"There is an Exception to every rule"
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"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?
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 Eebster the Great
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Re: "There is an Exception to every rule"
It is not paradoxical, it is simply false. There are rules which have no exceptions.
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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?
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 Soupspoon
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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.
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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".
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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.
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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.
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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
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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
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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
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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.
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 WibblyWobbly
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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
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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
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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
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Re: "There is an Exception to every rule"
And a squared example of a prime number.
 MartianInvader
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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}.
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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
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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
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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" ...
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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
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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
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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
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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!)
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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
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