It will be useful to take a look in broad-perspective at what we are about to do. I must therefore say a word about the historical role of the principle of Non-contradiction.
The usual view of logic is that “natural logic” – sometimes called “native wit” – is inborn, somehow “given” by nature (or Divine Providence) at the beginning of creation, and thenceforth handed on to us jointly by heredity and the permanent constitution of the universe. On this view, the pursuit of the subject of logic (orginated by the Greeks, notably Aristotle) consists largely in the recognition and formal presentation of these inherent, ready-made rules of thought to which we are, inscrutably, forever bound. Some such view of logic is very common, lingering even among the scientifically-educated, and is indeed a hangover from a literate but semi-scientific past.
To imply that the subject of logic sprang into existence with the Greeks – which, due to our scant attention to its origins, we are rather prone to do – is like thinking that a person’s life begins with a “pop” at the moment of birth! Sharply defined and fixed limits are useful in the early stages of learning, but often outlive their usefulness and become barriers to further advance. Let us take a wider view.
Looking back over the last two to three thousand years at the mainstream of man’s development, we can see clearly the growing importance of those rational techniques for manipulating our environment which, collectively, we call “technology.” We have, today, tremendous and ever-growing power to control material things and adapt them to our needs. Thus civilisation rests more firmly and squarely upon technology than ever. In turn, modern technology is enormously dependent upon the “natural” sciences, i.e. those which deal with matter: physics, chemistry, etc. The natural sciences, dependent as they are upon crucial techniques of measurement and calculation, themselves rest upon the firm foundation of mathematics. And, as we have already seen, mathematics is itself founded upon logic, the basic Principle of which is that of Non-contradiction.
Hence there is a sense in which it is true to say that in summary, (i) our material civilisation rests upon the foundation constituted by Aristotle’s Principle, and (2) the history of the last 2,000 years or so, is largely the history of the rise of our present civilisation upon that very foundation.
Three thousand or more years ago in some areas where nature was still abundant but mild – mostly around large, sub-tropical river basins and deltas – social evolution was following a pattern that was to become familiar. Civilisations were developing in Southern Asia, North-eastern Africa and Southern Europe, founded upon a simple technology and rule-of-thumb science which included early mathematics. Nations formed, societies grew, empires rose and fell – made possible by a combination of nature’s beneficence and man’s supreme cruelty to man: chattel-slavery. That institution, which did so much to make ancient civilisation and culture possible, arose out of small wars of conquest and was fed and extended by larger wars and larger conquests.
The systematic acquisition of prisoners-of-war as a source of “power” – slaves – was a basic part of the ancient “technology.” A typical power-slave was “outside” society, a chattel who performed work like that of a donkey-engine. Slave-power was viewed much as we view horsepower. Engineering projects merely required, apart from the knowhow and a few simple tools, the power represented by the appropriate number of slaves. The size of the job, however large, was measured in terms of the slave-power required to do it.
“Democracy,” that much-lauded flower of Greek culture, rested upon this hideous slave-powered technology. Masses of whipped, toiling humanity bore the Greek political freedom upon their suffering backs. Fortunately for us, the more intelligent of their politically-free masters had a strong taste for argument, for wide-ranging theoretical studies and the representation of practical knowhow in generalised form. The most important outcome of the Greek and Mediterranean schools was the formulation of logic and the development of mathematics as a logically-evolved discipline.
The next most important outcome of the Greek period was moral and philosophical – which, when it fused with certain religious developments of Jewish-Oriental origin, produced a religious movement identified with the oppressed, that was not only to engulf the Western World but also to do much, finally, to bring about the end of the chattel slave states.
Summarising, then, we may say that man’s great need around 3,000 years ago was to evolve the theoretical foundations and values which would (i) make possible modern technology, founded on “artificial” power, and (ii) make impossible the ancient technology, founded on chattel-slavery, i.e. “slave” power.
The need brought forth the men who were to supply it. They were many, but the most outstanding names are the familiar ones. It has taken more than 2,000 years of history to gather some of the ripe fruits of what they put into the ground, i.e. into our heads. And fundamental in that development has been Aristotle’s Principle (actually formulated earlier by others, e.g. Plato, “The Republic,” Book V, 436B). On its basis we have built the techniques which now enable us to “push matter about” almost as we wish.
Slowly and painfully we have learned that, if we are to control our material environment, we must think and act “logically” – that is, in a manner which is in harmony with the rigorously logical behaviour of matter itself. In sharp contrast to this logical behaviour of matter, the behaviour of humans is very often, perhaps mostly, capricious and illogical. It is no great wonder, then, that the rigorously logical techniques which have been so apt in controlling our material surroundings, have failed so dramatically, or been so unfitting, in our frantic efforts to control that part of our environment constituted by ourselves! If we are to fathom the secrets of self-control, then once again we must learn to think in a manner basically in harmony with the behaviour of that which we wish to control: namely, our own much more flexible, wide-ranging behaviour – which, though it can be rigorously logical at times, is not so restricted! Our task, in short, is to extend the territory of rationality, to find new realms where reason and sense can reign but where, up to the present, they are barred from reigning by the narrow scope of existing logic.
Aristotle’s Principle, so valuable in enabling us to build the techniques for the control of matter, thus becomes a restriction on building the techniques necessary for self-control. For this we need a wider logic.
It will now be evident that what we call “natural logic” is, basically, a set of restrictions upon our thinking which simply “grows up” with us as the result of each individual having to learn to cope, more or less successfully, with the world around him. That surrounding world is the familiar world of the child, of the home
Moreover, it is also clear that this natural logic is the general foundation upon which all formal systems of logic, however sophisticated and specialised, have to be built. Aristotle’s logic was the first successful formal specialisation its main role, as that of the mathematical systems erected on it, was to enable us to subdue matter and adapt it to our grosser, more urgent needs. It is, however, a specialised development. Control of matter without corresponding human self-control is dangerous, and can lead to social disaster. We are now in increasingly-urgent need of evolving the theoretical foundations and values which will make possible the rational techniques of social self-control.
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