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.
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.
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.”
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.
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.
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.
Okay, that makes sense.
This may seem like a silly question, but why isn’t not-not-dying the same thing as dying?
It is the same thing.
Oh.. Erm.. I read that wrong. >_>
Facepalm
Heh, and I misread your question to ask why it is the same thing, only realizing my mistake when I read this comment. :-)
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.”
Be careful with ordinary speech ;-)