Aristotle’s Principle, as we have just seen, implies the existence of some contradictions and denies the existence of others. This would account for the otherwise inexplicable fact that it is sometimes called “the Principle of Contradiction” and sometimes “the Principle of Non-contradiction,” titles which are in flat contradiction of one another! Strictly, of course, since it explicitly bans or denies self-contradiction, it should really be called “the Principle of Nonself-contradiction.” Let us look a little more closely at this somewhat peculiar phenomenon of “self-contradiction.” I have said that self-contradiction emerges when something implies that it is a contradiction of itself: namely, when something either is, or becomes, or implies that it becomes, identical with its opposite, or when two opposite poles or extremes meet and merge into one. The various modes by which opposites can fuse to identity need not occupy us here in any detail. Suffice it-to say at this stage that the process may be (1) a wholly-internal development, or (2) partly internal, i.e. an interpenetration of one pole by another from outside. The point I want to make here is that, in some cases, the fusion of the extremes results (as Aristotle’s Principle prescribes) in mutual cancellation – either zero meaning or zero existence, as we saw Bariler – but, in other cases, mutual cancellation does not occur, or does not occur completely or in the same sense.
It is these latter cases which constitute the sphere of meaningful self-contradiction and which, therefore, lie outside of the province of Aristotle’s Principle. I have now only to give an example, or examples, of meaningful self-contradiction to establish the existence of this sphere and its rationality.
Before I give these examples I must say something of the role of zero. At present the time-honoured method of dealing with self contradictions is, firstly, prophylactic: to take the greatest possible pains and care to avoid them completely. If, despite all, the pains, self-contradictions should appear in the course of scientific inquiry – as, indeed, they have in certain problems concerning logical and mathematical consistency – then the traditional attitude is as follows. Either we must eliminate them completely or, if this cannot be effected by normal methods, we must forthwith seek further ways or procedures for eliminating them (much gifted and brilliant work has been devoted in vain to the latter.) The normal methods are, of course, the standard logical or mathematical procedures, as e.g. x + (-x) = 0 or failing the existence of such built-in devices by abandoning the chain of reasoning, or even the whole line of enquiry, which led to self-contradiction. In sum, the orthodox attitude is that somehow or other the total self-contradiction must be equated to zero, either directly, inside the system of reasoning, or indirectly, outside, the system, i.e. by abandoning or expelling the offending parts of the system.
The time-honoured way of looking at this problem, however, is not the only possible way. We have seen that, from the orthodox viewpoint, “equation to zero” signifies “equation to complete or absolute nonexistence (namely, of any object or any meaning).” That is to say, mutual-cancellation of opposites equates to nothing whatever. Yet it is quite conceivable that, given certain conditions, the nonexistence (or absence) of a particular thing may signify, not just absolute nothing, but something. Consider, for instance, the old proverb, “No news is good news.” Quite obviously it is not universally true, but it could easily be true given certain conditions. Here, then, we have the equation of the nonexistence of something (namely, “all news, including good news”) with the existence of something (namely, “good news”). The statement is a meaningful one, yet it involves a self-contradiction. In other words, the self-contradiction is meaningful. Note well that its meaningfulness is conditional. This type of case depends on the condition that the absence of the whole of something signifies the presence of some part of it. So that what is asserted is: both the presence and absence of the same part at the same time. Here are further examples of the same type: “The best plan is to have no plan at all,” “My New-Year resolution is to make no resolutions,” “Do me a favour: please, don’t do me any favours!,” “Having no morality is the worst morality,” “As a complete failure he is a positive success!” etc. etc. There are many others, and everyday discourse is full of them.
The modern academic approach to this phenomenon stems from work pioneered by Russell and Whitehead, though its essential aspect goes back as far as the medieval Schoolmen. It is as follows: “The alleged self-contradiction does not exist. The two sides which are presumed to be in self-contradiction do not, in fact, belong together in the same universe of discourse. It is illegitimate and meaningless to speak of them as if they do. For example, if we insert the word “actual” in the statement above, the distinction between the two kinds of universe of discourse becomes quite clear: ‘This type of case depends on the condition that the actual absence of something signifies the presence of some part of it.’ The actual presence or absence of a thing cannot be equated with its signified presence or absence. They should not, therefore, be confused. One cannot choose one side of the contradiction from actuality and the other side from a signified actuality, and then equate them as if they both belonged to the same universe of discourse, i.e. the same logical type, the same order of reality. Hence, there is no self-contradiction, meaningful or otherwise.”
This somewhat plausible argument will not, however, hold up under critical examination – as a number of writers have shown. Bacon, criticising essentially the same thing in the early 1600’s, writes (First Book of Aphorisms): “… And if some opposite instance … chance to come in the way, the axiom is rescued and preserved by some frivolous distinction; whereas the truer course would be to correct the axiom itself.” But the overwhelming case against the orthodox attitude of “rescuing the axiom,” i.e. preserving Aristotle’s Principle intact, is the one we have already noted. It is, namely, that the orthodox approach assumes the exclusive universality of Aristotle’s Principle. This leads to an automatic ban upon any future development which may arise to challenge that universality. Hence, because of the very universality – and presumed eternity – of such a ban, the argument assumes virtual omniscience, and therefore, a species of divine ordinance. Such arrogance is against the modern spirit of enquiry, which latter has always had to do battle with Aristotelian tradition on the big issues.
It follows that we may regard any attempt to make universal and permanent – i.e. to fix – a fundamental distinction or ultimate difference between one universe of discourse and another, between one meaning and another, between one thing and another, as an attempt to make the principle of dichotomy and duality the basic principle of everything, forever. Moreover, we may regard it as quite arbitrary. Just as arbitrarily, we may set up a contrary principle, and this we do. To answer the orthodox argument above, then, we simply assert that “an actual absence which signifies presence, can, indeed, if we so choose, be included in the same universe of discourse.” There is no divine dispensation which can, declare it “illegitimate.” In fact, of course, the orthodox argument cannot be stated without itself doing what it declares to be illegitimate: namely, including the two sets of dichotomous “incompatibles” in the same universe of discourse!
The issue of the universality or nonuniversality of a principle is, as we have seen, bound up with (a) on the one hand, the existence of conditions under which the principle operates, and (b) on the other hand, the arbitrary assumption or choice of those conditions. All thought, all thinking, involves relation or comparison, and all comparison involves similarity (including identity) and difference (including opposition). When we compare any two things we are confronted with a series of choices.’ On the one extreme we can play down, or even entirely ignore, the differences between the two things and simply stress the similarities. On the other extreme, we can ignore the similarities and stress the differences. Which we choose depends largely on the purpose of our comparison. Our purposes may be such that, over any given period of time, we tend towards the one extreme rather than the other. Indeed, we may so occupy ourselves with one aspect of a given field of study, and for so long a time, that the tendency towards one extreme may become deeply habituated in our mental approach to certain things. Owing to its great value in the restricted, specialised field, this habitude of mind may even spread and become general in our approach to all fields. In our present age of excessive specialisation we could reasonably expect it to occur on a large scale. It is this tendency of specialists, especially academic specialists, to extrapolate these mental habituates (gained from successful experience in their own restricted fields) to all fields, which universalises such arbitrary principles as Aristotle’s.
This universalising of principles has the usual effect of deification: the universalised principle “takes up all the available room” so to speak, and leaves no place for a different, especially an opposite, principle. Thus, what amounts to a virtual taboo is put upon serious consideration of any principle which challenges, or appears to challenge, the universality of those established. It is one thing to concede all this in a general sort of way, as proponents of the so-called “postulational method” do in fact. However, to concede it in a particular concrete case, where actual practice belies abstract theory, is quite another thing. Hence, opposition to the de facto dethroning of Aristotle’s Principle (which, of course, does not mean abolition) could easily, and does, come from such quarters, even if that opposition be more silent than vocal.
Having dealt with the main initial objections encountered when considering cases of meaningful self contradiction, I will now go on to consider some further cases which, although similar to those already noted, are nevertheless a distinct variety. The first cases we met with involve, as we saw, the total absence of something, equated (or identified) with the partial presence of that thing. Now, whether the absence referred to is actual, while the presence is merely signalled by the absence, is a matter of irrelevance. We can and we do, as I have implied, identify two or more different things quite arbitrarily together – simply because it is very convenient and practical. We do it whenever we form the notion of a particular class of individuals or whenever we consider such things as the successive rotations of a planet. By sufficient abstraction – e.g. by ignoring actual differences between individuals or rotations we are able to regard two or more things as “identical” with one another.
The second group of cases involves the equation of the total absence of something, not with its partial presence, but with its total presence. As previously, we find zero (in “total absence”) playing a role. Often, when asserting a proposition, we ascribe an attribute to one or to some members of a whole class of things, where the truth or the proposition depends on the condition that the rest of the members (or at least one other) does not have the attribute. In other words, the comparison – implied by the comparative attribute – is within, and confined to, that total class of things. (The fact that the attribute is logically comparative does not necessarily mean that the term used is grammatically comparative; usually it is not). Of such a class we might say that some members are – e.g. tall – or warm, or heavy, etc. – tacitly implying that these members are tall compared, not with an outside standard of tallness, but simply with the rest of the members. We can separately express this governing, internal standard (unexpressed in our proposition about “some” members) by stating the principle that “if all the members are tall, then none are tall.” That is to say in succinct terms, “‘all tall‘ equals ‘none tall“,’ or even more succinctly, “all tall” = “none.” Here is another example. If in a book there are some words, or even some whole passages, printed in italics, then these words and passages are emphasised in relation to others which are not. However, if all the words in the book are so emphasised, the emphasis necessarily vanishes! Once again we can express this internal standard (or condition, or principle) by the proposition “All emphasis is no emphasis,” or more succinctly, “All emphasis = none.”
A Gilbert and Sullivan song tells us of a king who was so good that he promoted everyone to the top of the social scale. The song goes on to state the principle involved: “When everyone is somebody, then no one’s anybody!” Once more we can put “When all are top, then none are top,” or “all top = none.”‘ A BBC television play, “The Public Duck,” (519/66) contained a line spoken by a character in a pondside scene: ‘The ducks belong to everyone… ‘ Another character finished the sentence, ‘… and to no one.’ If all own, then none own, i.e. all owners = none. This immediately recalls the well-known proposal to abolish differential (i.e. private) ownership of property by making property-ownership common and undifferentiated, the principle of which is: common ownership = nonownership. Further examples of this second type of self-contradiction are furnished when something is said to “over-reach” itself, or to defeat its own purpose, namely, when it is self-defeating, self-refuting, self-negating, etc. For instance, someone may be over-cautious and bring upon himself the very disaster his caution is supposed to help him avoid. Such a high degree of caution is identical with a low degree of caution, because the victim is not cautious enough to see that his excessive caution leads to disaster. Hence, the highest caution = the lowest caution. Emerson, according to Lin Yutang (The Wisdom of Laotze) said, “The highest prudence is the lowest prudence.”
Further instances abound in philosophy, science, mathematics and even in logic itself. Take, for example, Descartes’ famous method of scepticism, or universal doubt. The self-contradictions involved in the development of this idea are many and devious. To doubt everything necessarily implies that we doubt the universal principle of doubt itself! Thus it is self-defeating or self negating. It is not even an indubitable principle! If we doubt everything, then the only thing we can be certain of is that nothing is certain. But this is something, so we cannot even be certain of that if we are to doubt everything. If nothing whatever is certain, then everything obviously – with complete and absolute certainty is uncertain. In other words, complete uncertainty, complete doubt, implies complete certainty. And vice versa, complete certainty, complete lack of doubt – since it implies something completely untested and untried – implies complete uncertainty. The two opposites, therefore, equate. Hence, we can doubt some things because we don’t doubt others; we can be certain of some things just because we cannot be certain of other things. The corresponding internal standard can thus be expressed in either of two equivalent ways “all doubt = none” or “all certainty = none.” It follows, of course, from these two connected proto-equations that “all doubt = all certainty.”
Other examples from philosophy are legion. I have space to note only two more. Kant is famous “antinomies” in certain arguments wherein Kant shows that, under given conditions, there arise identities of such opposites as the “beginning” and “nonbeginning” of the universe (i.e. its finite past and infinite past) etc.; and Hegel’s equally famous dictum, “Being is identical with nothing,” the most abstract expression of his dialectic principle.
Consider, now, a certain set of operations: namely, two or more successive operations each having an overt part and a hidden part. Any two overt parts are “identical” with one another; the hidden parts, however, are of two opposite kinds such that the effect of the one kind is to undo or cancel that of the other. Thus, the effect of any two successive, whole (and apparently identical) operations is for the second one to cancel the result of the first. A simple example is the “repeating” type of corded electric switch, where every pull undoes the effect of the preceding pull. We have here all the essential ingredients of our other examples. If it be objected that the “identical” overt parts only lend the “appearance” of identity or similarity to two successive operations – which are, in “reality” diametrically opposite – the answer is that this particular case is of the same type as our others. As I have previously stressed, what things we are able to consider as “identical” is arbitrary, a matter of purpose and sufficient abstraction. The point about this set of operations is that it represents one of the simplest types of the kind of system we have been considering, and moreover is becoming very common and of increasing importance in modern technology. The two “identical” and yet “opposite” operations can be symbolised arithmetically, where two successive operations add up to zero, implying that they are, though under one aspect, “identical,” nevertheless “opposites” 1 + 1 = 0, which implies 1 = -1. (I shall be dealing more fully with the mathematics of this set of operations later.)
Before going on to delineate in greater detail the common pattern of our examples, and present a visual model for the pattern, I must deal with some instances which do not seem to conform to either
(A) the complete absence of all X – i.e. no X, nothing, etc.
(B) the partial presence of all X (or non-X) – i.e. some X, some non-X, etc.
(C) the complete presence of all X – i.e. all X, everything, etc.
Glancing at these we can easily see that to obtain self contradictions we have only to equate any two in the following three possibilities: (1) A = B,(2) A = C, (3) B = C. The types of which I have given numerous examples correspond with possibilities (1) and (2). It remains, then, for me to deal with (3). As we can plainly see, possibility (3) involves the equating of “the partial presence of all X (or non-X)” with “the complete presence of all X.” The terms “some,” “something,” or their equivalents must, at this stage, be understood to mean “proper part.” In other words, (A), (B) and (C) are to be interpreted, in the first instance, as mutually exclusive of all X – i.e. no X, nothing, etc. of all X (or non-X) – i.e. some X.
Now, whether the self-contradictions so obtained are meaningful or not depends upon the context in which they appear. That is to say, self-contradictions are neither meaningful nor meaningless in themselves – i.e. they are not absolute – but depend upon certain conditions applying in the contexts to which they belong. In one kind of system, in one universe of discourse, in one context, a self-contradiction can be quite meaningless. In another system under another set of conditions, the same self-contradiction can be meaningful. Let us consider a convincing demonstration of this.
If I say, “All lobsters are no lobsters,” without giving a significant context, we get a self-contradiction which is just meaningless gibberish. But if I supply a suitable context, then the picture changes dramatically. The following quotation is from W.W. Sawyer’s Prelude to Mathematics (Pelican, 1963, p.96). The author solves a puzzle about lobsters, algebraically. As with all problems of this kind, before the mathematical working begins, there is a prior problem: how to cast the verbal problem into mathematical form. One aspect of it has a familiar ring. We are confronted with alleged “actual” lobsters and “signs” for alleged “actual” lobsters. Again, the difference is one we can ignore; in reality, it is between, simulated actuality and actual signification. Putting his puzzle into mathematical dress, the author says when he comes to the mathematical signification of “lobsters”): “As the whole question is about lobsters, we do not need to use a sign for ‘lobsters.’ We can work as if the universe contained nothing but lobsters, that is, regard lobsters as if they were ‘everything’…” In other words, since all are lobsters, then no lobsters need be distinguished, i.e. “All (actual) lobsters = no (signified) lobsters” – ignoring the familiar distinction-without-a-difference since all lobsters, actual or signified, are signified. The important thing to notice here is that this principle – the “internal standard” of the prior problem – helps to put the original problem into mathematical form, and thus contributes to the solution. Yet, as with the prior part of all such problems, it does not enter directly and explicitly into the mathematical working.
To return to possibility (3) let us consider a few examples. “Moderation in all things” is a well-known precept. It is, nevertheless, self-contradictory since it contains a contradiction of itself. Moderation in all things implies “moderation in moderation.” But it also equally implies “moderation to excess.” Here (C) – “the complete presence of all X” namely “moderation” – is equated with (B) “the partial presence of all non-X” namely “excess.” By a slight recasting of form, however, we can represent this as an example of equation (1): “moderation in all things” then becomes “excess in nothing” or “Nothing to excess,” and this is then equated with the implication “moderation to excess.” A similar instance is, “All generalisations are unsound.” By recasting we get “No generalisations are sound” and the implication “This is a sound generalisation.” The last example is very like the well-known one first given by the Cretan, Epimenides, and often called the Liar Paradox: “All Cretans are liars,” or more simply “I am lying” or “This statement is a lie.” The implication is a flat contradiction: “This statement is true.” Equating these two contradictory assertions shows a similar structure to that which we noted above in the equation concerning the repeating electric switch. (The equation was 1 + 1 = 0, or 1 = -1.) This equates “some X” (i.e. some truths, namely one) with “some non-X” (i.e. some lies, namely one). To keep things as simple as possible here, I have made no separate provision for this particular case since, by recasting, we can always represent it by equation (2). That is to say, given simple enough statements, we are always able to find a context such that “X is non existent,” “X is absent,” “no X…,” “non-X,” .”..not X,” “some non-X,” or any equivalent of any of these, may be represented in an equation by zero.
Just before turning to the question of infinity, I should say a word now on the appearance of self-contradictions in logic and mathematics over the last half-century or so. These include a number of paradoxes which have become famous, each being associated with names like Cantor, Burali-Forti, Hilbert, Richard, Frege, Russell, Goedel, and so on. Firstly, let me say that all these paradoxes are, in rock-bottom essentials, of the types of the simple everyday ones I have described above. I propose to deal with them in detail separately and elsewhere. Secondly, let me add that the important thing to notice, in the light of this discussion, is the orthodox attitude to the paradoxes. The attitude is that the paradoxes are exceptional, rather unaccountable freaks, or at least freakish oddities, not to be included in respectable class-room conversation (because, of course, they make nonsense of much that is said there). They are treated somewhat like disreputable outlaws – which, in a sense, they are of a nicely-conforming, consistent mathematical society, or as skeletons in an otherwise clean mathematical cupboard! This thinly veiled emotional approach is very like the Victorian attitude to sex. It has an obvious function. It serves to fix the paradoxes, keeping them, as it were, at arm’s length, at a distance, intellectually remote. Hence they are regarded, not as we are led to regard them here – i.e. as conditional, as relative to alterable, manipulable conditions – but as either illusory or (if they will not just “go away”) as static, fixed and absolute… an unfortunate monument to man’s falling-short of intellectual perfection.
The general effect of this approach is, naturally, to channel somewhat narrowly the efforts of the few individuals who do become interested in these “flaws” or “cracks” (as they are often called) in the foundations of mathematics and logic. Their energies are largely devoted to the discovery of ways and means for “bumping-off,” the offending “outlaws,” that is, of finding methods for detecting and getting rid of such paradoxes. As we shall see, there is a considerable difference between equating contradictions to zero in order to get rid of them – expelling them from the system of which they are a part – and equating them to zero or the purpose of using them as principles or “internal standards” of systems. At present, of tautologies and self contradictions, it is the former only which are used as laws. Tautologies are opposites of self-contradictions; if A
A = 0 and A + A = 1 are Aristotelian Laws of Non-contradiction and Excluded Middle, then AA = A and A + A = A are the corresponding Laws of Tautology. As Bell neatly put it “A proposition which is T for all the possible T/F choices (in Truth False tables) of the positions p, q, r,… from which it is constructed is called a tautology and is said to be a law of the system; one which is F for all choices is called a contradiction. Tautologies are more highly prized than contradictions.” (Mathematics: Queen and Servant of Science, p. 63)
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