The East Linthorpe Literary, Philosophical, and Debating Society
February, 1994

The Mathematician had just finished one of his familiar lectures on the power of rational thought, to which the rest of us had been giving as much attention as it deserved, when the Author roused us by asking, "Why do you always speak of rational thought, as if there were no other kind?"

"Well, in effect, there is not - none that truly deserves the name of thought, at least. Irrational thought is virtually a contradiction in terms." The Mathematician's reply received general assent, to judge by the number of nodding heads, although in truth some were nodding so far down on the chest as to suggest that the matter provoked as much somnolence as interest.

"I see," said the Author, reflectively. "Now tell me, your confidence in the validity of rational thought is not, I suppose, based upon a mere whim?"

"Certainly not," replied the Mathematician with some asperity. "It is derived from a logical analysis of the matter."

"In that case, I have some bad news for you." The Author's countenance was less grave than befitted his words. "You have committed one of the oldest fallacies in the history of logic, to wit, petitio principii, otherwise known as begging the question. You have, my dear old fellow, constructed an argument in which you have assumed, as your starting point, the very thing that you were intending to prove. You have assumed the validity of logic, and then used logic to deduce that logic is valid. You are trying to lift yourself off the ground by pulling on your own bootstraps. I might as well claim that hunches are superior to logic, on the grounds that I have had a hunch to that effect." The Mathematician was temporarily speechless at this impudent hijacking of his territory, and the Author took advantage of the fact to continue, "The application of logic so often leads to paradoxical conclusions as to suggest to the prudent that its powers are at best questionable, and at worst totally discredited." The Mathematician challenged him to provide some specific examples, and he continued, "I need look no further than my newspaper this morning to read that Mr William Waldegrave, a Minister in the British Government, has told a committee of enquiry that Ministers may tell lies when making official statements."

"Hardly a paradox, surely?" suggested the Philosopher. "Merely a fact of politics to which most of us have long been resigned."

"My point is, logic cannot help us to decide whether to believe him when he tells us that he lies."

"Oh, come, come, my dear chap, that is unworthy of you; no better than the sophomorish trick of writing, on one side of a card :

Side 1.The statement on the other side of this card is false
and on the other:
Side 2.The statement on the other side of this card is true

Such callow sophistries are of little consequence."

The Physicist whispered loudly to the Historian, "Captain James T Kirk finds callow sophistries of that sort to be invaluable for blowing the fuses of the power mad computers that attempt to take control of the starship Enterprise from time to time", but the Author chose to ignore this support, perhaps suspecting that it lacked gravity. Instead, he announced, with something of a sigh, "I see that I shall have to remind you of the tale of the Unexpected Hanging," and launched into the following narrative before anyone could demur.

The Unexpected Hanging

The prisoner was led away to the condemned cell with the judge's words ringing in his ears. "You are sentenced to be hanged. The sentence will be carried out at 4pm on one day next week, but to add to your discomfort in the meantime, you are not to know which day it is to be until the morning of the day in question, when the Governor of the prison will so inform you."

The prisoner related the sentence and its curious conditions to the radical journalist who occupied a neighbouring cell. "And so you see, I do not know when the fatal day will be; it could be any day from Sunday to Saturday."

The radical journalist tossed back the lank hair that fell across his forehead (he modelled himself on the Joseph Wiseman character in Viva Zapata) and said flatly, "Well, you know that it will not be Saturday, at least."

"How is that?" asked the condemned man.

"Think about it. If Friday noon comes and you still have not been notified of your execution, then you will be able to deduce immediately that it will take place on Saturday, because according to the terms of the sentence, you are to be hanged before the end of the week. But according to those same terms, you must not know what day it is to be until the morning of the day itself. Therefore, they cannot leave it until the last day, otherwise you will know the day before."

"Little comfort that is," said the prisoner. "That still leaves Sunday to Friday."

"But it cannot be Friday, can it? If they leave it until Friday, you will know the day before."

"I do not see that," said the prisoner. "If Thursday noon comes and I have not been informed of the day, it could still be either Friday or Saturday."

"No, it cannot. You will know that it must be one or the other, because they will be the only days left in the week; you will also know that it will not be Saturday, because you would know a day too soon if it were left until then. Therefore it has to be Friday, but you will be able to deduce as much Thursday noon. So they will be in contravention of the sentence if they leave it until Friday."

"So Thursday is the last day it can happen?" As the radical journalist smiled , a sudden comprehension flooded the prisoner's features. "But they cannot leave it until then, can they? Because I would know on Wednesday! And if Wednesday is the last day, they cannot leave it until then, because I would know on Tuesday! And they cannot leave it until Tuesday, because I would know on Monday! And they cannot leave it until Monday, because I would know on Sunday! So it will have to be Sunday! And I know that already, before Sunday morning, so there is not any day on which they can validly carry out the terms of the sentence!"

The prisoner's happy shouts attracted the attention of the warder, a kind-hearted man whose employment in his present position indicated a sorry lack of occupational guidance. When the prisoner had explained the reason for his joy, the warder pushed his cap back and scratched the top of his head in a gesture made hackneyed by a thousand B movies. "I cannot detect any flaw in your argument," he agreed, "but it is beyond my terms of reference to give a final adjudication." (He was definitely in the wrong job.) "I shall have to seek a ruling from the Governor."

Later that day the Governor called upon the prisoner. The gist of his message was that, while personally convinced of the validity of the prisoner's argument, he could not overturn the judge's verdict without the authority of the Minister of Justice, to whom he had already referred the question.

The following day, the Governor was back with the news that the Minister of Justice considered the matter to be of such importance that he was submitting it to a select committee of jurists, philosophers, and logicians for a definitive ruling.

Two days later, the Governor returned. "Great news!" he exclaimed. "The committee has ruled in your favour. They accept without reservation that your argument conclusively proves that it is logically impossible for the sentence to be carried out in accordance with the judge's conditions!"

The prisoner was overjoyed. "Does that mean that I shall now be released?" he asked.

"I fear not," replied the Governor. "As a matter of fact, the other thing I have to tell you is that you are to be hanged at 4 o'clock this afternoon."

"But you cannot do that!" the prisoner protested. "It is against the judge's conditions!"

"How so?" asked the Governor. "You did not know until this moment that it was to be today. On the contrary, you were convinced that it was never to be."

"But the committee has ruled that it cannot be done!"

"You were not paying sufficient attention," said the Governor. "The committee has ruled that you have validly proved that it is logically impossible. That is not the same as saying that it cannot be done, as I have now demonstrated. That is a valuable lesson that should serve you well for the rest of your life." He glanced at his watch. "All six hours of it."

The Philosopher was the first to break the silence that followed. "The tale is a familiar one in the journals of philosophical oddities."

"And what do the journals make of the paradox it poses?" asked the Physicist. "Is our friend the Author correct when he claims that there is no correlation between logic and real life?"

"Less correlation than we like to imagine, perhaps," conceded the Philosopher. "Concerning the unexpected hanging, there are usually three schools of thought. The first accepts the total validity of the prisoner's argument. Of course, their version of the story has a happy ending, with the prisoner released because of the impossibility of executing the sentence, thereby ignoring the most piquant aspect of the tale, namely that the Governor can vindicate the judge's conditions simply by naming a day. The Mensa Society, which purports to recruit its members only from the most intelligent members of the population, subscribes to this blinkered view of the question. It has published a book of puzzles containing a variant of the problem: two countries hostile to each other, one with an offensive missile which can be launched only at dawn within the next six days, and the other with a defensive anti-missile missile which is totally effective provided it is launched one hour before the incoming missile. Mensa's 'solution' to the problem is that détente is inevitable because of the impossibility of launching the offensive missile on a day which is not predictable."

"I have often wondered," said the Physicist, "why any intelligent person would wish to join a society for the intelligent."

The Mathematician intervened. "The Mensa definition of an intelligent person is one who scores highly in an IQ test. The inadequacy of that definition is amply demonstrated by their pathetic reliance on questions such as: What is the next number in this series ......., when anybody who has received an adequate education in mathematics knows that for any sequence of numbers there can be found a polynomial expression the evaluation of which for successive values will yield the given sequence. Thus to the question, What is the next number in the series: 2, 4, 6, 8? one may quite reasonably reply 99, or indeed any other number one fancies."

"As I recall," said the Author with evident delight, "the Mensa version of the problem insists that the countries in question are governed by skilled Mathematicians. That perhaps explains ..."

"The second school of thought," continued the Philosopher firmly, "accepts the validity of the prisoner's argument as far the elimination of Saturday is concerned, but holds that its extension thereafter is invalid, on the grounds that so long as more than one day remains, the Governor still has an open choice. They overlook the fact that at Thursday noon, when only two days are left, Saturday can be eliminated by precisely the same argument that was used to eliminate it from the whole week, leaving Friday as inevitable and thus predictable."

"The third school of thought rejects the whole argument as invalid, but with no agreement as to the nature of the fallacy. My own view of the matter is that the prisoner's argument suffers from the converse of petitio principii, which our friend the Author so aptly described as attempting to haul oneself up by one's own bootstraps. The prisoner's fallacy might be described in similarly homely terms as sawing off the branch whereon one sits. He finishes by destroying that which he relies upon. Every step of his reasoning assumes that the judge's conditions must prevail, but he concludes by discrediting them. If this fallacy has been given a name, I confess it escapes my memory. One might call it excisio principii, perhaps."

The Author, to his credit, did his best to conceal the wince that crossed his features. "Are not the logicians trying to have their cake and eat it? They accept the validity of rational argument when the results please them, but as soon as the same logic produces results that are plainly ludicrous, instead of conceding that the system is faulty, they start to make exceptions. Is it not the case that all arguments are essentially fallacious, but we do not usually scrutinise them critically unless we disagree with their conclusions?"

The Philosopher replied, after a moment's thought, "All positive arguments are fallacious, I believe. Let us look at the matter analytically. Arguments may be divided into two classes: circular and non-circular. Circular arguments may again be divided into two classes: those that conclude by proving their own premise, and those that conclude by invalidating their own premise. Both must be regarded as fatally flawed."

"You might mention", the Author intervened, "that the syllogism, the jewel in the crown of logic, is almost always a circular argument. Consider, for example:

All great classical composers are German
Beethoven is a great classical composer
Therefore Beethoven is a German.

The form of the argument pretends to have deduced Beethoven's nationality from the quality of his music, but that of course is nonsense. How did we come by the opening statement if not by listing all great classical composers, including Beethoven, and observing their nationality? So we already knew that Beethoven was German. That was one of the particular instances from which the generalisation in the premise was drawn. The syllogistic rigmarole that covers our return from the general to the particular is no more than the sleight of hand that appears to produce the coin from the victim's ear when it was in the conjuror's hand all the time. When it is not circular, the syllogism is mere tautology, as in:

All unmarried men are bachelors
John is an unmarried man
Therefore John is a bachelor.

Such an argument does not tell us anything we did not already know; it merely lays down some ground rules about terminology."

"Non-circular arguments," continued the Philosopher doggedly, "may also be divided into two classes, depending on whether the conclusion which they reach is one that we feel compelled to reject as self-contradictory, or is one that we are inclined to accept. The former is, of course, the familiar reductio ad absurdum, where the premise is set up with the intention of demonstrating that it leads to unacceptable conclusions. Whilst this is not fallacious, it is purely negative, convincing us that something is not so, never that something is."

"It grieves me," said the Mathematician, "that so few people are familiar with the most elegant example, namely the proof that the square root of 2 is an irrational number. The hypothesis that it is a rational number soon leads to the conclusion that there exists a number which is both odd and even at the same time, which is impossible. Ergo, the square root of 2 is an irrational number."

"Irrationality triumphs again," crowed the Author.

The Mathematician bridled. "In mathematics, a rational number is simply one which can be expressed as the ratio between two whole numbers. When we say that the square root of 2 is irrational, we mean only that it cannot be expressed as a vulgar fraction."

"All the same," said the Philosopher, "the Pythagoreans were so shocked at the discovery that they kept the proof secret, rather than reveal what they conceived to be a blemish on the face of Nature."

"At least they had the decency not to fudge the results to pretend that it was rational after all," remarked the Author. "A modern mathematician would be more inclined to claim that he had discovered a new class of numbers that were both odd and even at the same time." Observing the Mathematician's face, he went on, "You disagree? Consider then the implausibilities that scientists accept every day. When one set of experiments indicates that light consists of particles, and another that it consists of waves of energy, do they conclude that the experiments are unsoundly based? No, they prefer to swallow the camel and believe that light is both particle and wave at the same time. Subatomic particles, on the other hand, they claim to have divided and subdivided until there is no matter left at all, only zones of probability of electrical charges. However outrageous their findings, they never ask where they have gone wrong, but impudently claim ever more startling breakthroughs."

"Are you seriously suggesting", demanded the Mathematician, "that centuries of scientific advance be abandoned? Are you not overlooking the tangible results that benefit our everyday lives?"

"You are confusing science with technology," replied the Author. "There was a time when science and wisdom went hand in hand, but ever since Galileo, true science has languished behind an iron mask in the Chateau d'If, while the pretender technology reigns in his stead. Today, wisdom is the prerogative of the ignorant."

"Among whom you wish to be numbered, it seems."

"May I continue with my analysis?" The Philosopher was insisting rather than requesting. "You may remember that we were left with the last class of arguments, probably the largest and most important, namely non-circular arguments whose conclusions we are inclined to accept. What we must never forget about such arguments (but frequently do), is that their results are no more valid than their premises, and of those we have no proof at all."

"Nonsense!" protested the Mathematician. "The history of Science is one of incremental progress. What is proved today becomes tomorrow's premise, the fabric of knowledge growing from generation to generation, brick upon brick."

"Quite so," agreed the Philosopher, "but the first course must rest upon some foundation, which itself must rest upon....what? Somewhere there must be some initial assumption, which stands without proof. Hence the quest for a self-proving hypothesis: Descarte's Cogito ergo sum, or Anselm's ontological proof, for example, neither of which survives rigorous examination."

"The case of Descartes is a revealing instance of the way in which language moulds our thoughts." The Author looked around as he spoke. "He wrote his famous dictum in both Latin and French, you know, but I believe that he must have conceived it in Latin. If he had thought it in French he surely would have perceived his error." Seeing the puzzled faces that were turned towards him, he explained, "In French, the structure would be the same as in English: Je pense, donc je suis - I think, therefore I am. But this betrays the error as soon as the first word is uttered: the existence of the self is already assumed. In Latin however the personal pronoun is usually omitted, so the statement Cogito can be made without making explicit the assumption that there is an Ego."

"So much for Descartes," said the Physicist heartily. "Now who is going to dispose of St Anselm?" I took the question to be intended ironically, but the Author assumed the air of one preparing for an extended discourse. There was a flurry of suddenly remembered appointments and hasty departures until only the Author and I remained. It was no time for chivalry; only a body blow could save me now.

"Île Ste Marguerite!" I exclaimed.

"I beg your pardon?"

"Île Ste Marguerite; that was where the Man in the Iron Mask was imprisoned. The Chateau d'If was the Count of Monte Cristo." Seeing that he had been taken aback, I pressed home my advantage. "Rudolf Rassendyll, on the other hand, or rather his cousin, the King of ......"

"Please excuse me, my dear chap, but I do believe I left a tap running!" and the Author hurried away.