal presupposing which these syllogisms bring to closure is the turning back of this presupposing into itself – a presupposing that in this turning back forms a totality, and has the other to which every single syllogism refers, not outside by virtue of abstraction, but included within the circle'.

- 'The Science of Logic'

To summarize, across the Syllogism of Existence, the three terms each took a turn in the middle. Qualitative difference between the terms, however, disappeared in UUU (E). Nevertheless, we achieved a positive result - mediation [i.e., proof] is not effected through an individual qualitative determinateness of form, but through the concrete identity of the determinations. In other words, the fourth figure (UUU (E)) stands for the realization that neither P, I, nor U is properly the middle term. The unity of all the terms is the proper middle term - an external force that holds it together. Mediation has thus determined itself as the indifference of the immediate or abstract form determinations and as positive reflection of one into the other.

'Further, from the side of the single determinations of form it has been shown that in this whole of formal syllogisms each single determination has in turn occupied the place of the middle term. As immediate, this term was determined as particularity; thereupon, through dialectical movement it determined itself as singularity and universality. Likewise did each of these determinations occupy the places of both of the two extremes. The merely negative result is the dissolution of the qualitative determinations of form into the merely quantitative, mathematical syllogism. But what we truly have here is the positive result, namely that mediation occurs, not through any single qualitative determinateness of form, but through the concrete identity of the determinacies. The deficiency and formalism of the three figures of the syllogism just considered consists precisely in this, that one such single determinateness was supposed to constitute the middle term in it. –Mediation has thus determined itself as the indifference of the immediate or abstract determinations of form and the positive reflection of one into the other. The immediate syllogism of existence has thereby passed over into the syllogism of reflection'.

- 'The Science of Logic'

The entire course through the three figures presents the middle term in each of the 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.

'This formal syllogism is the contradiction that the middle term ought to be the determinate unity of the extremes – not, however, as this unity but as a determination qualitatively distinct from the terms whose unity it ought to be. Because the syllogism is this contradiction, it is inherently dialectical. Its dialectical movement displays it in the full range of the moments of the concept, so that not only the said relation of subsumption or particularity, but just as essentially the negative unity and the universality, are moments in the process of closing this unity. In so far as each of these equally is by itself 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 across the three figures displays this middle in each of these determinations, one after the other, and the true result that emerges from it is that the middle is not any single one of them but the totality of them all'.

- 'The Science of Logic'

We have therefore reached the Syllogism of Reflection.

'Inflexion', 1967, Richard Joseph Anuszkiewicz

Dedicated as always to my lovely One. My Love. Mon Amour. Meine Liebe. ❤️

Coming up next:

The Syllogism of Reflection.

To be continued ...