Paradoxes are self-contradictory statements; only careful scrutiny reveals their meaning. They have function in poetry, their function goes beyond mere wit or getting attention. Logical paradoxes are highly amusing and often tantalising and 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. If you take it to be true, it turns out to be false. If you treat it false, then it turns out to be true.
An English mathematician, PEB Jourdain gave a similar dilemma, in 1913, when he proposed the card paradox. On one side of the card, it was printed that ‘The sentence on the other side of this card is TRUE'. The print on the other side of the card was, ‘The sentence on the other side of this card is FALSE'.
The same is true of the barber paradox, proposed by Bertrand Russell. It is also known as the Russel's paradox. It goes like this. There was one barber in a village. He 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 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 in Principia Mathematica (1912) by introducing the concept of a hierarchy of logical types but were not successful.
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| Kurt Gödel |
Kurt Gödel was an Austrian-born U.S. Mathematician and logician. He ultimately resolved it in 1931. He proved that these paradoxes cannot 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 still continue to be felt and debated. Gödel's proof was written in German. The English translation of the title of his proof was 'On formally Undecidable Proposition of Principia Mathematica and Related Systems'. It first appeared in an article, in German in the Monatshefte für Mathematik und Physik, vol. 38 (1931) Pg. 173-98.
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. Justice Holmes rightly points out that ‘The life of law is not logic’. Just as all problems of Mathematics cannot be solved by logic, so is the case in Law; not all cases are decided on reason. 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 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 peculiar terms of the contract governing its payment. The junior was to pay the fee only when he won a case and he had not won any.
The article examines the connection between liar'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.’
(From Alice in Wonderland by Lewis Carrol)
