0704: "Principle of Explosion"

[joee]

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

P.S. Hi glasnt!

[A. Smith]

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

[TrueJournals]

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)

[chocolate.razorblades]

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!!!

[FishyFred]

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

[Steve the Pocket]

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

[acidcj]

Hey I just derived this post!

This is awesome.

[Nihiltres]

Randall, get out of my head!

I'm currently taking a course in deductive logic

Spoiler:

and averaging 100% so far

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.

[bradleyjx]

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...)

[Qaanol]

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.

[Omegaton]

The Principle of Explosion is exploding my head.

[VHBT]

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.

[Nihiltres]

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

[SocialSceneRepairman]

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."

[Turing Machine]

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:

[dennisw]

It's funny because it's true.

[notzeb]

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.

[Eternal Density]

Right, I was just thinking that today would not be complete without yet another your-mom joke...

[LuNatic]

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

[nathanmlevesque]

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.

[dunedain]

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).

[Cipherz]

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

[Red Hal]

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.

[merime]

This comic is talking about http://en.wikipedia.org/wiki/Ex_falso_quodlibet.

[phlip]

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.

[ijuin]

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!

['; DROP DATABASE;--]

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.

[phillipsjk]

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

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.

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

[Quicksilver]

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

[Arancaytar]

I've never seen mathematical logic drawn into a Your-Mom joke before. Thank you, Randall!

[akirjazi]

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.

[littlelj]

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!

[JurPic]

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

[rocketrat]

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.

[JeromeWest]

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.

[DT_]

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.

[DanielZKlein]

To P and not to P!

[phlip]

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.

[JeromeWest]

"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".

[phlip]

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.