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?
I think there is more than one cause of it. My ideas:
And maybe some wanted to confuse everybody :)
To add a bit:
Even with truth tables you see some publications where $0$ stands for true and others where $1$ stands for true. And that is just with two valued logic. If you are lucky you have a book that uses $T$ and $F$ or $\top$ and $\bot$. In either case the $T$ or $\top$ stands for true, and the $F$ or $\bot$ for false. But even so, be warned: always check the meaning first.
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!]
To paraphrase Bill Thurston, mathematics is just a way to organize human thought. The purpose of mathematical notation is to help us understand each other.
One reason that different notation is used is that the same concept may need to be compared or contrasted to different things in different situations. For instance, I recently discussed the various division symbols with someone:
The fraction notation $\frac{a}{b}$ is most useful when simplifying equations by hand.
The long division notation is most useful when calculating exact values.
The slash / is great for computers as it works with a regular keyboard.
The 'obelus' ÷ is used on calculators because the other notations are dissimilar from +,-, and x.
I feel that trying to standardize division would be counterproductive.
Now, mathematical logic is different from arithmetic, but the same truths hold. Russell and Whitehead used the notation most helpful for pure symbolic calculation, but this may not be the best notation for writing on the board or writing a computer program.
TL;DR Mathematical notation is designed to express thought as clearly as possible, and strict standardization makes this difficult.
The other answers don't seem to have addressed directly the point about implication $\rightarrow$ or $\implies$, so I will point out that (non-logician) mathematicians mostly use $\implies$ whereas logicians more often use $\rightarrow$. Possibly this is because logicians use it a lot more, so eventually one catches on that it is pointless to draw two lines when you can draw one. Since we are talking about two different scientific cultures, the difference in notation may be with us for the foreseeable future. For comparison, note that physicists often use different notation (than mathematicians are used to) for various mathematical concepts.
It's part of the evolution and change in language. And mathematics, besides what it is metaphysically, is also a language.
Certain parts of language, those in heavy and common use, tend to be standardised. For example, I do not think we will see new symbols for the natural numbers.
Those in less heavy use, tend to have variants which can be used by different parts of the community.
What could be useful is a standards committee. But only to a limited extent.
With regards to the ambiguity between implication $\rightarrow$ and entailment $\Rightarrow,$ the problem of course is that most theories have a deduction theorem, which has the consequence that the following judgements are equivalent.
$$\varphi \Rightarrow \psi,\quad\quad \Rightarrow \varphi \rightarrow \psi$$
Furthermore, every formula $\alpha$ can be seen as shorthand for the judgement $\Rightarrow \alpha$. So in some sense (as much as it pains me), the following are equivalent.
$$\varphi \Rightarrow \psi,\quad\quad \varphi \rightarrow \psi$$
This is probably the cause of a lot of ambiguity, but also suggests that we needn't worry; to claim that these strings of symbols "true" amounts to roughly the same meaning, so the ambiguity isn't too problematic. (On the other hand, to claim that the first string of symbols is false is very different to claiming that the second string of symbols is false.)