Hegel’s Science of Logic
Remark: A = A
The categories of reflection used to be taken up in the form of propositions, in which they were asserted to be valid for everything. These propositions ranked as the universal laws of thought that lie at the base of all thinking, that are absolute in themselves and incapable of proof, but are immediately and incontestably recognised and accepted as true by all thinking that grasps their meaning.
Thus the essential category of identity is enunciated in the proposition: everything is identical with itself, A = A. Or negatively: A cannot at the same time be A and not A.
In the first place, there is no apparent reason why only these simple determinations of reflection should be grasped in this particular form, and not also the other categories, such as all the determinatenesses of the sphere of being. We should then have the propositions, for example: everything is, everything has a determinate being, and so on, or: everything has a quality, quantity, etc. For being, determinate being, and so forth, are, simply as logical categories, predicates of everything. According to its etymology and Aristotle's definition, category is what is predicated or asserted of the existent. But a determinateness of being is essentially a transition into its opposite; the negative of any determinateness is as necessary as the latter itself; as immediate determinatenesses, each is directly confronted by the other. Consequently, if these categories are put in the form of such propositions, then the opposite propositions equally appear; both present themselves with equal necessity and, as immediate assertions, are at least equally correct. The one, therefore, would demand proof as against the other, and consequently these assertions could no longer be credited with the character of immediately true and incontestable propositions of thought.
The determinations of reflection, on the contrary, are not of a qualitative kind. They are self-related, and so are at the same time determinations removed from determinateness against an other. Further, in that they are determinatenesses which are in themselves relations, to that extent they already contain within themselves the prepositional form. For the difference between proposition and judgement is mainly that in the former the content constitutes the relation itself or is a specific relation. The judgement, on the contrary, transfers the content to the predicate as a universal determinateness which is for itself and is distinct from its relation, the simple copula. When a proposition is to be converted into a judgement, then the specific content — if, for example it is a verb — is changed into a participle, in order to separate in this way the determination itself and its relation to a subject. For the determinations of reflection, on the contrary, as positedness reflected into itself, the prepositional form itself lies immediately at hand. Only, since they are enunciated as universal laws of thought, they still require a subject of their relation, and this subject is: everything, or an A, which equally means each and every existent.
On the one hand, this prepositional form is a superfluity; the determinations of reflection are to be considered in and for themselves. Further, these propositions are defective in that they have for subject, being, everything. In this way, they resuscitate being and enunciate the categories of reflection-identity, and so on-of the something as a quality which something has in it, not in the speculative sense, but meaning that something as subject persists in such a quality as simply affirmative [als seiendes], not that it has passed over into identity, and so on, as into its truth and its essence.
But lastly, although the determinations of reflection have the form of equality-with-self and therefore of being unrelated to an other and without opposition, yet they are determinate against one another, as we shall find on closer examination of them, or as is immediately evident from the categories of identity, difference, and opposition; their form of reflection, therefore, does not exempt them from transition and contradiction. The several propositions which are set up as absolute laws of thought, are, therefore, more closely considered, opposed to one another, they contradict one another and mutually sublate themselves.
If everything is identical with itself, then it is not different, not opposed, has no ground. Or, if it is assumed that no two things are the same, that is, everything is different from everything else, then A is not equal to A, nor is A opposed to A, and so on. The assumption of any of these propositions rules out the assumption of the others. The thoughtless consideration of them enumerates them one after the other, so that there does not appear to be any relation between them; it has in mind merely their reflectedness-into-self, ignoring their other moment, positedness or their determinateness as such which sweeps them on into transition and into their negation.
1. Essence is simple immediacy as sublated immediacy. Its negativity is its being; it is self-equal in its absolute negativity, through which otherness and relation-to-other has vanished in its own self into pure equality-with-self. Essence is therefore simple identity with self.
2. This identity-with-self is the immediacy of reflection. It is not that equality-with-self that being or even nothing is, but the equality-with-self that has brought itself to unity, not a restoration of itself from an other, but this pure origination from and within itself, essential identity. Consequently, it is not abstract identity or has not arisen through a relative negating which had taken place outside it, merely separating off the distinguished moment but otherwise leaving it afterwards as simply affirmative [seiend] as it was before. On the contrary, being and every determinateness of being has sublated itself not relatively, but in its own self: and this simple negativity of being in its own self is identity itself. So far, then, identity is still in general the same as essence.
Remark 1: Abstract Identity
Thinking that keeps to external reflection and knows of no other thinking but external reflection, fails to attain to a grasp of identity in the form just expounded, or of essence, which is the same thing. Such thinking always has before it only abstract identity, and apart from and alongside it, difference. In its opinion, reason is nothing more than a loom on which it externally combines and interweaves the warp, of say, identity, and then the woof of difference; or, also, again proceeding analytically, it now extracts especially identity and then also again obtains difference alongside it, is now a positing of likeness and then also again a positing of unlikeness — likeness when abstraction is made from difference, and unlikeness when abstraction is made from the positing of likeness. These assertions and opinions about what reason does must be completely set aside, since they are in a certain measure merely historical; the truth is rather that a consideration of everything that is, shows that in its own self everything is in its self-sameness different from itself and self-contradictory, and that in its difference, in its contradiction, it is self-identical, and is in its own self this movement of transition of one of these categories into the other, and for this reason, that each is in its own self the opposite of itself. The Notion of identity, that it is simple self-related negativity, is not a product of external reflection but has come from being itself. Whereas, on the contrary, that identity that is aloof from difference, and difference that is aloof from identity, are products of external reflection and abstraction, which arbitrarily clings to this point of indifferent difference.
2. This identity is, in the first instance, essence itself, not yet a determination of it, reflection in its entirety, not a distinct moment of it. As absolute negation it is the negation that immediately negates itself, a non-being and difference that vanishes in its arising, or a distinguishing by which nothing is distinguished, but which immediately collapses within itself. The distinguishing is the positing of non-being as non-being of the other. But the non-being of the other is sublation of the other and therewith of the distinguishing itself. Here, then, distinguishing is present as self-related negativity, as a non-being which is the non-being of itself, a non-being which has its non-being not in another but in its own self. What is present, therefore, is self-related, reflected difference, or pure, absolute difference.
In other words, identity is the reflection-into-self that is identity only as internal repulsion, and is this repulsion as reflection-into-self, repulsion which immediately takes itself back into itself. Thus it is identity as difference that is identical with itself. But difference is only identical with itself in so far as it is not identity but absolute non-identity. But non-identity is absolute in so far as it contains nothing of its other but only itself, that is, in so far as it is absolute identity with itself.
Identity, therefore, is in its own self absolute non-identity. But it is also the determination of identity as against non-identity. For as reflection-into-self it posits itself as its own non-being; it is the whole, but, as reflection, it posits itself as its own moment, as positedness, from which it is the return into itself. It is only as such moment of itself that it is identity as such, as determination of simple equality with itself in contrast to absolute difference.
Remark 2: First Original Law of Thought
In this remark, I will consider in more detail identity as the law of identity which is usually adduced as the first law of thought.
This proposition in its positive expression A = A is, in the first instance, nothing more than the expression of an empty tautology. It has therefore been rightly remarked that this law of thought has no content and leads no further. It is thus the empty identity that is rigidly adhered to by those who take it, as such, to be something true and are given to saying that identity is not difference, but that identity and difference are different. They do not see that in this very assertion they are themselves saying that identity is different; for they are saying that identity is different from difference; since this must at the same time be admitted to be the nature of identity, their assertion implies that identity, not externally, but in its own self, in its very nature, is this, to be different.
But further, they do not see that, by clinging to this unmoved identity which has its opposite in difference, they thereby convert it into a one-sided determinateness which, as such, has no truth. It is admitted that the law of identity expresses only a one-sided determinatedness, that it contains only formal truth, a truth which is abstract, incomplete. In this correct judgement, however, it is immediately implied that truth is complete only in the unity of identity with difference, and hence consists only in this unity. When asserting that this identity is imperfect, the perfection one has vaguely in mind is this totality, measured against which the identity is imperfect; but since, on the other hand, identity is rigidly held to be absolutely separate from difference and in this separation is taken to be something essential, valid, true, then the only thing to be seen in these conflicting assertions is the failure to bring together these thoughts, namely, that identity as abstract identity is essential, and that as such it is equally imperfect: the lack of awareness of the negative movement which, in these assertions, identity itself is represented to be. Or, when it is said that identity is essential identity as separation from difference, or in the separation from difference, then this is directly the expressed truth about it, namely, that identity consists in being separation as such, or in being essential in separation, that is, it is nothing for itself but is a moment of separation.
Now as regards other confirmation of the absolute truth of the law of identity, this is based on experience in so far as appeal is made to the experience of every consciousness; for anyone to whom this proposition A = A, a tree is a tree, is made, immediately admits it and is satisfied that the proposition as immediately self-evident requires no further confirmation or proof.
On the one hand, this appeal to experience, that the proposition is universally admitted by everyone, is a mere manner of speaking. For it is not pretended that the experiment with the abstract proposition A = A has been made on every consciousness. The appeal, then, to actually carried-out experiment is not to be taken seriously; it is only the assurance that if the experiment were made, the proposition would be universally admitted. But if what were meant were not the abstract proposition as such, but its concrete application from which the former were supposed first to be developed, then the assertion of its universality and immediacy would consist in the fact that every consciousness would treat it as fundamental, even in every utterance it made, or that it lies implicitly in every utterance. But the concrete and the application are, in fact, precisely the connection of the simple identical with a manifold that is different from it. Expressed as a proposition, the concrete would at first be a synthetic proposition. From the concrete itself or its synthetic proposition, abstraction could indeed extract by analysis the proposition of identity; but then, in fact, it would not have left experience as it is, but altered it; for the fact is that experience contains identity in unity with difference and is the immediate refutation of the assertion that abstract identity as such is something true, for the exact opposite, namely, identity only in union with difference, occurs in every experience.
On the other hand, the experiment with the pure law of identity is made only too often, and it is shown clearly enough in this experiment what is thought of the truth it contains. If, for example, to the question "What is a plant?" the answer is given "A plant is a plant", the truth of such a statement is at once admitted by the entire company on whom it is tested, and at the same time it is equally unanimously declared that the statement says nothing. If anyone opens his mouth and promises to state what God is, namely God is — God, expectation is cheated, for what was expected was a different determination; and if this statement is absolute truth, such absolute verbiage is very lightly esteemed; nothing will be held to be more boring and tedious than conversation which merely reiterates the same thing, or than such talk which yet is supposed to be truth.
Looking more closely at this tedious effect produced by such truth, we see that the beginning, 'The plant is—,' sets out to say something, to bring forward a further determination. But since only the same thing is repeated, the opposite has happened, nothing has emerged. Such identical talk therefore contradicts itself. Identity, instead of being in its own self truth and absolute truth, is consequently the very opposite; instead of being the unmoved simple, it is the passage beyond itself into the dissolution of itself.
In the form of the proposition, therefore, in which identity is expressed, there lies more than simple, abstract identity; in it, there lies this pure movement of reflection in which the other appears only as illusory being, as an immediate vanishing; A is is a beginning that hints at something different to which an advance is to be made; but this different something does not materialise; A is—A; the difference is only a vanishing; the movement returns into itself. The prepositional form can be regarded as the hidden necessity of adding to abstract identity the more of that movement. And so an A, or a plant, or some other kind of substrate, too, is added which, as a useless content, is of no significance; but it constitutes the difference which seems to be accidentally associated with it. If instead of A or any other substrate, identity itself is taken — identity is identity — then equally it is admitted that also in its place any other substrate could be taken. Consequently, if the appeal is to be made to what experience shows, then it shows that this identity is nothing, that it is negativity, the absolute difference from itself.
The other expression of the law of identity: A cannot at the same time be A and not-A, has a negative form; it is called the law of contradiction. Usually no justification is given of how the form of negation by which this law is distinguished from its predecessor, comes to identity. But this form is implied in the fact that identity, as the pure movement of reflection, is simple negativity which contains in more developed form the second expression of the law just quoted. A is enunciated, and a not-A, the pure other of A; but it only shows itself in order to vanish. In this proposition, therefore, identity is expressed-as negation of the negation. A and not-A are distinguished, and these distinct terms are related to one and the same A. Identity, therefore, is here represented as this distinguishedness in one relation or as simple difference in the terms themselves.
From this it is evident that the law of identity itself, and still more the law of contradiction, is not merely of analytic but of synthetic nature. For the latter contains in its expression not merely empty, simple equality-with-self, and not merely the other of this in general, but, what is more, absolute inequality, contradiction per se. But as has been shown, the law of identity itself contains the movement of reflection, identity as a vanishing of otherness.
What emerges from this consideration is, therefore, first, that the law of identity or of contradiction which purports to express merely abstract identity in contrast to difference as a truth, is not a law of thought, but rather the opposite of it; secondly, that these laws contain more than is meant by them, to wit, this opposite, absolute difference itself.
Difference is the negativity which reflection has within it, the nothing which is said in enunciating identity, the essential moment of identity itself which, as negativity of itself, determines itself and is distinguished from difference.
1. This difference is difference in and for itself, absolute difference, the difference of essence. It is difference in and for itself, not difference resulting from anything external, but self-related, therefore simple difference. It is essential to grasp absolute difference as simple. In the absolute difference of A and not-A from each other, it is the simple not which, as such, constitutes it. Difference itself is the simple Notion. Two things are different, it is said, in that they, etc. 'In that' is, in one and the same respect, in the same ground of determination. It is the difference of reflection, not the otherness of determinate being. One determinate being and another determinate being are posited as falling apart, each of them, as determined against the other, has an immediate being for itself. The other of essence, on the contrary, is the other in and for itself, not the other as other of an other, existing outside it but simple determinateness in itself. In the sphere of determinate being, too, otherness 'and determinateness proved to be of this nature, to be simple determinateness, identical opposition; but this identity revealed itself only as the transition of one determinateness into the other. Here, in the sphere of reflection, difference appears as reflected difference, which is thus posited as it is in itself.
2. Difference in itself is self-related difference; as such, it is the negativity of itself, the difference not of an other, but of itself from itself; it is not itself but its other. But that which is different from difference is identity. Difference is therefore itself and identity. Both together constitute difference; it is the whole, and its moment. It can equally be said that difference, as simple, is no difference; it is this only when it is in relation with identity; but the truth is rather that, as difference, it contains equally identity and this relation itself. Difference is the whole and its own moment, just as identity equally is its whole and its moment. This is to be considered as the essential nature of reflection and as the specific, original ground of all activity and self-movement. Difference and also identity, make themselves into a moment or a positedness because, as reflection, they are negative relation-to-self.
Difference as thus unity of itself and identity, is in its own self determinate difference. It is not transition into an other, not relation to an other outside it: it has its other, identity, within itself, just as identity, having entered into the determination of difference, has not lost itself in it as its other, but preserves itself in it, is its reflection-into-self and its moment.
3. Difference possesses both moments, identity and difference; both are thus a positedness, a determinateness. But in this positedness each is self-relation. One of them, identity, is itself immediately the moment of reflection-into-self; but equally, the other is difference, difference in itself, reflected difference. Difference, in that it has two moments that are themselves reflections-into-self, is diversity.
1. Identity falls apart within itself into diversity because, as absolute difference, it posits itself as its own negative within itself, and these its moments, namely, itself and the negative of itself, are reflections-into-self, are self-identical; or, in other words, precisely because identity itself immediately sublates its negating and in its determination is reflected into itself. The distinguished terms subsist as indifferently different towards one another because each is self-identical, because identity constitutes its ground and element; in other words, the difference is what it is, only in its very opposite, in identity.
Diversity constitutes the otherness as such of reflection. The other of determinate being has for its ground immediate being in which the negative subsists. But in reflection it is self-identity, reflected immediacy, that constitutes the subsistence of the negative and its indifference.
The moments of difference are identity and difference itself. They are [merely] diverse when they are reflected into themselves, that is, when they are self-related; as such, they are in the determination of identity, they are only relation-to-self; the identity is not related to the difference, nor is the difference related to the identity; as each moment is thus only self-related, they are not determined against one another. Now because in this manner they are not different in themselves, the difference is external to them. The diverse moments are, therefore, mutually related, not as identity and difference, but merely as simply diverse moments, that are indifferent to one another and to their determinateness.
2. In diversity, as the indifference of difference, reflection has become, in general, external to itself; difference is merely a posited or sublated being, but it is itself the total reflection. When considered more closely, both identity and difference, as has just been demonstrated, are reflections, each of which is unity of itself and its other; each is the whole. Consequently, the determinateness in which they are only identity or only difference, is sublated. Therefore they are not qualities, because through the reflection-into-self, their determinateness is at the same time only a negation. What is present, therefore, is this duality, reflection-into-self as such, and determinateness as negation or positedness. Positedness is the reflection that is external to itself; it is the negation as negation-and so therefore in itself or simplicity, the self-related negation and reflection-into-self, but only implicitly; it is relation to the negation as something external to it.
Thus the reflection that is implicit, and external reflection, are the two determinations into which the moments of difference, namely, identity and difference, posited themselves. They are these moments themselves in so far as they have now determined themselves. Reflection in itset is identity, but determined as being indifferent to difference, not as simply not possessing difference, but as being self-identical in its relationship with it; it is diversity. It is identity that has so reflected itself into itself that it is really the one reflection of the two. moments into themselves; both are reflections-into-self. Identity is this one reflection of both, which contains difference only as an indifferent difference and is simply diversity. External reflection, on the other hand, is their determinate difference, not as an absolute reflection-into-self, but as a determination to which the [merely] implicit reflection is indifferent; difference's two moments, identity and difference itself, are thus externally posited determinations, not determinations in and for themselves.
Now this external identity is likeness, and external difference, unlikeness. Likeness, it is true, is identity, but only as a positedness, an identity that is not in and for itself. Similarly, unlikeness is difference, but as an external difference that is not in and for itself the difference of the unlike itself. Whether or no-, something is like something else does not concern either the one or the other; each of them is only self-referred, is in and for itself what it is; identity or non-identity, as likeness or unlikeness, is the verdict of a third party distinct from the two things.
3. External reflection relates what is diverse to likeness and unlikeness. This relation, which is a comparing, passes to and fro between likeness and unlikeness. But this relating to likeness and unlikeness, back and forth, is external to these determinations themselves; also, they are related not to one another but each, by itself, to a third. In this alternation, each stands forth immediately on its own. External reflection is, as such, external to itself; the determinate difference is the negated absolute difference. Therefore it is not simple, not reflection-into-self; on the contrary, it has this outside it. Its moments, therefore, fall asunder and are related also as mutually external to the reflection-into-self confronting them.
In the self-alienated reflection, therefore, likeness and unlikeness appear as mutually unrelated, and in relating them to one and the same thing, it separates them by the introduction of 'in so far', of sides and respects. The diverse, which are one and the same, to which both likeness and unlikeness are related, are therefore, from one side like one another, but from another side are unlike, and in so far as they are like, they are not unlike. Likeness is related only to itself, and similarly unlikeness is only unlikeness.
But by this separation of one from the other they merely sublate themselves. The very thing that was supposed to hold off contradiction and dissolution from them, namely, that something is like something else in one respect, but is unlike it in another - this holding apart of likeness and unlikeness is their destruction. For both are determinations of difference; they are relations to one another, the one being what the other is not; like is not unlike and unlike is not like; and both essentially have this relation and have no meaning apart from it; as determinations of difference, each is what it is as distinct from its other. But through this mutual indifference, likeness is only self-referred, and unlikeness similarly is self-referred and a reflective determination on its own; each, therefore, is like itself; the difference has vanished, since they cannot have any determinateness over against one another; in other words, each therefore is only likeness.
This indifferent point of view or external difference thus sublates itself and is in its own self the negativity of itself. It is the negativity that belongs to the comparer in the act of comparing. The comparer goes from likeness to unlikeness and from this back to likeness, and therefore lets the one vanish in the other and is, in fact, the negative unity of both. This unity, in the first instance, lies beyond the compared and also beyond the moments of the comparison as a subjective act falling outside them. But, as we have seen, this negative unity is, in fact, the very nature of likeness and unlikeness. The independent self-reference which each of them is, is in fact the self-reference that sublates their distinctiveness and so, too, themselves.
From this side, likeness and unlikeness, as moments of external reflection and as external to themselves, vanish together in their likeness. But further, this their negative unity is also posited in them; they have, namely the [merely] implicit reflection outside them, or are the likeness and unlikeness of a third party, of an other than they. And so likeness is not like itself; and unlikeness, as unlike not itself but something else unlike it, is itself likeness. The like and the unlike are therefore unlike themselves. Consequently each is this reflection: likeness, that it is itself and unlikeness, and unlikeness, that it is itself and likeness.
Likeness and unlikeness formed the side of positedness as against the compared or the diverse which had determined itself as the [merely] implicit reflection contrasted with them. But this positedness as thus determined has equally lost its determinateness as against them. But likeness and unlikeness, the determinations of external reflection, are just this merely implicit reflection which the diverse as such is supposed to be, the merely indeterminate difference of the diverse. The implicit reflection is self-relation without the negation, abstract self-identity, and so simply positedness itself. The merely diverse, therefore, passes over through positedness into negative reflection. The diverse is the merely posited difference, therefore the difference that is no difference, and therefore in its own self the negati on of itself. Thus likeness and unlikeness themselves, that is, positedness, returns through indifference or the implicit reflection back into the negative unity with itself, into the reflection which the difference of likeness and unlikeness in its own self is. Diversity, whose indifferent sides are just as much simply and solely moments of one negative unity, is opposition.
Remark: The Law of Diversity
In opposition, the determinate rejection, difference, finds its completion. It is the unity of identity and difference; its moments are different in one identity and thus are opposites.
Identity and difference are the moments of difference held within itself; they are reflected moments of its unity. But likeness and unlikeness are the self-alienated reflection; their self-identity is not merely the indifference of each towards the other distinguished from it, but towards being-in-and-for-self as such, an identity-with-self over against the identity that is reflected into itself; it is therefore the immediacy that is not reflected into itself. The positedness of the sides of the external reflection is accordingly a being, just as their non-positedness is a non-being.
Closer consideration shows the moments of opposition to be positedness reflected into itself or determination in general. Positedness is likeness and unlikeness; these two reflected into themselves constitute the determinations of opposition. Their reflection-into-self consists in this, that each is in its own self the unity of likeness and unlikeness. Likeness is only in the reflection that compares on the basis of unlikeness, and therefore is mediated by its other, indifferent moment; similarly, unlikeness is only in the same reflective relationship in which likeness is. Therefore each of these moments is, in its determinateness, the whole. It is the whole in so far as it also contains its other moment; but this its other is an indifferent, simple affirmative moment; thus each contains reference to its non-being, and is only reflection-into-self or the whole, as essentially connected with its non-being.
This self-likeness reflected into itself that contains within itself the reference to unlikeness, is the positive; and the unlikeness that contains within itself the reference to its non-being, to likeness, is the negative. Or, both are a positedness; now in so far as the differentiated determinateness is taken as a differentiated determinate self-reference of positedness, the opposition is, on the one hand, positedness reflected into its likeness to itself and on the other hand, positedness reflected into its unlikeness to itself — the positive and the negative. The positive is positedness as reflected into self-likeness; but what is reflected is positedness, that is, the negation as negation, and so this reflection-into-self has reference-to-other for its determination. The negative is positedness as reflected into unlikeness; but the positedness is unlikeness itself, and this reflection is therefore the identity of unlikeness with itself and absolute self-reference. Each is the whole; the positedness reflected into likeness-to-self contains unlikeness, and the positedness reflected into unlikeness-to-self also contains likeness.
The positive and the negative are thus the sides of the opposition that have become self-subsistent. They are self-subsistent in that they are the reflection of the whole into themselves, and they belong to the opposition in so far as this is the determinateness which, as a whole, is reflected into itself. On account of their selfsubsistence, they constitute the implicitly determined opposition. Each is itself and its other; consequently, each has its determinateness not in an other, but in its own self. Each is self-referred, and the reference to its other is only a self-reference. This has a twofold aspect: each is a reference to its non-being as a sublating of this otherness within it; thus its non-being is only a moment in it. But on the other hand positedness here has become a being, an indifferent subsistence; consequently, the other of itself which each contains is also the non-being of that in which it is supposed to be contained only as a moment. Each therefore is, only in so far as its non-being is, and is in an identical relationship with it.
The determinations which constitute the positive and negative consist, therefore, in the fact that the positive and negative are, in the first place, absolute moments of the opposition; their subsistence is inseparably one reflection; it is a single mediation in which each is through the non-being of its other, and so is through its other or its own non-being. Thus they are simply opposites; in other words, each is only the opposite of the other, the one is not as yet positive, and the other is not as yet negative, but both are negative to one another. In the first place, then, each is, only in so far as the other is; it is what it is, through the other, through its own non-being; it is only a positedness; secondly, it is, in so far as the other is not; it is what it is, through the non-being of the other; it is reflection-into-self. But these two are the one mediation of the opposition as such, in which they are simply only posited moments
Further, however, this mere positedness is simply reflected into itself; in accordance with this moment of external reflection the positive and negative are indifferent to that first identity in which they are only moments; in other words, since that first reflection is the positive's and negative's own reflection into themselves, each is in its own self its positedness, so each is indifferent to this its reflection into its non-being, to its own positedness. The two sides are thus merely different, and in so far as their being determined as positive and negative constitutes their positedness in relation to one another, each is not in its own self so determined but is only determinateness in general. Therefore, although one of the determinatenesses of positive and negative belongs to each side, they can be changed round, and each side is of such a kind that it can be taken equally well as positive as negative.
But thirdly, the positive and negative are not only something posited, not merely an indifferent something, but their positedness, or the reference-to-other in a unity which they are not themselves, is taken back into each. Each is in its own self positive and negative; the positive and negative are the determination of reflection in and for itself; it is only in this reflection of opposites into themselves that they are positive and negative. The positive has within itself the reference-to-other in which the determinateness of the positive is; similarly, the negative is not a negative as contrasted with an other, but likewise possesses within itself the determinateness whereby it is negative.
Thus each [the positive as well as the negative] is a self-subsistent, independent unity-with-self. The positive is, indeed, a positedness, but in such wise that for it the positedness is only positedness as sublated. It is the not-opposite, the sublated opposition, but as a side of the opposition itself. As positive, something is, of course, determined with reference to an otherness, but in such a manner that its nature is to be not something posited; it is the reflection-into-self that negates the otherness. But the other of itself, the negative, is itself no longer a positedness or moment, but a self-subsistent being; thus the negating reflection of the positive is immanently determined as excluding from itself this its non-being.
The negative, as such absolute reflection, is not the immediate negative but the negative as a sublated positedness, the negative in and for itself, which is based positively on itself. As reflection-into-self it negates its relationship to other; its other is the positive, a self-subsistent being; consequently, its negative relation to it is to exclude it. The negative is the independently existing opposite contrasted with the positive, which is the determination of the sublated opposition-the self-based whole opposition opposed to the self-identical positedness.
The positive and negative are therefore not merely implicitly [an sich] positive and negative, but explicitly and actually so [an undfiir sich]. They are implicitly positive and negative in so far as one makes abstraction from their exclusive relation to other and only takes them in accordance with their determination. Something is in itself positive or negative when it is supposed to be so determined not merely relatively to an other. But when the positive and negative are taken, not as positedness, and therefore not as opposites, then each is the immediate, being and non-being. But the positive and negative are moments of opposition; their in-itself constitutes merely the form of their reflectedness-intoself. Something is in itself positive, apart from the relation to the negative; and something is in itse@ negative, apart from the relation to the positive;' in this determination, one clings merely to the abstract moment of this reflectedness. But the positive or negative in itse@ essentially implies that to be an opposite is not merely a moment, does not stem from comparison, but is a determination belonging to the sides of the opposition themselves. They are therefore not positive or negative in themselves apart from the relation to other; on the contrary, this relation-an exclusive relation-constitutes their determination or in-itself; in it, therefore, they are at the same time explicitly and actually [an undfiir sichl positive or negative.
Remark: Opposite Magnitudes of Arithmetic
Here is where we must take a look at the notion of the positive and negative as it is employed in arithmetic. There it is assumed as known; but because it is not grasped in its determinate difference, it does not avoid insoluble difficulties and complications. We have just found the two real determinations of the positive and negative — apart from the simple notion of their opposition — namely, first, that the base is a merely different, immediate existence whose simple reflection-into-self is distinguished from its positedness, from the opposition itself. This opposition, therefore, is not regarded as having any truth in and for itself, and though it does belong to the different sides, so that each is simply an opposite, yet, on the other hand, each side exists indifferently on its own, and it does not matter which of the two opposites is regarded as positive or negative. But secondly, the positive is the positive in and for itself, the negative in and for itself the negative, so that the different sides are not mutually indifferent but this their determination is true in and for itself. These two forms of the positive and negative occur in the very first applications of them in arithmetic.
In the first instance, +a and -a are simply opposite magnitudes; a is the implicit [ansichseiende] unity forming their common base; it is indifferent to the opposition itself and serves here, without any further notion, as a dead base. True, -a is defined as the negative, and +a as the positive, but the one is just as much an opposite as the other.
Further, a is not merely the simple unity forming the base but, as +a and -a, it is the reflection of these opposites into themselves; there are present two different a's, and it is a matter of indifference which of them one chooses to define as the positive or negative; both have a separate existence and are positive.
According to the first aspect, +y - y = 0; or in -8 + 3, the 3 positive units are negative in the 8. The opposites are cancelled in their combination. An hour's journey to the east and the same distance travelled back to the west, cancels the first journey; an amount of liabilities reduces the assets by a similar amount, and an amount of assets reduces the liabilities by the same amount. At the same time, the hour's journey to the east is not in itself the positive direction, nor is the journey west the negative direction; on the contrary, these directions are indifferent to this determinateness of the opposition; it is a third point of view outside them that makes one positive and the other negative. Thus the liabilities, too, are not in and for themselves the negative; they are the negative only in relation to the debtor; for the creditor, they are his positive asset; they are an amount of money or something of a certain value, and this is a liability or an asset according to an external point of view.
The opposites certainly cancel one another in their relation, so that the result is zero; but there is also present in them their identical relation, which is indifferent to the opposition itself; in this manner they constitute a one. Just as we have pointed out that the sum of money is only one sum, that the a in +a and -a is only one a, and that the distance covered is only one distance, not two, one going east and the other going west. Similarly, an ordinate y is the same on which ever side of the axis it is taken; so far, +y - y = y; it is only the one ordinate and it has only one determination and law.
But again, the opposites are not only a single indifferent term, but two such. For as opposites, they are also reflected into themselves and thus exist as distinct terms.
Thus in -8 + 3 there are altogether eleven units present; +y and -y are ordinates on opposite sides of the axis, where each is an existence indifferent to this limit and to their opposition; thus +y - y = 2y. — Also the distance travelled cast and west is the sum of a twofold effort or the sum of two periods of time. Similarly, in economics, a quantum of money or of wealth is not only this one quantum as a means of subsistence but is double: it is a means of subsistence both for the creditor and for the debtor. The wealth of the state is computed not merely as the total of ready money plus the value of property movable and immovable, present in the state; still less is it reckoned as the sum remaining after deduction of liabilities from assets. For capital, even if its respective determinations of assets and liabilities nullified each other, remains first, positive capital, as +a - a = a; and secondly, since it is a liability in a great number of ways, being lent and re-lent, this makes it a very much multiplied capital.
But not only are opposite magnitudes, on the one hand, merely opposite as such, and on the other hand, real or indifferent: for although quantum itself is being with an indifferent limit, yet the intrinsically positive and the intrinsically negative also occur in it. For example, a, when it bears no sign, is meant to be taken as positive if it has to be defined. If it were intended to become merely an opposite as such, it could equally well be taken as -a . But the positive sign is given to it immediately, because the positive on its own has the peculiar meaning of the immediate, as self-identical, in contrast to opposition.
Further, when positive and negative magnitudes are added and subtracted, they are counted as positive or negative on their own account and not as becoming positive or negative in an external manner merely through the relation of addition and subtraction.
In 8 - (-3) the first minus means opposite to 8, but the second minus (-3), counts as opposite in itself, apart from this relation.
This becomes more evident in multiplication and division. Here the positive must essentially be taken as the not-opposite, and the negative, on the other hand, as the opposite, not both determinations equally as only opposites in general. The textbooks stop short at the notion of opposite magnitudes as such in the proofs of the behaviour of the signs in these two species of calculation; these proofs are therefore incomplete and entangled in contradiction. But in multiplication and division plus and minus receive the more determinate meaning of the positive and negative in themselves, because the relation of the factors-they are related to one another as unit and amount-is not a mere relation of increasing and decreasing as in the case of addition and subtraction, but is a qualitative relation, with the result that plus and minus, too, are endowed with the qualitative meaning of the positive and negative. Without this determination and merely from the notion of opposite magnitudes, the false conclusion can easily be drawn that if -a times +a = -a2, conversely +a times -a = +a2. Since one factor is amount and the other unity, and the one which comes first usually means that it takes precedence, the difference between the two expressions -a times +a and +a times -a is that in the former +a is the unit and -a the amount, and in the latter the reverse is the case. Now in the first case it is usually said that if I am to take +a -a times, then I take +a not merely a times but also in the opposite manner, +a times -a; therefore since it is plus, I have to take it negatively, and the product is -a2. But if, in the second case, -a is to be taken +a times, then -a likewise is not to be taken -a times but in the opposite determination, namely +a times. Therefore it follows from the reasoning in the first case, that the product must be +a2. And similarly in the case of division.
This is a necessary conclusion in so far as plus and minus are taken only as simply opposite magnitudes: in the first case the minus is credited with the power of altering the plus; but in the second case the plus was not supposed to have the same power over the minus, notwithstanding that it is no less an opposite determination of magnitude than the latter. In point of fact, the plus does not possess this power, for it is to be taken here as qualitatively determined against the minus, the factors having a qualitative relationship to one another. Consequently, the negative here is the intrinsically opposite as such, but the positive is an indeterminate, indifferent sign in general; it is, of course, also the negative, but the negative of the other, not in its own self the negative. A determination as negation, therefore, is introduced solely by the negative, not by the positive.
And so -a times -a is also +a2, because the negative a is to be taken not merely in the opposite manner (in that case it would have to be taken as multiplied by -a), but because it is to be taken negatively. But the negation of negation is the positive.
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