Paradox

 Editor
Vázquez, Margarita  mvazquez@ull.es
 Incorporated contributions
F. Salto (20/08/2009)
 Usage domain
transdiciplinary, philosophy
 Type
concept
 French
paradoxe
 German Paradox, Paradoxon

A paradox is a conflict between reasons: those grounding it and those refuting it. The more solid the reasons in conflict, the greater the philosophical interest of the paradox. In this general sense there are paradoxes of very different genres: (a) paradoxes challenging the intelligibility of particularly basic notions, such as: infinite, time, space, identity, etc.; (b) paradoxes challenging the rationality of our action or decision strategies: Newcomb's, Gaifman's paradoxes, prisioner's dilemma, etc.; (c) paradoxes challenging the rationality of our bodies of belief: selfdeceiving paradoxes, Goodman's , knower's paradoxes, etc.; among other many paradoxes, more or less important and more or less funny.

Logical paradoxes or antinomies are logically valid reasonings with non reasonable conclusions. Therefore we call antinomy any deductively valid reasoning driving to a contradiction from rationally justified, highly acceptable or assertable premisses.

 
References
  • McGEE, V. (1993). Truth, Vagueness and Paradox. Indianapolis: Hackett.
  • QUINE, W. (1976). The Ways of Paradox and other essays (rev. ed.). Cambridge (Mass.): Harvard University Press.
  • SALTO, F. (2005). "Verdad y recursividad", in J.M. MÉNDEZ (ed.). Artículos de Segunda Mano. Salamanca: Varona, pp.51-156.
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Incorporated entries

Francisco Salto (20/08/2009)

A paradox is a conflict between reasons: those grounding it and those refuting it. The more solid the reasons in conflict, the greater the philosophical interest of the paradox. In this general sense there are paradoxes of very different genres: (a) paradoxes challenging the intelligibility of particularly basic notions, such as: infinite, time, space, identity, etc.; (b) paradoxes challenging the rationality of our action or decision strategies: Newcomb's, Gaifman's paradoxes, prisioner's dilemma, etc.; (c) paradoxes challenging the rationality of our bodies of belief: selfdeceiving paradoxes, Goodman's , knower's paradoxes, etc.; among other many paradoxes, more or less important and more or less funny.

Logical paradoxes or antinomies are logically valid reasonings with non reasonable conclusions. Therefore we call antinomy any deductively valid reasoning driving to a contradiction from rationally justified, highly acceptable or assertable premisses.


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Victor Omura Kawai (29.12.2020, linked to the seminar "A Journey through philosophy" facilitated by J.M.Díaz at HM)

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Abstract: Paradoxes are defined as irrefutable, but counter-intuitive phenomena and [I think it has to be removed] appear mostly in the mathematical environment. There are three main forms of paradoxes: veridical paradoxes, falsidical paradoxes and antinomies. The first consists of true-telling paradoxes and is normally discarded by assuming reductio ad absurdum, meaning that such situations described in the paradox don’t exist in our reality. The second are the ones that contain fallacies in it and can be discarded by identifying the ones in the speech, which automatically invalidate the argument. The last one is the most famous and consists of contradictions between two laws, which are assumed as correct and logical. In order to eliminate them, Russell’s subscripts theory should be applied, and the text thereafter reviewed. Russell itself also elaborated an antinomy, which is different from the others and can’t be solved by the subscript’s method. It is different, because it doesn’t treat terms like ‘true’ or ‘true of’, but about self-membership of a class. Some interesting extra paradoxes are briefly summarized at the end of this work.

1. Introduction

What is a paradox? From the Greek, para means “distinct form” and doxa means ”opinion”. In other words, it results in “contrary opinion”, which is basically the meaning of paradox: any conclusion that, at first sounds absurd, confronts the common truth and the common logic, but at the same time exists and cannot be unproved. In the meantime, as it will be discussed through this present text, not everything that looks logical and exists can be classified as a paradox, because it can be sometimes a fallacy.

Through the development of history, the conception of paradox, antinomy and what is true or not has been changing and what we consider today as a paradox may not be in the coming years or centuries. 

Before Bertrand Russell, for example, the great majority of antinomies were unresolvable and were therefore logically unjustifiable, making us just wonder why something was wrong. So happened before Copernicus: he was taken as insane and the world used to structure their knowledge around geocentrism, nowadays our common reasoning makes its roots in heliocentrism, leading one of the authors used in this article to think if in a few years the use of subscripts (created in Bertrand Russell’s and Alfred Tarski’s theory) won’t be how we write in our day-to-day life. Is it everything that comes to the opposite common believed truth? Well, not really.

2. Ways of Paradox

There are three main forms of paradox: veridical paradoxes, falsidical paradoxes and antinomies. One example used by (Quine,1976) of a veridical paradox is the statement:

Frederic, protagonist of The Pirates of Penzance, has reached the age of 21 after passing only five birthdays.”

In the moment you hear this phrase, you instantly say it's absurd. But after giving the explanation he is born on 29th of February, the case is explainable and seen as true. The likelihood of this happening is so low that we forget about it. 

Another example, but with a different solution used by (Quine,1976) is one first introduced by Russell in 1918 (whose authorship is unknown) about a barber. Suppose the following sentence:

“In a certain village there is a man, so the paradox runs, who is a barber; this barber shaves all and only those men in the village who do not shave themselves. Query: Does the barber shave himself?”

Well, if a barber shaves only the men who don't shave themselves, the only way for him to shave himself is not shaving himself. This problem is solved by reductio ad absurdum, meaning that if we reduce it to an absurd, saying there’s  no possibility of the existence of this situation, disproving the barber.

On the other hand, the falsidical paradox is a paradox that contains a fallacy in the middle, meaning it’s false, typically propositions 2=1. A falsidical paradox must contain a fallacy, but can’t be reduced to so, because it also must seem absurd and to be false. The most famous example of this type is the paradox of Zeno about Achilles and the tortoise. This paradox talks about how Achilles will never overtake a tortoise that started ahead in a run, no matter how slow she travels. The explanation given is that when Achilles reaches the previous position of the tortoise, she will have already walked a bit further. The fallacy lies in the misconception that an infinite succession of intervals of time will lead to infinity. As matter of fact, the time intervals will become smaller and smaller every time, which results in a convergent series, which nowadays we can solve.

As the last case, we have antinomy, which is the current unresolvable problem in the world, and for which will be used more subtopics to be discussed.

3. Antinomy

We can define at first antinomy as the simultaneous conflict of two logical laws, seeming absurd, while they are reasonable. This is a topic that intrigued thinkers since ancient times, like Socrates.

The Liar Paradox for (Beall, Glanzberg and Ripley, 2020), also known as “Epimenides Paradox”, is one specific type of paradox related to the self-affirmation of being a lie. There are many paradoxes of this type, antinomy, for example “Simple-falsity Liar”, “Simple-untruth Liar’, Liar Cycles” “Grelling’s Paradox”. The common feature of all of them is the creation of a reflexive affirmation, that leads to a logical loop, yielding a truth and a lie concomitantly.

The Liar Paradox resides in the affirmation: “I am lying”. We have then an antinomy: it is true that he is lying, if and only if he is saying the truth. In other words, it is true, if and only if it is false. But well, how can something be true and false at the same time?

In the same way of thinking, Kurt Grelling in 1908 defined two terms: autological and heterological. Autological is a term that confirms it’s meaning and is true of itself. For example: the word “short” is in fact short and therefore autological. On the other hand, we have heterological words that conflict with their meaning. For example, the word “long” is not indeed long.

The paradox arises from the reflection: is the adjective “heterological” autological or heterological? Well, if it is true that it is autological, then it is “true of itself”, but that makes it rather heterological. And there it is the paradox: It is if, and only if, it is not, meaning that it is and it is not at the same time.

4. Bertrand Russell

[Since you're not going to speak about Russell himself, but on his work regarding paradoxes, the title should refer more specifically what you will lead with in the section]

Bertrand Russell solved this kind of antinomy with his theory of the subscripts. Basically, expressions related to truth, such as: “true”, “false”, “true of” in a sentence S possesses a numerical subscript (0,1,2…) increasing according to the hierarchy. The solution resides in taking care that the subscript of a truth locution T is always higher than any subscript of a sentence S. If this rule is violated, then the phrase is taken as meaningless, non-sense or grammatically incorrect. Example from (Quine, 1976):

“We could meaningfully ask whether the adjectives `long' and `short' are   of themselves; the answers are respectively no and yes. But we could not meaningfully speak of the phrase ‘not  of self ‘ as  or  of itself; we would have to ask whether it is  or  of itself, and this is a question that leads to no antinomy. Either way the question can be answered with a simple and unpenalized negative.”

At this point, (Quine, 1976) demonstrates one important opinion about Russell: during the Copernican Revolution, for instance, the theory that the Earth revolves around the sun was held as a paradox, even by the people who believed in it. With that analogy, the author tries to make a relation between our current reality and Russell’s theory, since it seems so awkward to put subscripts in terms of a text. Maybe in the future, it will seem as normal as heliocentrism seems to us now. Or in another quote of the author:

“One man's antinomy is another man's falsidical paradox, give or take a couple of thousand years.”

Russel’s Antinomy

Of course, that the solver of the newest problem introduced a new one. Russel’s antinomy, or Russell-Zermelo Paradox (David and Deutsch, 2020), is the most famous puzzle of the current logic, since it’s a special one, it doesn’t fit in the other types of antinomies, because it is not about ‘true’ or ‘true of’, it is about a self-membership of a class. Although Ernst Zermelo discovered the enigma before Russell, it didn’t receive much importance, before Russell found it independently.

Russel discovered the puzzle during the late spring of 1901. Interesting to note is that Gottlob Frege was just publishing his second volume of the foundations of arithmetic, when he received a letter from Russell stating that the axioms he used to deduce his theory were false. Due to his acceptance that Russell was right, he felt forced to abandon many of his ideas about logic and mathematics. The episode demonstrated mutual respect between both. Frege readily assumed the mistake and made a public declaration, saying that it was extremely painful to be wrong after having his work done. Russell, on the other hand, said that he assumed the mistake with fortitude, and it takes a great man to do this in benefit of human knowledge, instead of getting famous and known. 

Let’s look at what the antinomy looks like, before we discuss the reason for its inadequacy. (Quine, 1976) stated:

“For example, the class of all classes that have more than five members clearly has more than five classes as members; therefore, the class is a member of itself. On the other hand, the class of all men is not a member of itself, not being a man. What of the class of all classes that are not members of themselves?”

The answer, as typical of an antinomy, is that it’s a member, if and only if it is not. Or in the words of (David and Deutsch, 2020), if we define a function ϕ(x), whose members are the ones that satisfy ϕ(x). So, if we define a set R = {x: ∼ϕ(x)} and ϕ(x) stand for x ∈ x, it means that R is the set of the members that are exactly the ones that are not members of themselves. 

As Russell itself said, if there is one class that embraces everything, then this class must embrace itself. But normally a class it’s not a member of itself. For example, “Mankind” is not “man” (member of Mankind). This is a class, is it a member of itself or not? Well, if it is, then it is one of the classes that are not members of themselves, which means it is not a member of itself. If it is not, then it is not one of the classes that are not members of themselves, meaning it is a member of itself.

The method to solve this antinomy, so as the others, is to eliminate the sets of self-membership and, at the same time, retain the ones needed for mathematics. If, for instance, we define R = {x ∈ S:x ∉ x}, the antinomy is gone, since it consists of the members found inside of S. In this situation, the set R fails to include as a member of itself. Groucho Marx (1895-1977) made a humorous commentary about the topic, when he was resigning of a club he belonged to:

Please accept my resignation, I don’t want to belong to any club that will accept me as a member.

5. Famous Paradoxes

This section is destined to show a few others famous paradoxes, although there are countless numbers of them. They will be briefly spoken about, just in order to incite curiosity in the reader, due to its complexity and necessity of a much longer description. Although they seem unlikely, going against the intuition, they don’t contain contradictions. Therefore, it divides the scientific society between what is true.

Newcomb Paradox

This is a game theory paradox, created by William Newcomb, first published in a philosophy paper in 1969.

The paradox consists of a game between two players, in which one must choose between two options. The problem proposed is as follows: it is given two boxes A and B to a player and the winnings depend on the prediction of the other player about which option will be chosen. If the person predicts the other player is going to open only box A, it puts $1,000,000 into box A and $1,000 into box B. If the person predicts you are opening both, then puts $0 into box A and $1,000 into box B.

And there we have the paradox again: if the player chooses to pick box A, because it has more money, then of course that taking both will just add money and therefore will result in the highest amount earned. But the condition is that if he chooses both boxes, there will be no money in box A and it will result in the lowest amount earned. Which option should he choose? (Darling, 2004)

Banach-Tarski Paradox

This paradox proves the possibility of getting a single sphere, dividing it into 5 parts and rearrange its parts forming in the end 2 spheres.

The paradox is based on the set divisions of numbers into infinite numbers. The reason we can’t reproduce this theorem in reality is that we can’t divide a sphere in this way. Between 0 and 1, for instance, there are infinite subdivisions. 

It intrigues specially the physical and mathematical societies because it generates the doubt: is it/will it be possible to reproduce it into reality? As a matter of fact, mathematics represents a huge bunch of events with great accuracy (Darling, 2004).

6. Conclusion

The present work showed some types of the most common and well-known antinomies, fallacies and paradoxes. It serves to the reader as a guide in how to better identify and interpret the present logical errors in a discourse and therefore fallacies. 

As a rule, every time a phrase stated by someone involves the use of “self-lie”, “self-ownership” or anything related to itself, the listener/reader should pay extra effort, because probably a logical contradiction lies in it.

In the best scenario, the person can make use of Russell’s theory of the subscripts, which tells us that whenever someone applies one self-statement, this statement should be of a higher subscript than the current one. At any moment, the declaration is only valid if, and only if, the subscript is higher than of which it talks about. And with this, antinomies are not a problem anymore.

Worthy of being remembered are also the veridical and falsidical paradoxes, which differ by the existence of a fallacy in the second. In this case, the paradox is disproved simply by identifying the fallacy and mentioning that a theory can’t be true containing a lie. In the first one, many can be redacted to absurd, for example in the case of the barber who just cuts the hair of the ones who don’t do it themselves.

References

Beall, Jc, Michael Glanzberg, and David Ripley, Liar Paradox, The Stanford Encyclopedia of Philosophy (Fall 2020 Edition), Edward N. Zalta (ed.) [Online] <https://plato.stanford.edu/entries/liar-paradox/> (accessed: 13/12/2020)

Irvine, Andrew David and Harry Deutsch, Russell’s Paradox, The Stanford Encyclopedia of Philosophy (Winter 2020 Edition), Edward N. Zalta (ed.) [Online] <https://plato.stanford.edu/entries/russell-paradox/> (accessed: 13/12/2020)

Strandberg, A. et al., Newcomb's Paradox, Brilliant.org [Online] <https://brilliant.org/wiki/newcombs-paradox/> (accessed: 13/12/2020)

Quine, W. (1976). The Ways of Paradox and other essays (rev. ed.). Cambridge (Mass.): Harvard University Press.

Darling, D. (2004). The universal book of mathematics: from Abracadabra to Zeno's paradoxes. John Wiley & Sons.


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