The limits one may set to the term “contradiction” are to some extent arbitrary, since the word is normally used in different senses. I shall use the term in the widest sense compatible with my immediate object. This follows excellent precedent. For example, Aristotle’s Principle is generally applied to “contraries” (such as “black” and “white”) which logicians distinguish sharply from “contradictories,” (such as “black” and “nonblack”) as well as to contradictories themselves. Thus the term “contradiction” may include by implication – or even by definition – the notion of “contrariety.”
Contradictions may be divided into three main types.
(I) Static contradiction: this type is of the “black-versus-white” and “black-versus-nonblack” varieties. It is identical with a type of opposition sometimes called “fixed opposition.” Any two things or two sets of things, may be said to be in a state of “static contradiction” when they mutually exclude each other while joined in a complemental relation – which latter may be explicit or implicit, i.e. obvious or subtle. (“The knowledge of opposites is one” – Aristotle.) The emphasis in this type of contradiction is upon the mutual exclusion- the separateness, the opposition: the duality – of the two “sides” (sometimes called “poles” or “extremes”) and, because there is normally no traffic or interpenetration between the two sides, contradictions or oppositions of this type are variously called “rigid,” “uncompromising,” “external,” “extrinsic,” etc. Static contradiction is well typified in a simple “lumps-of-matter” environment i.e. a Euclidean environment – and is the sphere in which Aristotle’s Principle applies par excellence; for when the two sides of the contradiction are forced together to become one, either in thought or in “fact,” the net result is respectively zero meaning or zero existence; that is, the proposition becomes meaningless or the “thing” becomes non-existent. “Positive and negative,” we say, “cancel each other out when added together.” Hence, x + (-x) = 0 is at one and the same time, the typical example of the operation of Aristotle’s Principle, and also of static contradiction. Static contradictions are thus conceived in terms of the co-existence and the mutual exclusion or incompatibility) of opposites – here there is a sharp line dividing the two sides. Even the succession of opposites (in time) may be, and often is, so conceived like a stick or a straight line with two opposite (yet co-existent) ends. Thus the concept of static contradiction may be said to be a “linear” concept, i.e. typical of an environment described by Euclidean geometry, in which the line plays the major role as bounding or dividing two opposite sides, or joining, uniting two opposite ends.
(II) Dynamic contradiction: this type is more concerned with time and with processes than is the static type. As soon as movement, or reaction, or interchange occurs between the two opposites we find that they tend to interpenetrate – that is, the separate identities of the two begin to break up and merge – it becomes difficult to define sharply the boundary between them. With increasing reciprocal activity between the two sides, there is, of course, a corresponding increase in their mutual dependence. Moving or changing physical systems, including living organisms, with their many opposite reactions and functions, well exemplify this dynamic type of contradiction. Aristotle’s Principle still applies to this sphere, of course, but it is beginning to apply less well. Rather, the law which applies in this sphere par excellence is the Newtonian Third Law of Motion: “Action and reaction are equal and opposite.” In many cases the functional interdependence of the two sides is so great that it is difficult to conceive of them as two “actually-separate” things. The duality which is so obvious and dominant in the static type of contradiction becomes, in the dynamic type, subordinate to a dominant unity. The two opposite “things” tend to become two opposite functions or aspects of one “thing” – or in some cases, even, to become two opposite aspects of one function, – or opposite parts of one aspect. Hence contradictions or oppositions of this type are often called “internal,” “intrinsic,” “inherent” “living” “mobile” etc. Nevertheless, the interpenetration of opposites still leaves each side with its own independent identity, even though that independence has been eroded by the “higher” unity, that is, by the overall identity of the system formed by the mutually-dependent pair. The independence of the two sides has changed from the apparent absoluteness and fixity promoted by Aristotle’s Principle (i.e. as dualism, dichotomy, bifurcation, etc.) to their relative independence – “relative” because limited by their equally relative mutual-dependence. The simple “linear” concept, which is so appropriate to static contradiction, is now modified to become the path of a moving point – a “locus.” Sets or masses of loci thus represent to us “forces” and “lines of force,” “mechanical motion,” “chemical change” and “organic process.” Even time itself is abstracted from experience by means of the concept of locus; we conceive it as the motion of a point representing “the present,” along a line representing “past” and “future” – i.e. as the path of a moving point. We construct “graphs” and “vectors” on this basis to aid our understanding of process and change. As a further development of this, when we wish to represent to ourselves – in the time dimension – processes involving repetition (and therefore “to-and-fro” motion) we further modify the linear concept and construct it in typical “wave-form.” Loci, graphs, vectors, periodic functions, etc. – all are capable of mathematical development. They thus become, together with the mathematics of the static linear concept, the foundation of those modern scientific techniques for changing our material environment. Just as static contradiction is the ruling type of contradiction in Aristotle’s logic and Euclid’s geometry – i.e. in “logico-geometry” or “static geometry” – so dynamic contradiction is the ruling type in Descartes’ geometry and Newton’s mechanics – i.e. in “mechanico-geometry” or “dynamic geometry.” (It was, of course, Rene Descartes’ geometry of moving points and loci badly named “analytic geometry” because it is more unitive than separative – which led Newton to the calculus and Leibniz to the calculus and to functions etc., just as Aristotle’s logic had led to Euclid’s “Elements” and to further mathematical development.)
(III) Self-contradiction: this type can be regarded as the rational outcome of the continued change of viewpoint which we detected as we moved from consideration of the static to the dynamic types of contradiction. Starting with static contradiction, we noticed that it presupposed a fundamental duality in the contradictory system. That is to say, we could acknowledge the “actual” or “real” existence of the contradiction so long as we could regard the two opposites as basically independent of each other, i.e. such that the elimination or destruction of the one did not necessarily involve the same of the other. It is this presupposed duality or dichotomy, of course, which is implied in Aristotle’s Principle. In the Euclidean environment the dichotomy is explicit, obvious, and moreover, “real”; but the complemental relation of the opposites – their unity – is implicit, subtle and, “mental” (“the knowledge of opposites is one”). Because of this lack of real mutual dependence of the two sides, they can be said to be “indifferent” in their opposition. When we turn to consider dynamic contradiction, we note an important change in the relation of the two sides: the sharp line of their division begins to blur; the opposites start to fuse or interpenetrate. Movement occurs involving reaction, reciprocity and response. The dichotomy is no longer fixed and absolute; the unity is no longer merely implicit, or of the superficial, “side-by-side,” extrinsic sort. The one side can no longer exist without the other; they have become mutually dependent. The fusion or interpenetration of the opposites, however, remains partial and incomplete. Self-contradiction arises when the movement towards greater interpenetration and mutual dependence reaches its theoretical limit – when the two sides or opposites come into coincidence with one another, i.e. when they become “identical.” When do they become identical? Under what conditions? Aristotle’s Principle says unequivocally and without qualification, “never” and “under no conditions.” But we have seen that, if taken as meaning “universally never” – i.e. if we make the universality (implied in the unqualified “never”) explicit – then the Principle itself becomes a self-contradiction! Hence, it would appear that there are, indeed, conditions under which self-contradictions “exist” – i.e. under which the two sides or extremes of the opposition move into coincidence, into identity. What are those conditions? We have seen that static contradiction is typical of an Aristotelian-Euclidean environment; that dynamic contradiction is typical of a Cartesian-Newtonian environment. What sort of environment is it of which self-contradiction is typical?
continue reading The Paradox Principle by Harold Walsby (1967):
Dedication | Aristotle’s Principle | The Role of Logic | Do Self-Contradictions Exist? | Three Types of Contradictions | Meaningful Self-Contradictions | Infinity and Self-Contradictions | Models for Self-Contradiction | The Paradox Principle and Applications | Appendix