Friday, May 05, 2006


Paradoxes are self-contradictory statements, the meaning of which is revealed only by careful scrutiny. They have function in poetry, however, that go beyond mere wit or getting attention. Highly amusing and often tantalising, logical paradoxes generally lead to searching discussions on the foundations of mathematics.

The most talked about paradox of all times is Epimenides or the liar’s paradox. Epimenides was a Greek poet who lived in Crete in 6th century BC. He remarked, ‘All Cretans are liars’. It is self-contradictory for Epimenides was a Cretan himself.

An English mathematician, PEB Jourdain gave a similar dilemma, in 1913, when he proposed the card paradox. This was a card on one side of which was printed: ‘The sentence on the other side of this card is TRUE.’ On the other side of the card the sentence read: ‘The sentence on the other side of this card is FALSE.’

The same is true of the barber paradox, offered by Bertrand Russell. The only barber in the village declared that he shaved everyone in the village who did not shave himself. On the face of it, this is a perfectly innocent remark until it is asked ‘Who shaves the barber?’

Russell’s paradox hinges on the distinction between those classes that are members of themselves and those that are not members of themselves. Bertrand Russell and Alfred North Whitehead attempted to resolve the paradox of the class of all classes by introducing the concept of a hierarchy of logical types but without much success. Principia Mathematica (1912) was the result of their labour.

Kurt Gödel, an Austrian-born U.S. mathematician, logician, and author of Gödel's proof ultimately solved this conflict, in 1931. He proved that it could not be solved. According to him one may start with any set of axioms and yet there will be propositions (or questions) which can neither be proved nor disproved on the basis of the axioms within that system and therefore, the basic axioms of arithmetic will give rise to contradictions. His proof ended nearly a century of attempts to establish axioms that would provide a rigorous basis for all mathematics. This proof has become a hallmark of 20th-century mathematics, and its repercussions continue to be felt and debated. Gödel's proof first appeared in an article in the Monatshefte für Mathematik und Physik, vol. 38 (1931) Pg. 173-98, 'On formally Undecidable Proposition of Principia Mathematica and Related Systems I.' His paper, together with a commentary, is available in translation under the original title Meltzer and Braithwaite, published by Oliver & Boyd, 1962.

There is a parallel in law. It is commonly assumed that all judicial decisions are taken on the basis of reason. Well, it is not true: At least not for most important decisions. ‘The life of law is not logic’ declared Holmes. Just as all problems of Mathematics cannot be solved by logic, so also in Law. Often decisions are taken first, reasons are found later.

The article below, an inter disciplinary study, is written in the form of judgement of a Court. Here a senior lawyer has sued his junior for his ‘Guru Dakshina’ (a fee given by the student to the teacher) and difficulties faced by the judges due to the special terms of the contract. The junior was to pay the fee only when he won a case and he had not own any. The article examines the connection between lair's or Epimenides' paradox' and the decision making process. It also explains the paradox and its impact in the field of Mathematics, Literature, and Philosophy as well as on jurisprudence. It is an extension of the earlier article ‘Decisions Are From Heart Rather Than the Head'.
'These are old fond Paradoxes to make fools laugh in the alehouse’. Unfaithfully faithful Desdemona in Othello Act I Scene I
‘Let the jury consider their verdict’
the King said, for about the twentieth time that day.
‘No, no!’ said the Queen
‘Sentence first - verdict afterwards.’ (Lewis Carrol; Alice in Wonderland)

(This case is reported in 1931 TR 173. One does not know what TR stand for. If it is not trash, it could be tripe. Though others insist it is Tommy Rod. The year and the page number may not be confused with the publication of the godel's proof.)

Who does not know the parties of this case? At least those who have any concern with this court know them well. Both are lawyers.

The plaintiff, a senior lawyer, was at his best when we entered the bar and we followed him wherever he argued. His command over language, grasp over the law, and his court craft yet to be surpassed by any one. He is kind and generous as a man should be, but today he is destitute no one to lookafter him. He is in the end, a failure in this material world. We would like him to win if we can.

The defendant, a junior lawyer, if we do not use the four-letter word, is a s.o.b. He exemplifies what Rumpole1 says,
‘Being a lawyer has almost nothing to do with knowing the law.’
But then he is rich. You know what that means - successful.

This is the way of the world. May be they depict,
‘The things we admire in men, kindness and generosity, openness, honesty, understanding and feeling are the concomitants of failure in our system. And those traits we detest, sharpness, greed, acquisitiveness, meanness, egotism and self-interest are the traits of success. And while men admire the quality of the first they love the produce of the second’ (Cannery Row; John Steinbeck).

The defendant took his training with the Plaintiff. Here it is obligatory to give fee to the senior after the training. We call it training fee or fee only. It is sacrosanct. The defendant did not have any money. He entered into a contract with the Plaintiff. This was a peculiar contract. We have never seen one like this one before. The defendant was to pay the fee whenever he won his first case. The defendant has not paid his fee to this date. That is why this suit. According to the defendant he has yet to win a case and this is the excuse for his non-payment. The plaintiff would not have filed the suit but for the fact that he had no money. It will be no harm if he gets some. You know those who are supposed to look after the lawyers' interests look after their interest only. Lawyers do not have any old age benefits.

We would like to say a few words about the training fee. It exists in all parts of the world. Known by different names and has many dimensions. In India it is known as ‘GURU DAKSHINA’. We know that modern Archery was born because of it. If the teacher Drona in the Mahabharat, in India had not taken the right thumb of one of his alleged pupil Eklavya as Guru Dakshina, he would not have started archery with his middle and fore finger, thus giving birth to modern archery. But we are not here to talk about archery. We have to decide the case. We cannot help it. We love to go off on a tangent off the point. This is our second nature. The first being - giving sermons. Now let’s go back to the case.

The defendant, though wealthy and rich, says he has never won a case. We searched all the cases in which he had filed his power of attorney. Our search was in vain. He had lost every such case. One does not have to file a power of attorney in order to win a case or to earn any money. And lawyers may lose the case and yet earn money. We, grudgingly presume that the defendant is yet to win a case.

The plaintiff does not mind for he is sure of his getting the money. He says,
‘If I win the case I get my money. If I lose then the defendant wins his first case and under the contract he has to pay the fee/money. I always get my money’.
The defendant does not agree with this. According to him he does not have to pay anything. He says
‘If I lose the case I will not have yet won my first case and under the contract I do not have to pay. And if I win, then obviously I don’t pay’.

This case reminds us of another case. You know why the world comes here (to our country). We do not have to elaborate it. It is certainly not tourism. We have to check it. We don’t want any one to come here. We made this law that every visitor has to make a statement if the statement is correct then he is burned to death. But if the statement is false then he is to be hanged to death.

One day a visitor came. We forget his name. Was it Gödel? He made a simple statement,
‘I have come here to be hanged.’
We did not know what to do with him. Whatever we decided was sure to be against law. You know - whenever we have a problem we search for precedents. And search we did, despite Jonathan Swift’s reminder,
‘It is a maxim among these lawyers that whatever has been done before may legally be done again, and therefore they take special care to record all the decisions formerly made against the common justice and general reason of mankind. These under the name of precedents, they produce by authorities to justify, the most iniquitous opinions and the judges never fail to directing accordingly’ (Gulliver’s Travels A Voyage to Houyhnhnm Book IV).
Our research was fruitful.

You remember Sancho Panza don’t you? A short, pot-bellied peasant whose gross appetite, common sense, and vulgar wit served as a foil to the mad idealism of his master Don Quixote. He later became the governor. We decided to do exactly as was done by him in the same situation, namely to let the visitor go free (Don Quixote; Cervantes Chapter 51 Second book).

In the process we took so much of time, a normal feature here, that he (Gödel) went away without doing what he wanted to do. Mathematics, was his love. This was, contrary to what we thought that he would do. He was a good man. Well, G.K. Chesterton may be right,
‘We do not get good laws to restrain bad people. We get bad people to restrain good laws’ (All things considered 1908).

Of course you must be remembering the case of the barber. We remember him well. Russell was his name. He had come to us with a special problem. He wanted to know who shaves him for there was notice in his shop;
‘I shave those who do not shave themselves.’
If he shaved himself then according to the notice he could not have shaved. And if he did not then he should have. What a notice! We ultimately held that he was not a he but a she; not requiring the shave.

These questions have troubled men (well we are male chauvinists) of all times. It was Epimenides, a Greek poet, who lived in Crete in the 6th century BC, who created it for the first time by saying,
‘All Cretans are liars.’
If you take it to be true it comes out as false and if you presume it to be false it boomerangs at you as true.

This problem has surfaced at different times; in different ways; in different fields. If Bach was its manifestation in music then Escher was its manifestation in painting. Can you guess what came first the hen or the egg? Guess, who is sketching who in the picture?

MC Escher: Lithograph, January 1948.

Many have tried to solve this. You know what happened to Bertrand Russell, one of the greatest philosophers and mathematicians of our time when he tried to solve this.
‘I set to work to write out the logical deduction of mathematics, which afterwards became Principia Mathematica. I thought the work was nearly finished, but in the month of May I had an intellectual set back … At first I supposed that I should be able to overcome the contradiction (Epimenides or liar's paradox) quite easily, and that probably there was some trivial error in the reasoning. Gradually, however, it became clear that this was not the case … A contradiction essentially similar to that of Epimenides can be created by giving a person a piece of paper on which is written: ‘The statement on the other side of this paper is false.’ The person turns the paper over, and finds on the other side, “The statement on the other side of this paper is true”. It seemed unworthy of a grown man to spend his time on such trivialities, but what was I to do? There was something wrong,, since such contradictions were unavoidable on ordinary premises. Trivial or not, the matter was a challenge. Throughout the latter half of 1901 I supposed the solution would be easy, but by the end of that time I had concluded that it was a big job. I therefore decided to finish The Principles of Mathematics, leaving the solution in abeyance. … It was clear to me that I could not get on without solving the contradictions, and I was determined that no difficulty should turn me aside from the completion of Principia Mathematica, but it seemed quite likely that the whole of the rest of my life might be consumed in looking at that blank sheet of paper. What made it more annoying was that the contradictions were trivial, and that my time was spent in considering matters that seemed unworthy of serious attention’ (The Autobiography of Bertrand Russell page 149-154).

We all know it has been solved. Kurt Godel2, solved the problem. He proved that it cannot be solved.

These questions have manifestation in law too. We know them well
‘The golden rule is that there are no rules.’ (GB Shaw)
Holmes would often say,
‘No general proposition is worth a damn.’
Is it a general proposition?

You know about the Allahabad High Court; the first Sadar Adalat to be elevated to the status of the High Court (1866); the first court to comprehend a woman as a person (1921) (see here). The rest of the world followed it; the first High Court to have a row with democratically elected legislature (In re Under Article 143, Constitution of India AIR 1965 SC 745); the first High Court to unseat a sitting Prime Minister (Indira Gandhi 1975); and of course the first High Court to restore a dismissed government (see here). Yes, we are talking about the same High Court.

It has three full benches of the same strength (3 Judges). The first two UPSRTC Vs STA , AIR 1977 Allahabad 1 and Gopal Krisihna Vs DJ, AIR 1981 Allahabad 300 have held that in the case of a conflict in two decisions of equal strength of judges then it is the later one that prevails. The third one Ganga Saran Vs. Civil Judge, AIR 1981 Allahabad 300 without referring to the earlier ones, says that in the same situation one should follow the one which one thinks to be right. Imagine what one should now do in a similar situation. If he takes the first two to be correct that he should follow the third one, which in principle negates the earlier ones. If he takes the first two as not laying down the correct law and accepts the third (last) one as correct, he ultimately ends up following the first two namely to follow the last one; A logical contradiction. Legally this problem would not have arisen if the third one had referred to the first two. But unfortunately it did not. Surprisingly, how were two full benches missed? The lawyers must not have done their homework properly. We never blame ourselves.

Many have equated law with reason.
‘Reason is the life of law; nay the common law is nothing else but reason. The law is perfection of reason’ (Sir Edward Coke).
‘Let us consider the reasons of the case. For nothing is law that is reason (Coggs Vs Bernard 2 Ld. Raym 911; Sir John Powell).
We never believed in this kind of fairy tale. If we ever had then we would have done the same as our philosopher friend; spend rest of our life looking at the facts of the case. We long ago solved such problems; even before they were raised. Like in Mathematics, where all problems can not be solved by logic, so in law. Most of the cases are not decided on reasons. They are formulated later to support the decisions already taken. Important cases are decided from heart and not from the head.

We still believe in Aristotle the heart is the seat of emotions. Science now says that emotions lie on the right side of the brain and logic on the left side.3 We without science always believed in being right with right rather being left with the left.

It is not something new that we say. It is that we now admit it; in so many words. Wasn’t it Holmes who said it,
‘The life of the law has not been logic; it has been experience. The felt necessities of the time, the prevalent moral and political theories, intuitions of public policy, avowed or unconscious, even the prejudices which judges share with their fellow-men, have had a good deal more to do than the syllogism in determining the rules by which men should be governed. The law ... can not be dealt with as it contained only the axioms and corollaries of a book of mathematics.’ (The common law page - 5).

Many others have said it; in their own style.
‘A court invokes whichever of the rules produces a result that satisfies its sense of justice in the case before it.’ (Statute Interpretation in a Nutshell’ 1938; Wills `16 Canadian BR- Dias, Jurisprudence 4th ed. page 241).
‘We may try to see things as objectively as we please. Nonetheless we can never see with any eyes except our own.’ (Nature of judicial process; Benzamin Cardozo; page 13).

It is for this reason that Harold J. Laski says;
‘There are judges whose sentences in sexual cases are notoriously light, there are others who, in similar cases, inflict punishment of the utmost rigour. There are benches of magistrates in England where conviction for violation of the Factory Acts is punished severely; there are others where the penalty inflicted is almost always nominal. The only possible answer I think is that each will decide by his conception of … what ought to be the law. And that conception will be determined by what William James called his sense of the `total push and pressure of the cosmos’. Law therefore, is always made in terms of what life has meant to those who make the law. Nor is that conclusion invalidated by the fact that great judges, like Mr. Justice Holmes, are able in rare instances to transcend the limitations of experience and see the issue in a wider perspective.’ (A Grammar of Politics page 543).

Laski mentioned the name of Holmes for not that Holmes could transcend the limitations but for the fact that his philosophy was same as Holme's. Who does not want to say that his views are objective?

Enough of the sermons. We can not help it. It is our first nature. We had already decided that the plaintiff should get the amount. It was the reasons that we were finding. We hereby decide this case in favour of the defendant as he has not won any case till today but as soon as we say that he has won a case and has to pay fee under the contract. And if the plaintiff sues for the second time he should get the amount. The second time will only delay the result. If our existence has any meaning, any relevance, delay has to be avoided. Otherwise it will crush us. We may and should adopt unorthodox methods. There is no point delaying it till second time. So formally the defendant wins but the plaintiff gets his fee. With this we consign the records of the case to the dustbin.

P…ssst. Originality is not original but, more often than not, is copied. So is the present opinion. It is copied from.

  • GöDEL, ESCHER, BACH: An Eternal Golden Braid A Metaphorical Fugue On Minds and Machines in the Spirit of Lewis Carroll; Douglas R Hofstadter.
  • THE EMPEROR’S NEW MIND. Concerning Computers, Minds, and the Laws of Physics; Roger Penrose.
  • VICIOUS CIRCLES AND INFINITY. An Anthology of Paradoxes; Patrick Hughes & George Brecht.
  • WHAT IS THE NAME OF THIS BOOK? The Riddle of Dracula and Other Logical Puzzles; Raymond Smullyan.
  • AHA GOTCHA, Paradoxes to Puzzle and delight; Martin Gardner

1Rumpole is a fictional character created by John Mortimer. He has written many books featuring Rumpole, published by Penguin. These stories are about cases in the law courts conducted by Rumpole the Barrister.

2Gödel also spelled GOEDEL (b. April 28, 1906, Brünn, Austria-Hungary--d. Jan. 14, 1978, Princeton, N.J., U.S.), Austrian-born U.S. mathematician, logician, and author of Gödel's proof, which states that within any rigidly logical mathematical system there are propositions (or questions) that cannot be proved or disproved on the basis of the axioms within that system and that, therefore, it is uncertain that the basic axioms of arithmetic will not give rise to contradictions. This proof has become a hallmark of 20th-century mathematics, and its repercussions continue to be felt and debated.
A member of the faculty of the University of Vienna from 1930, Gödel was also a member of the Institute for Advanced Study, Princeton, N.J. (1933, 1935, 1938-52); he immigrated to the United States in 1940 (naturalized 1948) and from 1953 served as a professor at the institute.
Gödel's proof first appeared in an article in the Monatshefte für Mathematik und Physik, vol. 38 (1931), on formally indeterminable propositions of the Principia Mathematica of Alfred North Whitehead and Bertrand Russell. This article ended nearly a century of attempts to establish axioms that would provide a rigorous basis for all mathematics, the most nearly (but, as Gödel showed, by no means entirely) successful attempt having been the Principia Mathematica. Another well-known work is Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory (1940; rev. ed., 1958), which has become a classic of modern mathematics. Courtesy Britanica.

3For details on Right -Left side of brain, see The Dragons of Eden; Carl Sagan.

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