TITLE: What are reasons why some symbols in mathematical logic are not standardized?
QUESTION [21 upvotes]: Why is so hard to find a standardisation regarding symbolism and/or terminology in Mathematical Logic ?
We see again and again students asking if e.g. $\rightarrow$ and $\implies$ means the same thing : somebody answer : "yes", somebody answer : "no".
The same thing happens with "tautology" and "validity", with "logical consequence" and "logically implies", and so on ...
Why is this problem still with us, and what can we do about it?

REPLY [8 votes]: Why is so hard to find a standardisation regarding symbolism ... in Mathematical Logic ?

Well, if you compare the situation with (say) fifty to sixty years ago -- the time when the books I was looking at while I was a student were written -- I would have said that there has been quite a considerable standardisation in symbolism, at least in mainstream mathematical logic books/articles. And one reason for that, surely, has been the universal adoption of LaTeX, which makes it so easy to type \land [for 'logical and'] and always get '$\land$' (and not '&', or a dot, etc.) and to type \forall x and get '$\forall x$' (and not e.g. '$(x)$' or '$\Pi x$'). So, let's be duly grateful for all the standardisation there in fact now is!
True there is the annoying business with $\to$ vs $\implies$. This too is partly to down to LaTeX I guess, as \implies yields the second. Now 'A implies B' gets used in informal talk both as variant on 'if A then B' and as a variant of 'A logically entails B', i.e. as both what me might regiment as $A \to B$ and as $A \vdash B$ [or $A \vDash B$]. And low and behold, we find $\implies$ being confusingly used both ways [in the object language, or in the metalanguage]. Conservatism in symbolism is a Good Thing, so I think the use of $\implies$ is to be deprecated: I'd say, use $\to$ for an object language conditional, and the appropriate turnstile in in metalanguage. The exception might be in a sequent calculus. [Indeed, it was only when I started regularly visiting math.se that I really registered that the double arrow was used in non-sequent-calculi so widely outside the very narrow logic community I was most familiar with -- though that might just show I wasn't really paying attention!]