I have spoken before of the unreliability of formal logic. The Author clearly intended to take his favourite hobby-horse for another canter.
Those of you who remain unconvinced should pay particular attention to the tale of:
Ho Ping walked by the bamboo grove, his mind occupied, not for the first time, with schemes that would allow him to retire from active work with an assured source of income. To be a venerable Chinese sage was an honourable occupation, but he was finding it increasingly difficult to sympathise with his pupils' interests and activities, and looked forward to spending more time cataloguing his collection of brightly coloured beetles. His chief hope of realising this modest ambition lay in the commercial exploitation of a curious discovery which he had recently made. A combination of charcoal, sulphur, and saltpetre, he had found, would, when ignited, create an explosion of flame, smoke, light, noise, and energy which was at once terrible and fascinating. His experiments had been confined to small quantities of the powder, which he had named The Highly Combustible Mixture of Finely Ground Substances, but he assumed that for larger quantities the results would be proportionate. At first, he had contemplated offering the formula to the Imperial Army for use in warfare, but further consideration convinced him that this would be futile. Although the powder had the potential to create great damage and consternation among the ranks of the enemy, it could never be so used, for only the finest of men were fitted to be warriors, and such men would never descend to the use of so ignoble a weapon — an impasse of the nature referred to by many of his colleagues as an entrapment of the 22nd kind.
The problem of devising an alternative use for the mixture was exercising his mind as he took his daily walk, when his attention was distracted by the sight of one of his pupils behaving in an outlandish manner. Li Lao had long been something of an enigma to the elderly professor. His fellow students often referred to him as the Gay Li Lao, but Ho did not detect in him the blitheness of spirit that such a description would seem to imply. Li was now comporting himself in a furtive manner and a crouching posture among the undergrowth at the river's edge. As Ho watched, Li caught a bird in a large net, examined it closely, made an entry in a small notebook taken from his sleeve, and then released the bird, after tying a short length of silken thread to one of its legs.
Greetings, Li. May this unduly inquisitive older person enquire whether your present activity leads toward a practical end, or is it, perhaps, the latest exercise programme advocated by some well-known person formerly active in the theatrical profession?
Greetings, Master. Your unworthy pupil Li is seeking to find the answer to a question posed recently in a wayside hostelry that he occasionally visits for the purpose of sampling exotic beverages and indulging in recreational pastimes of a trivial but entertaining nature. The question that was asked is: are all crows black?
This short-sighted individual would have thought that the answer might best be found by consulting the volumes bequeathed to us by the wise ones of old, wherein all knowledge is to be found.
My Master, as always, is a sure guide to the well-trodden paths of learning. On this occasion, however, his humble pupil has dared to presume that there might be an alternative method of proceeding, namely to examine crows and observe what colour they are. The results I have been recording in my notebook. After a half day's assiduous enquiry, I have observed 23 crows, and found them all to be black. According to my method, which I call the Empirical Technique, such an outcome lends weight to the proposition that all crows are black. When I have amassed a sufficiency of confirming instances, it will be possible to take it as proved. As it would be misleading to observe the same bird more than once, I tie a thread of silk to each bird to identify it as already recorded.
Ho Ping listened impassively to Li's explanation, masking his inward thoughts beneath a bland visage.
This ancient individual had thought that all things had already come to his ears. Tell me, my dear Li, is it necessary to your method that you alone bear the burden of this research, or may others assist you to find confirming instances?
Therein lies another of its advantages, O Master. Not only may others assist, but the communications that will be necessary between persons pursuing the same project may be eligible for the the Emperor's Grand Award.
The prize whereof Li spoke was the chief topic of interest in the Imperial City at that time. The circumstances were as follows. A generation earlier, someone of innovative disposition had invented paper, and soon many persons had become employed in its manufacture. The task of transcribing all written texts to the new medium and lodging a copy of each in the Great Library of the Imperial University was, however, now completed. The demand for paper was naturally much diminished, and many of those who worked in paper mills had been informed that the proprietors were desolated to discover that it was necessary to dispense with their services. The Celestial Society of Paper Pressers, Rollers, and Packers, and Allied Crafts and Mysteries, which represented the interests of those who now found that they could spend with their families the long hours that they had formerly been wont to pass in manual drudgery, submitted a petition to the Emperor that this was an affront to the dignity of labour, and a likely cause of unrest in certain quarters of the city of whose existence he was in fact blissfully unaware.
In the ordinary course of events the petition would have been accorded the disdain that its authors fully expected, but the Emperor had cause to take it more seriously. When the paper industry had begun to flourish, the Keeper of the Imperial Purse had devised a scheme whereby a regular percentage of the earnings of those engaged therein was diverted to the Royal Counting-House. This income was now indispensable to the maintenance of national interests, such as the upkeep of the Emperor's concubines and the provision of suitable dowries for his less personable daughters. Accordingly, the Emperor had announced that he would award a great prize to any person who could devise a scheme that would ensure the continued consumption of paper in generous quantities.
There had been a number of unsuccessful applications. An early claimant had suggested that scribes could record the more significant events of each day upon large sheets of paper, sufficient copies of which could be made to be delivered each morning to the households of those desirous of being kept informed. This scheme had been discontinued when it became evident that the strain of writing such copious amounts of script sapped the intellect of those engaged in the task until they were incapable of distinguishing fact from fiction, knowledge from opinion, investigation from persecution, or revelation from innuendo. Young female companions of leading Mandarins had also brought the daily sheets into disrepute by inducing the scribes to devote much space to their own uninteresting and not always truthfully related activities, especially one, Bim Bo by name, who was renowned for the almost inconceivable disparity between her physical and mental attributes.
Another promising idea had been to issue persons of deficient memory with pads of small pieces of paper, treated on the back with a resinous compound so that they might adhere easily to many surfaces, and coloured a bright yellow so as to be most conspicuous to the sight. The purpose of the pads was to be for the writing of brief messages, not only to others, but even to oneself. They fell victim to their own success. A husband would return home after a hard day's philosophical discussion at the local tea-house, to find a message on such a paper advising him that his dinner was in the wok, but the wok itself would be impossible to locate beneath the multitude of other messages, concerning the collection of the offspring from the neighbourhood academy of learning, submitting the rickshaw for examination by a person of mechanical aptitude, shortening the verdure in front of the house, and similar items of agenda prepared by his conscientious spouse. When the Emperor learned that the proliferation of the message pads had resulted in visitors from other countries describing Cathay as The Land of the Yellow Peril, he prohibited their further use.
The Great Award had thus remained unwon, and was now exciting Li's expectations.
For if a number of persons are all engaged upon the same enquiry, it will be necessary for them to share their results. It is this hopeful person's vision that each of them would write his findings upon a paper, and that they would send copies of these papers to one another at regular intervals. If such a manner of proceeding became common among scholars, His Imperial Majesty might consider doing this poor student the honour of awarding him some modest sum.
With an expression that one astuter than Li might have interpreted as being capable of more than one meaning, Ho announced that he himself would be glad to assist in the current research, and invited Li to call upon him that very evening for the purpose of continuing the project together. Li demurred at first, on the grounds that the hours of darkness and an urban milieu were unconducive to the finding of crows, but finally accepted Ho's assurance that total confidence in his professor would not be unrecognised at the time of assessing students' final grades.
On Li's arrival, Ho was impatient to begin.
You have your notebook, ink, and brush? Then I shall make the observations while you record them." Whereupon the professor began to examine objects in the room, reciting as he did so. "Here is a non-black object. It is brown. Is it a crow? No, it is a sandal. Good; note that as a confirming instance. Here is another non-black object. It is blue and white. Is it a crow? No, it is the vessel from which this person habitually drinks his chocolate-flavoured beverage before retiring for the night. Good; another confirming instance. Here is a non-black object. It is yellow. Is it a crow? No, it is a chysanthemum. Good; another confirming instance. Here ...
Forgive my uncultured interruption, Master, but it seems that this unworthy student's incoherent description has misled your noble self as to the nature of the Empirical Technique. In order to verify the proposition that all crows are black, it is necessary to observe crows that are black.
Or non-black objects that are non-crows. That amounts to the same thing, does it not?
Li confessed that such a contention did not present itself to his inward eye as transparently self-evident.
This inadequate mentor has failed in his duty, then. However, the deficiency can be speedily remedied. You are familiar, of course, with the principles of Yin and Yang, and their celebrated symbol.
Perhaps, however, you are less familiar with the equally powerful diagrams of Venn." The professor drew upon a blackboard as he spoke. "Imagine, my dear Li, that this circle represents all crows. Every crow is contained within the circle; every thing within the circle is a crow; every thing that is not a crow is outside the circle; every thing outside the circle is not a crow. Is that clear so far?
It is plain even to this ignorant person's limited intellect that making such a drawing recommends itself as exceeding in practicability the alternative course of actually confining all crows within one circular enclosure.
Ho projected a sharp-edged glance towards his pupil.
Similarly, we may draw a circle that represents all black objects. Every black thing is within the circle; every thing within the circle is black; every thing that is not black is outside the circle; every thing outside the circle is non-black. Is my wearisome discourse still honoured by the accompaniment of Li's incisive comprehension?
As usual, my Master makes his teaching easy to digest, by diluting its powerful intellectual essence with the copious flow of his limpid prose.
Finally, (Ho gave the impression of speaking between clenched teeth)
we may draw both circles together. In order to represent the condition that all crows are black, we must place the circle of crows within the circle of black objects. In this way, everything within the inner circle, namely all crows, must also be within the outer circle, that is, must be black. But it also follows that everything outside the outer circle, namely all non-black objects, must also be outside the inner circle, that is to say, must be non-crows.
Thus it appears that the statement: All crows are blackis precisely equivalent to the statement: All non-black things are non-crows.
They are two ways of saying the same thing, just as x is greater than y says the same as y is less than x .
I must yield to the inexorable logic of your incontrovertible argument, Master.
"So, to summarise the situation:
And that, my dear Li, is what we were doing when you requested that the validity of the procedure should be the subject of my inadequate exegesis.
You will now appreciate the invaluable addition that this makes to the Empirical Technique. The occasions upon which you can pursue your enquiry are greatly increased. At any time, and in any place, you may continue to find confirming instances, unhampered by the non-availability of the feathered species in question. You will be able to make as many observations between one gong stroke and the next as formerly occupied your labours for a day. In fact, this extension to the Technique is so important, you must make it the subject of your first Paper. I have one request to make in that regard however, namely that this person's humble contribution should not be mentioned. I have no desire, dear Li, to take any share of the reputation that shall surely accrue to one advocating this method of proceeding. But I see a qualm of unease troubling your otherwise symmetrical features. You did not, I hope, partake of shellfish of debatable wholesomeness for your evening meal?"
The vacillation of my physiognomy mirrors the inconstancy of my resolve, Master. It is beginning to occur to this indecisive person that he should allow another the honour of formulating the Empirical Technique, and should seek a lowly paid post in the University Library, where he might find ample opportunity to peruse the recorded knowledge of the ancient wise ones.
When Li had left, Ho repaired to his bed. Fragments of the day's events passed before his inner eye. The syllables of the words "Empirical Technique" arranged and rearranged themselves, until, as the approach of sleep liberated his thoughts from the control of his intellect, they suddenly formed the word "Pyrotechnics." Ho's drift from consciousness was halted. Could not small packets of the Highly Combustible Mixture be sold for the purpose of producing agreeable effects on the occasion of notable anniversaries? Once formed, the idea grew swiftly. Traces of various impurities could be added to influence the colour of the displays. If the powder were confined in a tightly rolled tube of thick paper, the conflagration issuing from the end might be of sufficient force to lift the device into the air, where the results would be visible from a distance. The scheme had certain innate advantages. Nobody would ever find a way to re-use the product, for example. Nor would it be difficult to make its purchase almost mandatory. Only a person of unpatriotic instinct could refuse to ignite several packets of the powder every year on the fourth day of the seventh moon, when the populace was wont to celebrate the accession of the first Emperor and the passing of the bad old days when the nation had been ruled by any person of low birth who could win popular acclaim. A suitable name would have to be found, one which would come swiftly to the mind and trip easily from the tongue in the commercial hurly-burly of the market-place - something like Ho's Flagrant Sources of Extravagant Brilliance and Diverting Coruscation. It would be necessary to devise a means of igniting the packets without injury, but that would not be difficult. A spill of paper of well-judged length would suffice, of some distinctive colour so that the user would know where to apply the match. Blue would be quite suitable. Ho did not doubt that his future prosperity was assured as the proprietor of a concern manufacturing a constant stream of the Flagrant Sources, each one clearly labelled "Light the blue touch paper." He would be able to retire immediately.
With this beguiling thought, Ho floated into a blissful sleep, enlightened by visions of brightly coloured beetles and young women with limbs of pleasant proportions.
There was a respectful pause before the Historian opened the discussion.
The tale has certainly created confusion in my mind. Li's method appears reasonable enough, and indeed, as I understand it, has been the basis of scientific enquiry ever since Galileo. Ho's method, on the other hand, is plainly insane. Yet Ho's proof that the two are equivalent appears to have no flaw in it, and is according to the strict tenets of formal logic. Li's method must be wrong, or Ho's method must be right, or the proof of their equivalence must be wrong, but I cannot bring myself to accept any one of these three options.
Let us construct an alternative example, suggested the Physicist.
Suppose that you join a large corporation, and a colleague tells you, All the women who work here are married. In order to verify the statement, you consult the personnel department, and they offer to supply you either with a list of all female employees, showing their marital status, or with a list of all unmarried employees, showing their sex. Which of these lists would suit your purpose? Why, either of them. Ho was quite right: testing all females to verify that they are married, and testing all unmarried persons to verify that they are not female, are absolutely equivalent.
Your example may be distinguished from the Author's in at least one vital respect, said the Mathematician.
Your universe of discourse is limited to employees of the corporation - Ho's was infinite. In Ho's terms, the complementary process to observing female employees to verify that they are married would be to observe any object that was not married and verify that it was not a female employee of the corporation. Ho would say that the nearest lamp-post would provide a confirmatory instance: it is not a married person, and it is not a female employed by the corporation.
That is a pertinent point, the Philosopher interposed,
but it only deepens the mystery. The law of logic that Ho relies on is supposed to apply to an unrestricted universe of discourse. It insists that All A is B is absolutely equivalent to All Non-B is Non-A, for any A and for any B. Why then do we find it totally convincing in certain cases, and utterly unacceptable in others?
Another point of distinction, arising out of the first, continued the Mathematician,
is that, with regard to the employees of the corporation, it will be possible to check either list in its entirety. Thus one will know for certain whether the proposition is true. With regard to the crows, however, it is inconceivable that one will be able to observe every crow, still less every non-black object. Therefore, one will be making an inference from a sample, rather than having the certainty of a total census.
Ah, now we are approaching the nub of the matter. The Author's tone of subdued eagerness put me in mind of a cat stalking a bird.
No matter how many black crows one examines, one will never know for sure that all crows are black. Ho's method is insane; examining non-black objects cannot confirm the proposition - but neither can examining crows! It is all this talk of confirmatory instances that throws the dust in our eyes. The only useful search is for a disproving instance. Both Li and Ho are really looking for the same thing, a non-black crow, the finding of which will disprove the proposition that all crows are black. Obviously, either a crow that is non-black, or a non-black thing that is a crow, will suffice. The observation of crows that are black, or of non-black things that are not crows, confirms nothing ; it merely constitutes a failure to find a disproving instance. No scientific proposition has ever been proved; the most one can say is that it has not yet been disproved.
The question is, said the Philosopher,
if a continued search fails to turn up a disproving instance, to what extent should that persuade us of the truth of a proposition? When a complete census has been possible, then obviously the failure is conclusive. In other cases, however, we might take into account how assiduously the search for a disproving instance has been carried out. Many may think, perhaps justifiably, that Li was making a more serious attempt to find a non-black crow than Ho, but the matter is debatable. You may be familiar with Goldberg's principle, which says, Never ask a barber whether you need a haircut. If you want to know where you are most likely to find a non-black crow, should you ask a man who believes that all crows are black? I think not. Consideration of this question leads us to an uncomfortable conclusion. Scientific theories that are hotly disputed will be well tested; their opponents will see to that. But what of theories that are generally accepted? Who is motivated to find the hiding place of the disproving instance? We are forced to conclude that we should have least faith in those theories upon which scientists are unanimously agreed.
Both the Mathematician and the Physicist were beginning to fidget, but the Author took up the discussion, as though he and the Philosopher were playing as a team determined not to lose the ball.
It goes even further than that, he said.
Some things are not merely unprovable, they are unknowable, at least in the absolute terms that scientists insist upon. Take for example the present unseemly wrangle about what has been called nuclear fusion in a test-tube. One group of scientists claims that in their experiments they have produced more energy from a simple apparatus than they put into it. They ascribe the difference to a process of nuclear fusion. Other scientists seek to replicate the experiment, and say that they get no energy gain. In accordance with modern scientific dogma, if it cannot be replicated, it does not count. But that leaves them chasing their tail, for if the later experiments invalidate the earlier, by the same token the earlier experiments invalidate the later. The ignorant man in the street might think: Well, sometimes it happens, and sometimes it doesn't. But scientists will not allow that. They would rather make accusations of bad faith and falsified results, because Galileo's mechanistic view of the universe demands that if you press the same buttons, you must get the same results. Good Heavens, even machines do not actually behave like that, as anyone who has ever owned an automobile knows. Our everyday experience teaches us that results are not always consistent. Some gardeners have green fingers; some cooks have a light touch with pastry. There are people who can detect underground water with a twig of hazel; when another performs precisely the same actions with precisely the same equipment, he gets no reaction. Scientists either deny that such effects occur, or they suggest that there are factors at play as yet unidentified. Just give them time, and they will find the hidden buttons. And so they go on proliferating the exceptions and the caveats, the conditions and the qualifications, insisting that their colander is really a bowl - there are just one or two leaks that need repairing. William of Occam put his finger on it over six hundred years ago: a theory that is not simple is a theory not worth having.
The Author paused at last, having exhausted either his arguments or his breath, and the rest of us looked towards the Physicist and the Mathematician. The latter wore an air of smug anticipation that belied his opening words.
There is much in what the Author says. Scientists are making grievous errors all the time. For example, at this moment, in the vicinity of Ernest Bramah's burial place, seismologists are convinced that a serious earth tremor has occurred. We, however, all know that the perturbations of their instruments have been caused by the illustrious author of the Kai Lung stories spinning in his grave.
It is usually considered a matter for regret when our discussions descend into personal abuse, but on this occasion most of those present thought the Author's comeuppance well merited.