Negation and Contexture

Gotthard Günther*1)

To begin with we shall have to distinguish between categorial context and universal contexture. Everybody is familiar, from the normal use of conventional language, with the idea of a context. We refer, for instance, to human beings within such different contexts as are denoted by law, by biology, by politics or by history. Within each of these contexts we assign to a person different properties. Within the context of (criminal) law a person may be guilty or not guilty. Within the context of biology we may consider a person healthy or sick, and within the context of politics an individual may be considered conservative or progressive. All these cases have one thing in common: wherever we perform a predication - as e.g. in the proposition: "this person is guilty"- we assign to the object of the predication not only a predicate but also a context within which the predicate is relevant, or not relevant. We are not permitted to ignore this relation between predicate and context. And it makes no sense to say that a sin is triangular or may be octagonal. In other words, the Tertium Non Datur (TND) which decrees that a given datum of experience must either have the property a or non-a (exclusively) normally refers to a stateable context. Such contexts may be very narrow or extremely comprehensive; but their stateablity is always required in order to make Logic applicable to the empirical world. On the other hand, this world displays such a fan-tastic amount of contexts and demonstrates such an impene-trable incommensurability between uncountable groups of them that it was necessary from the very beginning of the history of logic to introduce a "metaphysical" postulate with regard to the disparity and incommensurablity of certain contexts. It was assumed that all contexts are ultimately capable of well ordering and forming a universal system in the sense of the Platonic pyramid of Diairesis. This led to two con-clusions which are closely connected with each other. The first is that a statement like "a sin is triangular or not triangular" is meaningful in the sense of the TND and the second that we have to stipulate that the TND may be used in two ways: either with referring to a stateable context or in the sense that it is in principle impossible to indi-cate the context to which the alternative of position and negation may refer. The history of logic has not always clearly distinguished between the two ways of applying the TND. The context which determines the operational field of the excluded middle in the first case may be of such practically unlimited generality that it may be difficult to find a negation for it which would establish a material viewpoint outside of the proposed context. However, this practical difficulty should not be confused with the prin-cipal absence of a context. This latter case has, in the history of logic, found its most famous expression in the coincidentia oppositorum of Nicolaus Cusanus.

This raises the question: is the universal system of all conceivable contexts which is denoted by the index of the Platonic pyramid also a context or is it not? The answer is rather obvious. A system which integrates all possible contexts cannot itself be interpreted as a context because if it were a context it would have to be stateable as such and materially differ from the other contexts. But this means it would be a potential object of integration itself which precludes that it could take over the function of integrating concepts.

If we still insist on the logical meaningfulness of the idea of a total integration of all stateable contexts it must be something that - although it is governed by the TND - cannot be defined as a context with positive properties. We shall call such a domain without positive properties a universal contexture and want to add that it can only be interpreted as an empty dimension which may either be filled with "objects" ( scil. contexts) or not.

This means that the TND is still relevant, even under circumstances where its relevancy does not belong to a stateable context. In other words: we have to distinguish between two entirely different functions of the TND which, in the history of logic, have not always been clearly separated: the TND referring to stateable (positive) contexts on one hand, and the TND referring to a universal contexture on the other. In order to illustrate the difference and also the case where the TND is not relevant at all we shall go back to our example about the predication of sin. If we say ´sin is triangular or rainy´ the TND is totally inapplicable, because 'sin', 'triangle' and 'rain' belong to three different contexts. On the other hand if we say ´sin is permissable or not permissable´ the TND is applicable because sin refers to a context which is positively stateable and which is meaningful for the term to be affirmed or negated. But there is a third case which may be exemplified by the proposition ´sin is triangular or not triangular´. This latter statement should never be confused with our first one that ´sin is triangular or rainy´ because in this former case we have arbitrarily chosen for predication two contexts which do not form an alternative in the sense of the TND and which exclude positively other contexts. However, if we state ´sin is triangular or not triangular´ our alter-native does not exclude any context at all because ´not trian-gular´ may encompass all conceivable contexts except the one to which the term triangular belongs. Thus we are permitted to say that the statement ´sin is not triangular´ is in a peculiar and limited way true insofar as this negative pre-dicate implies all possible affirmative predicates which may be assigned to the subject of predication. But if we say, that, owing to the character of implication, there is some sense in saying that such seemingly absurd statement like ´sin is not triangular´ covers some hidden logical meaning, the same must also apply to the other predicate of the alternative. What is meant is this: the term triangular is only an empirical index of some hidden ´metaphysical´ property. Therefore it could be re-formulated in a way that the total alternative of triangular or not triangular would be applicable to our propositional subject called ´sin´. However, it should be undersstood that such a re-formulation could not be produced by a finite number of steps. Ergo it can never lead to a context which can be stated in positive terms. What this postulate of re-formulation really designates is what we have called a universal contexture. In other words: an empty domain in which operations may be performed.

Thus we have described two modi of operation for the TND. First it may operate within a stateable context which can be described in positive terms of this empirical world. Second the TND may operate in such way that it encompasses all positive contexts and puts them into relation to something that is not a positive context at all. It stands to reason that in the second case no context can be given for the operation of the TND. It designates a universal contexture. The tradition has old names for the two modes of operation in which the TND may be activated. In the first case where it is concerned with a positive context it applies itself to Existence. In the second case it refers to Essence. Existence has frequently been identified with the particular forms of Being and Essence with Being-in-general as the underlying substratum for all empirical contexts of Existence. Another historical form in which universal contexture has made itself felt in the history of Logic is the coincidentia oppositorum of Nicolaus Cusanus. It is highly significant that it is impossible to interpret the coincidentia oppositorum as a material context because what coincides in it is the alternative of affirmation and negation. Thus the coincidentia is not negateable. But a context has to be negateable in order that it can be exchanged against a different one. This leads us to the conclusion that, if the TND is applied in such a way that no concept can be given as the range of its application, then the result will always be the coincidentia oppositorum. At this point Logic transcends into Metaphysics. This is incontestable in the case of Nicolaus Cusanus because he expressly identifies the coincidentia oppositorum with God, and since Christianity is a monotheistic religion this identification implies that there is only one universal contexture.

It goes without saying that this sort of argumentation is of little use to mathematics and exact science. For in the classic tradition a universal contexture can only denote a metaphysical entity and it is not our intent to lose ourselves in metaphysical speculations. It seems we have been led astray by following the classical argument. We shall therefore retrace our steps in order to find out whether we have not overlooked something that will permit us to remain with our logical analysis in this world instead of being transported into a mystical Beyond.

We repeat: two interpretations of the TND are extant in the history of Logic. It can be either assumed that the TND operates in a definable positive context or that it is effective although it is on principle impossible to state any positive context to which it may refer. In the first case it is capable of a material interpretation, in the second case it denotes the purest expression of formality. What has been overlooked, however, is the fact that the second interpretation of the TND is ambiguous and can be understood in a twofold way. We may either assume that the exclusive alternative which the formal TND represents may be understood as an alternative between context and contexture, in other words between material content and that which does the containing. But another interpretation is also possible. The ultimate TND may not refer to a positive context because it represents an alternative between two universal contextures. It is evident that the introduction of this ambiguity is incompatible with the total of classic tradition and especially with the philosophy of Nicolaus Cusanus. If we assume that the TND is originally directed by positive contexts which follow each other in a hierarchi-cal arrangement of ever increasing generality, then it follows that the separating power of the TND which keeps an affir-mation and its total negation apart grows weaker and weaker the more general the individual contexts become till finally the point is reached where the context becomes so general that the separating power of the TND completely disappears and nothing is left but the coincidentia oppositorum. To put it differently: the classic tradition postulates an ulti-mate collapse of the TND and at the point of the collapse the Physical transcends into the Meta-physical.
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