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.

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. :)

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

So... Does anyone know of any helpful presentations? My brain likes pictures. This was probably the most helpful thing I've come across on here. I'm 16, not the best at math and complex equations, so this sort of helped a lot of stuff click in my mind. :)

There are definitely approaches to QM that smack of subjective reality ("subjective" describing this one you of the many near-clones of you, one in each possible worlds each quantum mechanical outcome involving you creates, if you believe the MWI the way EY does). However, it is indeed best to stay away from the topic unless you are well versed in it.

The topic of quantum mechanics should probably be avoided, unless you can expect the audience to have taken a course or read a book on quantum mechanics. Long story short, though, the "what if we create our own reality?" debate was indeed revived by quantum mechanics. The debate was then concluded, long ago, because this was science and not a debate club. Objective reality was not overturned.

Although I'm not sure why I get the correct answer when I'm working with the actual numbers and not percentages when I do the math wrong.

I know it now makes more sense to you now, but I want to point out that reality isn't school, and nobody is going to take marks off for using actual numbers or ratios instead of percentages (the 'pure' way that the teacher prefers or what-have-you).

A calculator more reliably gets me the answer than mental arithmetic, and so I use a calculator at work even though it seems lazier than doing it in my head - in the same way, if ratios and actual numbers more reliably let you use Bayes Theorem than percentages, use actual numbers and all the people who think it's purer to use percentages be damned.

You're not. Remember, you're not taking 80 of the 10,000 women in the population. You're only taking 80 of the 100 women with breast cancer. Likewise, it's not 9.6% of all the women, it's 9.6% of the women who don't have breast cancer, or 950/9900. The wording of the problem already took the base rates into count, so when you're plugging the real numbers in, you are automatically taking the base rates into account. By giving you 80/100 and 950/9900, Eliezer already did the division for you.

You're forgetting the "base rate" in your calculation: the actual rate of cancer in the population. What you should really be taking the ratio of is (the fraction of all women that have cancer and test positive) / (the fraction of all women that test positive, whether or not they have cancer). In percentages, that's

(80% of the 1% of women who have cancer, who correctly test positive) = 0.8 * 0.01.

divided by

(80% of the 1% of women who have cancer, who correctly test positive) together with (9.6% of the 99% of women who don't have cancer, who test positive anyway) = 0.8 * 0.01 + 0.096 * 0.99.

So the ratio is (0.8 * 0.01) / (0.8 * 0.01 + 0.096 * 0.99), and that does equal 0.078.

Yes.