0704: "Principle of Explosion"

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0704: "Principle of Explosion"

Postby joee » Fri Feb 19, 2010 5:09 am UTC

Image

http://xkcd.com/704/

Title: You want me to pick up waffle cones? Oh, right, for the wine. One sec, let me just derive your son's credit card number and I'll be on my way.

Lenhart -- which comic have we heard that in before?

Also, I read that as P ^ NP and was confused

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Re: "Principle of Explosion" Discussion

Postby A. Smith » Fri Feb 19, 2010 5:12 am UTC

As I'm currently reading Gödel's Proof (by Nagel and Newman), I find this completely hilarious.

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Re: "Principle of Explosion" Discussion

Postby TrueJournals » Fri Feb 19, 2010 5:15 am UTC

joee wrote:Lenhart -- which comic have we heard that in before?


Multiple places, it would appear:
http://xkcd.com/135/ http://xkcd.com/416/ http://xkcd.com/499/ (at least, that's all I found with the search)

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Re: "Principle of Explosion" Discussion

Postby chocolate.razorblades » Fri Feb 19, 2010 5:16 am UTC

Two comics in a row alluding to logic...I wonder what Randall's doing. It's a kind of "get out of my head" moment since I have a Discrete Math exam on Monday...eh, not really.

Haha, P^NP -- you've solved the most important problem in Computer Science!!!

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Re: "Principle of Explosion" Discussion

Postby FishyFred » Fri Feb 19, 2010 5:17 am UTC

I don't have anything close to the math expertise to understand the joke here. Can someone translate for the laypeople?

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Re: "Principle of Explosion" Discussion

Postby Steve the Pocket » Fri Feb 19, 2010 5:18 am UTC

Huh. When I use the Principle of Explosion, I usually derive large craters. I guess I'm doing it wrong.

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Re: "Principle of Explosion" Discussion

Postby acidcj » Fri Feb 19, 2010 5:18 am UTC

Hey I just derived this post!

This is awesome.

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Re: "Principle of Explosion" Discussion

Postby Nihiltres » Fri Feb 19, 2010 5:22 am UTC

Randall, get out of my head!

I'm currently taking a course in deductive logic
Spoiler:
and averaging 100% so far :mrgreen:
and I've been thinking about pretty much exactly this comic (but not your mother).

Of course, there are worse logical principles that can be abused. If you know that a given statement is true, you can make arbitrary true if-then statements with the true statement as the consequent, since if-then statements are only false when the antecedent is true and the consequent is false. It may not make the antecedent true, but it sure sounds crazy. :D

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Re: "Principle of Explosion" Discussion

Postby bradleyjx » Fri Feb 19, 2010 5:23 am UTC

FishyFred wrote:I don't have anything close to the math expertise to understand the joke here. Can someone translate for the laypeople?


The idea is that if I have something that is both true and not true (which that equation is), I can do logical things to the equation to make it seem like anything is anything. Or, "If I can prove that anything is both true and not true, I can never prove anything to not be either")

Taken into this comic, they turned that equation into the later things; this is impossible in real life, however, as just because you have something which you can extrapolate anything from, you can't actually extrapolate anything from it. (this is beginning to sound like the last comic...)

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Re: "Principle of Explosion" Discussion

Postby Qaanol » Fri Feb 19, 2010 5:25 am UTC

FishyFred wrote:I don't have anything close to the math expertise to understand the joke here. Can someone translate for the laypeople?

The statement P^¬P means "Statement P is true, and the negation of statement P is true." In particular, P represents some arbitrary logical statement which has a boolean value (true/false). The ^ symbol means AND. The ¬ symbol means NOT. So P^¬P can be read "P and not P", which is a false statement since there's no way P and its negation can both be true.

Furthermore, a statement of the form "If A then B" is only ever false when A is true and B is false. In particular, if A is false then the statement "If A then B" is true. Thus, if we write "If P^¬P then B" it doesn't matter what statement B is, the whole quoted statement is true.

An old anecdote, almost certainly apocryphal but foundhere, goes:

A story is told that the famous English mathematician G.H. Hardy made a remark at dinner that falsity implies anything. A guest asked him to prove that 2 + 2 = 5 implies that McTaggart is the Pope. Hardy replied, "We also know that 2 + 2 = 4, so that 5 = 4. Subtracting 3 we get 2 = 1. McTaggart and the Pope are two, hence McTaggart and the Pope are one."

The joke in the comic stems from the fact that, in formal logic, the statement "If P^¬P then X is your mom's phone number" is true for any value of X.
Last edited by Qaanol on Fri Feb 19, 2010 5:33 am UTC, edited 2 times in total.
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Re: "Principle of Explosion" Discussion

Postby Omegaton » Fri Feb 19, 2010 5:26 am UTC

The Principle of Explosion is exploding my head.

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Re: "Principle of Explosion" Discussion

Postby VHBT » Fri Feb 19, 2010 5:29 am UTC

For those who don't understand, the expression "P ^ ~P" translates to "P and not P" ("not P" being the logical contradiction of "P"). The principle of explosion refers to the idea that, if you assume that one false statement is true, you can derive any other false statement.
The comic's joke requires that "any other false statement" be replaced by the word "anything." The second character then misinterprets the word "anything" (quite successfully, as it turns out).

My guess is that he actually looked up the phone number at some other point in time and was saving it for something like this.

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Re: "Principle of Explosion" Discussion

Postby Nihiltres » Fri Feb 19, 2010 5:32 am UTC

Omegaton wrote:The Principle of Explosion is exploding my head.


Damn it, I just came back to say 'More like "Principle of Head Explosion", eh?' but I've been beat to it. :p

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Re: "Principle of Explosion" Discussion

Postby SocialSceneRepairman » Fri Feb 19, 2010 5:45 am UTC

FishyFred wrote:I don't have anything close to the math expertise to understand the joke here. Can someone translate for the laypeople?


It's really pretty simple. If you can come up with a coherent statement, no matter how blatantly false, in a formal discourse, you can prove it, rigorously, by starting from any coherent statement and its opposite.

The joke is that this doesn't allow you to divine specific knowledge, much less true knowledge, it just lets you get to absurd conclusions, but the guy seized on an alternate sense of "derive."

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Re: "Principle of Explosion" Discussion

Postby Turing Machine » Fri Feb 19, 2010 5:58 am UTC

chocolate.razorblades wrote:Two comics in a row alluding to logic...I wonder what Randall's doing. It's a kind of "get out of my head" moment


Same thing here! But I've been reading An Introduction to Non-Classical Logic, so I'm coming at it from a different perspective...

First degree entailment does not suffer from explosion! Just throwing that out there. I guess I should also throw out there that whether "if...then..." is represented accurately by the material conditional is an object of some debate.

I seriously had to register just to say this, logic in xkcd :wub:

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Re: "Principle of Explosion" Discussion

Postby dennisw » Fri Feb 19, 2010 6:05 am UTC

It's funny because it's true.
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Re: "Principle of Explosion" Discussion

Postby notzeb » Fri Feb 19, 2010 6:17 am UTC

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.
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 Eternal Density » Fri Feb 19, 2010 6:18 am UTC

Right, I was just thinking that today would not be complete without yet another your-mom joke...
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Re: "Principle of Explosion" Discussion

Postby LuNatic » Fri Feb 19, 2010 6:26 am UTC

I thought the principle of explosion is what occurs when you put Jamie and Adam in a shed together?

Hmmm...

X = ( B * ( √ ( T - W - R ))) / 10

where
X = Probability of explosion
B = Budget of project
T = Cumulative time spent in shed by Adam and Jamie
W = Time wasted by Adam on silliness
R = Time wasted by Jamie reprimanding Adam for silliness
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Re: "Principle of Explosion" Discussion

Postby nathanmlevesque » Fri Feb 19, 2010 6:44 am UTC

I like how this is situational, the phrasing 'and derive anything' is something you could expect to hear. And the joke is about taking that literally such that it applies to anything, which of course it doesn't. And then I check the forum's and everyone's hung up on the math. Really unnecessary to understanding and appreciated the comic.

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Re: "Principle of Explosion" Discussion

Postby dunedain » Fri Feb 19, 2010 7:03 am UTC

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).

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Re: "Principle of Explosion" Discussion

Postby Cipherz » Fri Feb 19, 2010 7:07 am UTC

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

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Re: "Principle of Explosion" Discussion

Postby Red Hal » Fri Feb 19, 2010 7:09 am UTC

LuNatic wrote:I thought the principle of explosion is what occurs when you put Jamie and Adam in a shed together?

Hmmm...

X = ( B * ( √ ( T - W - R ))) / 10

where
X = Probability of explosion
B = Budget of project
T = Cumulative time spent in shed by Adam and Jamie
W = Time wasted by Adam on silliness
R = Time wasted by Jamie reprimanding Adam for silliness
That could lead to probabilities greater than one, still funny though.
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Re: "Principle of Explosion" Discussion

Postby merime » Fri Feb 19, 2010 7:22 am UTC


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Re: "Principle of Explosion" Discussion

Postby phlip » Fri Feb 19, 2010 7:25 am UTC

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

If [imath]P \land \lnot P[/imath] then the lotto numbers will be 4, 19, 23, 194[imath]\pi[/imath], the fine structure constant, and Birmingham.

You can thank me later.

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Re: "Principle of Explosion" Discussion

Postby ijuin » Fri Feb 19, 2010 7:34 am UTC

Red Hal wrote:
LuNatic wrote:I thought the principle of explosion is what occurs when you put Jamie and Adam in a shed together?

Hmmm...

X = ( B * ( √ ( T - W - R ))) / 10

where
X = Probability of explosion
B = Budget of project
T = Cumulative time spent in shed by Adam and Jamie
W = Time wasted by Adam on silliness
R = Time wasted by Jamie reprimanding Adam for silliness
That could lead to probabilities greater than one, still funny though.

If the probability is greater then one, then that means you can get multiple explosions! :wink:

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Re: "Principle of Explosion" Discussion

Postby '; DROP DATABASE;-- » Fri Feb 19, 2010 7:38 am UTC

Why is it that only now that I'm taking a class on boolean logic, where I recently saw most of these symbols and terms for the first time ever (despite being familiar with the concepts from coding), do I start to see all sorts of comics and jokes using it? O.o It's like "Randall get out of my head", but instead of Randall, it's Randall and a bunch of random people.

I must have seen like 4 comics/puns using this stuff since I started taking this class, and none before... and I'm not the kind to simply forget I ever saw it because i didn't get it at the time. Hmm.
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Re: "Principle of Explosion" Discussion

Postby phillipsjk » Fri Feb 19, 2010 7:44 am UTC

Is Mr. Munroe making fun of the discussion the last comic generated?

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

If [imath]P \land \lnot P[/imath] then the lotto numbers will be 4, 19, 23, 194[imath]\pi[/imath], the fine structure constant, and Birmingham.

You can thank me later.


That's funny, I got: 14-35-45-55-56 Mega ball: 13
I could be wrong because all the numbers were equally valid each time. I drew 55 twice (rejected the second one). I even have some entropy left over: 0b26891273f (I took the md5 hash of a "news site" and used Round(12 bit chunks/FFF*3716)+1)

Edit: I also realized I don't know which drawing that is. Deriving that from the remaining entropy is an exercise left to the reader. :mrgreen:

Edit: All that work avoiding an off-by-1 error, an I made an off by 0.5 error. To chose a random number between 1 and 56, based on a random number from 0 to 1, you would use: Ceiling(55*(Random#))+1) or (Floor(55*(Random#))+2). Except, in typing this, I noticed that those are not equivalent in the case the random number is equal to 0 or 1. Maybe I should ask in the Computing Science forum. In the above formula I can guarantee that 0 an 1 don't happen by doing: (12bit chunks +1)/100116
Last edited by phillipsjk on Sat Feb 20, 2010 7:14 am UTC, edited 1 time in total.
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Re: "Principle of Explosion" Discussion

Postby Quicksilver » Fri Feb 19, 2010 8:00 am UTC

I've done Computational Mathematics, so I love these kinds of jokes. but that's not a tautology, so the inference is invalid.
Proof:
P ^ ¬p
1 0 01
0 0 10

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Re: "Principle of Explosion" Discussion

Postby Arancaytar » Fri Feb 19, 2010 8:15 am UTC

I've never seen mathematical logic drawn into a Your-Mom joke before. Thank you, Randall!
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Re: "Principle of Explosion" Discussion

Postby akirjazi » Fri Feb 19, 2010 9:11 am UTC

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.

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Re: "Principle of Explosion" Discussion

Postby littlelj » Fri Feb 19, 2010 9:25 am UTC

Qaanol wrote:
FishyFred wrote:I don't have anything close to the math expertise to understand the joke here. Can someone translate for the laypeople?

The statement P^¬P means "Statement P is true, and the negation of statement P is true." In particular, P represents some arbitrary logical statement which has a boolean value (true/false). The ^ symbol means AND. The ¬ symbol means NOT. So P^¬P can be read "P and not P", which is a false statement since there's no way P and its negation can both be true.

Furthermore, a statement of the form "If A then B" is only ever false when A is true and B is false. In particular, if A is false then the statement "If A then B" is true. Thus, if we write "If P^¬P then B" it doesn't matter what statement B is, the whole quoted statement is true.

<funny anecdote>

The joke in the comic stems from the fact that, in formal logic, the statement "If P^¬P then X is your mom's phone number" is true for any value of X.


*applause* Excellent explanation for those of us whose logic is a bit ... rusty.

Liked this comic lots. Math(s) + "your mom" joke = Classic XKCD. Win!
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Re: "Principle of Explosion" Discussion

Postby JurPic » Fri Feb 19, 2010 9:35 am UTC

Oh I do so love Logic/Mathematics in my comics! :D
This one was great.

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Re: "Principle of Explosion" Discussion

Postby rocketrat » Fri Feb 19, 2010 9:53 am UTC

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.



Also, I was burned by cheese, even though I wasn't exactly sure what the equation was, and hadn't heard the term "Principle of Explosion" before. I prefer the Mythbusters definition, though having Adam wasting time on silly stuff often also produces explosions.

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Re: "Principle of Explosion" Discussion

Postby JeromeWest » Fri Feb 19, 2010 10:12 am UTC

Qaanol wrote:Furthermore, a statement of the form "If A then B" is only ever false when A is true and B is false. In particular, if A is false then the statement "If A then B" is true.


I'm still lost. I don't see how A being false makes the statement "If A then B" automatically true. Doesn't it just mean that we don't know if the statement is true or not? Sort of like this: -

Code: Select all

A       B     if A then B
true   true   true
true   false  false
false  true   unknown
false  false  unknown


Help out a formal logic noob here, and point out the flaw in my thinking.

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Re: "Principle of Explosion" Discussion

Postby DT_ » Fri Feb 19, 2010 10:53 am UTC

I'm still lost. I don't see how A being false makes the statement "If A then B" automatically true. Doesn't it just mean that we don't know if the statement is true or not? Sort of like this: -

"If A then B" or "A implies B" means that B must be true if A is true, but we place no restriction on B if A is not true. So if A is false, then what is unknown is the value of B (it could be either true or false), but the value of "if A then b" is true.

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Re: "Principle of Explosion" Discussion

Postby DanielZKlein » Fri Feb 19, 2010 11:04 am UTC

To P and not to P!

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Re: "Principle of Explosion" Discussion

Postby phlip » Fri Feb 19, 2010 11:07 am UTC

JeromeWest wrote:I'm still lost. I don't see how A being false makes the statement "If A then B" automatically true.

It's called "material implication", and the truth table looks more like:

Code: Select all

A       B     if A then B
true   true   true
true   false  false
false  true   true
false  false  true


It makes a bit more sense if you talk in the sense of "All things that have property A also have property B"... [imath]\forall x : Ax \to Bx[/imath]
The only way you could prove this statement wrong would be to find a thing that had property A, but didn't have property B... this would be the only thing that would make the implication false. For all other combinations (things with both properties, things with neither, and things with just property B not not property A), the implication must be true.

There's also the old chestnut: "If I told you 'If it rains tomorrow, I'll wear a raincoat', and then it doesn't rain, would you call me a liar?"

A implication where the antecedant is never true (for instance, to use my pattern from two sentences ago, "All things that are both on fire and not on fire also have property B") is still true... there's still no counterexample (find me an object that's both on fire and not on fire but doesn't have property B). It's sometimes called a "vacuous truth"... but it's still truth.
Last edited by phlip on Fri Feb 19, 2010 11:11 am UTC, edited 1 time in total.

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Re: "Principle of Explosion" Discussion

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

DT_ wrote:"If A then B" or "A implies B" means that B must be true if A is true, but we place no restriction on B if A is not true. So if A is false, then what is unknown is the value of B (it could be either true or false), but the value of "if A then b" is true.


But if A is false then where is the evidence of whether "If A then B" is true or false? Regardless of whether B is true or false, we have no information to help us decide whether "If A then B" is true.

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".

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Re: "Principle of Explosion" Discussion

Postby phlip » Fri Feb 19, 2010 11:13 am UTC

JeromeWest wrote:They say "but if A was true, B would be true too!",

To which I would say "that's irrelevant, whether it's true or not... if the truth value of A changes, then naturally the truth value of any statement involving A (such as 'if A then B') may also change."

The statement "if A then B" is not a statement about the fundamental structure of the universe... it's a statement about A and B. You're confusing it with "For all states the universe could be in, if A was true then B would be true"... an altogether different statement.
Last edited by phlip on Fri Feb 19, 2010 11:15 am UTC, edited 1 time in total.

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