"There is an Exception to every rule"
Moderators: gmalivuk, Moderators General, Prelates
"There is an Exception to every rule"
Is the sentence "There is an Exception to every rule" a version of the liars paradox? Because for it to be true, it must itself have an exception, but that would make it false. Is this the same as the liars paradox, or something different?
Please be gracious in judging my english. (I am not a native speaker/writer.)
http://decodedarfur.org/
http://decodedarfur.org/
 Eebster the Great
 Posts: 2652
 Joined: Mon Nov 10, 2008 12:58 am UTC
Re: "There is an Exception to every rule"
It is not paradoxical, it is simply false. There are rules which have no exceptions.
 stopmadnessnow
 Posts: 83
 Joined: Thu Mar 03, 2016 10:58 pm UTC
 Location: Somewhere in that country with Big Ben in it.
Re: "There is an Exception to every rule"
If it is a physical law, would it necessarily be the same just after the Big Bang?
For comic writing or short story writing posthaste, contact the offthe peg joke chap. He has the nous.
http://plwimsett.simplesite.com/ for my site.
https://www.upwork.com/freelancers/~01110048bacb43eae6 to hire me at Upwork.
http://plwimsett.simplesite.com/ for my site.
https://www.upwork.com/freelancers/~01110048bacb43eae6 to hire me at Upwork.
 Soupspoon
 You have done something you shouldn't. Or are about to.
 Posts: 2297
 Joined: Thu Jan 28, 2016 7:00 pm UTC
 Location: 531
Re: "There is an Exception to every rule"
"There is an exception to every rule" is not itself a Rule, it is an Aphorism. Rules, themselves, are not Laws, and Laws are not Truths, which are in turn different from Facts that may or may not be backed up with Data. This post is a Summary, and as such contains Innacuracies and Assumptions for the sake of Brevity. Including any claim that it exhibits anything other than Verbose Waffling largely uninfluenced by Analysis of Facts for the possible Purposefulness of conveying Humour.
Hope That Helps. Have A Nice Day.
Hope That Helps. Have A Nice Day.
Re: "There is an Exception to every rule"
Epimenides of Crete is claimed to have said "All Cretans are liars". If true, then it means that all Cretans are liars, but Epimenides being a Cretan himself must then be a liar, meaning his statement cannot be true. If the statement is false, then it means that Epimenides was lying, and that not all Cretans are liars. It is still consistent that Epimenides is a liar, as long as there is at least one Cretan who tells the truth.
Similarly, if we assume that "There is an exception to every rule" is meant to be a rule, then if it's true it must have an exception, and if that's the case then there is a rule without an exception and hence the original rule is false. So therefore "There is not an exception to every rule" is the true rule, and there must be at least one rule that has no exception.
tl;dr The negation of "For all X, Y" is "There exists X such that not Y", not "For all X, not Y".
Similarly, if we assume that "There is an exception to every rule" is meant to be a rule, then if it's true it must have an exception, and if that's the case then there is a rule without an exception and hence the original rule is false. So therefore "There is not an exception to every rule" is the true rule, and there must be at least one rule that has no exception.
tl;dr The negation of "For all X, Y" is "There exists X such that not Y", not "For all X, not Y".
pollywog wrote:I want to learn this smile, perfect it, and then go around smiling at lesbians and freaking them out.Wikihow wrote:* Smile a lot! Give a gay girl a knowing "Hey, I'm a lesbian too!" smile.
 doogly
 Dr. The Juggernaut of Touching Himself
 Posts: 5183
 Joined: Mon Oct 23, 2006 2:31 am UTC
 Location: Somerville, MA
 Contact:
Re: "There is an Exception to every rule"
ConMan wrote:Similarly, if we assume that "There is an exception to every rule" is meant to be a rule, then if it's true it must have an exception, and if that's the case then there is a rule without an exception and hence the original rule is false.
"There is an exception to every rule" can be true AND have an exception  it is a rule, A, about rules. If the exception to A is A itself, then we're fine!
Though most people take the definition of a rule to be a thing with no exceptions, so it would just be a stupid nonmathematical statement rather than a rule at all, really.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Re: "There is an Exception to every rule"
"Every even number is divisible by 2 with no remainder"  there's your exception to "There's an exception to every rule".
Re: "There is an Exception to every rule"
That's not a rule, that's a tautology.
Mighty Jalapeno: "See, Zohar agrees, and he's nice to people."
SecondTalon: "Still better looking than Jesus."
Not how I say my name
SecondTalon: "Still better looking than Jesus."
Not how I say my name
Re: "There is an Exception to every rule"
That's not a rule, that's a tautology.
Fine, for every prime p, and every integer a prime to p, a raised to power p minus a is divisible by p. Not a tautology, definitely no exceptions. Happy now?
 Eebster the Great
 Posts: 2652
 Joined: Mon Nov 10, 2008 12:58 am UTC
Re: "There is an Exception to every rule"
DavCrav wrote:That's not a rule, that's a tautology.
Fine, for every prime p, and every integer a prime to p, a raised to power p minus a is divisible by p. Not a tautology, definitely no exceptions. Happy now?
If you mean "For every prime p and integer a coprime to p, (a^{p}  a)  p," then that is also a tautology.
 WibblyWobbly
 Can't Get No
 Posts: 506
 Joined: Fri Apr 05, 2013 1:03 pm UTC
Re: "There is an Exception to every rule"
Eebster the Great wrote:DavCrav wrote:That's not a rule, that's a tautology.
Fine, for every prime p, and every integer a prime to p, a raised to power p minus a is divisible by p. Not a tautology, definitely no exceptions. Happy now?
If you mean "For every prime p and integer a coprime to p, (a^{p}  a)  p," then that is also a tautology.
Pardon my stupidity, but why is that a tautology? Does it not have to be proven?
Re: "There is an Exception to every rule"
Eebster the Great wrote:DavCrav wrote:That's not a rule, that's a tautology.
Fine, for every prime p, and every integer a prime to p, a raised to power p minus a is divisible by p. Not a tautology, definitely no exceptions. Happy now?
If you mean "For every prime p and integer a coprime to p, (a^{p}  a)  p," then that is also a tautology.
To be perfectly pedantic, this seems to be a logical validity, not a tautology.
I think a nice rule without exception is "I think", in as much as it can be considered a rule. ("you may not kill someone" is considered a rule, so every "<entity> <action>" can be considered a rule, right?)
No one* having that thought can deny they* think and are an entity.
*for lack of a better word.
 Eebster the Great
 Posts: 2652
 Joined: Mon Nov 10, 2008 12:58 am UTC
Re: "There is an Exception to every rule"
Flumble wrote:Eebster the Great wrote:DavCrav wrote:That's not a rule, that's a tautology.
Fine, for every prime p, and every integer a prime to p, a raised to power p minus a is divisible by p. Not a tautology, definitely no exceptions. Happy now?
If you mean "For every prime p and integer a coprime to p, (a^{p}  a)  p," then that is also a tautology.
To be perfectly pedantic, this seems to be a logical validity, not a tautology.
I guess so. The tautology would be "(p is a prime) ∧ (a is an integer coprime to p) → ((a^{p}  a)  p)."
Re: "There is an Exception to every rule"
Eebster the Great wrote:Flumble wrote:Eebster the Great wrote:DavCrav wrote:That's not a rule, that's a tautology.
Fine, for every prime p, and every integer a prime to p, a raised to power p minus a is divisible by p. Not a tautology, definitely no exceptions. Happy now?
If you mean "For every prime p and integer a coprime to p, (a^{p}  a)  p," then that is also a tautology.
To be perfectly pedantic, this seems to be a logical validity, not a tautology.
I guess so. The tautology would be "(p is a prime) ∧ (a is an integer coprime to p) → ((a^{p}  a)  p)."
No. That's a falsehood.
Re: "There is an Exception to every rule"
If a is not coprime to p, then it's a multiple of p, so a^{p} and a are also multiples of p, and so is their difference.
Re: "There is an Exception to every rule"
The difference being a multiple of p would be denoted p(a^pa), though.
No, even in theory, you cannot build a rocket more massive than the visible universe.
 WibblyWobbly
 Can't Get No
 Posts: 506
 Joined: Fri Apr 05, 2013 1:03 pm UTC
Re: "There is an Exception to every rule"
rmsgrey wrote:If a is not coprime to p, then it's a multiple of p, so a^{p} and a are also multiples of p, and so is their difference.
Forgive me, but isn't 6 not coprime to 9? They have a factor in common, but neither is a multiple of the other.
Re: "There is an Exception to every rule"
WibblyWobbly wrote:rmsgrey wrote:If a is not coprime to p, then it's a multiple of p, so a^{p} and a are also multiples of p, and so is their difference.
Forgive me, but isn't 6 not coprime to 9? They have a factor in common, but neither is a multiple of the other.
Last time I checked, 9 isn't a prime...
 WibblyWobbly
 Can't Get No
 Posts: 506
 Joined: Fri Apr 05, 2013 1:03 pm UTC
Re: "There is an Exception to every rule"
rmsgrey wrote:WibblyWobbly wrote:rmsgrey wrote:If a is not coprime to p, then it's a multiple of p, so a^{p} and a are also multiples of p, and so is their difference.
Forgive me, but isn't 6 not coprime to 9? They have a factor in common, but neither is a multiple of the other.
Last time I checked, 9 isn't a prime...
Well maybe you should check again. Or maybe I should stop posting in this thread.
 MartianInvader
 Posts: 765
 Joined: Sat Oct 27, 2007 5:51 pm UTC
Re: "There is an Exception to every rule"
9 is a prime example of a square number.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!
 Eebster the Great
 Posts: 2652
 Joined: Mon Nov 10, 2008 12:58 am UTC
Re: "There is an Exception to every rule"
And a squared example of a prime number.
 MartianInvader
 Posts: 765
 Joined: Sat Oct 27, 2007 5:51 pm UTC
Re: "There is an Exception to every rule"
This got me thinking  18 is a really boring, uncool number. Like, unusually so, so much that nothing could be more boring or uncool. I'd definitely choose 18 as an ususual example of a number of unparalleled uncoolness.
In other words, 18 is a prime odd perfect square.
In other words, 18 is a prime odd perfect square.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!
Re: "There is an Exception to every rule"
MartianInvader wrote:This got me thinking  18 is a really boring, uncool number. Like, unusually so, so much that nothing could be more boring or uncool. I'd definitely choose 18 as an ususual example of a number of unparalleled uncoolness.
In other words, 18 is a prime odd perfect square.
Here's a reasonably cool property of 18.
18² + 1 = 325 = 13 x 5², so 18 / 5 is the best approximation of sqrt(13) in that neighbourhood. Note that 13 = 2² + 3², so sqrt(13) is the length of the diagonal of the smallest rectangle with unequal (integer) sides > 1.
It's an interesting "coincidence" that 18  5 = 13 as well as 18 / 5 ≈13^{½}.
My favourite rule about primes:
"All primes are odd, except 2, which is the oddest of them all." — Graham, Knuth, Patashnik, Concrete Mathematics, second edition, page 129.

 Posts: 594
 Joined: Fri Feb 07, 2014 3:15 pm UTC
Re: "There is an Exception to every rule"
ConMan wrote:Epimenides of Crete is claimed to have said "All Cretans are liars". If true, then it means that all Cretans are liars, but Epimenides being a Cretan himself must then be a liar, meaning his statement cannot be true. If the statement is false, then it means that Epimenides was lying, and that not all Cretans are liars. It is still consistent that Epimenides is a liar, as long as there is at least one Cretan who tells the truth.
Similarly, if we assume that "There is an exception to every rule" is meant to be a rule, then if it's true it must have an exception, and if that's the case then there is a rule without an exception and hence the original rule is false. So therefore "There is not an exception to every rule" is the true rule, and there must be at least one rule that has no exception.
tl;dr The negation of "For all X, Y" is "There exists X such that not Y", not "For all X, not Y".
The Oxford Dictionary defines 'liar' as "A person who tells lies." In other words, there is a nonzero probability that a statement by a liar is false. This probability does not have to be one, so a liar can make a say a statement that is true. Therefor, Epimenides of Crete's statement can be true without creating a paradox.
 Soupspoon
 You have done something you shouldn't. Or are about to.
 Posts: 2297
 Joined: Thu Jan 28, 2016 7:00 pm UTC
 Location: 531
Re: "There is an Exception to every rule"
If it were "are compulsive liars", it may be a problem, if that assures the required absolute.
Or, then again, Epimenides of Crete may have assumed that being told that he was Cretan is a lie, by his habitually lying parents, and his statement is a perceived truth but, due to his misapprehension of the truth (that he and/or his parents actually did not (think they made a) lie) he is effectively lying anyway.
Or Epimenides of Crete actually was from Athens. Those habitually lying Athenians, right?!?
Or, then again, Epimenides of Crete may have assumed that being told that he was Cretan is a lie, by his habitually lying parents, and his statement is a perceived truth but, due to his misapprehension of the truth (that he and/or his parents actually did not (think they made a) lie) he is effectively lying anyway.
Or Epimenides of Crete actually was from Athens. Those habitually lying Athenians, right?!?
 Eebster the Great
 Posts: 2652
 Joined: Mon Nov 10, 2008 12:58 am UTC
Re: "There is an Exception to every rule"
I think it is pretty clear that while Epimenides was trying to get across a genuinely interesting paradox, he did a spectacularly poor job of it.
Re: "There is an Exception to every rule"
So if Epimenides had said "All Cretans are poor expressers of logical statements" ...
pollywog wrote:I want to learn this smile, perfect it, and then go around smiling at lesbians and freaking them out.Wikihow wrote:* Smile a lot! Give a gay girl a knowing "Hey, I'm a lesbian too!" smile.
Re: "There is an Exception to every rule"
Perhaps Epimenides did formulate the paradox correctly, but it was lost in translation: we're speaking of attributions to a semimythical[1] person in a dialect used 2 millenia ago.
 Soupspoon
 You have done something you shouldn't. Or are about to.
 Posts: 2297
 Joined: Thu Jan 28, 2016 7:00 pm UTC
 Location: 531
Re: "There is an Exception to every rule"
Don't be silly. If it is written down, regardless of how many intermediate translations were made, it is True.
(Unless it was ever written by a Cretan, obviously!)
(Unless it was ever written by a Cretan, obviously!)
Re: "There is an Exception to every rule"
That rule only applies to the internet!
...which makes for a interesting paradox now that news agencies publish the same articles on the internet and in the newspaper itself, which we know is full of lies.
...which makes for a interesting paradox now that news agencies publish the same articles on the internet and in the newspaper itself, which we know is full of lies.
 Eebster the Great
 Posts: 2652
 Joined: Mon Nov 10, 2008 12:58 am UTC
Re: "There is an Exception to every rule"
Flumble wrote:That rule only applies to the internet!
...which makes for a interesting paradox now that news agencies publish the same articles on the internet and in the newspaper itself, which we know is full of lies.
Mainstream Media is all Fake Lies though. You can only get real lies on alt media like Breitbart.
 Soupspoon
 You have done something you shouldn't. Or are about to.
 Posts: 2297
 Joined: Thu Jan 28, 2016 7:00 pm UTC
 Location: 531
Re: "There is an Exception to every rule"
(Not really germaine to the mathematical logic, but on UK keyboards, the right Alt key is "Alt Gr", thus confirming that everything under the banner of AltRight was always supposed to be aggressive and threatening... Grrrrr!)
 Xanthir
 My HERO!!!
 Posts: 5198
 Joined: Tue Feb 20, 2007 12:49 am UTC
 Location: The Googleplex
 Contact:
Re: "There is an Exception to every rule"
PM 2Ring wrote:18² + 1 = 325 = 13 x 5², so 18 / 5 is the best approximation of sqrt(13) in that neighbourhood. Note that 13 = 2² + 3², so sqrt(13) is the length of the diagonal of the smallest rectangle with unequal (integer) sides > 1.
It's an interesting "coincidence" that 18  5 = 13 as well as 18 / 5 ≈13^{½}.
18  5 = 13
18 / 5 ≈ √13
18 + 5 = 23
2 + 3 = 5
MATH ILLUMINATI CONFIRMED
(defun fibs (n &optional (a 1) (b 1)) (take n (unfold '+ a b)))
Who is online
Users browsing this forum: No registered users and 8 guests