# Harold Walsby: Three Types of Contradictions

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