Logic: '≮' vs '≥' "$g(x)$ is greater than or equal to $0.5$ for some $x$" is equivalent to "$g(x)$ is not less than $0.5$ for all $x$".
In mathematical symbols, the former translates to "$g(x) \geq 0.5$ for some $x$". I'm confused about the latter. Should it be "$g(x) \not≮0.5$ for all $x$"? When I speak aloud it sounds correct as I am saying the whole thing as "not less than for all $x$". But then I'm thinking that $\not<$ and $\geq$ are the same thing and the statement is really saying that "$g(x) \geq 0.5$ for all $x$" which is not true.
 A: When speaking aloud (or even writing something in words) I would normally say "at least..." rather than "not less than...". I would definitely still use $\geq$ rather than $\not<$; you normally only see the latter symbol in situations where two quantities might actually be incomparable (i.e. neither $x<y$ nor $x\geq y$ is true - obviously this can't happen with real numbers but it can in some other settings).
[edit] For a concrete example of this, suppose you are comparing vectors, and have the (quite natural) rule that $\underline v\leq\underline w$ if the components satisfy $v_i\leq w_i$ for each $i$. Then $\underline v\not<\underline w$ and $\underline v\geq\underline w$ have quite different meanings.
[further edit] I think I misunderstood the question actually. The first two statements you give are not equivalent - or at least, the second is not clear. When you say "$g(x)$ is not less than $0.5$ for all $x$" this could be interpreted as "$g(x)$ is not (less than $0.5$ for all $x$)" or "($g(x)$ is not less than $0.5$) for all $x$". The first is what you meant, but the second is I think how most people would interpret it. 
A: $\exists x: g(x) \geq 0.5 \Leftrightarrow \neg ( \forall x: g(x) < 0.5) $
Please tell me if you need further explanation.
