Sunday, June 19, 2011

Top 10 Paradoxes

We can express paradox in different ways.  In literature, paradox is a mere figure of speech.  In theology, a paradox is distinct from a contradiction; paradoxes in theology are truths that only seem contradictory, but further and careful scrutiny would show that they are not, whereas contradictions are impossibilities and untruths.  In the mindblowing world of mathematics, paradoxes are contradictions that are possible and true (e.g. 1 = 0.99999…).

Indeed there are different treatments of paradox among different disciplines.  But what is paradox in general?  A paradox, according to wiki, is “a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition.”  So, as I want to interpret it, paradoxes can either be possible (which happens in theology and math) or impossible.   Paradoxes can be seemingly contradictory but actually not or they can be outright contradictory.    

I love paradoxes because they are awesome mindblowers. 

Warning:  This may happen while reading this list 

Once you are ready for some massive brain-beating, meet me after the jump.

10.) Omnipotence Paradox

“Can God create a stone that he won’t be able to lift?”

If God can create and then creates a stone that he won’t be able to lift, therefore he ceases to be omnipotent since there will be something that he can’t do, which is lifting that stone.  But if he can’t create such a stone that he won’t be able to lift, then he is not omnipotent in the first place since there is something he can’t do, which is to create such stone.  Either way, it is implied that omnipotence is something impossible.  Therefore, if omnipotence is an impossibility, then an omnipotent God is an impossibility.  And since a God can’t be a God without omnipotence, then a God does not exist.  That’s how the logic of it works.  And this argument is a favorite of atheists. 

Personally, I am already tired of this argument.  But I have to add this in this list since, I admit, it had stumped me for a while before I learned the solution to this dilemma (with the help of two great apologists, C.S. Lewis and R.C. Sproul).  I already wrote about it in my “Top 10 Thought Provokers” post.  If you care to learn it, just read it there.  The “Omnipotence Paradox” item is also a number 10 in that list.      

9.) Schrödinger's Cat

Schrödinger's Cat is a thought experiment in Quantum Physics that can be related to several quantum topics, which ultimately arrive at many worlds theory (see number 2 of my “Top 10 Thought Provokers” post).  This is a very notoriously complex thesis (like all other topics that Quantum Physics touches).  I don’t really understand it totally.  But here’s the scenario in a nutshell...
A cat is sealed inside a box with this device: in a Geiger counter, there is a tiny bit of radioactive substance, tiny enough that there is a 50/50 chance that one of the atoms will decay, which means that there’s the same 50/50 chance that no atom will, but if one does, it is designed to relay a signal for a hammer to break a flask of hydrocyanic acid.  If no atom will decay, the cat lives after an hour.  But if one does, then the cat dies because of the release of the poison.  We will never know if the cat lived or died unless we open the box.  But according to the theory, until then, the cat is paradoxically both alive and dead inside the box!  The argument is since nobody is around to witness what is happening inside the box, then the cat exists in all its possible states.  Yah, Quantum Mechanics is that weird – actually, weirder than we can ever imagine. 

In simpler words, the Schrödinger's Cat paradox’s notion is alike with the question: “If a tree falls in the woods and there’s no one to hear it, does it make a sound?”

Try answering that.

8.) This geometrical paradox…

Awesome, right?  According to this pic’s source, this is a modified version of Curry’s paradox by Martin Gardner.  Actually, the most mindblowing paradoxes exist in Math, but I’m afraid my head will explode if I force myself to grasp or study fully all of those. 

7.) Crocodile Paradox

There are different variations of this dilemma.  But here’s the basic: a crocodile kidnaps a child of a father.  The crocodile promises that he will return the child if the father correctly predicts whether the croc will return the child or not.  No logical problem will occur if the father guesses, “You will return my child.”  The crocodile now has the power to make the father’s guess correct or wrong.  If the crocodile chooses to return the child, then the father’s guess is correct, and if the crocodile chooses to keep the child – which the croc will definitely do in such scenario since it’s the beneficial choice for itself – then the father’s guess is incorrect.  Regardless of what the croc’s choice is, either way, there will be no violation of the crocodile’s stipulation.  However, a logical headache happens if the father guesses, “You will not return my child.”  Now, if the crocodile chooses to keep the child (since, as I’ve said, it’s the beneficial choice for itself) then the father’s guess is correct, and under the stipulation, the croc has to return the child.  But how can the croc keep and return the child at the same time?!  Even if the crocodile chooses to return the child to the father, the father’s guess would be incorrect then, thus under the stipulation, the croc should not return the child. 

Bang! We’re only on number 7.  Still more to come.

6.) Paradox of the Court

This is a logical problem dating back to ancient Greece.

According to legend, the famous sophist Protagoras agreed to take Euathlus as his pupil, with the contract that the latter will pay the former for his lessons after he – Euathlus (let’s make this clear since these paradoxes already are confusing on their own) – wins his first court case.  According to some accounts, Protagoras demanded the money as soon as Euathlus finished his schooling under Protagoras.  On other accounts, Euathlus, after finishing his schooling, failed or made no effort on taking on clients.  Regardless of which is true, Protagoras decided to sue Euathlus for the amount owed to him. 

In Protagoras’ perspective, the logic was, if he won the case, he would be paid the money due to him.  And, on the other hand, if Euathlus won the case, then as their contract stipulates, Euathlus still had to pay Protagoras since he won his first case.  Either way, he gets his money.

However, Euathlus’ argued that if he won the case, by the court’s decision, he would not have to pay Protagoras.  And, on the other hand, if Protagoras won the case, it meant that Euathlus was still unable to win his first case, thus he wouldn’t have to pay Protagoras.

If you were the judge of the case, who would you rule in favor for?

Be thankful you were not a judge.  
Rumor has it this is also what happened to him.  

5.) Catch-22

This logical paradox was coined by Joseph Heller in his novel “Catch-22” (a hilarious and must read book!).   This refers to a situation in which an individual needs something that can only be acquired by not being in that very situation.  In the novel, the protagonist Captain John Yossarian, a US Army Air Forces bombardier, who as much as possible tries to avoid situations that endanger his life, wishes to be grounded from combat flight.  This is only possible if he is evaluated by the flight surgeon as “unfit to fly”.  However to be branded as “unfit”, a pilot must volunteer for extremely dangerous missions, i.e. to be mad enough to be willing to fly to possible death.  But to be evaluated, a pilot should request an evaluation – an act that is considered sufficient proof to be declared sane.  Therefore, it is impossible for someone to be declared “unfit to fly”, since a pilot who request mental fitness evaluation is deemed sane, thus should fly in combat, and at the same time, if he does not request an evaluation, he will not receive one and thus can never be found insane and must fly in combat.   That is the “Catch-22”. 

I can give another example of a “Catch-22” scenario, which you surely will see is true:  In applying for a job, a job experience is required for it.  But how can one earn a job experience without having a job in the first place?

4.) Barber Paradox

There’s a town with just one male barber.  Every man in the town keeps himself clean-shaven.  Some shave themselves, while some by going to the barber.  Now, by this, we can logically assume that the barber shaves all and only those men who do not shave themselves

However, one question would lead us to a paradox… “Who shaves the barber?”

From the assumption we established, the answer is either he shaves himself or the barber of the town does, which also happens to be himself as well.  Which is impossible!  If the barber does shave himself, then according to the “barber only shaves those who does not shaves himself” rule, he must not shave himself.  And if the barber does not shave himself, according to the rule, he – being the barber of the town – must shave himself.

3.) Exception Paradox

“If every rule has an exception, then there must be an exception to the rule that every rule has an exception.”

Glad you brought an extra head with you?

Let’s further analyze:

The assumption is “Every rule has an exception.”  From what the assumption establishes, even “Every rule has an exception” has an exception itself.   Thus, a rule without an exception can exist.  There would now be a contradiction, since according to the initial assumption, “Every rule has an exception”, but if the assumption is applied on the assumption itself, we conclude that some rule with no exception is possible.            
Since the assumption implodes in itself, then we conclude then that such assumption, “every rule has an exception”, will always be a false premise since it’s impossible (Note: if you have noticed, this structure of argument was what was used in the Omnipotence Paradox).

There are other variations to the Exception Paradox but my favorite is “If everything is possible, then it is possible for anything to be impossible.”

There goes your extra head

2.) Unexpected Hanging

A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week, but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.

Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the “surprise hanging” can’t be on a Friday, as if he hasn’t been hanged by Thursday, there is only one day left – and so it won’t be a surprise if he’s hanged on a Friday. Since the judge’s sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.

He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn’t been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.

The next week, the executioner knocks on the prisoner’s door at noon on Wednesday — which, despite all the above, will still be an utter surprise to him. Everything the judge said has come true.

1.) Liar Paradox

A line in a poem of Epimenides goes: “The Cretans, always liars, evil beasts, idle bellies!”

He, being a Cretan himself, unwittingly called himself a liar.  Thus, a dilemma arises.  If all Cretans, including himself, are liars, then his statement that all Cretans are liars is a lie.  Then, if his statement was a lie, then it means that all Cretans are truthful.  However, if all Cretans are truthful, then he being a Cretan himself had told the truth that all Cretans are liars.  And we have ourselves an infinite logical loop. 

That is the mechanics of the liar paradox.  We can make it comprehensive with the simple paradoxical sentence of “This statement is false.”  If the statement “this statement is false” is true, then it is false.  If the statement is false, then it is true.  Either way, it is both true and false.  Which is a contradiction. 

The best way the Liar Paradox is presented was by using Pinocchio as the case study.  

When not writing about mindblowing stuff in the "Experimental Theatre", Bernel is writing about mindblowing stuff in "The Bernel Zone"...


  1. hahaha...great one nel! ^_^ i love the geometric paradox one hehe

  2. It's been 47 years since I sold my first piece of Art. Admittedly, I have made a mess of the one opportunity I've had at Life. Art, I've crushed.