Logic and Quantifiers

A lot of mathematical statements are really claims about how many things satisfy some property, and precision there matters enormously. Two shorthand symbols carry that weight: $\forall$ means "for all" (every single one, no exceptions), and $\exists$ means "there exists" (at least one). "$\forall x, x^2 \geq 0$" says every real number squares to something non-negative — true. "$\exists x, x^2 = 4$" says at least one number squares to 4 — also true, and it doesn't claim every number does.

The order of quantifiers is not cosmetic — swapping it changes the claim entirely. "For every trader, there exists a strategy that beats the market" ($\forall$ trader $\exists$ strategy) just says each trader has some winning strategy, possibly a different one each. "There exists a strategy that beats the market for every trader" ($\exists$ strategy $\forall$ trader) claims one single strategy works for everyone — a far stronger and, in this case, false claim. Getting the order backwards is one of the most common errors when a proof turns out to be wrong.

Negating a quantified statement flips it and swaps the quantifier: the negation of "$\forall x$, P(x)" is "$\exists x$, not P(x)" — if it's false that every strategy is profitable, then at least one isn't.