0704: "Principle of Explosion"

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-.Mateo.-
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Re: "Principle of Explosion" Discussion

Postby -.Mateo.- » Sat Feb 20, 2010 7:33 pm UTC

bmonk wrote: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.


You can prove that <<(P^~P)>ↄ Q>
--------------------------
[
(P^~P)
P
~P
[
~Q
P
~~P
]
<~Q ↄ~~P>
<~P ↄ Q>
]
<<P^~P>ↄQ>
--------------------------------
I took this out of "Gödel, Escher, Bach" (I'm halfway through it).
Magus wrote:If history is to change, let it change. If the world is to be destroyed, so be it. If my fate is to die, I must simply laugh.

Just as you touch the energy of every life form you meet, so, too, will will their energy strengthen you.

hamsterpoop
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Re: "Principle of Explosion" Discussion

Postby hamsterpoop » Sat Feb 20, 2010 9:47 pm UTC

There are a couple basic themes on xkcd, and some of the comics in this theme have become retarded.

This one fits in the "random science and math that Randall just learned". It seems like he doesn't even have anything to say about things; he just wants to tell us all that he learned something new.

Basically, he's trying too hard. I blame the fact that he forces himself to post 3 times a week.

Maybe he should do like Buttercup Festival and only post new comics when he has content to do it.
.

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Re: "Principle of Explosion" Discussion

Postby Okapi » Sat Feb 20, 2010 10:45 pm UTC

No! He must post every single day, so that he will come up with more good ones!

Also, whether or not you feel like being cynical and hypercritical, and whether or not Randall does have an entire theme of math he's just learned, the comic is good. This particular episodic strip is good. Xkcd is good.

You are not good. Bad person. BAD! :evil:

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Re: "Principle of Explosion" Discussion

Postby dunedain » Sat Feb 20, 2010 10:47 pm UTC

hamsterpoop wrote:There are a couple basic themes on xkcd, and some of the comics in this theme have become retarded.

This one fits in the "random science and math that Randall just learned". It seems like he doesn't even have anything to say about things; he just wants to tell us all that he learned something new.
Actually, hard as this may be for you to believe, a lot of us were already familiar with the principal of explosion; some already knew jokes about it (as this threat attests). I think you may have mistaken what you don't know for what isn't widely known...

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Re: "Principle of Explosion" Discussion

Postby LTK » Sat Feb 20, 2010 11:08 pm 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.

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.


I laughed a lot more at that anecdote than at the comic. :D

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Re: "Principle of Explosion" Discussion

Postby MrGuy » Sat Feb 20, 2010 11:10 pm UTC

hamsterpoop wrote:There are a couple basic themes on xkcd, and some of the comics in this theme have become retarded.

This one fits in the "random science and math that Randall just learned". It seems like he doesn't even have anything to say about things; he just wants to tell us all that he learned something new.

Basically, he's trying too hard.


I find your premise unsound. While I don't think Randall is a towering genius, he's clearly reasonably familiar with math, science, and the philosophy behind them. I find it hard to believe that he's just learned about the Principle of Explosion (it's not exactly obscure in a philosophy of science course). Put another way, it's considerably less obscure than the in-joke here: http://xkcd.com/468/

I think Randall is just someone who enjoys a good "your mom" joke, and sometimes realizes some principle that he happens to be thinking about would make a good "your mom" joke. Consider http://xkcd.com/89/, http://xkcd.com/563/, and a bunch of others I'm too lazy to search for.

Not everyone likes "your mom" jokes. http://xkcd.com/366/

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Re: "Principle of Explosion" Discussion

Postby EmotionallyTonedGeometry » Sun Feb 21, 2010 1:24 am UTC

Warning:

The following will ask you to expand the paradigms that you are currently using. This may be uncomfortable.

The Diamond Sutra of Mahayana Buddhism has been using a logical formulation similar to the one posited in this cartoon. However, it reaches different conclusions. As it goes, there have been those who have been questioning the nature of logic for 2500 years.

For those who have had their curiosity piqued by this, please check out the links below. For those who instantaneously dismiss this as nonsense, please disregard.

http://www.thezensite.com/ZenEssays/Phi ... _Sutra.pdf

This is an exposition of the logic of the Diamond Sutra which will ask you to re-examine the structure of Aristotelian logic.

http://en.wikipedia.org/wiki/Prat%C4%AB ... tp%C4%81da

This is the lame wikipedia article that addresses this issue further.

Thanks to those of you who wish to re-examine the narrow confines of logic. For those who charge in and refuse to critically examine yourselves other than by your own self-affirming means, you'll get it some day.

As a side note, Levine's "Materialism and Qualia: The Explanatory Gap" plays directly into the conclusion of the Diamond Sutra. Note the logic that applies to theory and the logic that applies to first-hand, subjective experience. Can the experiencer experience itself? This seems to open up a can of worms that most seem to gloss over.

Good luck and good night.

-EG

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Re: "Principle of Explosion" Discussion

Postby hamsterpoop » Sun Feb 21, 2010 2:37 am UTC

dunedain wrote:Actually, hard as this may be for you to believe, a lot of us were already familiar with the principal of explosion; some already knew jokes about it (as this threat attests). I think you may have mistaken what you don't know for what isn't widely known...


My post said absolutely nothing about what I know or don't know. Please reread.

MrGuy wrote:I find your premise unsound. While I don't think Randall is a towering genius, he's clearly reasonably familiar with math, science, and the philosophy behind them. I find it hard to believe that he's just learned about the Principle of Explosion (it's not exactly obscure in a philosophy of science course). Put another way, it's considerably less obscure than the in-joke here: http://xkcd.com/468/

I think Randall is just someone who enjoys a good "your mom" joke, and sometimes realizes some principle that he happens to be thinking about would make a good "your mom" joke. Consider http://xkcd.com/89/, http://xkcd.com/563/, and a bunch of others I'm too lazy to search for.


Don't get me wrong, I like xkcd a lot. I've been reading it since a little after it started.

But ok. Maybe I was exaggerating a little by saying that he just learned something like moment of inertia or gravity. My point is that the jokes are becoming less about the joke and more about the reference to a certain scientific or mathematical concept.

Just because I get the inside joke, it doesn't mean the joke was funny. And sometimes I get a feeling that there's canned laughter in the background. All I'm saying is that some of the comics suck, like it is to be expected from any comic strip. Most of them are good. But some are kind of lame. My problem is that the ratio of good to lame has been decreasing. I suspect that it's because of the unconditional praise for every single comic.
.

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Re: "Principle of Explosion" Discussion

Postby Chariot » Sun Feb 21, 2010 2:43 am UTC

This is interesting, as I have been reading this for years, and this goes right up there as one of my favorite comics.

I'm sorry you don't like it, but if you believe that the quality is going down or something perhaps you should re-read some of the older ones.

Example: discussion about centrifugal/centripetal force that most high schools discuss. Punch line is a cliche bond quote.

There are plenty of other early examples.

I love how people constantly claim he's losing touch etc. as he is putting up still hilarious comics after years.

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Re: "Principle of Explosion" Discussion

Postby dunedain » Sun Feb 21, 2010 4:54 am UTC

hamsterpoop wrote:
dunedain wrote:Actually, hard as this may be for you to believe, a lot of us were already familiar with the principal of explosion; some already knew jokes about it (as this threat attests). I think you may have mistaken what you don't know for what isn't widely known...


My post said absolutely nothing about what I know or don't know. Please reread.
I realize that you didn't say anything about your own knowledge. I was obliquely pointing out the same thing as MrGuy: that the subject matter in question wouldn't be new to anybody with a background in formal logic or advanced math, which Mr. Monroe clearly has, as do many of his fans. It occurred to me that only someone unfamiliar with the material at hand would think otherwise.

It also struck me as ironic to word it as I did: I had at least as much evidence to impute ignorance to you as you had to impute it to Mr. Monroe.

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Re: "Principle of Explosion" Discussion

Postby NumberFourtyThree » Sun Feb 21, 2010 11:04 am UTC

Consider this statement:

If this statement is true, then this statement is false.
The world is imperfect because it has to be. If everything were perfectly fair and without problems we would all live the exact same pointless life, with no possible meaning to it.

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Re: "Principle of Explosion" Discussion

Postby MrGuy » Sun Feb 21, 2010 1:22 pm UTC

NumberFourtyThree wrote:Consider this statement:
If this statement is true, then this statement is false.


No. I refuse to consider your statement. So there.

I win!

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Re: "Principle of Explosion" Discussion

Postby Malina » Sun Feb 21, 2010 2:19 pm UTC

Hadn't heard of the principle of explosion, so had to read the Wikipedia entry. Then I had to read the paraconsistent logic entry. Trying to figure out the difference between paraconsistent and classical logic is like trying to figure out the difference between bayesian and frequentist statistics, or quantum and classical computing. My brain hurts. I'm going back to bed and I'm going to hide there all day. STAY AWAY paraconsistent logic, I deny you!

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Re: "Principle of Explosion" Discussion

Postby dunedain » Sun Feb 21, 2010 3:27 pm UTC

NumberFourtyThree wrote:Consider this statement:

If this statement is true, then this statement is false.
That looks like a paraphrase of the simple liar paradox: "This statement is false." The conditional adds a bit of complication, but it comes to the same thing in the end.

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Re: "Principle of Explosion" Discussion

Postby DCB » Mon Feb 22, 2010 5:08 am UTC

Haven't taken any logic courses yet, just some vague references in geometry class last year (U.S. curriculum FTW).

"If this statement is true, then this statement is false"

Would I be correct in assuming the statement is false and then principle of explosion my way out of the conundrum? Or is the only solution to assume each statement is self-referential?

As for, "This statement is false", I feel like that is a tougher problem and I don't feel like googling the answer.

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Re: "Principle of Explosion" Discussion

Postby phlip » Mon Feb 22, 2010 5:11 am UTC

DCB wrote:Would I be correct in assuming the statement is false and then principle of explosion my way out of the conundrum?

You could try it, but that would make the statement true (and you assumed it to be false).

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 NumberFourtyThree » Mon Feb 22, 2010 5:13 am UTC

Malina wrote:Hadn't heard of the principle of explosion, so had to read the Wikipedia entry. Then I had to read the paraconsistent logic entry. Trying to figure out the difference between paraconsistent and classical logic is like trying to figure out the difference between bayesian and frequentist statistics, or quantum and classical computing. My brain hurts. I'm going back to bed and I'm going to hide there all day. STAY AWAY paraconsistent logic, I deny you!

You're lucky there. I ended up with over a dozen pages open such as "large cardinal axioms" and "the axiom of determinacy" as illustrated here:
Image
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Re: "Principle of Explosion" Discussion

Postby zbigbox » Mon Feb 22, 2010 5:30 am UTC

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


As I'm currently dating women with sons old enough for me to say this too, I too find this hilarious. 8)


Too bad they won't get the math... :?

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Re: "Principle of Explosion" Discussion

Postby bmonk » Tue Feb 23, 2010 1:30 am UTC

-.Mateo.- wrote:
bmonk wrote: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.


You can prove that <<(P^~P)>ↄ Q>
--------------------------
[
(P^~P)
P
~P
[
~Q
P
~~P
]
<~Q ↄ~~P>
<~P ↄ Q>
]
<<P^~P>ↄQ>
--------------------------------
I took this out of "Gödel, Escher, Bach" (I'm halfway through it).



Yes, all that is true--but you can also prove ~Q, or Q for any value of Q. Which means, supposing your Mom's number is 555-1234, you can derive that, but also that her number is 555-6789, or 555-4321 or 314-159-2653, or 271-828-1828...so, how can you distinguish which one it the actual value? You can't.
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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Re: "Principle of Explosion" Discussion

Postby notzeb » Tue Feb 23, 2010 1:49 am UTC

bmonk wrote:...so, how can you distinguish which one it the actual value? You can't.
Speak for yourself. Just because you aren't good enough to figure out how to do it doesn't mean everyone else is an idiot.
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Re: "Principle of Explosion" Discussion

Postby Mazzula » Thu Feb 25, 2010 12:46 pm UTC

JeromeWest wrote:
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!


I'd like to give it a try.

"(P IMPLIES Q)" is equivalent to "(NOT Q IMPLIES NOT P)". So the statement "if it is raining tomorrow, I will be wearing a raincoat" can be restated as "If I am not wearing a raincoat tomorrow, it will not be raining". Thus if a raincoat is worn, there is no more information about the implication than if it does not rain. So maybe it will help to think about why you think that "it rains and you wear a raincoat" says something about the implication, but "it does not rain and you do not wear a raincoat" says nothing. I think it has to do with the idea that "implies" involves some kind of causation, so we first test for causes and only test for effects if we find those causes, but causation doesn't come into the formal logic. In formal logic "(P IMPLIES Q)" is equivalent to "(NOT P OR Q)".

In a part of your posts about this you used a truth table. The truth table interpretation of propositional logic is that if we can derive some proposition R from some set of propositions S, this means that any truth table rows (assignments of truth values to proposition variables) that satisfy all of the propositions in S will also satisfy proposition R. So the principle of explosion follows immediately and trivially in that any set of propositions S that contains a contradiction will be satisfied by no rows in the truth table, and thus all zero of those rows will satisfy any R. So all states of affairs that satisfy "(P AND NOT P)" will also satisfy "Q", where Q means "X is your Mom's phone number" (and they will also satisfy "X is not your Mom's phone number", i.e. "NOT Q"). So the statement that "if there is any contradiction, then we can prove anything" really just means "all [zero] states of affairs that satisfy a contradiction also satisfy every other proposition". It just says that if your range of possible states of affairs is the empty set, then all of those possible states of affairs will satisfy any proposition.

In real life, we want to consider formalisms that are useful and informative, so we rule out sets of propositions that aren't satisfied by any truth table rows, and thus we rule out sets of propositions that contain contradictions. But it is a very useful feature of propositional logic that it allows contradictions because this is what allows us to use it to examine an arbitrary set of propositions to look for contradictions. If contradictory propositions weren't allowed in the formalism, then the formalism couldn't be used in that way. And the nice thing about the principle of explosion is that once you have found any contradiction in your accepted propositions, you don't have to worry about whether the contradiction is trivial or essential because all contradictions are essential.

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Re: "Principle of Explosion" Discussion

Postby rkitect » Wed Mar 03, 2010 5:18 pm UTC

EmotionallyTonedGeometry wrote:...

The Diamond Sutra of Mahayana Buddhism has been using a logical formulation similar to the one posited in this cartoon....

-EG


I believe the Kama Sutra would give more insight to this formula. Just my $.02

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Re: "Principle of Explosion" Discussion

Postby sourgrass » Thu Apr 15, 2010 10:01 pm UTC

Yes, all that is true--but you can also prove ~Q, or Q for any value of Q. Which means, supposing your Mom's number is 555-1234, you can derive that, but also that her number is 555-6789, or 555-4321 or 314-159-2653, or 271-828-1828...so, how can you distinguish which one it the actual value? You can't.


Yes, also true. But the point of the principle of explosion is that when you have a contradiction in the system then you can show any statement to be true. Thus any of those numbers is in fact your Mom's number. I will accept that 555-1234 might be your Mom's number, but I can also prove that 555-6789 is her number and thus when I call 555-6789 I will be talking to her. (And she wants to go out with me, and she is going to buy the wine, and . . .) No need to distinguish between numbers when I have proven that my number is correct (this is a cartoon after all).



(. . . and we should fight a war to spread democracy throughout the world.)

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Re: "Principle of Explosion" Discussion

Postby Mazzula » Fri Jul 30, 2010 3:35 pm UTC

sourgrass wrote:Yes, also true. But the point of the principle of explosion is that when you have a contradiction in the system then you can show any statement to be true. Thus any of those numbers is in fact your Mom's number.

No, that isn't the point of the principle of explosion (although it may be the point of the comic by that title). The point of the principle of explosion is that any state of affairs that satisfies the contradiction will satisfy any other statement as well.

This does not mean that "any of those numbers is in fact your Mom's number", rather it means that the original proposition, which was (P AND NOT P), is not a possible in any state of affairs at all. It is allowed in symbolic logic because symbolic logic is allowed to express propositions that cannot be satisfied by any states of affairs.

So you make a truth table. The original propositions will be satisfied by some (possibly empty) set of rows in the truth table. The rules of symbolic logic allow you to extend those propositions in any way such that the set of rows in the truth table that satisfy the extended set of propositions is identical to the set of rows that satisfy the original propositions.

If the truth table contains a column for P and a column for "555-1234 is your Mom's Phone number" and another column for "555-6789 is your Mom's Phone number", then any rows that satisfy (P AND NOT P) will also satisfy both of those other statements, and they will also satisfy the negation of those other statements, simply because there are no rows at all that satisfy (P AND NOT P) and making the other assertions doesn't change that. Whether the entry in the truth table contains a T or F for P, the row will get excluded by (P AND NOT P) regardless of whether it has a T or F in any other column.

The trick is that having a proposition in symbolic logic does not mean that the proposition is true, it only means that it is asserted to be true, in other words that it will be true in any state of affairs that satisfies the proposition. But if there is no row in the truth table that satisfies the proposition, it is still allowed to be asserted in symbolic logic, even though it will never be true.

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Re: "Principle of Explosion" Discussion

Postby Pfhorrest » Fri Jul 30, 2010 10:13 pm UTC

This is the best explanation of the principle of explosion I have ever read.

It reminds me of a way I like explaining to people that "all" does not imply "some" in the way that those are used in formal logic (and upon reflection, there's a good reason; these are closely related issues):

1) All of my children are over the age of five.
2) None of my children are over the age of five.

There is no formal contradiction in the above two statements. How is that possible? Simple: I have no children. None (that is to say, zero) of my zero children are over the age of five; and since zero of my zero children are over the age of five, all zero of my zero children are over the age of five. "All of my children are over the age of five" just means "None of my children are five or younger", which is true; I have no children who are five or younger, because I have no children at all.

You can generalize that the only condition upon which some contradiction applies to all members of a set (they all have some property and none of them have that property) is when the set is empty. One case of that general principle is a set of possible worlds all and none of which have the property of some proposition being true in them; the only condition that satisfies that is if said set of possible worlds is empty; which is just another way of saying, there is no possible world where something is both true and false; or in other words, it is not possible for something to be both true and false.
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