Belkar and the Unexpected Hanging

[davidbofinger]

Every time Belkar is in danger, people expect him to die. When he does die, some will say they could see it coming but nobody will really have been certain. It's an example of a famous logical paradox, The Unexpected Hanging (http://en.wikipedia.org/wiki/Unexpected_hanging_paradox).

This has nothing to do with actually hanging Belkar, which has been done (http://www.giantitp.com/comics/oots0165.html).

[BaronOfHell]

Thanks for the link, great read.

I'm not certain it is an example of said paradox, though.

In Belkar's case, everyone is at any time predicting the next strip will be his end, like if someone claims all possible statements to be true, they'd be right about everything, and way more wrong than right.

In the hanging paradox, the prisoner rules out the possibility of predicting any date as the right date, thereby falsely concluding that no date will be the right date, because every day ruled out will per. definition then be the right date (that is, what is the right date depends on the prisoner's own belief, for it to be possible, it must be concluded impossible, else it's impossible).

[hagnat]

this good read got me by surprise... and its not even a wednesday

[littlebum2002]

It's not an example of the Unexpected Hanging, because no one said Belkars death wouldn't be expected.

[Turgon9357]

Reading paradoxes and other such logic traps is perhaps my favorite thing.

So is it an Expected Nonhanging, then?

[Cerussite]

It's not an example of the Unexpected Hanging, because no one said Belkars death wouldn't be expected.

It's been implied by the Giant himself that he enjoys to prove all our expectations to be wrong (http://www.giantitp.com/FAQ.html#faq8). I say it qualifies.

[Zerter]

I never really see logic traps that work. If you convince yourself the hangman will never come, than any day is unexpected. If you convince yourself he can come at any time, any time is expected. God can do anything, so he can make a rock he can not lift, and then he can lift it, because he can do anything, even though it is impossible to lift.

I mean, sure, it might be possible to construct a loop somehow, but that is because language is limited and you are abusing that limit and probably are wilfully interpreting it a certain way (much like clicking on any random topic in the optimization forum), not because there is an actual logical trap.

[BaronOfHell]

I suppose one major point of these logic traps is that some sentences seems to make perfect sense, but actually does not upon further scrutiny.

Like with math, for an omniscient being every math result is obvious because it follows logically from whatever definition/axiom, yet we math is a field with much development.

An example of a sentence which sounds fine, but is not possible to do is the barber who shaves everyone except those who shave themselves. Taking the sentence literally, the barber can only be shaved if he does not get shaved. A common sense understanding of the sentence includes without mention that the barber also shaves himself.

Some like to claim this is a paradox, because it's neither true nor false (or both), whereas I prefer to simply list it under statements which is not possible to do. Similar is a true <=> false loop, which again is simply a statement which is not possible to do.

Example:
The sentence below is true.
The sentence above is false.

Neither of those are true or false. But it does not mean it's a paradox.

[Mammal]

Potatoes have skin. I have skin. Therefore, I am a potato.

I don't know. I've never been a fan of formal reasoning. It starts with "what if," and then ends with trying to redefine the entire sentence based on archaic alternate definitions for indefinite articles. Not my thing.

I did have to take an epistemology course last year. I think I like the Unexpected Hanging more than the ones we had to learn about. No true scholar likes formal logic :smallwink:

[BaronOfHell]

Yes for everything with skin is a potato :smalltongue:

I do enjoy math, a lot, so maybe I'm not a true scholar :smalltongue:

[Cerussite]

I suppose one major point of these logic traps is that some sentences seems to make perfect sense, but actually does not upon further scrutiny.

Like with math, for an omniscient being every math result is obvious because it follows logically from whatever definition/axiom, yet we math is a field with much development.

An example of a sentence which sounds fine, but is not possible to do is the barber who shaves everyone except those who shave themselves. Taking the sentence literally, the barber can only be shaved if he does not get shaved. A common sense understanding of the sentence includes without mention that the barber also shaves himself.

Some like to claim this is a paradox, because it's neither true nor false (or both), whereas I prefer to simply list it under statements which is not possible to do. Similar is a true <=> false loop, which again is simply a statement which is not possible to do.

Example:
The sentence below is true.
The sentence above is false.

Neither of those are true or false. But it does not mean it's a paradox.

Under formal logic (predicate logic, first-order logic, among other non-fuzzy logic systems), a statement is either true or false. The fact that any possible assignment of truth-values to the variables of the statements can derive a logical contradiction given the axioms of a logical system (where 'can derive' is mathematically well-defined) is exactly what a paradox means. Under other logical systems, the paradox might not exist, but that doesn't make the statement any less paradoxical within the constraints of the other system.

[rodneyAnonymous]

Potatoes have skin. I have skin. Therefore, I am a potato.

That is unsound logic.

[hopeful1212]

Potatoes have skin. I have skin. Therefore, I am a potato.

That's a formal logical fallacy (http://en.wikipedia.org/wiki/Fallacy_of_the_undistributed_middle) and not a paradox.

[hopeful1212]

Or more accurately I think it's just a non sequitur (http://en.wikipedia.org/wiki/Non_sequitur_(logic)).

[BaronOfHell]

Under formal logic (predicate logic, first-order logic, among other non-fuzzy logic systems), a statement is either true or false. The fact that any possible assignment of truth-values to the variables of the statements can derive a logical contradiction given the axioms of a logical system (where 'can derive' is mathematically well-defined) is exactly what a paradox means. Under other logical systems, the paradox might not exist, but that doesn't make the statement any less paradoxical within the constraints of the other system.

Then formal logic as the above is obviously insufficient in describing the world. :smallsmile:

I don't get the need for a system reduced to a point where it's no longer self-consistent, as it then becomes useless when applied to more general problems.

It's like trying to describe the world by assuming everything is linear.

Anyway what I meant was that there exists no actual paradoxes. You can limit a system to a degree where it can't solve a problem and yell paradox, but that's like you write only within said system. A proper system is one that's globally applicable, which obviously then would not have any such inconsistencies.

But really, it's not "logic" for statements to only be true or false. It's easy to make statements which are neither, or even both.

[Ghost Nappa]

Then formal logic as the above is obviously insufficient in describing the world. :smallsmile:
False.
I don't get the need for a system reduced to a point where it's no longer self-consistent, as it then becomes useless when applied to more general problems.

Flawed Premise. The purpose of reducing the system to a binary condition is precisely to attain a sort of self-consistency. See: The Very Machine you are using to read this message, which runs on BINARY.

It's like trying to describe the world by assuming everything is linear.
...Go on.
Anyway what I meant was that there exists no actual paradoxes. You can limit a system to a degree where it can't solve a problem and yell paradox, but that's like you write only within said system. A proper system is one that's globally applicable, which obviously then would not have any such inconsistencies.

The issue with "inconsistencies" is that you are implying they are ALL coming from paradoxes, but that is HARDLY the case. From a technical standpoint, you can confuse a program or a machine by attempting to use a value that is explicitly defined as not being valid input, and seeing what happens. See something like Division by Zero: Sure, you can calculate x/x = 1, but if X=0, you have a big problem. You can get around the problem by responsibly having your calculator or program check to see if x=0 prior to doing the division or use a catch to report the mistake, but this is very much an instance where you are breaking a contract with the program. It gives you explicitly defined parameters, and will return a value. If you try to go outside those parameters, its model will break down and it can't accurately tell you what you want to know. This is less of an issue for more global systems like "What is the truth-value of X (where the options are either 'true' or 'false')?" but pushing them all on paradoxes is silly.

But really, it's not "logic" for statements to only be true or false. It's easy to make statements which are neither, or even both.

False. Within mathematical logic, having a truth-value of 'true' or 'false' is a pre-equisite for being a statement. In order words:

Bolded mine.

All statements are sentences.
All paradoxes are sentences.
Not all sentences are statements.
Not all sentences are paradoxes.
If a sentence is 'true', it is a statement.
If a sentence is 'false', it is a statement.
If a sentence is neither 'true' nor 'false', it is a paradox.
If a statement is 'true', it is not false.
If a statement is 'false', it is not true.

[Porthos]

Logic merely allows one to be wrong with authority.
-Some dude with a Doctorate :smallwink:

[Cerussite]

Then formal logic as the above is obviously insufficient in describing the world. :smallsmile:

There's no mathematical construct that can completely describe the physical reality of the world as of now. Congratulations, captain obvious.


I don't get the need for a system reduced to a point where it's no longer self-consistent, as it then becomes useless when applied to more general problems.

Except it's not useless. Formal logic is widespread in its applications to computer science (http://en.wikipedia.org/wiki/Logic_in_computer_science).


Anyway what I meant was that there exists no actual paradoxes. You can limit a system to a degree where it can't solve a problem and yell paradox, but that's like you write only within said system. A proper system is one that's globally applicable, which obviously then would not have any such inconsistencies.

There's no such thing as a globally applicable logical system. By Gödel's incompleteness theorems, any system that is stronger than basic arithmetic can't be provably consistent, and needs axioms outside of its scope to "demonstrate" its correctness.



But really, it's not "logic" for statements to only be true or false. It's easy to make statements which are neither, or even both.
It depends on the logical system you're using: in predicate logic and first order logic (which are the systems more used in logical programming, automated theorem proving and other areas), it's easily derivable that P or ~P (through natural deductions).

In fuzzy logic, a statement has a 'probability of being true', but, as that probability rises, the probability of being false decreases: something can't at the same time be likely to be true AND likely to be false.

And so it is in real life. Either something is true, or not, or we have a paradox within our set of axioms, or someone is making clever use of sophistry.

[Doug Lampert]

One thing to remember is that a false statement can have true implications in formal logic. This is not a contradiction or paradox.

A=>B is false means that A can be true, and B can be true or false, and A can be false, and B can be true or false. Given a true statement about A=>B then I CAN derive information from knowing not B (for example), but that's knowledge I get from the true implication.

But you can not in formal logic derive ANYTHING from a statement that is false. False statements can have true implications.

The statement below is false.
The statement above is true.

Is not any sort of contradiction. The second statement is false. Thus it tells me nothing about the first statement. The first statement can thus be true without any contradiction.

This sort of so called paradox is trying to confuse the issue by treating the implication statements themselves as the "A" and "B" that the implications are about and PRETENDING that the fact that not A and not B are allowed statements that I can draw conclusions from means that I can draw an implication from a false statement. But this is not the case.

[Cerussite]

But you can not in formal logic derive ANYTHING from a statement that is false. False statements can have true implications.

The general idea of your post is right, but I feel like nitpicking this. In logic, you can derive anything and everything from a false statement, as per the principle of explosion (http://en.wikipedia.org/wiki/Principle_of_explosion).

[BaronOfHell]

The point is that obviously there exists sentences which are neither true, nor false, and if applied if real world, they're impossible to do. That is not a paradox. A paradox is something which is both possible and impossible simultaneously.

Not to mention there are also sentences which are both true and false, etc.

Secondly there do exist a system of logic which is capable of describing nature, this system is nature itself, otherwise you could arrive at a situation where nature itself breaks down, which would put you outside of nature similar to when a program breaks down you're put to a point outside of it. (PS: I'm not implying a program only breaks down due to a paradox).

Everything can be described, because for every set of inputs, there's a set of outputs, it's similar to say you can draw any figure you wish.


And so it is in real life. Either something is true, or not
Absolutely not.

PS: I'm without statements such as "Congratulations, captain obvious.", thank you. I'm here to have a good time, don't try to ruin it like that.


The statement below is false.
The statement above is true.

Is not any sort of contradiction. The second statement is false. Thus it tells me nothing about the first statement. The first statement can thus be true without any contradiction.

If the second statement is False, and you limit your system to True or False, then it tells you that the first statement is not True, or the second statement would have been True.
If the first statement is Not True, it is False, which means the second statement is Not False, thereby it is True.

You're very welcome to include more information to a False statement so that True is not Not False, but I think it's important to have a precise formulation where you can take advantage of True is Not False, and have more terms for what are statements which are neither True, nor False.

[TheWolfe]

The point is that obviously there exists sentences which are neither true, nor false, and if applied if real world, they're impossible to do. That is not a paradox. A paradox is something which is both possible and impossible simultaneously.

Not to mention there are also sentences which are both true and false, etc.


You're confusing statements with sentences. A statement, or declarative sentence, is either true or false. A sentence can be a statement, but doesn't have to be. I would like you to give me a declarative sentence which is not either true or false.



Secondly there do exist a system of logic which is capable of describing nature, this system is nature itself, otherwise you could arrive at a situation where nature itself breaks down, which would put you outside of nature similar to when a program breaks down you're put to a point outside of it. (PS: I'm not implying a program only breaks down due to a paradox).


I agree that nature is consistent with nature, just like any x=>x. But the whole point of mathematics is to describe this nature in abstract terms so that we can more easily deduce and conclude things in a correct and consistent way. We can for example have the following logical deductions:

All men are mortal.
I am a man.
Therefore I am mortal.

All cats are lazy.
Garfield is a cat.
Therefore Garfield is lazy.

We can conclude such things individually every time we come across such a case, or we can define rules that fit all cases, and discover theorems that let us conclude more things, by reducing the above statements to the general case of:

All p are q.
x is p.
Therefore, x is q.

I do not pretend formal logic holds the answers to the universe, but it does help to define what we can and can not conclude in our search for those answers.



If the second statement is False, and you limit your system to True or False, then it tells you that the first statement is not True, or the second statement would have been True.
If the first statement is Not True, it is False, which means the second statement is Not False, thereby it is True.

You're very welcome to include more information to a False statement so that True is not Not False, but I think it's important to have a precise formulation where you can take advantage of True is Not False, and have more terms for what are statements which are neither True, nor False.

You misunderstand what a false statement is. When we conclude an ENTIRE STATEMENT is false, it does not mean the opposite of what the statement said is automatically true. It simply means that we cannot be sure, this we can not conclude anything from it. For example, let's say we have the statement A: p => q. Let's say we find A is false. That does not mean it's reverse, q => p, is true. It does not say anything about the truth of either p or q themselves. It simple means that if p is true, q is not automatically true.

[BaronOfHell]

I am aware of the strength of logic, also of some of its weaknesses when it comes to True/False limitations.


A statement, or declarative sentence, is either true or false. A sentence can be a statement, but doesn't have to be. I would like you to give me a declarative sentence which is not either true or false.
Thank you, but it's not very relevant to the discussion. I'm only talking about statements here, that I may sometime call them sentences doesn't mean I'm not talking about the limitations of truth/false logic.

In regard to no-information. I agree, but you can easily construct a statement which holds only one possible outcome and from that deduct more than for statements of several possible outcomes. Which is the case with the above circular example.

[TheWolfe]

I don't see what's the problem with the supposed contradicting example. One statement is false, and the other is unknown. Let me show you. We have two statements. A says: B is not true. B says: A is true. In other words:

A => not B
B => A

Let's examine it with the rules of formal logic. B => A => not B, thus B => not B.
This means that B must be false, for if B were true, B would be false, and a contradiction would occur.

The fact that B is false, does not have any other impact on the situation because not B does not imply anything. The premise 'B => A' does not automatically imply 'not B => not A', though the wording does suggest it.

Strictly speaking, A can be either true or false, because we don't have the premise 'not A => B' either.(The inverse of the first statement), thus neither A or not A lead to contradictions.

[littlebum2002]

It's been implied by the Giant himself that he enjoys to prove all our expectations to be wrong (http://www.giantitp.com/FAQ.html#faq8). I say it qualifies.

OH, that's actually pretty brilliant then. So every strip we predict he will die, so every time Rich keeps him alive just to prove us wrong!

[Turgon9357]

But now we've vocalized that we figured it out and so are expecting him to live!

[Unisus]

When we conclude an ENTIRE STATEMENT is false, it does not mean the opposite of what the statement said is automatically true.

That's not correct - if a statement is false, then its opposite is true and vice versa. The problem is, that most people have difficulties with finding the opposite of a statement. The opposite of "all cats are lazy" is not "No cat is lazy" but "Not all cats are lazy".



I don't see what's the problem with the supposed contradicting example. One statement is false, and the other is unknown. Let me show you. We have two statements. A says: B is not true. B says: A is true. In other words:

A => not B
B => A

Let's examine it with the rules of formal logic. B => A => not B, thus B => not B.
This means that B must be false, for if B were true, B would be false, and a contradiction would occur.

The fact that B is false, does not have any other impact on the situation because not B does not imply anything. The premise 'B => A' does not automatically imply 'not B => not A', though the wording does suggest it.

Strictly speaking, A can be either true or false, because we don't have the premise 'not A => B' either.(The inverse of the first statement), thus neither A or not A lead to contradictions.

Actually even though we don't know if a statement is true or false, it has to be one of those two.

The two statements are:
A: ~B
B: A

If A is true, then B must be false. But as B states that A is true, to be false A must be false also.

Remember, the statements were "A: B is false" and "B: A is true". Your transscription is for "A: if this is true, B is false" and "B: if this is true, A is true". To confuse these two is a mistake easily made.