Before one can deliberately, systematically sit back to contemplate life and its problems, one must have the more urgent preoccupations of the struggle for life taken off one’s mind (and hands). The institution of chattel-slavery made possible the creation of a class society with the more intelligent members of the leisured class roughly divided into executives and theorists.
The real problems of life under such conditions were simple compared with those of our own day. They consisted largely of “pushing lumps of matter around” and developing the power and the manipulative skills to do it. The top “theorists” became preoccupied with the more general and abstract aspects of just these problems. The need to solve them had, in the, first place, given rise to the need to “push humans around,” i.e. to evolve chattel-slavery – and slaves, in that age, were viewed as hardly much more than lumps of animate matter. The top “executives were the hierarchy of military and civil governors, etc. who saw to it that humans were, indeed, “pushed around” in accordance with the ruling conceptions of the common welfare.
The modern child, in the course of its education, to some extent recapitulates the essential phases of this simple pattern of development, with parents and teachers sharing the functions of executive and theorist. One of the earlier things which a child is taught is to distinguish the fundamental difference between what is so and what is not so. “Don’t contradict!,” “‘Tis so,” “‘Tisn’t” are familiar phrases a child. Exploring the surrounding, finite world of material things supplements and confirms the implied dictum; “A thing cannot both be so-and-so and not be so-and-so at the same time.” On this basis the child succeeds in acquiring manipulative skills leading to mental manipulation of signs, tokens, etc. – to logical calculation and simple arithmetic. The child’s thought is thus guided by this dictum in proportion as it becomes educated and “rational” – i. e. logical.
But education not only enables us to “see” certain things; it also helps to make, us “blind” to other things, especially to the things which oppose or contradict those we have come to see so clearly. For example, the more we have learned to speak our native tongue, the more we find the learning of a foreign tongue just that much harder. Hence, since our education is largely founded on the Principle of Non-contradiction, it is hardly surprising that the more “educated” we are the more we tend to be blind to the many instances of self-contradiction to be found almost anywhere within and around us!
Aristotle’s Principle tells us that self-contradictions do not exist! “Nothing,” it says, “can both be and not be at the same time.” If it is taken as a universal law, if we assume it to be unconditionally true, then it is equivalent to asserting the complete non-existence of self-contradiction altogether! Why then, if self-contradiction is unconditionally non-existent and therefore impossible, do we need the Law of Non-contradiction? It is rather like Parliament solemnly legislating against an utterly impossible crime! Thus, if we take Aristotle’s Principle as universal (as unconditional or absolute) then it follows that it is absolutely unnecessary! But to assert that a rule or law is universal, unconditional, or absolute, is to assert that its application is always necessary – i.e. that its necessity is not conditional or contingent but complete and absolute. Hence, taken as a universally applicable law, Aristotle’s Principle implies a contradiction of itself. Hence – because anything which implies a contradiction of itself is said to be a “self-contradiction” – Aristotle’s Principle, as a universal law, is a self-contradiction.
Evidently, then, some very awkward consequences follow from, assuming that the Principle of Non-contradiction applies without restriction. This seems at first sight to be in accordance with the postulational method – in which the basic principles of a mathematical system are taken as postulates only; i.e. as essentially non-universal or arbitrarily restricted, and as necessarily holding only within the postulate system itself. However, to think that my analysis is four-square with prevailing postulate theory would be a grave mistake. The fact is, there are rules for constructing postulate systems, and here is one rule which must always apply to a set of postulates – and which therefore applies throughout all postulate systems. That rule is: “The postulates must never lead to an inconsistency.” (These words, including the word “never,” are Professor Bell’s own). And, of course, the ultimate test of consistency is the Principle of Non-contradiction!
Thus, despite appearances to the contrary, “the postulational method” includes rules which insist that at least one postulate is not arbitrary, not restricted, but is universally valid and necessarily applicable to all – and throughout all – viable postulate systems. But we have just demonstrated that this very universality is self-contradictory and self-negating. The truth must therefore be faced: the much, vaunted postulational method is not postulational enough!
We are now left with the following alternative position
(1) that the Principle of Non-contradiction is of limited or restricted application;
(2) therefore, since it cannot apply throughout all systems, there must exist at least one system in which self-contradictions, in some form or other, are “allowed”;
(3) therefore there may be a way of specifying and developing such a system;
(4) therefore the present criterion of consistency must be shifted from the Principle of Non-contradiction to a wider, more general basis;
(5) this does not mean the contradiction – merely and its inclusion as a principle.
What is that more general principle?
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