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.