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Zack_M_Davis13y50

A number of authors speak of "de Morgan's laws for quantifiers," and I think this is a wise choice of terminology. A universal (respectively, existential) quantifier behaves just like a conjunction (respectively, disjunction) over all the objects in the universe, so, aesthetically and pedagogically, I think it's much more elegant to speak of ¬∃x(P(x)) <---> ∀x(¬P(x)) and ¬∀x(P(x)) <---> ∃x(¬P(x)) as generalized de Morgan's laws, rather than to reserve the term "de Morgan's laws" for ¬(A ∧ B) <---> (¬A ∨ ¬B) and ¬(A ∨ B) <---> (¬A ∧ ¬B) and have a separate term like "quantifier negation laws" for the tautologies involving quantifiers. Because, you know, it's the same idea in slightly different guises. Some authors may prefer different terminology, but I stand by my comment.

printing-spoon13y20

Okay, that makes sense.

3

How to write a mathematical formula on the fear of death?

by TheatreAddict

20th Nov 2011

1 min read

3

I was reading the Methods of Rationality, and I was reading the part about how it's irrational to fear death. Well I came across "All x: Die(x) = Not Exist x: Not Die(x)" I really don't get this.. I'm sorry, I'm not good at math. But does "x" here represent an unknown variable? If so, is it being like, multiplied when it's put in parenthesis? Could this be put into a simpler equation?

Because I totally get the part where you either have to want to keep living, because I want to live right now, I'll want to live tomorrow, so therefore I'll want to live forever. And then if I want to not live forever, it would mean that I don't really want to live very much.. Right?

This is what happens when someone who hasn't a clue about math and science reads a smart fanfiction. But if someone could either verify the part about "All x: Die(x) = Not Exist x: Not Die(x)" being the correct formula, and then explaining why, that would be like, really cool.

Thanks! :D

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[-]printing-spoon13y90

Red(x) means "x is red." X = Y in this case means that X and Y are either both true or both false. All x : Bouncy(x) means that everything under consideration is bouncy. Exists x : Fluffy(x) means a fluffy thing exists. The sentence says "if everything dies, then nothing doesn't die" and vice versa. This is http://en.wikipedia.org/wiki/First-order_logic .

edit: And here are the fruits of google: http://math.stackexchange.com/questions/52323/how-do-you-read-this-logical-statement-aloud-and-how-do-you-notate-it-in-symbol

[-]TheatreAddict13y00

Your first link doesn't work, but I'll check out the second one. I don't completely understand, but I understand more than I did before you commented, so thanks! :]

[-]Manfred13y70

So basically, two statements are being compared (and said to be equivalent) by the equals sign.

The first statement is "All x: Die(x)." For every x, x dies. Or, replacing the label 'x' by the label 'that person': Every person dies.

The second statement is "Not exist x: Not Die(x)." There does not exist a person who doesn't die.

This equation describes the fact that if nobody's immortal, everyone dies.

[-]Zack_M_Davis13y50

In more detail: the underlying principle here is called De Morgan's law. De Morgan's law is our name for the fact that to say that a cat is not both furry and white, is the same as saying that the cat is either not-furry or not-white (or both).

(More generally: the negation of a conjunction (respectively, disjunction) is the disjunction (respectively, conjunction) of the negations.)

Suppose we lived in a world with twenty cats. We could make a statement about all of the cats by saying "The first cat is furry and the second cat is furry and the third cat is furry and [...] and the twentieth cat is furry." But that would take too long; instead we just say, "Every cat is furry." Similarly, instead of "Either the first cat is white or the second cat is white or [...] or the twentieth cat is white," we can say, "There exists a white cat." Thus, the same principles that we use for and-statements ("conjunctions") and or-statements ("disjunctions") can be used on ("quantified") for every-statements and there exists-statements. "There does not exist a winged cat" is the same thing as "For every cat, that cat does not have wings" for the same reason that "It is not the case that either the first cat has wings or the second cat has wings" is the same thing as "The first cat does not have wings and the second cat does not have wings." That's de Morgan's law.

So, suppose there does not exist a person who does not die. De Morgan's law tells us that this is equivalent to saying that for every person, that person does not-not-die. But not-not-dying is the same thing as dying. But this is that which was to be proven.

[-]printing-spoon13y00

This is not De Morgan's law. There is no conjunction or disjunction involved, only quantification:

Ax:P(x) = ¬Ex:¬P(x)

I'm not sure if there's a name for this type of tautology.

[-]Zack_M_Davis13y50

A number of authors speak of "de Morgan's laws for quantifiers," and I think this is a wise choice of terminology. A universal (respectively, existential) quantifier behaves just like a conjunction (respectively, disjunction) over all the objects in the universe, so, aesthetically and pedagogically, I think it's much more elegant to speak of ¬∃x(P(x)) <---> ∀x(¬P(x)) and ¬∀x(P(x)) <---> ∃x(¬P(x)) as generalized de Morgan's laws, rather than to reserve the term "de Morgan's laws" for ¬(A ∧ B) <---> (¬A ∨ ¬B) and ¬(A ∨ B) <---> (¬A ∧ ¬B) and have a separate term like "quantifier negation laws" for the tautologies involving quantifiers. Because, you know, it's the same idea in slightly different guises. Some authors may prefer different terminology, but I stand by my comment.

[-]printing-spoon13y20

Okay, that makes sense.

[-]TheatreAddict13y00

This may seem like a silly question, but why isn't not-not-dying the same thing as dying?

[-]Zack_M_Davis13y20

It is the same thing.

[-]TheatreAddict13y00

Oh.. Erm.. I read that wrong. >_>

Facepalm

[-]Kaj_Sotala13y10

Heh, and I misread your question to ask why it is the same thing, only realizing my mistake when I read this comment. :-)

[-]Kaj_Sotala13y00

I'm not sure if this helps, but: you can think of it this way

Dying = Someone dies.
Not-dying = It is not so that someone dies.
Not-not-dying = It is not so that (it is not so that someone dies).

The first "it is not so that" cancels out the second "it is not so that".

Similarly, if someone said (in ordinary speech) "I'm not ungrateful", that would mean that they were grateful, while "I'm not grateful" or "I'm ungrateful" would mean that they weren't. "I'm not-not-grateful = I'm grateful."

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Similarly, if someone said (in ordinary speech) "I'm not ungrateful", that would mean that they were grateful...

Be careful with ordinary speech ;-)

[-]Vladimir_Nesov13y60

I was reading the Methods of Rationality, and I was reading the part about how it's irrational to fear death.

Death is well worth fearing. You must have mixed something up...

[-]Normal_Anomaly13y00

Perhaps he was referring to Dumbledore's opinion?

[-]TheatreAddict13y20

I was, but it would've made more sense to refer to Harry's, sorry, my bad.

*She

[-]daenerys13y20

Thanks for posting this TheatreAddict! I also didn't understand the equation, but didn't even think to ask what it meant.

I learned something new today because of you. :)

[-]TheatreAddict13y30

Awhh! :D You're welcome! It makes me happy knowing I helped someone.. Albeit inadvertedly. :]

[-][anonymous]13y00

It's a made-up notation based on first-order logic. It simply represents the statement "If all people were to die this would be equivalent to there existing no one who does not die".

It's a tautology.

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