I often see in papers something like:

1) This is in general not true

2) This is not true in general

1. $\forall x: \neg p(x)$
2. $\exists x: \neg p(x)$

But I wonder whether this is generally what is meant and if the mathematical community is careful about how they use the word "general" or if it used in a more colloquial sense. Being somewhat of an outsider I find this hard to judge. Partly as it is often used as an aside and rarely a formalisation of the statement is present to check it against.

It's the sort of thing you just can't look up.