By Félix Lambert

*Relativity* by M.C. Escher

Logical twists and games have always seemed to intrigued thinkers from all times and civilizations. As being mostly curiosities, they appeared in an unorganised fashion in many disciplines, arts and games. One such logical twists is the paradox. From Antiquity’s philosophy to modern mathematics, paradoxes have brought various questions and, in some cases, answers about human knowledge. The main goal of this paper is to demonstrate how paradoxes have proven useful in various cases in sciences and narrative arts, therefore justifying them as proper object of study instead of being considered simply as odd singularities. Different definitions of paradoxes will first be discussed and a series of paradoxes will be presented. We will then use Octavio Paz’s discussion over dual concepts to approach paradoxes. Finally, we will come back to some examples where a dualistic study of some paradoxical structures has been useful. This will show how paradoxes now constitute a significant part of contemporary knowledge, art and to a certain extent, mythology.

The first common point found in various definitions of paradoxes is the self-contradictory aspect of its claim. One of the most common paradoxes that clearly exemplifies this fact is the liar’s paradox. It seems to first have been proposed by Epimenide when saying that all Cretans lie. The problem appears when we realise Epimenide is himself native from Crete and therefore two options are possible: first, if he lies then his statement is right and therefore we cannot trust his saying, secondly if he tells the truth we because he is Cretan then he tells a lie. A more condensed version of this paradox was expressed by Eubulibe of Millet (Vidal-Rosset, 31): ‘’I’m lying’’ of which the literary equivalent is ‘’This sentence is false’’. The common point of all these statements is that they all appear as contradicting themselves, leaving us incapable of deciding the rightness of their claim.

Many authors have proposed classifications for paradoxes. One interesting proposition has been Willard Van Orman Quine’s tripartite classification. Quines distinguishes falsifical paradoxes, veridical paradoxes and antinomy (Vidal-Rosset, 105). The first category includes paradoxes that are finally proved to be false. Such an example is given during the Renaissance by Guido Ubaldus finding that *0=1*, which was interpreted at the time by implying that matters can be creating out of nothing. The claim seems to contradict itself by giving two different values to an integer, one being equal to zero and of course to itself. The proof uses infinitely many addition of ones and zeros by reorganising them in such a way to obtain the result (Stewart, 578). The paradox is false because it does not use the allowed operations between infinite series (Labelle, 262-264). Another common example is *1=2* obtained by a division by zero, which is of course prohibited. The second type of paradoxes, the veridical ones, contains paradoxes that seem false but end up being true. The most common example is the Monty Hall paradox. A player is offered three choices of doors behind which one of them a price is hidden. The player picks a door. After the choice, one of the two remaining doors is opened and shows no price. The player is then asked to choose again. Although it is commonly believed that the chances on the last pick are even, it is in fact false: it seems that there is a 50% chance of winning when there is actually a higher chance to win if the player switches their choice. We can compute all possible options and the results shows the player stands better chances, in fact 2 out of 3, if he doesn’t change his mind. This paradox is a veridical paradox because we can prove it to be true. (Figure 1)

Figure 1: The Monty Hall paradox

Another veridical paradox has been used by Edgar Allan Poe in one of his short story “Three Sundays in a Week*’’. *In this story, two young lovers want to get married but the uncle in charge decides it will only happen when a week will have three Sundays. A year later on a Sunday afternoon, the couple meet the uncle with two captains that traveled the globe in opposite directions at such speed that they respectively lost and gained a day, therefore thinking that Sunday was the day before or the day after and fulfilling the uncle’s condition (Poe, 225-232). Yet again, the statement seems contradictory but an analysis on the matter shows in fact in it is a veridical paradox; Poe constructs the story around the fact that the referential for the day was not specified.

Finally, there are paradoxes that can be both, true or false. Grelling’s paradox falls into this category: let us divide all adjectives in two sets, the autological and heterological ones. The first ones are those that describes themselves, for instance short is a short word. Heterological are those that does not describes themselves: long is a short word. The paradox arises when we try to classify the adjective heterological: if its autological then it describes itself, it is then heterological which contradicts the statement. On the other hand, if it is heterological, then it cannot be in its own category, it must then be autological, but we already showed that it can’t be (Vidal-Rosset, 26). We will see later that another paradox by Bertrand Russell is similar to this case.

Trying to classify paradoxes into one of these categories is the first way to turn paradoxes into useful abstract objects enlarging the scope of knowledge. The categorisation implies gathering enough knowledge and understanding of the paradigm in which the paradox is stated to be able to classify it into the proper box, but this procedure might redefined the paradigm or lead to new theories. Although, it is quite natural to take some assumption as true -as axiomatic- in part of building knowledge, but the interrogation about these keystones of knowledge really comes unavoidable when paradoxes are found. This thought is well express by many great thinkers in sciences and philosophy. As Niels Bohr said when working on theoretical model for the atom ‘’How wonderful that we have met with a paradox. Now we have some hope of making progress.’’ (Moore, 196). The idea of meditating on the meaning of paradoxes have even brought to an almost mystical perspective in Kierkegaard’s’ writings: ‘’ The paradox’s really the pathos of intellectual life and just as only great souls are exposed to passions it is only the great thinker who is exposed to what I call paradoxes which are nothing else than grandiose thoughts in embryo’’ (Slaatté, 64). In these cases, paradoxes technically mean self-contradiction, but it is seen as well as a possible extension of knowledge, and wider possible range for the concerned paradigm. An interesting and humoristic quote sometimes attributed to Mark Twain expresses this state of mind about knowledge: ‘’All generalisations are false, even this one’’.