0704: "Principle of Explosion"

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DT_
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Re: "Principle of Explosion" Discussion

Postby DT_ » Fri Feb 19, 2010 11:15 am UTC

In other words, if someone tells me "A implies B", and I say "fair enough, but look - A is false". They say "but if A was true, B would be true too!", to which I reply "maybe so, but perhaps if A was true B would be false - since A is false, it's impossible to tell".

If A implies B, and A is false then "If A was true then B would be true" is still true - that is the definition of A implies B. "Perhaps if A was true B would be false" is a contradiction of our assumption that A implies B. We do know that if A was true, then B would also be true. "since A is false, it's impossible to tell" <-- that statement is incorrect.

edit: Consider phlips' raincoat example. I tell you that "If it will rain tomorrow, then I will wear a raincoat." Now suppose it does not rain tomorrow - then you say "Perhaps if it had rained today, you would not have worn a raincoat. Since it didn't rain, it's impossible to tell." But we can tell - I told you that if it would have rained, I would have worn a raincoat, no matter what.
Last edited by DT_ on Fri Feb 19, 2010 11:21 am UTC, edited 2 times in total.

serrath
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Re: "Principle of Explosion" Discussion

Postby serrath » Fri Feb 19, 2010 11:19 am UTC

Someone please explain the waffle cone joke to me.

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SirMustapha
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Re: "Principle of Explosion" Discussion

Postby SirMustapha » Fri Feb 19, 2010 11:24 am UTC

chocolate.razorblades wrote:Two comics in a row alluding to logic...I wonder what Randall's doing.


Been reading Logic for Dummies, vol. 1, I guess.

JeromeWest
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Re: "Principle of Explosion" Discussion

Postby JeromeWest » Fri Feb 19, 2010 11:40 am UTC

DT_ wrote:edit: Consider phlips' raincoat example. I tell you that "If it will rain tomorrow, then I will wear a raincoat." Now suppose it does not rain tomorrow - then you say "Perhaps if it had rained today, you would not have worn a raincoat. Since it didn't rain, it's impossible to tell." But we can tell - I told you that if it would have rained, I would have worn a raincoat, no matter what.


You assure me you'll wear a raincoat tomorrow if it rains. If it rains and you wear a raincoat, your statement was true. If it rains and you don't wear a raincoat, your statement has been proven false. If it doesn't rain, I have no idea whether you would have worn a raincoat or not. I'll happily admit that the statement has not been disproven, but it hasn't been proven either.

I'm obviously missing the point somewhere along the line here. Thanks to you both for taking the time to try and enlighten me!

JeromeWest
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Re: "Principle of Explosion" Discussion

Postby JeromeWest » Fri Feb 19, 2010 11:44 am UTC

SirMustapha wrote:Been reading Logic for Dummies, vol. 1, I guess.


I think I need to look up a copy of that myself!

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phlip
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Re: "Principle of Explosion" Discussion

Postby phlip » Fri Feb 19, 2010 11:47 am UTC

JeromeWest wrote:If it rains and you wear a raincoat, your statement was true.

But by your standards, this isn't true either! Maybe I was going to wear a raincoat regardless, and it had nothing to do with the fact that it rained? Correlation is not causation and all that.

Consider instead the statement "if I'm not wearing my raincoat tomorrow, then it wasn't raining"... clearly it's the same thing - both of them are only false if I show up tomorrow, sans raincoat, on a rainy day. The technical term for this is a "contrapositive"... the statements [imath]A \to B[/imath] and [imath]\lnot B \to \lnot A[/imath] are equivalent.
Now, by your standard, if I am wearing a raincoat, then we can't test the conditional, because the antecedant is false. You can say "but if I didn't wear the raincoat, then it wouldn't've rained!" but there's no way to test that, because I did wear it.

Putting it around in this direction shows the problem - it's obvious that whether I wear a raincoat doesn't control the weather... there's no causal link there at all. But the material implication doesn't talk about causality... just about what's true and false. Just correlation, in other words, not causation.

Code: Select all

enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};
void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}
[he/him/his]

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Re: "Principle of Explosion" Discussion

Postby Spaztaztic » Fri Feb 19, 2010 11:55 am UTC

Putting it around in this direction shows the problem - it's obvious that whether I wear a raincoat doesn't control the weather... there's no causal link there at all. But the material implication doesn't talk about causality... just about what's true and false. Just correlation, in other words, not causation.

by phlip


Ah thank you, I finally understand. Didn't really affect the joke of the comic, but at least the explosion stuff makes more sense

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Re: "Principle of Explosion" Discussion

Postby ghostman » Fri Feb 19, 2010 12:24 pm UTC

I'm also stumped by the waffle cone and wine. Any insight? My google-fu has failed me for the last time.

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Re: "Principle of Explosion" Discussion

Postby jimbob_barnes » Fri Feb 19, 2010 12:39 pm UTC

JeromeWest wrote:
You assure me you'll wear a raincoat tomorrow if it rains. If it rains and you wear a raincoat, your statement was true. If it rains and you don't wear a raincoat, your statement has been proven false. If it doesn't rain, I have no idea whether you would have worn a raincoat or not. I'll happily admit that the statement has not been disproven, but it hasn't been proven either.

I'm obviously missing the point somewhere along the line here. Thanks to you both for taking the time to try and enlighten me!


Formal logic isn't about evidential proof in the sense you are understanding it. Formal logic is "topic neutral" which means that the arguments deal purely with the form of the statements involved, not their interpretation. For instance from a&b derive a, now I could put some silly nonsense in the place of a and b but the inference would still be correct, because it is "truth preserving" meaning that we don't go from something true to something false. Now in the particular case where b = ¬a we have something which can never be true, i.e.it is not the case for any statement that the statement is true and its negation is true. So it is entirely truth preserving to go from something which is always untrue to anything else.

This is the "semantic interpretation" which means we talk in terms of truth and truth preservation. You can talk in syntactic terms i.e. purely from proof but that requires some background knowledge. Anyway I hope that clears that up.
N.B. there are logicians who reject this type of inference, but they reject it for odd reasons, such as the belief that there can be true contradictions, or a 3rd truth value or by rejecting reducto ad absurdum none of which is the line you are taking I think

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neoliminal
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Re: "Principle of Explosion" Discussion

Postby neoliminal » Fri Feb 19, 2010 12:52 pm UTC

Randal!!!

Get out of my head!!!

How did you know my Mom drinks box wine from waffle cones!?!?

PM me for her phone number. Oh wait, you already have it.
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Re: "Principle of Explosion" Discussion

Postby javahead » Fri Feb 19, 2010 1:51 pm UTC

As a non-physics/mathematical engineer type, I am totally blown away by this concept


Box-wine and waffle cones...wow

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Re: "Principle of Explosion" Discussion

Postby raisedbyanother » Fri Feb 19, 2010 2:21 pm UTC

Lol, I have my first Discrete Math test today, Randall get out of my head!!

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Exüberance
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Re: "Principle of Explosion" Discussion

Postby Exüberance » Fri Feb 19, 2010 3:13 pm UTC

When I read 'wine' I first thought of the Windows "emulator" rather than the drink. What does that say about me?

Also, a slightly simpler explanation of how this works for those who don't understand it:
Saying if P and !P then X is guaranteed to never be evaluated since if P is false then [P and not P => false and true => false] else if P is true then [P and not P => true and false => false], so this is basically saying:

if FALSE then X;

This statement is always true, not matter what you put in for X. Why? Because when you evaluate that expression, you never get to X since the conditional is always false, so you never get to the result. Since there is no result (well, you never get to it) there cannot be a contradiction since there's nothing there to make a contradiction, so the entire statement is true.

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Re: "Principle of Explosion" Discussion

Postby cparker15 » Fri Feb 19, 2010 4:01 pm UTC

notzeb wrote:I imagine the derivation went something like this:

Code: Select all

Assume P ^ ~P.
- Then 7+2 = 9, and 72/9 = 8.
- Now assume for contradiction that Mrs. Lenhart's phone number is not 978-448-6872.
-- But then P ^ ~P, contradiction!
- Thus Mrs. Lenhart's phone number must be 978-448-6872.

I think you have the wrong Mrs. Lenhart. That one is a high school counselor, not a teacher.

How do I know this? P^¬P, my friends.
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A Derivation using the Principle of Explosion

Postby pscottdv » Fri Feb 19, 2010 4:37 pm UTC

From P and Not P is true, we can derive that 1) P is true and Not P is true. Therefore the following is true: 2) P or "Your mom's number is 867-5309". From Not P is true we can derive that 3) P is False. From 2) and 3) we can derive that "Your mom's number is 867-5309" QED

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Re: "Principle of Explosion" Discussion

Postby Carteeg_Struve » Fri Feb 19, 2010 5:24 pm UTC

1) P AND Not P is true.
2) Invert (1): Not P AND P is false.
3) Commutativity applied to two: P AND Not P is false.
4) So, from (1) and (3), P AND Not P is true, and P AND Not P is false.
5) Thus: true is false and false is true.

This leads to the final result that:
If a Wookie lives on Endor, then AND is not Commutative.

Here. Look at the monkey. Look at the silly monkey.

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Re: "Principle of Explosion" Discussion

Postby daenku32 » Fri Feb 19, 2010 6:18 pm UTC

Carteeg_Struve wrote:1) P AND Not P is true.
2) Invert (1): Not P AND P is false.
3) Commutativity applied to two: P AND Not P is false.
4) So, from (1) and (3), P AND Not P is true, and P AND Not P is false.
5) Thus: true is false and false is true.

This leads to the final result that:
If a Wookie lives on Endor, then AND is not Commutative.

Here. Look at the monkey. Look at the silly monkey.


While this is all true, the comic had OR. "^"

In symbolic logic class you couldn't explode P OR ~P forwards.
But you could P AND ~P.

hackers238
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Re: "Principle of Explosion" Discussion

Postby hackers238 » Fri Feb 19, 2010 6:23 pm UTC

I'm not sure if this is the place for it, but the website seems to be broken around this comic.

http://www.xkcd.com links to the 703 comic from wednesday, and the "next" button doesn't browse to this new one.

http://xkcd.com works fine.

Someone move this if it's in the wrong place!

pscottdv
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Re: "Principle of Explosion" Discussion

Postby pscottdv » Fri Feb 19, 2010 6:31 pm UTC

While this is all true, the comic had OR. "^"


Yeah, except "^" is AND; "v" is OR.

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Re: A Derivation using the Principle of Explosion

Postby pscottdv » Fri Feb 19, 2010 6:33 pm UTC

pscottdv wrote:From P and Not P is true, we can derive that 1) P is true and Not P is true. Therefore the following is true: 2) P or "Your mom's number is 867-5309". From Not P is true we can derive that 3) P is False. From 2) and 3) we can derive that "Your mom's number is 867-5309" QED


And from this entire proof we can derive that I had terrible taste in music back in the '80s.

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Re: "Principle of Explosion" Discussion

Postby Technical Ben » Fri Feb 19, 2010 6:49 pm UTC

dunedain wrote:Has anyone here ever heard of Curry's Paradox? It's a personal favorite way to use explosion to get crazy results. It runs something like this (as a semi-formal proof):

Statement A: If statement A is true then I am the King of Spain.

1. assume statement A is true
2. then I am the King of Spain. (from 1 and A)
3. so, if statement A is true then I am the King of Spain. (from 1 and 2)
4. We just proved 3, which is a restatement of A, so A is true. Consequently I am the King of Spain!

The cool thing is that it really is deductively valid, and it can be really tough to figure out what's wrong with it. Just thought I'd share (this seems like a place where I won't be the only one who knows a little formal logic).


Is the self referencing the hole in the "logic" argument? That in any self contained system, you can create any rules, or not, as you like? It does not prove anything to be true outside of the statement. For example you are not the Kind of Spain. However within your "Statement A" or the written list, you are the king of spain.
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Re: "Principle of Explosion" Discussion

Postby LakatosIstvan » Fri Feb 19, 2010 7:18 pm UTC

Man, Logic always blows away my mind. This is a perfect example to show to people that although something is logical, it doesn't mean that it can necessarily be applied to our experienced reality :) So the next time kids someone asks you to logically prove them a belief, an idea, you know what to do ;)

By the way, I think that I found a real-world case of p and non-p. Take a pink object. The funny thing is that although it's pink, in the same time it's also green, your mind only decided to show you it's pink colour, not the green one. The object actually reflects back red and violet rays of light, which can be combined into two different colours.
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Re: "Principle of Explosion" Discussion

Postby g30dud3 » Fri Feb 19, 2010 8:05 pm UTC

akirjazi wrote:
Cipherz wrote:Can someone derive the mega million lottery numbers for me plz? I'll share some I promise.


I would but the margin of this page is too small to contain the derivation.


I've been reading these forums for a while now. I registered just because I understood this, having read about Fermat's last theorem only yesterday.

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Re: "Principle of Explosion" Discussion

Postby RogueCynic » Fri Feb 19, 2010 8:10 pm UTC

The concept of finding someone's phone number as Randall described is not new. Douglas Adams postulated it in the HitchHiker's Guide to the Galaxy. A funny cartoon, though not origional.
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Re: "Principle of Explosion" Discussion

Postby CrazyPirateNinja » Fri Feb 19, 2010 8:21 pm UTC

LakatosIstvan wrote:Man, Logic always blows away my mind. This is a perfect example to show to people that although something is logical, it doesn't mean that it can necessarily be applied to our experienced reality :) So the next time kids someone asks you to logically prove them a belief, an idea, you know what to do ;)

By the way, I think that I found a real-world case of p and non-p. Take a pink object. The funny thing is that although it's pink, in the same time it's also green, your mind only decided to show you it's pink colour, not the green one. The object actually reflects back red and violet rays of light, which can be combined into two different colours.


That only becomes p and not p if "green" is equal to or a subset of "not pink". You essentially just proved that there are green things which are not "not pink", so there is no logical contradiction. If you think of it in terms of a venn diagram, "green" and "pink" are two circles that overlap, while "not pink" consists of everything in the diagram that isn't inside the "pink" circle, meaning it excludes the part of "green" that overlaps with "pink", thus showing that "green" and "not pink" are not equal.

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tpow
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Re: "Principle of Explosion" Discussion

Postby tpow » Fri Feb 19, 2010 9:49 pm UTC

serrath wrote:Someone please explain the waffle cone joke to me.



It seems that Mrs. Lenhart likes to drink her box wine out of waffle cones (which would be a bad idea since anyone who has ever eaten ice cream out of waffle cones knows that once the ice cream melts it begins to leak)
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Re: "Principle of Explosion" Discussion

Postby kernelpanic » Fri Feb 19, 2010 10:15 pm UTC

...to the Wikipediamobile!

...

...Oh...

OK, I didn't get it at all. But the comic is still funny.
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Re: "Principle of Explosion" Discussion

Postby rateguard » Fri Feb 19, 2010 10:36 pm UTC

I love how P^~P is bolded so that all of us nerds have something more fit for nerd wankery :)

wankwankwankwankwank

mPasCa
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For those having trouble with the logic P^~P=>(your mom)

Postby mPasCa » Fri Feb 19, 2010 11:48 pm UTC

Terry Tao just buzzed this (you know, the Fields medal winning Mathematician). He has some insights into logic that may help those who are confused on this board. He's certainly more of an authority than anyone here can claim to be, and he's quite a bit more clear. (Hey look, its my first post and I've already insulted 60 people!)

Here's a link to the profile:
http://www.google.com/profiles/114134834346472219368#buzz

MrGuy
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Re: "Principle of Explosion" Discussion

Postby MrGuy » Sat Feb 20, 2010 12:28 am UTC

Let this be a lesson to all of you people who insist that .9999 repeating is NOT equal to one. See what can happen?

And you thought you were just being clever...

daenku32
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Re: "Principle of Explosion" Discussion

Postby daenku32 » Sat Feb 20, 2010 1:12 am UTC

pscottdv wrote:
While this is all true, the comic had OR. "^"


Yeah, except "^" is AND; "v" is OR.


Doh. My bad. Here's my nerd card.

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Re: "Principle of Explosion" Discussion

Postby BioTube » Sat Feb 20, 2010 2:59 am UTC

Psshhh. Everybody knows and is &, or is | and ^ is xor. Double for boolean instead of bitwise.
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Re: "Principle of Explosion" Discussion

Postby Superuser » Sat Feb 20, 2010 3:27 am UTC

Mrs Lenhart, are you trying to seduce me?
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dunedain
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Re: "Principle of Explosion" Discussion

Postby dunedain » Sat Feb 20, 2010 4:43 am UTC

Technical Ben wrote:
dunedain wrote:Statement A: If statement A is true then I am the King of Spain.

1. assume statement A is true
2. then I am the King of Spain. (from 1 and A)
3. so, if statement A is true then I am the King of Spain. (from 1 and 2)
4. We just proved 3, which is a restatement of A, so A is true. Consequently I am the King of Spain!


Is the self referencing the hole in the "logic" argument? That in any self contained system, you can create any rules, or not, as you like? It does not prove anything to be true outside of the statement. For example you are not the Kind of Spain. However within your "Statement A" or the written list, you are the king of spain.
Not exactly: this doesn't really create a self-contained system. Step one just uses normal assumption introduction, which is valid in formal logic as long as anything you remember that anything you derive from it only follows "if [assumption] is true". So that's what I did in step three--as I said before, the logic here is formally valid.

So no, I can't think of any way that the argument is "inside" the definition of statement A; but the trick does have to do with self-reference in that there's a liar paradox hidden in it. Here's how it works: A liar paradox is something like "this statement is false". Such a statement can't consistently be true or false--you can't assign it a truth value. That's where the explosion is, in a sense a liar sentence is simultaneously both true and false.

Now, build that into a disjunction (a statement that joins two other statements together using "or"): "Either this statement is false or I am the King of Spain". You can't say that's false either, that would just lead to the paradox again, so the only thing you can call it is true, and if it's true it can't be because it's false, so the part about me being the King must be true.

If I wrote it that way somebody might spot it though, so you change it a bit. In logic "~P or Q" is the same as "P then Q", so I get "This statement is true or I am the King of Spain". To simplify things call it "statement A" and there you have it: a statement that looks fairly harmless but that can be proved true no matter what it claims.

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Re: "Principle of Explosion" Discussion

Postby lcdrambrose » Sat Feb 20, 2010 5:18 am UTC

My personal favorite use of this kind of proof is from a book called "Zero: The Biography of a Dangerous Idea". In the appendix, he uses the statement 1+1=1 (which he derived from dividing by 0) to conclusively prove that Winston Churchill was, in fact, a carrot.
On a similar note, 2+2=5. At least according to Radiohead.

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Re: For those having trouble with the logic P^~P=>(your mom)

Postby notzeb » Sat Feb 20, 2010 10:32 am UTC

mPasCa wrote:Terry Tao just buzzed this (you know, the Fields medal winning Mathematician). He has some insights into logic that may help those who are confused on this board. He's certainly more of an authority than anyone here can claim to be, and he's quite a bit more clear. (Hey look, its my first post and I've already insulted 60 people!)
Bullshit. I claim to be the emperor of the solar system. Off with his head!
Zµ«V­jÕ«ZµjÖ­Zµ«VµjÕ­ZµkV­ZÕ«VµjÖ­Zµ«V­jÕ«ZµjÖ­ZÕ«VµjÕ­ZµkV­ZÕ«VµjÖ­Zµ«V­jÕ«ZµjÖ­ZÕ«VµjÕ­ZµkV­ZÕ«ZµjÖ­Zµ«V­jÕ«ZµjÖ­ZÕ«VµjÕ­Z

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Re: "Principle of Explosion" Discussion

Postby axilog14 » Sat Feb 20, 2010 11:05 am UTC

rocketrat wrote:What's with this Mrs Lenhart anyway? Scantron, Zealous Autoconfig, the one with Randall teaching about raptors, and now this.

She could be an actual person.

Holy carp, I think we may have stumbled on a breakthrough here: a whole FAMILY of recurring characters!

Wait, are the Mrs. Lenhart in this strip and Miss Lenhart from the old teacher strips the same person? Or Is "Miss Lenhart" in fact Mrs. Lenhart's older daughter? Does that make Randall's friend in this strip Teacher Miss Lenhart's brother? And does this mean one of the recurring members of stick-Randall's posse is a Lenhart?

So confused.
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Re: For those having trouble with the logic P^~P=>(your mom)

Postby MrGuy » Sat Feb 20, 2010 1:20 pm UTC

notzeb wrote:
mPasCa wrote:Terry Tao just buzzed this (you know, the Fields medal winning Mathematician). He has some insights into logic that may help those who are confused on this board. He's certainly more of an authority than anyone here can claim to be, and he's quite a bit more clear. (Hey look, its my first post and I've already insulted 60 people!)
Bullshit. I claim to be the emperor of the solar system. Off with his head!


Premise: If this statement is true, then notzeb is emperor of the solar system and Terry Tao has been beheaded.

It may not be immediately clear that "notzeb is the emperor of the solar system and Terry Tao has been beheaded" is true or not. However, we CAN agree that if it WERE true, then that would make the premise statement true.

But the last sentence above is exactly what the premise statement claims--IF the sentence is true, then notzeb is emperor of the solar system and Terry Tao has been beheaded. This means we've proven that the premise actually IS true. And SINCE the sentence is true, the notzeb really is emperor of the solar system, and Terry Tao has been beheaded. Poor guy. Vanished in a puff of logic.

If this sentence is true, then playing with the Curry paradox is fun for the whole family.

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Re: "Principle of Explosion" Discussion

Postby Okapi » Sat Feb 20, 2010 5:18 pm UTC

tpow wrote:
serrath wrote:Someone please explain the waffle cone joke to me.



It seems that Mrs. Lenhart likes to drink her box wine out of waffle cones (which would be a bad idea since anyone who has ever eaten ice cream out of waffle cones knows that once the ice cream melts it begins to leak)



In fact, I believe this is an allusion to "white trash" culture, which is often transposed on Your Mother jokes (which, ironically, came from Adolf Hitler--Your Mother jokes were a political rhetorical device employed by the dictator, which is ironic because one of the common associations now is with something (white trash culture) that looks down upon Hitler's "Aryan Race" of blonde-haired, blue-eyed whites, now "rednecks," "hicks," or "white trash."

Box wine is popular among many groups, and has many uses, but the usual stereotypical consumer of box wine is a low-life living in the trailer park, often the butt of "Your Mother" jokes, and such people often are also construed as buying into generic consumerism, having skewed ideas of "delicious" and "good ideas," and being cheap.

Therefore, it is funny and fits the stereotype of calling the subject's immediate family white trash to suggest that they not only think that box wine is classy enough for a first date, but that they also think it is proper to drink it out of waffle cones. It works on multiple levels, as well, because it suggests that the subject's friend (whom I suppose is Randall) is so desperate or so wants to one-up his friend that he would actually tap that, as well as the obvious surface joke about the Principle of Explosion.

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bmonk
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Re: "Principle of Explosion" Discussion

Postby bmonk » Sat Feb 20, 2010 5:25 pm UTC

I'm not sure that this is how the Principle of Explosion works. It's true that he could end up with Mom's phone number--but there would be no way to be sure it was right...

Still, I can't prove he didn't derive it. In fact, with P & ~P, I can't prove anything, just because I can prove anything.

Someone once said that Logic is the Science of going wrong with utter assurance. It's also been called the way to know that what you already know is true is in fact true. But it's still fun, especially when it's fuzzy.
Having become a Wizard on n.p. 2183, the Yellow Piggy retroactively appointed his honorable self a Temporal Wizardly Piggy on n.p.1488, not to be effective until n.p. 2183, thereby avoiding a partial temporal paradox. Since he couldn't afford two philosophical PhDs to rule on the title.


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